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Assessing Connectivity in Simulated, in-Vitro, and in-Vivo Neural Networks Using Granger Causality

Permanent Link: http://ufdc.ufl.edu/UFE0021659/00001

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Title: Assessing Connectivity in Simulated, in-Vitro, and in-Vivo Neural Networks Using Granger Causality
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Cadotte, Alex J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: array, causality, electrophysiology, engineering, epilepsy, extracellular, granger, microelectrode, modeling, network, neural, plasticity
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Since the early 1980s, interest in synchronization between regions in the brain has led to the adaptation of forecasting measures such as Granger Causality from fields like economics to neuroscience. The basic principle behind Granger Causality is that a directional causal relationship can be inferred from a time series X to a time series Y when the introduction of previous measurements of X reduces the variance of the error time series in the linear modeling of Y. Granger Causality methods are used in the Neural Hybrid Computing Laboratory to investigate functional relationships between neural units in simulation, in-vitro, and in-vivo. Computer simulations using biologically plausible neural-network models have shown that the direct component of pairwise Granger Causality (PGC) scales linearly with synaptic efficacy within biologically relevant ranges. This suggests that the PGC values obtained from living neural networks may be useful in calculating the underlying connectivity of living networks. Thus, PGC was used in-vitro to identify connectivity within neural pathways in a living neural network of 50,000 neurons grown on a microelectrode array (MEA). An MEA is an 8x8 grid of electrodes embedded in a cell culture dish. Data recorded from two plastcity experiments using these MEA cultures was analyzed using PGC. The resulting difference in PGC connectivity before and after the induction stage of this experiment reveals that PGC was able to describe plasticity within neural pathways in-vitro. PGC was also applied in-vivo to spontaneous epileptic seizures recorded from a 32-electrode array implanted in a rat?s hippocampus. These rats were electrically stimulated causing limbic status epilepticus and later began to develop spontaneous seizures consistent with Lothman's animal model of chronic limbic epilepsy. Results from the analysis of spontaneous limbic seizures reveal the dynamics of the directional causal influences during seizure. Interestingly, they indicate a sudden change in direction of causal influence between the CA1 and dentate gyrus regions of the hippocampus near the end of status epilepticus.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alex J Cadotte.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Demarse, Thomas.

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Permanent Link: http://ufdc.ufl.edu/UFE0021659/00001

Material Information

Title: Assessing Connectivity in Simulated, in-Vitro, and in-Vivo Neural Networks Using Granger Causality
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Cadotte, Alex J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: array, causality, electrophysiology, engineering, epilepsy, extracellular, granger, microelectrode, modeling, network, neural, plasticity
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Since the early 1980s, interest in synchronization between regions in the brain has led to the adaptation of forecasting measures such as Granger Causality from fields like economics to neuroscience. The basic principle behind Granger Causality is that a directional causal relationship can be inferred from a time series X to a time series Y when the introduction of previous measurements of X reduces the variance of the error time series in the linear modeling of Y. Granger Causality methods are used in the Neural Hybrid Computing Laboratory to investigate functional relationships between neural units in simulation, in-vitro, and in-vivo. Computer simulations using biologically plausible neural-network models have shown that the direct component of pairwise Granger Causality (PGC) scales linearly with synaptic efficacy within biologically relevant ranges. This suggests that the PGC values obtained from living neural networks may be useful in calculating the underlying connectivity of living networks. Thus, PGC was used in-vitro to identify connectivity within neural pathways in a living neural network of 50,000 neurons grown on a microelectrode array (MEA). An MEA is an 8x8 grid of electrodes embedded in a cell culture dish. Data recorded from two plastcity experiments using these MEA cultures was analyzed using PGC. The resulting difference in PGC connectivity before and after the induction stage of this experiment reveals that PGC was able to describe plasticity within neural pathways in-vitro. PGC was also applied in-vivo to spontaneous epileptic seizures recorded from a 32-electrode array implanted in a rat?s hippocampus. These rats were electrically stimulated causing limbic status epilepticus and later began to develop spontaneous seizures consistent with Lothman's animal model of chronic limbic epilepsy. Results from the analysis of spontaneous limbic seizures reveal the dynamics of the directional causal influences during seizure. Interestingly, they indicate a sudden change in direction of causal influence between the CA1 and dentate gyrus regions of the hippocampus near the end of status epilepticus.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alex J Cadotte.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Demarse, Thomas.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021659:00001


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ASSESSING CONNECTIVTY IN SIMULATED, IN-VITRO, AND IN-VIVO
NEURAL NETWORKS USING GRANGER CAUSALITY






















By

ALEX JOSEPH CADOTTE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































2007 Alex Joseph Cadotte




























To my Aunt Claire & Uncle Bill Parker.









ACKNOWLEDGMENTS

I thank Tom DeMarse for his guidance and patience for the past five years. I thank the

other graduate students I have worked with in Neural Hybrid Computation Laboratory including,

Ping He, Vinh Trin, Karl Dockendorf, Nicky Grimes, Pamela Douglas, Hany Elmariah, Crystal

Stephens, Memming Park, and Amber Scarlatos for the many interesting conversations and

collaborations. I specifically thank Ping He for the data provided from the Jimbo replicate

experiment. Also, I specifically thank Amber Scarlatos, for the extensive collaborations we had

during the past 2 years.

I thank Mingzhou Ding for his valuable teachings, time, and conversations and also

members of his laboratory including Yonghong Chen, Anil Bollimunta, Yan Zhang, and Raj

Rajagovindan. I thank Paul Carney, William Ditto, and all those associated with the Evolution

into Epilepsy project for the work behind the data that was provided me for analysis. I thank all

of the staff and faculty within the Biomedical Engineering department. I thank Yanni

Constantinidis for his influence on my teaching style and for his career advice. His presence in

the Biomedical Engineering department has been missed, may he rest in peace.

I thank my family and friends for their support over the past four years of graduate school.

I thank my parents and my brothers for believing in me. Thanks to Shane Calfee and Courtney

Calfee for suggesting UF for grad school, GO Gators! I'd also like to thank the many friends I

have made here at the University of Florida including Cameron Siggs, John Azeke, Neel

Bhatavadekar, Andres Argibay, Ning Jiang Argibay, John Quinlan, Amber Pouncy, and Lance

Gardner. Finally, I'd like to thank Theresa McGuirk for her patience, love, and support while I

was hard at work producing this document.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IST O F T A B L E S ................................................................................................... . 8

LIST OF FIGURES .................................. .. ..... ..... ................. .9

A B S T R A C T ......... ....................... ............................................................ 1 1

CHAPTER

1 INTRODUCTION AND BACKGROUND OF CAUSAL METHODS IN
N E U R O SC IE N C E ............................................................ .......... ............ ... ....... ... 13

In tro d u ctio n .................. ......... .... ................... ................... ................ 13
Review of Causal M ethods in N euroscience................................... .................................... 15
N on-P aram etric M ethods....................................................................... ........ ........... 16
C ross C orrelation ......... .............................................................. ... ...... ....... 16
C o h e ren c e ...................................... ................................................ 17
M mutual Inform action .................. ................................................ .. 19
P aram etric A pproaches.......................................................................... ....................20
Likelihood M methods ................................. ................ ................ .........20
M ethods using Autoregressive M odeling ...................... ........................... ....20
Partial Directed Coherence, the Directed Transfer Function, and Granger
C a u sa lity ...............................................................................................................2 1

2 MATHMATICAL BASIS FOR GRANGER CAUSALITY ................................................28

M them atical Basis for Tim e D om ain M measures ........................................ .....................28
Granger Causality ........................ ............... .................. ......28
A utoregressive M modeling ........................................................................ .................. 28
Time Domain Granger Causality............................ ................. ....................... 1
Spectral and Conditional Granger Causality ........................................ ....... ............... 32
Spectral G ranger C ausality .............................. ...................................... ............... 33
Conditional Granger Causality ........................................................ ............... 35
Lim stations of C ausal M ethods ............................................... ............................ 37

3 RECOVERY OF SYNAPTIC WEIGHTS FROM SPIKING ACTIVITY USING
G R A N G E R C A U SA L IT Y ............................................................................ ...................4 1

Introduction ................. ................ ................. ... ............ .......................... .4 1
Conditional Granger Causality and Mediated vs. Direct Relationships..............................44
Simulation and Analysis .................................................... .......... .45
M ono-directional Sim ulation...................................................... ........................ 47
Relationship between Synaptic Weight and Granger Causality.............. ............. 49









Mediated vs. Direct Causal Relationships under Serial Connectivity.............................51
Monte Carlo Simulations with Random Serial Topologies ...................................55
100 Neuron Neural Network .......................................................57
Application to Spike Timing Dependent Plasticity....................... ............... 59
C onclu sions.......... .............................. ................................................62

4 ASSESSMENT OF PLASTICITY IN-VITRO USING GRANGER CAUSALITY ..............65

In tro d u ctio n ............................................................................................. .. 6 5
W hat do PGC values m ean in neural netw works? ........................................ .....................66
M methods .............................................................67
D associated C ortical Cultures .............................................. .............................. 67
M icroelectrode A rrays...................................................... .. .... 68
Stim ulation Induced Plasticity Protocol ........................................ ............................. 69
R results ............... .................................................................................................. ....... 71
Spike R ate versu s P G C ...................................................................... .............. 7 1
Surrogate Analysis............................... ....... .... ..... ........... 74
D isc u ssio n ............. ....... ........ ... ....... ..................................................7 7
A m plitude D dependent Plasticity........................ ... .............................. ............... 78
D irectional C au sal Influences............................................................... .....................80
Spontaneous/E licited C onnection........................................................................ .. .... 85
C onclu sions.......... ............................... ................................................87

5 DIFFERENTIAL TETANAZATION OF SINKS AND SOURCEs IN DISSOCIATED
C O R TIC A L C U L TU R E ............................................................................... ... ............ 92

In tro d u c tio n ....................................................................................................................... 9 2
Plasticity by Tetanization .............................. ............................... ....... 95
Experimental Design .................. .......... ..................... ........ ................... 96
Spontaneous vs. Elicited Activity as a Measure of Plasticity ............................................98
R results and D discussion ........... .. ...... ................................. ......... ........... 100
R e su lts ........................................................................1 0 0
C o n c lu sio n s .............................................................................10 6

6 ANALYSIS OF TEMPORAL LOBE SEIZURES USING GRANGER CAUSALITY......108

In tro d u ctio n ................... ...................1.............................8
T em p oral L ob e E pilep sy ...............................................................................................109
Animal Model of Chronic Limb Epilepsy ........... ...............................................110
Seizure Analysis with Pairwise Granger Causality ......................................... 114
R results ........... ... .... ......... ....... .............. ...................... ........... 116
D discussion ........... .... ..... ......... ............................................126

7 SUMMARY OF RESULTS AND CONCLUSSIONS ................................................132

S u m m ary o f R esu lts............................................................................................... .... 13 2
Overall Conclusions......... .................... .. .... ..... ....... ........ 137



6









L IST O F R E F E R E N C E S .............................................................................. ..........................140

B IO G R A PH IC A L SK E T C H ............................................................................... ............... ..... 150









LIST OF TABLES


Table page

3-1 A table of the Pairwise Granger Causal influences calculated between the 5 neurons
in the sim ulation displayed in Figure 3-5 ............................................... ............... 53

3-2 Pairwise Granger Causal influences calculated between the 5 neurons in the
simulation displayed in Figure 3-5 after the use of significance thresholds,
conditional granger causality analysis, and removal of mediated influences .................53

5-1 Global culture causality changes post-tetnanization....................................................... 102

5-2 Tetanized channel total causality changes post-tetanization. ........................................103









LIST OF FIGURES


Figure page

2-1 Conditional Pairwise Granger Causality Example. ................................... ..................... 36

3-1 Topology diagram of a simple five neuron network...................................................46

3-2 Power spectra and cross-spectra calculated from the 5 neuron model simulation ............48

3-3 G ranger C ausality Spectra. ...................................................................... ....................49

3-4 The PGC results from a mono-directional simulation with the weight from neuron 4
to 5 varied from 0-75mV in increments of 5mV. ................................... ............... 50

3-5 Five neuron simulation.. ................................. ... .. ........... ........ ..... 52

3-6 Direct influences between N2 and N3 were recovered by both methods to allow a
comparison of Geweke's subtraction method to CGC. ................................................55

3-7 DGC values can be recovered at each of the synapses in the serial chain.......................56

3-8 100 N euron Sim ulation. ............................................................................. 58

3-9 A simulation to determine if changes in synaptic weight consistent with stimulation
timing dependent plasticity (STDP) at a synapse between two biologically relevant
neurons could be detected after a paired pulse stimulation was applied to the
neurons.......................................................... 60

3-10 Statistical Analysis of Plasticity Simulation........................................... ............... 61

4-1 Comparison of spike rate to Pairwise Granger Causality ............................................72

4-2 A comparison of spike rate and PGC analysis for enhanced channel 26 and depressed
channel 41. .......................................................................74

4-3 Surrogate analysis of the PGC analysis. .....................................75

4-4 Amplitude dependent plasticity (ADP) can be explored in-vitro using PGC.
According to amplitude dependent plasticity (ADP), changes in synaptic weight due
to correlated time dependant spiking are a function of initial synaptic strength. ..............79

4-5 Stimulus-Response difference plot of PGC where the EGC difference matrix is
collapsed by source.. ..........................................................................80

4-6 Stimulus- Source difference plot of PGC where the EGC difference matrix is
collapsed by response.. ........................ ...... ................................... .... .. ....81









4-7 Source-Response difference plot of PGC collapsed by probe before and after
application of tetanus. ......................... .......... .. .. ... ...... .. ............83

4-8 Source-response plots by spatial probe location..................................... ............... 84

4-9 Spontaneous Source-Response plot calculated from windowed bursting events
during spontaneous (no stimulation probes) activity. ................. .............................. 86

5-1 Topographical map of sinks and sources spatially laid out and superimposed on a
picture of the m icroelectrode array....................................................... ............... 96

5-2 Source response plot for culture 9059 tetanized on a sink............... ..........................99

5-3 Source-response plot for culture 9449, which was tetanized on a source. ....................101

5-4 Culture wide Amplitude Dependent plasticity............................................................103

5-5 Tetanized channel amplitude dependent plasticity. ................................. ...............104

6-1 Diagram of microelectrode placement in the rats brain..............................................111

6-2 Placement of the electrodes during surgery................................................................. 112

6-3 Example of the rat environment and recording chamber................................................ 113

6-4 Initial Pairwise Granger Causality analysis of four 60-second seizures for a single
an im al......... ....... ........... ................ ................................................ 1 14

6-5 Example of data and Granger analysis during the interictal (pre-ictal) stage................18

6-6 Beginning of the seizure in the high resolution movie classified as Stage 1.................19

6-7 Example of a directional transfer during Stage 2................................. ..................120

6-8 Example from Stage 3 -Late intra-hemisphere.. ..................................................121

6-9 Stage 4 Transition........... ...... ............................ ...... ...... .. .. ............. 122

6-10 Stage 5 Post transition ........................................... .......................... ............... 123

6-11 Quantitative analysis of the 5 seizure stages......................................... ............... 130









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ASSESSING CONNECTIVTY IN SIMULATED, IN-VITRO, AND IN-VIVO NEURAL
NETWORKS USING GRANGER CAUSALITY

By

Alex Joseph Cadotte

December 2007

Chair: Thomas DeMarse
Major: Biomedical Engineering

Since the early 1980s, interest in synchronization between regions in the brain has led to

the adaptation of forecasting measures such as Granger Causality from fields like economics to

neuroscience. The basic principle behind Granger Causality is that a directional causal

relationship can be inferred from a time series X to a time series Y when the introduction of

previous measurements of X reduces the variance of the error time series in the linear modeling

of Y. Granger Causality methods are used in the Neural Hybrid Computing Laboratory to

investigate functional relationships between neural units in simulation, in-vitro, and in-vivo.

Computer simulations using biologically plausible neural-network models have shown that

the direct component of pairwise Granger Causality (PGC) scales linearly with synaptic efficacy

within biologically relevant ranges. This suggests that the PGC values obtained from living

neural networks may be useful in calculating the underlying connectivity of living networks.

Thus, PGC was used in-vitro to identify connectivity within neural pathways in a living neural

network of 50,000 neurons grown on a microelectrode array (MEA). An MEA is an 8x8 grid of

electrodes embedded in a cell culture dish. Data recorded from two plastcity experiments using

these MEA cultures was analyzed using PGC. The resulting difference in PGC connectivity









before and after the induction stage of this experiment reveals that PGC was able to describe

plasticity within neural pathways in-vitro.

PGC was also applied in-vivo to spontaneous epileptic seizures recorded from a 32-

electrode array implanted in a rat's hippocampus. These rats were electrically stimulated causing

limbic status epilepticus and later began to develop spontaneous seizures consistent with

Lothman's animal model of chronic limbic epilepsy. Results from the analysis of spontaneous

limbic seizures reveal the dynamics of the directional causal influences during seizure.

Interestingly, they indicate a sudden change in direction of causal influence between the CA1

and dentate gyms regions of the hippocampus near the end of status epilepticus.









CHAPTER 1
INTRODUCTION AND BACKGROUND OF CAUSAL METHODS IN NEUROSCIENCE

Introduction

Multivariate time series analysis is increasingly used during the last 20 years in

neuroscience to determine the relationships between simultaneously recorded signals. Granger

Causality, a predictive forecasting tool originating from the field of economics (1), is one of the

latest examples of multivariate time series analysis that has been adapted for use with neural

systems (2). According to Granger Causality, a causal relationship is said to exist when, by

introducing previous measurements of the first time series into the linear regression of the second

time series, the error of the linear regression of the second time series is reduced. In essence, this

says that the second time series is better predicted with information about the first time series

because the second time series contains time delayed elements of the first time series when a

causal relationship exists. This is the basic principle behind the calculation of Granger Causality.

Expanding on this basic principle, the relationship between higher order multivariate systems can

also be explored. However, reducing the multivariate system to pairwise relationships to

determine causal relationships is often insufficient to sort out causal relations when those

relations are mediated through another element rather than direct. While it is necessary to use

Granger Causality to determine the interdependency and directionality of relationships (e.g.,

neural connections), Conditional Granger Causality is required to resolve whether relationships

(e.g., neural pathways) are direct or mediated. Conditional Granger Causality does this by

comparing error terms from bivariate and trivariate linear regressions. If the addition of a third

time series to the linear regression that is already known to influence a first time series does not

add to the prediction of a first, it can be inferred that its influence is already accounted for by

mediation through the second time series.









This body of work focuses on the use of Granger Causality based methods as a tool to

determine the effective connectivity in neural systems where the anatomical connectivity may be

unknown or unclear. Included is a review of current causal methods, an explanation of the

mathematical basis for Granger Causality, and four experiments where outputs from neural

networks, both simulated and living, are analyzed using Granger Causality to determine effective

connectivity. These include modeled, in-vitro, and in-vivo paradigms. First, a simple neural

network model (3) will be used to determine the relationship between Granger Causality and

synaptic weights used to drive the model. A modeled system is ideal because the synaptic

weights are known and can be compared with those weights recovered using Granger Causality.

Understanding this relationship is a prerequisite to experiments in living systems because it will

define what kind of relationship between Granger Causality and the underlying anatomical

synaptic efficacy underlying the living neural network.

Next, data from cortical cultures grown on microelectrode arrays will be the focus of two

in-vitro experiments in Chapters 4 and 5. The first experiment based on the results from the

modeling section determines the ability of using Granger methods to recover the effective

connectivity of a dissociated neural network. Granger Causality was used to measure changes in

effective connectivity observed after treatment with a protocol designed to induce plasticity (4).

Granger's results are compared to estimates based on changes in firing rate. The second in-vitro

experiment investigated Granger Causality's ability to provide information of effective

connectivity prior to experimental manipulations. Using this effective connectivity sinks, where

the balance of effective connectivity is directed inward, and sources, where the balance of

effective connectivity is directed outward will be identified. Sinks and sources will be

differentially subject to the same high frequency plasticity induction stimulation protocol, or

tetanization, with the prediction that tetanization of sinks would yield depression while









tetanization of sources would yield potentiation. This prediction is based on the differential

orientation of the connectivity of these pathways with the rest of the culture.

Finally, Chapter 6 examines the ability of Granger Causality to measure progressive

changes in effective connectivity during epileptic seizures measured in a chronic limbic epileptic

rat model. This data analysis will be conducted on in vivo 32-channel microelectrode arrays data

in rat hippocampus originally recorded by the Evolution into Epilepsy project team to study the

mechanisms of epileptigenesis. The goal of this work is to determine if Granger Causality

methods can successfully be used to describe the changes in effective connectivity before and

during seizures to learn more about in effective connectivity during seizure.

Review of Causal Methods in Neuroscience

Many experimental platforms are used in the neuroscience community to answer

scientific and practical questions about the behavior and function of the nervous system. As

these platforms have evolved over the last half a century, so to has the quality, content, and

availability of data sets from these systems. These platforms have been adapted to better answer

questions of causality and binding between various elements requiring multichannel data to be

acquired. Subsequently, the methods used to analyze multichannel data have evolved alongside

or even been borrowed from other disciplines, providing meaningful analysis of this data. Some

examples of these methods are cross correlation, coherence, mutual information, partial directed

coherence, the directed transfer function, and Granger causality. These methods fall into two

broad categories, those that are non-parametric (calculated directly from data) and those that are

parametric (the data is fit to a model and parameters from the model are used to calculate a

result). The following sections will provide a brief background and examples using each of these

methods.









Non-Parametric Methods

Cross Correlation

Cross correlation is one of the most basic and longest used algorithms that has been used to

determine the interdependence between two time series. The cross correlation function measures

the linear dependence between time series as a function of their time delay, r. A significant

value of the cross correlation function at a time delay r may indicate a causal relationship.

Consider two time series x(t) and y(t) that have been normalized to have a zero mean and unit

variance. The cross correlation function of these two time series is shown in equation (1.1),

where N is the length of the time series and ris the time delay.


1 Nr
C(r) x(k + r)y(k) (1.1)
N k=l

The output of the cross correlation function ranges from -1 to 1 for each value of r, where

-1 is complete inverse linear correlation and 1 is complete linear direct correlation. Values of r

where the cross correlation function is equal to zeros represent circumstances where there is no

linear correlation. In practice, a statistical threshold is calculated around zero perhaps using a

surrogate data set to determine when significant correlations exist.

The cross correlation function has been used in neuroscience for over 50 years, since it was

used to measure correlation between EEG signals (5). Use of correlation based methods is

widespread in neuroscience and a full elaboration of it use is beyond the scope of this research

(see (6-9) for an excellent review). This measure has more recently been used to describe

correlations among neurons (10-17) and specifically in studies of plasticity (4, 18-20). For

example, Jimbo (4) found that neural activity that was highly correlated with a tetanic

stimulation pulse train tended to be potentiated (an increase in effective connectivity). In

contrast, neural activity that was weakly correlated with tetanic stimulation typically resulted in









depression (a decrease in effective connectivity). Jimbo suggests that the strength of correlation

between these channels during tetanization may be representative of the total number of synapses

in the pathway between the electrodes. Tateno and Jimbo (21) use a normalized cross correlation

analysis for all neuron pairs during a similar plasticity experiment. The difference in cross

correlation (post-tetanization correlation pre-tetanus correlation) between all neuron pairs after

tetanic stimulation highlighted changes in the correlation that was indicative of plasticity within

a living neural network. Segev et al (14) has used cross correlation to describe connectivity

patterns in multichannel data from similar living neural networks during synchronized spiking

events known as bursts. Segev demonstrated that bursting activity contains statistically separable

subgroups that he suggests may be a template for storage and retrieval of information.

Overall, cross correlation has been one of the primary methods used in neuroscience to

demonstrate relationships especially in multichannel data. It has the advantage of being a widely

understood, straightforward, and simple method. However, it does have some limitations. The

results from cross correlation can often be subject to interpretation and be ambiguous (7).

Additionally, for results to be statistically significant often a large amount of data is necessary.

Directional information can be ascertained using cross correlation but this can become

problematic and ambiguous over large data sets because manual interpretation is often required.

Additionally, cross correlation, as presented here, is strictly a time domain measure. To examine

correlations in the spectral domain known as coherence additional manipulations are required.

Coherence

Coherence is the linear correlation between two time series as a function of frequency.

Also known as squared coherence or the coherence spectrum, this measure is simply the

normalized Fourier transform of equation (1.1). Skipping several steps of this derivation, the

resulting equation for squared coherence is shown in equation (1.2).









S (f)
C(f) = (1.2)
S(f ) S (if)

The spectral matrix S, used to calculate coherence, is non-parametric calculating using

Welch's method to create a periodogram. An estimate of the average spectra can be estimated

over several trials of event related data by averaging the resultant periodograms. However, this

method only provides an estimate of coherence and significance would need to be established by

calculating confidence intervals. The resulting coherence values for each frequency vary from 0

to 1, where 0 would indicate no correlation at that frequency and 1 would indicate maximum

linear correlation at that frequency.

Like correlation, coherence is also widely used in neuroscience. Recent reviews detail the

use of coherence in neuroscience (9) and in human epilepsy (22) over the past 40 years. While

coherence shares many of the weaknesses of correlation, a critical weakness of this method is the

lack of directionality. This specific weakness spurred the development (23) of a method called

directed coherence also known as Gersch causality (24) that was initially used identify epileptic

foci using three electrodes. Gersch stated "one channel is said to drive the other channels if the

first channel explains or accounts for the linear relation between the two." This method has also

found various neural applications (25) including EEG (26) (27, 28), fMRI (29) and spike trains

(27, 30). Unfortunately this method has been shown to be sensitive to noise contamination (25),

which can potentially lead to misidentification of directional influences. While coherence is

often calculated directly from data (non parametrically) it can also be calculated using

parameters from autoregressive models fit to the data (31). These parametric approaches will be

elaborated later in this review. Mutual information, a non-parametric method that uses

probabilistic estimates, is discussed next.









Mutual Information

This measure falls under the auspices of information theory and probability. The primary

measure used in information theory is entropy. Entropy is defined for a variable X as the sum

over all possible outcomes M ofpi (probability that an outcome occurs) multiplied by the base 2

logarithm ofpi as shown in equation (1.3) as Shannon entropy. Consequently, the Shannon

entropy will be high for a uniform system where all states have an equal probability and lower

for a system with a Poisson distribution where certain states have much higher probability.

M
I= p, log, p, (1.3)
1=]

It is this concept that is applied to the calculation of Mutual Information (MI) between two

variables. MI is defined as:


M -- =-p, log p' (1.4)
P, P,

where pi is the joint probability that the outcomes for x and y occur. Essentially, this measure

describes how much extra information is known about one signal by knowing the outcomes of

the other one. MI will equal zero if the two variables are independent because pi would be equal

to pipj.

Mutual Information has been used to estimate the relationship between dynamic stimuli and the

response in corresponding sensory areas in-vivo for various animals (32), including the response

motion sensitive neurons in the visual cortex of primates to visual stimuli (33). It has also been

used to estimate synaptic efficacy in vitro between pyramidal neurons in-vitro (34). Joint

probability density can also be estimated between spike trains using Mutual Information (17, 35).

Like cross correlation, Mutual Information also requires large amounts of data (9) in order to









produce a reliable representation of the probability distribution. Thus, the use of Mutual

Information on experimental time-scales can sometimes be problematic.

Parametric Approaches

Likelihood Methods

Maximum Likelihood is a parametric approach that creates statistical models of a system

for analysis. A parametric probability model is assumed for a process under study. Methods

associated with Likelihood methods are very case dependent of the system being analyzed and

the statistical model chosen (17) and cannot be elaborated on completely within this review. The

likelihood is represented as joint probability density of the experimental data as a function of the

unknown or free parameters in the model. Thus, the models parameters are estimated from the

experimental data. If the experimental data fits the chosen model well, this is thought to be an

optimal way of representing the experimental process (17). This method has been used to

analyze the interactions of simultaneously recorded spike trains in the cat's auditory system (36,

37) and place cells within the hippocampus of rats (38). While this methods shows promise,

extreme care must be taken in choosing an appropriate model to estimate the probability

distribution. Similar to Mutual Information, this process often requires large amounts of data

that can make analysis of experimental systems problematic.

Methods using Autoregressive Modeling

Several parametric methods make use of autoregressive modeling (AR) in order to assess a

system's causality. These methods, including Partial Directed Coherence (PDC), the Directed

Transfer Function (DTF), and Granger Causality (GC) each use different parameters from an AR

model fit to the system in question. Additionally, it should be noted that the parameters of an

AR model have been used on their own to assess changes in neural systems over time (39, 40).

For example, Sanchez et. al. specifically highlights the changes in AR coefficients calculated









from a spiking time series over the development of spontaneous limbic seizures in-vivo. A

general form for the AR model is shown in equation (1.5), where Xis a vector consisting ofp

time series, A is ap xp coefficient matrix and E is the uncorrelated noise vector with a

covariance matrix of Y.


X(t)+ fA(m)X(t- m)= E(t) (1.5)
m=l

AR modeling makes use of the autocorrelation function to model the current value of a

variable from past instances of that same variable (p=l). A multivariate AR (MVAR) model

(p>l) uses this same framework to model the current value of a variable using not only past

instances of that same variable, but past instances of the other variables in that multivariate

system. It is easy to see why autoregressive modeling has become such a popular tool in

neuroscience where multichannel recordings are common. A more comprehensive mathematical

review of AR modeling will be expanded upon in Chapter 2.

Several more advanced methods collect information from the AR model and perform

operations on them to determine a measure for causality or direction. Often these methods

employ two or more AR models, which are calculated including and excluding a variable of

interest. This methodology calculates a particular variable's effect on AR model coefficients

and linear regression error that is used to calculate a measure of causality.

Partial Directed Coherence, the Directed Transfer Function, and Granger Causality

Three methods have emerged that use AR models to determine directional connectivity.

Partial Directed Coherence (PDC), the Directed Transfer Function (DTF), and Granger Causality

all use aspects of an AR model including and excluding a time series hypothesized to have a

driving relationship with a second time series. This basic principle is common to all three

methods. While these methods are often considered to be the same or similar, there are subtle









differences between these methods. For example, PDC and DTF are exclusively frequency

domain measures that rely on the Fourier transform of the AR model resulting in new frequency

domain coefficients. In contrast, Granger Causality has formulations in the time domain and the

frequency domain. While PDC and DTF are closely related, both being derived from this new

coefficient matrix, DTF differs from PDC in that the coefficient matrix is inversed,

H, (f) = A (f), and the DTF is squared when compared to the PDC. The equations for PDC

and the DTF are shown below in equations (1.6) and (1.7) respectively. Notice the similar

formulation for both equations.


ifr, (f)= (1.6)
k, A,(f)f




H,, (f)2
3,l (f ) a(1.7)


Another major difference between Granger Causality and the other two methods is that

Granger Causality is a statistically relevant quantity. For example, the time domain version of

Granger Causality, shown in equation (1.8) is the ratio of the variance from two error time series.

The DTF and PDC are ratios of spectral coefficients from the AR model, which don't have any

meaningful statistical interpretation. Granger Causality is simply a more statistically rigorous

measure. This does not mean that the DTF and PDC are not useful tools. In fact, these methods

have produced interesting results in the analysis of neural time series including EEG (41-43),

neural simulation (43, 44) and living neural networks (45-47). Fanselow et al. (46) uses PDC to

reveal causal influences from the sensory cortex to the thalamus during rhythmic activity

associated with whisker twitching behavior in rats. This suggests that sensory cortex driven

thalamic bursting, associated with whisker twitching behavior during exploratory behavior, may









prime the thalamo-cortical loop for sensory signal detection. Schelter et al (43) calculated PDC

between EEG and EMG signals and determined that the motor cortex likely participates in the

generation of essential tremor. Ginter et al uses the DTF to determine if the signature of

imagined movements of the index finger can be detected reliably in the cortex using EEG. This

information is relevant in the creation of a brain machine interface (BMI). This research area

seeks to help amputees and those suffering from various forms of paralysis (due to spinal cord

injury for example) control prosthetic devices by imagining movement of the affected limb.

These are just a few examples of the use PFC and the DTF. These methods were often

developed to improve spectral information content but are only empirical measures that don't

have a firm statistical foundation. Thus, they often are used in parallel with the Granger

Causality metric.


var(ex(t))
Fyx = Invar(x(t)) (1.8)
var(rx(t))

Granger Causality has found similar applications within neuroscience. The original

concept of Granger causality has played a significant role in the field of economics since the

1960s. Within the last 15 years, it is now increasingly used in neuroscience to understand the

dynamics of brain circuitry. In addition to the time domain version of Granger Causality, shown

in equation (1.8), Granger Causality also has formulations in the frequency domain and can be

used to separate mediated and direct influences (31). The math behind these concepts will be

explored further in Chapter 2.

There are a wide variety of systems that Granger Causality and associated methods have

been applied to with success. For example, Zhu et al (48, 49) explores causal interactions

between neurons during learning in the primary motor cortex of rhesus monkeys. Specifically,

Zhu monitored the interactions between 8 electrode channels in the primary motor cortex. The









monkeys used in this experiment were trained in a reaching task prior to surgery implanting the

electrode array into motor cortex. A string was then attached to the arm of the monkey that

interfered with the completion of previously learned motor task. The monkey adapts or

essentially learns to complete the same task with this perturbing string attached. Recordings

from the 8 electrodes during this re-learning phase were associated with changes in the strength

and direction of causal interactions over time. Coupling strength between the electrodes

increased during the learning phase, but was attenuated to pre-learning levels once adaptation

had been achieved. In this experiment the network appears to be essentially rewired before and

after learning and may reflect plasticity associated with the re-learning process.

Functional Magnetic Resonance Imaging (fMRI) allows for three-dimensional brain

activity mappings to be collected during a behavioral task over time. The dynamics observed

during fMRI recordings can be used to determine functional connectivity between brain areas

using Granger Causality (50, 51). An fMRI time series can be considered a filtered and sampled

version of a local field potential (LFP) generated from the observed neuronal population. The

fMRI time series for each area is functionally a measure of the blood oxygen levels in that area.

This can be used as an indicator for brain activity based on consumption of blood oxygen.

Casual analysis of a simulated fMRI time series between two brain areas shows a nearly identical

relationship to that of the original LFPs generated in the simulation (50). A cognitive task where

various stimuli were delivered requiring a "left" or "right" response was observed using fMRI

(50) and analyzed using Granger Causality. Frontal areas were found to drive the parietal areas

during these tasks. These results were interpreted as neural correlates of executive control and

working memory. A similar study (51) showed that pre-motor areas could be used to predict

activity in the motor cortex.









A variety of neural systems are addressed in Kaminski et al (52), including simulated local

field potentials, simulated spike trains, human sleep analysis, and visuomotor integration in

macaque monkeys using LFPs (local field potentials). Analysis of human EEG patterns during

sleep reveals a causal influence from anterior P3 and P4 and the posterior region P3 electrode

(the international 10-20 system was used for electrode placement). These patterns were found to

be consistent with anatomical projections from the pre-frontal cortex. Additionally, these effects

were observed to fade during the transition from sleep to wakefulness. In-vivo LFP data was

analyzed from macaques that were trained to perform a visuomotor pattern discrimination task.

The data was collected from 15 bipolar twist electrodes over 888 trials of a go no-go task.

Using these trials to reduce the length of the data sample necessary, a 50 ms moving window of

Granger Causality values was used to reveal causal interaction through the 500 ms trials. This

analysis reveals that two cortical areas, a striate and prestriate area, shown to be coherent during

the task, show a directional transfer from the striate to a prestriate which is followed by feedback

in the opposite direction 50 ms later. Note that these directional interactions were unattainable

using coherence. A more detailed analysis of this experimental paradigm can be found in

Brovelli et al (53) and Chen et al (54). In Chen et al (54) a Spectral (15-30Hz) Conditional

Granger Causality analysis was used to show that some cortical areas were mediating effects

from other areas while other relationships were direct. Note that a conditional analysis is

necessary to ascertain when mediation occurs in certain circumstances. Kaminski et al also

introduces a method to address the statistical significance of results using surrogate data analysis,

a method that will be used in all studies in this research. Additionally, Kaminski's methods for

preprocessing point process data (A first order Butterworth set at 10% of the sampling

frequency) for causal analysis between simulated spike-trains are used in this research.









Seth and Edelman (55, 56) employed granger causality to distinguish patterns of

connectivity and causality within complex neural network simulations. Seth (56) defines a set of

tools to analyze the output of causal analysis of a multivariate system. While calculation of

pairwise granger causality in a large neural system can be straight forward, making sense of the

large amount of data resulting from these data sets can be challenging. This paper defines

several concepts that can help reduce the complexity of results. These include causal density,

causal flow, causal reciprocity, causal disequilibrium, casual sources, and causal sinks. Causal

density is defined as "the fraction of interactions among nodes that are causally significant".

Causal flow is a measure of the difference between casual inflow and causal outflow from a

node. Causal reciprocity is a measure of feedback between nodes. Causal disequilibrium refers

to an imbalance in reciprocity and flow between nodes. Causal sinks refer to nodes where the

balance of information flows into the node, while causal sources refer to nodes where the balance

of information flows out of the node. These basic definitions help lead to tools that can be used

to understand the effects of causal flow within a network. Seth demonstrates the use of the tools

in a 32-neuron neural network model of target fixation using a head/eye system. Results from

this model and analysis using these tools allow new hypothesis including being able to predict

the effects of "lesions" within simulated networks. Seth's further work with Edelman (55)

includes the idea of a causal core within large neural populations during behavior. A causal core

is defined as neural interactions within the network present when a reference neuron, such as an

output motor neuron, fires. This idea allows the linkage of neural interactions with specific

outputs from the network. Seth and Edelman use the Darwin X system, an autonomous robot

with a simulated neural network brain based on cortical and hippocampal anatomy. The robot

performed a dry version of the water maze test for rates where the robot's goal was to find a

hidden platform in a static environment with spatial clues. Over time the robot learned to find









the platform more quickly similarly to rats that have performed the water maze. The behavioral

output from the Darwin X system was analyzed using the causal core idea and Granger

Causality. Consistent causal cores could be identified for specific outputs of reference neurons.

Changes in the causal core during learning sequences were also observed. The causal core was

observed to reduce in size and become more refined as Darwin X learned to move more directly

to the hidden platform.

These examples show that Granger Causality has proven to be a useful tool for uncovering

effective and functional relationships in various neural systems. These tools and associated

methods will be used in several experiments in simulated, in-vitro, and in-vivo neural networks.

We will begin with a more comprehensive review of the autoregressive modeling and the

properties of Granger Causal tools presented in the next chapter.









CHAPTER 2
MATHEMATICAL BASIS FOR GRANGER CAUSALITY

Mathematical Basis for Time Domain Measures

Granger Causality

The goal of Granger Causality (GC) is to provide a strong mathematical basis to determine

causal influence and direction of neural interaction. The basic idea can be traced back to Wiener

(57). He proposed that, for two simultaneously measured time series, one series can be called

causal to the other if we can better predict the second series by incorporating past knowledge of

the first one. This concept was later adopted and formalized by Granger in 1969 (1) for linear

regression models of stochastic processes. Specifically, if the variance of the prediction error for

the second time series at the present time is reduced by including past measurements from the

first time series in a linear regression model, then the first time series can be said to have a causal

(directional or driving) influence on the second time series. The advantage over cross correlation

metrics is that GC is more robust with respect to changes in overall rate of activity, provides a

measure of the "causal" influence, is directional, and often requires less data for analysis. A

description of this technique is provided in the following sections describing autoregressive

modeling as well as time-domain, spectral, and conditional methods. A more detailed review by

Ding et al in a recently published handbook on causal methods can be found here (2). The

purpose of this chapter is to describe what Granger Causality does and does not do. This is

accomplished by describing how GC produces a measure of effective connectivity and

discussion of several of its inherent weaknesses.

Autoregressive Modeling

Autoregressive (AR) modeling of neural time series is the foundation of parametric

Granger Causality methods. Time series in multivariate neural data are typically recorded from









multiple electrodes or channels. While AR models are often presented in a theoretical context,

the examples reported here will be provided in the context of a multichannel recording from a

neural process. Consider the multivariate random process, X(t), consisting ofp independent

channels of neural data.


X, (t)

X(t) =X2 (t) (2.1)

X, (t)

A recording of time series from these channels would be considered a single realization of

this neural process. Multiple realizations of the neural process are advantageous to creating a

more accurate model describing this process. X(t)is described using an AR model of model

order m assuming that X(t)is a stationary process.

X(t) + A(1)X(t -1) +...+ A(m)X(t -m)= E(t) (2.2)

A(i) are p by p coefficient matrices where i = 1, 2,..., m and E(t)is a zero mean

uncorrelated noise vector with a covariance matrix of Y. A(i) and Y are unknown matrices that

can be estimated from the realizations of X(t). This is done by right multiplying Equation (2.2)

by XT (t k), where T indicates a transposed matrix. The expectation is then taken on the

resulting equation yielding the Yule-Walker equations for k = 1, 2,..., m containing a total of

mp2 model coefficients to be solved for.

R(-k)+ A(1)R(-k +1)+....+ A(m)R(-k +m)= 0 (2.3)









R(n) =< X(t)X (t + n) > is the auto covariance function of X(t). Note that lag n is

n = \-k + 1 ranging from 0 to m-1. The auto-covariance function is then calculated from each

realization, x(t),of X(t) of length N.

1 N-n
R(n) x(i)XT (i + n) (2.4)
N- n ,

If multiple realizations of the data are available the auto covariance functions are averaged

across realizations. Once the auto covariance function has been calculated the mp2 model

coefficients of the A(i) coefficient matrix can be solved for because there are mp2 equations

within equation (2.3). There are several methods that can be used to solve this including simply

solving for each of these coefficients or using methods such as the Levinson, Wiggins, and

Robertson method or Morf s method, used for this analysis. Once A(i) is known, the noise

covariance matrix Y is obtained using the following relationship.


S= R(0) + A(i)R(i) (2.5)


This preceding section describes the estimation of a MVAR for any model order m.

However, it is necessary determine the correct model order for use with each system being

studied. The correct model order can be determined using several methods including the Akaike

Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The BIC is more

often used for neural applications as it compensates for the large N (number of data points)

common to neural data sets. Thus, the BIC will be the primary method used to determine model

order in these studies.

2mp2
AIC(m) = -2log[det(X)] + 2- (2.6)
N










2mp2 logN
BIC(m) = -2 log[det(X)] + '(2.7)
N

The BIC is plotted as a function of the model order m. The correct model order usually

corresponds to a minimization of BIC. However, when the model order becomes too large this

may result in excessive computation time. Often, a smaller model order with a similarly

minimized value of BIC is used rather than the absolute minimum BIC for this reason.

The resulting time domain MVAR model is the basis for the calculation of both time and

spectral domain granger causality, which will be discussed in the following section.

Time Domain Granger Causality

With the core AR modeling tools now fully described, Parametric Granger Causality can

be calculated from the resulting models. GC methods make use of the variance of prediction

errors from various combinations of AR models. Pairwise Granger Causality (PGC) is used to

investigate the directional interactions between two time series. Two AR models are required to

determine for each direction of influence. Consider two time series X(t) and Y(t) where separate

AR models are created to predict current value of the X series from m (a previously determined

model order) previous X values and:

m
X(t)= b((j)X(t j)+ s(t), =var(E(t))


Y(t)= byr(j)Y(t j)+(t), F, =var(/(t))
J-1

The variance of the error series e(t), ,1, is a gauge of the linear prediction accuracy of

X(t) as F, is for Y(t). Now consider a bi-variate multivariate autoregressive (MVAR) model,

W(t), where both X(t) and Y(t) are calculated from the previous values of the X and a Y time

series:










X() = ia(j)X(t- j) + j)Y(t j) + (t)
W(t) = J (2.9)
Y(t)= ax(j)X(t- j) + (j)Y(t j) + (t)
J1= J1=

The variance of the new error series is a gauge of the prediction accuracy of the new

expanded predictor.

(e2 2) var(q(t)) cov(q(t),y(t))'
Y2 F2) J cov(r(t), (t)) var((t)) (2.10)

Based on Wiener's idea, Granger (1) formulated that if the X prediction is improved by

incorporating past knowledge of the Y series, the Y time series can then be said to have a

Granger-causal influence on the X time series. This is the basis for the time domain version of

Granger Causality where the variance of the linear prediction error of X alone, Y1, is compared

to the variance of linear prediction error of X including Y, 2 :


Fr=x = In (2.11)
Y2


Note that when Y, = 2 (i.e. the linear prediction error is not improved by including Y)

that this relationship will yield a PGC value of zero. Driving in the opposite direction is

addressed by simply reversing the roles of the two time series. It is clear from this definition that

timing plays an essential role in directional causal influences.

Spectral and Conditional Granger Causality

Two additional developments of Granger's causality are important for the proposed

research. First, natural time series, including ones from neurobiology, contain oscillatory aspects

in specific frequency bands. A spectral representation of Granger Causality was developed by

Geweke (58), allowing casual interactions to be quantified at specific frequencies. Second,









pairwise Granger Causality can often yield false direct influences due to parallel driving by or

mediation through elements not included in the AR model. When data from at least 3 or more

time series is available, such as from neurobiological multi-electrode arrays, conditional Granger

causality (54, 59) can be used to address these limitations. These methods are described in the

following section.

Spectral Granger Causality

Geweke (58) published his groundbreaking paper in 1982 where he identified a novel way

to decompose the time domain causality measure into its spectral content. His method compares

the intrinsic power (defined as the power of a time series generated by the stochastic processes

intrinsic to that time series) of one time series and the total power of the same time series, which

may contain a causal influence from a second time series, in the frequency domain. When there

is no causality between the two time series (i.e. the total power is equal to the intrinsic power)

the log ratio of the total to intrinsic power is equal to zero. The frequency domain function

W(f) is obtained by taking the Fourier transform of the previously described MVAR W(t):


A(f)W(f)= E(f) (2.12)


A(f) is the spectral coefficient matrix and E(f) is the spectral noise vector resulting from

the Fourier Transform performed on W(t). Rearrangement of (2.12) by setting the matrix

inversion A(f)- equal to H(f) (commonly known as the transfer function) yields:


W(f)= H(f)E(f) (2.13)


The spectral matrix S(f) can now be calculated that contains the auto spectra and cross

spectra of W(t).









S(f)= H(f) 3H*(f) (2.14)

H (f)is the transpose and conjugate of H(f). X3 is the covariance matrix of the noise

terms from W(t).

This spectral matrix, containing the total power of the X time series, is retained. Spectral

Granger Causality is based on decomposing the total power (variance) at a given frequencyffor

one time series into the sum of an intrinsic part and a causal part from the other series. A new

spectral matrix is calculated in the same way except that both sides of equation (2.14) are left

multiplied by Geweke's normalization transform:

( 1 0 ( 1 0
P= _Y2 cov(r(t),y(t)) 1 (2.15)
Y 2 var(q(t))

Geweke's normalization transform isolates the intrinsic power of X by allowing the

intrinsic power component of the total power to be calculated separately from the causal power

component of X(59). The X component of the new transfer function, shown in equation (2.16) is

then used to calculate the intrinsic power of the x time series.


H,(f)= H (f)+ 2 H (f) (2.16)

Taking the logarithm of the ratio of the total power over the intrinsic power gives the

Granger causality spectrum.

K ( S ((f) "
Fy, (f)= In / (f)2 = (f) (2.17)









From the value of the causality spectrum, one can evaluate how much of one signal's

variance atf is explained by the second signal exerting causal influence. Geweke showed that

the integration of this spectral quantity over frequency is the time domain Granger causality.

Conditional Granger Causality

Highly multivariate data can be analyzed by PGC, reducing the problem to a bivariate case.

This approach is straightforward and often effective for simple systems. However, this

methodology can yield distorted results in more complex systems. Consider the following two

situations. In one case, causal influence from Z to X is mediated through Y. For the second case,

Z influences X through Y, but Z also has a direct influence on X. These two cases are

indistinguishable using a bi-variate GC analysis; in fact the analysis will identify both cases as

the second case. Conditional GC (CGC) is an algorithm that can be used in both in the time

domain and in the frequency domain for three time series (31, 60) to overcome this shortcoming.

We will only describe the time domain version of CGC, as the core idea of spectral CGC is very

similar to the previously described Spectral Granger Causality.


X(t) = A (j)X(t -j) + A(j)Y(t -j) + A (j)Z(t j) + A(t)
J1 J1 J1
p p p (2.18)
U(t)= Y(t)=IA,(j)X(t-j)+A,(j)Y(t-j)+A(j)Z(t-j)+(t) (2.18)

Z(t)= A (j)X(t j)+ A (j)Y(t j)+ A (j)Z(t j) + v(t)
J-1 J-1 J-1
Time domain CGC compares the prediction of X including Y in a bi-variate MVAR model

(from the previously described W(t)), with the prediction of X including Y and Z in the tri-

variate MVAR model U(t) shown in figure 2-1..


( YT YT x' var(A(t)) cov(A(t),p(t)) cov(A(t), v(t)))
u = Yx y FY Z = cov(A(t), p(t)) var(/(t)) cov(/(t), v(t) (2.19)
Yyz Azz cov(A(t), v(t)) cov(U(t), v(t)) var(v(t)) )









a) b)







x x

Figure 2-1 Conditional Pairwise Granger Causality Example. If Pairwise Granger Causality were
applied to determine the connectivity of both of these network configurations the
results for both would resemble panel a). Using Pairwise Granger Causality alone, it
is not possible to differentiate between these network configurations. Conditional
Granger Causality is required to determine if the connection from Y to X is real.
Conditional Granger Causality does this by determining if Y affects X directly or if
the effect of Y mediated through Z by determining how well X is predicted by Z with
and without the influence of Y.

The influence of Z on X is entirely mediated through Y when the prediction of X is not

improved in the trivariate MVAR model over the bivariate MVAR model. Specifically, CGC is

calculated by taking the log of the ratio of the variance of the prediction error of X from W(t) to

the variance of the prediction error of X from U(t). This is formalized in the equation for CGC:


Fzx = In (2.20)


Conditional Granger Causality calculated using this equation is greater than zero when

some of the power from Z is directly causal on X. The result would be equal to zero when the

influence of Z on X is entirely mediated through Y. Thus, this method allows the two examples

stated earlier to be easily differentiable. While Conditional Granger Causality addresses specific

deficiencies in the original formulation of Granger Causality, there remain several further

weaknesses that need to be considered when drawing conclusion from a causal analysis.









Limitations of Causal Methods

Since the time of Aristotle, philosophers have reached little consensus on a common

definition for the concept of causality (61). However, causation is a concept critical to many

disciplines including law, medicine, and the sciences. A functional definition of this concept is

necessary to develop tests of causality that are useful in the context of these disciplines.

Interestingly, the debate over the definition of causality has provided insights into several

weaknesses of causal methods including Granger Causality that must be considered when

designing sound experiments.

Clive Granger best describes the inherent limitations of his original method in his accounts

of the origins of Granger Causality (62) (61). He describes how several of his peers noted that his

method "is not real causality, it is only Granger causality". Granger also describes several

axioms that must be assumed to be valid. The first among these is that "the past and present may

cause the future, but the future cannot cause the past" (61). Second, everything in the universe is

described and this set of information includes no redundant information. In this hypothetical

universe, it would be easy to talk about the relationship between a variable X and a variable Y.

However, these assumptions are difficult to satisfy in real systems. In reality, causality

algorithms can be effected by many factors including sampling rate, time scale selection,

seasonal influences, signal to noise ratio, variable selection, variable availability, and even

chance.

Time scale and sampling rate are two variables that can have intertwined yet diametrically

opposed effects on a causal analysis. Selection of time scale and sampling rate is dependent on

the processes being observed. If the sampling rate is too low, it can be difficult to observe time

delayed influences in the system because the system may be interacting at time scales of a higher

frequency than the sampling rate. Conversely, if the sampling rate is too high it can be









computationally intractable to model far enough into the future (i.e. high model order m) to

capture interactions. Additionally, it is possible that two time series may interact differently on

two largely different time scales that may require separate analysis. These examples demonstrate

that proper consideration must be given to selection of time scale and sampling rate depending

on the system being analyzed. The data in these experiments are often recorded at very high (up

to 25kHz) sampling rates that must be low passed filtered to provide a proper sampling rate and

time scale to study causal interactions. All data is down sampled to 1kHz prior to analysis.

Spike train point processes are further low pass filtered to 50Hz (10% of nyquist for 1kHz) to

create a continuous time series so that AR models can be used.

It is also not possible to describe all variables in an experimental system. Consequently,

unobserved time series are a factor that must always be considered when talking about causal

influences. Implementation and subsequent interpretation of Granger Causality inherently

depends entirely on the availability and selection of variables for analysis. Specifically, the

causal relationship between two neurons may be influenced by a third and likely many other

neurons that are not recorded from (e.g., a neuron too distant from an electrode to be measured).

The likelihood of this situation increases with the complexity of the system being observed.

Consequently, the contribution of these unobserved neurons would not be accounted for. This

can have several effects on the relationships between observed time series. One possibility is

that a common driving input at differing time delays could create a false causal influence where

no real connectivity even exists. A priori knowledge of possible anatomical structure can help in

this type of situation.

So called seasonal influences also present an interesting possibility. Two time series may

be causal because of a similar common driving by an unobserved seasonal process.









Subsequently, the causality between these time series may wax and wane dependent of this

seasonal influence or possibly dissapear altogether. Another possible scenario is that the

influence between two time series could include mediated influences that cannon be conditioned

out using previously mentioned conditional methods. This would tend to distort the magnitude

of the influences reported from analysis. This is a likely possibility in complex neural systems

that cannot be easily remedied. This problem impacts not only GC analysis but also every other

multivariate statistic used in neuroscience today. Thus, it is clear that results from this type of

analysis could potentially suffer from poor experimental designs where the important variables

are unobserved or uncontrolled.

Additionally, there is another specific weakness of GC relating to the analysis of point

processes. Often, the output from an observed neuron is described in terms of when that neuron

fires action potentials. Point processes are not a continuous function over time. AR models can

only describe continuous processes. In order to compensate for this, spike files from individual

neurons must be low pass filtered prior to causal analysis. Subsequently, high frequency

information above the cutoff for the low pass filter is not included in analysis. However,

unpublished results from a non-parametric version of Granger Causality for point processes

being developed by Mingzhou Ding and colleagues are very similar to parametric results.

Specifically, the non-parametric method does not show any significant interactions over the

cutoff of the low pass filter used for the parametric method.

These strengths and weaknesses of Granger Causality must be considered carefully before

application to neural systems. In the following chapters, these methods will be used to describe

the effective connectivity using the observable outputs of these systems to better understand the









underlying systems. These weaknesses will be carefully considered during experimental design,

analysis, and subsequent interpretation.









CHAPTER 3
RECOVERY OF SYNAPTIC WEIGHTS FROM SPIKING ACTIVITY USING GRANGER
CAUSALITY

Introduction

Recent advances in multichannel recording techniques enable simultaneous access to the

electrophysiological activity of hundreds to thousands of neurons. These advances allow

investigators to address one of the primary challenges in neuroscience, linking the structure of a

network to its potential functions. For example, in cognitive neuroscience, by recording the

activations of brain structures during various behavioral performances, investigators now have

the technology for understanding how different brain structures interact to produce behavior.

This experimental advance has been supported by a simultaneous advance in the quality of

analytical tools that quantify neuronal interactions, such as the classical cross-correlation (14, 18,

21, 63-65) and coherence functions (66). At their core, however, these methods do not provide

directional or "causal" relationships between different channels whether those channels are

recorded from individual neurons or from different brain regions. In contrast, Granger Causality

(GC) (1) has emerged in recent years as a technique of choice to provide a mathematically sound

method for estimating causal relations.

Pairwise Granger causality (PGC) was initially developed in the economics community to

describe and quantify the "causal relationship" between data from two different economic time

series. The foundation for this technique can be traced back to Wiener (57), who proposed that

for any two simultaneously recorded time series, one series could be called causal to another if

incorporating past knowledge of the first time series permits more accurate prediction of the

second series. Granger formalized this idea in the context of linear regression models of

stochastic processes (1). Specifically, consider two simultaneously recorded time series xj, x2,

x3... xn and yl, Y2, y3... yn. Suppose one would like to construct a linear predictor of the current









value of the x series based on m prior values: x, = alxl + a2xz2 + amn-m + En. This is nothing

more than a single variable autoregressive model in which standard procedures can be applied to

yield the model order and model coefficients, where the variance of error series, En, is a gauge of

the prediction accuracy. Now consider a predictor of the current values of the x series by

including both the previous values of the x series and the previous values of they series, namely:

Xn blxn-i + b2Xn-2 ... + bmxn-m + cyn-1 + C2yn-2 ... + cn-m + qn. The variance of the error series,

rq, is then a gauge of the prediction accuracy of the new expanded predictor that includes

information about they series. Based on Wiener's idea, Granger formulated that if the variance

(r7i) variance (En) is less than one in some suitable statistical sense (i.e. the prediction ofx is

improved by incorporating past knowledge of y) then we say that they series has a causal

influence on the x series.

The concept of Granger causality has played a significant role in the field of economics

since 1960s. Only recently has its potential for understanding the dynamics of brain circuitry

been realized in neuroscience. For example, Zhu et al (48, 49) explores causal interactions

between neurons during learning in the primary motor cortex of rhesus monkeys. Fanselow et al.

(46) reveals causal influences from the sensory cortex to the thalamus during rhythmic activity

associated with whisker twitching behavior in rats. It is expected that Granger causality based

methods will experience significant growth in applications to a wide variety of neural systems

where information transfers exist among individual neurons or different brain areas, including

data from neural simulations, in vivo electroencephalogram (EEG), and in vivo microwire arrays

(27, 30, 37, 43-47, 49, 50, 52, 56, 67-74).

These studies have mainly focused on revealing structures of interaction in the recorded

circuits. The physiological meaning underlying the statistical values of Granger causality









remains unexplored. The goal of this study is to examine whether values of Granger causality

can be used to recover synaptic weights between interacting neurons in a complex network.

While intracellular recording methods are able to discern the synaptic weight from postsynaptic

potentials, they cannot reveal network level activity in vitro and are of very limited use in vivo.

Thus, to understand the synaptic relationships that regulate network activity, a technique to

reconstruct the weights from only spike train outputs must be devised.

For a simple two-neuron network in which neuron A is synaptically connected to C, our

first result is to show that the aforementioned PGC can easily discriminate a direct causal

relationship from A to C when the strength of that coupling is even moderately high and, more

importantly, the values of PGC can be related to synaptic efficacy. For more complex networks

the pairwise approach encounters significant methodological limitations when direct, mediated,

and serial influences exist between neurons. For example, consider a three-neuron case in which

neuron A is synaptically coupled to a mediating neuron, B, who is then connected to neuron C.

First, PGC would identify an erroneous causal connection between A and C, even though this

pair is not directly coupled. The reason is that the activity of A, does in fact, causally influence

the activity of C, but does so only through the mediating relationship involving B.

Second, when trying to recover the synaptic efficacy from B to C, due to the serial

influence from A, the values of PGC from B to C are no longer a true reflection of the synaptic

weight between the two neurons. These problems can become even more acute in more complex

network with many interacting neurons or brain areas. The second result of this paper is to

describe a technique to recover the Direct Granger Causality (DGC) and the synaptic efficacy

between neurons within networks with more realistic topologies, including direct, mediated, and

serial influences.









The organization of this chapter is as follows. Existing methods such as Pairwise Granger

Causality and Conditional Granger Causality are described in terms of their application to

spiking neural networks. A computational shortcut that allows faster recovery of DGC is also

explored. These methods are then systematically expanded to accommodate increasingly

complex network conditions including direct, mediated, and serial influences. Each solution is

illustrated with biologically plausible, simulated networks. Finally, we examine how DGC could

be used to estimate both synaptic weights within a network and how it might provide a useful

measure for describing changes in network wide plasticity.

Conditional Granger Causality and Mediated vs. Direct Relationships

In Chapter 2, the mathematical basis for both Pairwise and Conditional Granger causality

was described. One of the limitations of PGC was that when it is used in a multivariate system,

often these results do not reflect the true connectivity. False directed influences and cases where

direct and mediated influences are confounded within the same pathway are the two primary

problems. In the following section we will describe how these relationships can be untangled

using Granger Methods. Chapter 2 also shows that CGC can be used to identify and remove

erroneous direct connections that are actually mediated though other elements. However, at this

point, we are left with a second problem. The effect of Y on X is embedded in the relationship

reported between Z and X. This results in a stronger PGC value from Z to X when compared to

the PGC value from Y to Z, even though the actual synaptic weights between these pairs may be

identical. The direct influences need to be separated from mediated influences in order to more

accurately quantify synaptic interactions within this network. Direct influences, after being

recovered from an entangled pathway, will be referred to as Direct Granger Causality (DGC) to

avoid confusion with values recovered using only PGC. Note that in neuron pairs that do not

contain mediated influences, PGC already represents the DGC between these neurons.









One way to recover DGC from an entangled pathway would be to make the relationship

between X and Z conditional on Y using CGC. This was the solution described by Geweke (59)

shown in equation (3.21). However, this method is computationally expensive in small networks

and possibly intractable in large networks, due to the exponential number of potential

combinations that must be addressed. Interestingly, Geweke also noted that it might be possible,

when the influence from Y to X is entirely mediated through Z, to directly subtract the PGC

calculated from Y to X from the PGC value from Z to X. This would recover the same quantity

as calculating CGC from X to Y conditional on Z, without having to perform the

computationally intensive CGC analysis. This concept will be evaluated empirically later in the

paper.


Fz = In var( (t)) (3.21)
var( (xy,z)(t))

Simulation and Analysis

In this section, we illustrate the analytical solutions described earlier within a simple,

biologically plausible neural network model. We have chosen Izhikevitch's simple neuron model

(3, 75), a stochastically driven neural network with modifiable synaptic weighting that provides

biologically relevant spikes that can then be analyzed using GC(76, 77). Briefly, the model is a

reduction of the Hodgkin-Huxley model (78) to a simple two-dimensional system of differential

equations. One equation models membrane voltage, v, equation (3.22), while the other models

and a recovery variable, u, shown in equation (3.23).


v' = 0.04v2 + 5v + 140 u + I (3.22)
u' = a(bv u) (3.23)

The auxiliary after spike resetting of the neuron is mediated by equation (3.24).








if,v> 30mV,thent v +dC (3.24)
u <--u+d

The variable u is a membrane recovery variable that accounts for K+ activation and Na+
inactivation in the neuron. The variables a, b, c, and d are dimensionless parameters that effect
the temporal characteristics of the action potential, while v' and u' are the first time derivatives
ofv and u. These variables remain constant and are the default values used by Izhikevich for
regular spiking cortical neurons (3) Each time step of the model represents 1 ms, equivalent to a


Synaptic Weights
from simulation


30mV


b) Granger Causality
from spike trains


0.53


1.00


(N3)





Figure 3-1 Topology diagram of a simple five neuron network. In panel a) the synaptic weights
from N1 to N2 and N4 to N5 have been synaptically coupled. N3 is left uncoupled to
demonstrate that PGC will indicate null connectivity. Equivalent but independent
random processes drive each of the five neurons. 100 realizations of this network
using Izhikevich's simple neuron model with these weights yields the results shown
in panel b) using Pairwise Granger Causality. These results demonstrate that
Pairwise Granger Causality can, not only resolve the difference between null and
actual connectivity, but also determine the directionality of those influences.


45mV









1kHz sampling rate of a realistic neural system. During simulation, neurons can spontaneously

fire based on spontaneous super-threshold fluctuations of membrane noise injected into each

neuron and from synaptic inputs from other neurons in the network.

Mono-directional Simulation

The first model, illustrated in Figure 3-1, consisted of a five neuron network composed of

mono-directional connectivity (i.e., one way singular direction for causal relations) from Neuron

4 (N4) to 5 (N5) and from Neuron 2 (N2) to 1 (N1), with neighboring Neuron 3 (N3) uncoupled

(zero synaptic weight). Hence, the activity of N2 and N4 will have a causal influence on firing of

N1 and N5, respectively. In contrast, only the equivalent independent random process intrinsic to

N3 drives N3. It is important to note that the intrinsic random processes that drive these neurons

are equivalent, but independent. The cross-power spectra for all neurons during simulated

activity are shown in Figure 3-2. During the simulation, all five neurons were active, indicated

by the auto-spectra (along the diagonal) for each neuron. As expected from the configuration of

the synaptic weights, the cross-power spectra for N1 to N2 and N4 to N5 is larger than that of N3

with its neighbors. The spectra resulting from Granger causality is shown in Figure 3-3. This

figure shows causality by frequency from 0 to 50 Hz between N4 to N5 (top panel) and the

converse relation, N5 to N4 (bottom panel).

Peak causality between N4 to N5 occurred at frequencies near 20 Hz. In contrast, spectral

causality within the reverse relationship, N5 to N4, was very low. This verifies the absence of a

bi-directional relationship between the pair. Similar results using other modeling systems can be

found elsewhere (48, 49, 76).

















S05 c05 05 05 L 05

0 0 0 0 0
0 20 40 0 20 40 0 20 40 0 20 40 0 20 40
Hz Hz Hz Hz Hz
1 I 1 1

S05 \05 05 05

0 0 0 0
0 20 40 0 20 40 0 20 40 0 20 40
Hz Hz Hz Hz
I 1 1

2 05 \ 05 05

0 0 0
0 20 40 0 20 40 0 20 40
Hz Hz Hz
1 i ------- i 1 i ------

S05 05

0 O
0 20 40 0 20 40
Hz Hz


i 05


0 20 40
Hz


Figure 3-2 Power spectra and cross-spectra calculated from the 5 neuron model simulation.
These plots are organized in matrix format mirroring that of the spectral matrix so
that that row 1 column 1 represents the intrinsic power spectra for N1, while row 1
column 2 represents the cross-spectra between N1 and N2. Thus, the diagonal
contains the individual neurons spectral power while the upper triangular graphs
represent Cross-spectra between neurons. Notice that Neuron 3 (Row 3, Column 3)
has intrinsic power but demonstrates no significant cross-spectra (Other plots in both
Row 3 and other plots in Column 3) with its neighbors as it is independent and
uncoupled to the other neurons. Neuron pairs 1,2 and 4,5 both demonstrate
significant cross-spectra (Row 1 Column 2 and Row 4 Column 5) consistent with
initial synaptic weights. However, it is not possible to determine directionality from
this information.














1.5




0.5


0 5 10 15 20 25 30 35 40 45 50
Hz




1.5




0.5


0 5 10 15 20 25 30 35 40 45 50
Hz


Figure 3-3 Granger Causality Spectra. The causality from N4 to N5 is apparent, with a maximum
value of approximately 0.92 around 20Hz, while the reverse direction of N5 to N4
approaches zero at all frequencies and is not statistically significant. This verifies that
PGC provides directional information that cannot be discerned from power spectra,
shown in Figure 3-2. The Pairwise Granger Causality for this directional neuron
coupling would be reported as 0.92.

Relationship between Synaptic Weight and Granger Causality

At this point, we have shown how PGC can provide an estimate of the relationships

between elements. One important consideration, however, is how PGC values correspond to

actual synaptic weights, especially when those weights might be in constant flux. In other words,

how is an estimate of causality using PGC related to the actual underlying synaptic weight

especially when those weights may be changing?























M
m
O.6



0.4



0.2




-10 0 10 20 30 40 50 60 70
Synaptic Weight (mrnV)


Figure 3-4 The PGC results from a mono-directional simulation with the weight from neuron 4
to 5 varied from 0-75mV in increments of 5mV. Synaptic weight is plotted on the x-
axis while the resulting PGC values calculated from the spike timing is plotted on the
y-axis. The relationship between synaptic weight and PGC is sigmoid described by
equation 12. Notice the region approximately between 15mV and 45mV where the
relationship between synaptic weight and PGC is linear. Each increment in the graph
was simulated 100 times to provide the mean and variance shown in the plot.

The simulation was modified such that the connectivity between N1 and N2 was

systematically incremented from 0 mV to 75 mV to observe the effects on our estimate of

causality. Each increment was simulated 100 times to determine both the average estimate by

PGC and variance of that estimate. The results of those simulations indicate a sigmoid

relationship between the synaptic weight from N1 to N2 (S1,2) and resultant GC, Flz2 shown in

Figure 3-4. There is a region between 15mV and 45mV where the relationship between S1,2









and F1,2 is linear with a slope of 0.033 causal units/mV. Synaptic weights above this range do

not produce significant changes on F1,2, suggesting that 45mV is the maximum effective

synaptic weight. Similarly, weights below 15 mV produce only slight changes in the estimate of

causality. The data from this experiment were fit with a sigmoid equation, shown in Equation

(3.25), to quantify the functional relationship between Granger estimates and actual synaptic

weights. An example of estimated synaptic weights using this equation is presented in panel d of

Figure 3-5.

1
Frx + 38402124S(3.25)
1 + 384e 2124S

Mediated vs. Direct Causal Relationships under Serial Connectivity

PGC works well for recovering synaptic weights between two neurons using Equation

(3.25). However, there are situations where PGC values will not adhere to this relationship due

to added complexity within the network. Often PGC values include not only the direct

information between nodes but also include mediate influences that could distort the relationship

between PGC and synaptic weights. This becomes more common as the number of elements and

connectivity increase within a network. This issue is addressed in the second simulation, where a

serial topology was created to demonstrate discrimination between direct and mediated

connectivity as well as separation and recovery of individual synaptic weights within a "chain"

of serially connected neurons. The five-neuron topology, shown in Figure 3-5 a), consisted of

causal















15m 15mV 2.11\ 1.37
V 0--4 N N3
15mV 2.02

c) NI d) NI 0.92
0.92 14.2mV


N2 N5N2

2.11 1.37 0.86 0.82
2N3 13.6mV N3 13.mV
2.02.08
2.2 15.6mV

Figure 3-5 Five neuron simulation. This was carried out where a) shows the synaptic weights
before simulation. Calculation of PGC for all possible connections for the 5 neurons
yields the plot shown in b). Notice that using PGC alone many new false connections
are shown. When CGC is used to eliminate the false connection the plot is reduced to
what is shown in c). After CGC plot c) begins to resemble the connectivity pattern as
shown in a), however, the values associated with c) do not scale with the synaptic
weights shown in a). A further step using CGC a second time or using Geweke's
subtraction method is required to detangle direct and mediated influences. The
results after the use of Geweke's subtraction method to calculate DGC are shown in
d) along with corresponding approximations of synaptic weight.

synaptic weights of 45 mV from N1 through N5 similar to a synfire chain (79, 80). The original

PGC estimate of causality, shown in panel (b), indicates erroneous connections among the entire

pool of neurons reflecting the complex causal relationships between each element. Table 3-1

shows the PGC results for each relationship. Note also that the PGC values, shown along the

elements of the serial chain, steadily increase from N1 to N5 (diagonal in Table 3-1), reflecting

the embedded mediated causal influences cascading throughout the chain.









Table 3-1 A table of the Pairwise Granger Causal influences calculated between the 5 neurons in
the simulation displayed in Figure 3-5. The right column indicates what the from
neuron is and the top row indicates what the to neuron. For example, to locate the
influence of neuron 4 on neuron 1 a value of 0.01 is reported in From N4, To N1.
To NI To N2 To N3 To N4 To N5
From NI 0.92 0.55 0.38 0.48
From N2 0.01 1.38 0.94 0.78
From N3 0.01 0.01 2.02 1.25
From N4 0.01 0.01 0.02 2.11
From N5 0.00 0.01 0.02 0.01

CGC was applied to discriminate between false direct from mediated connections in order

to determine the network's true connectivity. Panel c of Figure 3-5 shows the results of CGC,

where the topology has been correctly estimated but includes the systematic errors in estimating

the strength of coupling inherent within this method due to the causal cascade inherent in serial

chains. The problem arises because values remaining after calculating CGC represent a

combination of direct and mediated values that correspond to an increase in total information

flow down the chain. This is due to the accumulation of mediated influences towards the end of

the chain. Thus, the direct components between any two neurons must be separated from

mediated influences to relate the direct component to a meaningful synaptic weight.

Recovery of direct synaptic weights could be achieved using CGC for a second time on the

mediated pathway. However, a simpler solution may exist. IfPGC simply sums the effects

Table 3-2 Pairwise Granger Causal influences calculated between the 5 neurons in the
simulation displayed in Figure 3-5 after the use of significance thresholds, conditional
granger causality analysis, and removal of mediated influences. This table contains
only direct causal influences. All other values reported in Table 3-1 were found to be
false or insignificant. The right column indicates what the from neuron is and the top
row indicates what the to neuron. For example, to locate the influence of neuron 4 on
neuron 1 a value of 0.01 is reported in From N4, To N1.
To N1 To N2 To N3 To N4 To N5
From NI 0.92
From N2 0.82
From N3 1.08
From N4 0.86
From N5 -









of previous elements along a serially connected chain, then it may be possible to simply subtract

the previously conditioned out PGC values that represent the mediated component of the

distorted PGC value. For example, by subtracting N113 (0.55) from N213 (1.37), a value of 0.82

is obtained, which better estimates the actual synaptic weights incorporated into the model. This

process was repeated for each pair and the resulting estimates are shown in panel (d) of Figure 3-

5 and Table 3-2. The relationship between CGC and the simple subtraction of N1-3 to N2-3 is

derived from the properties of both PGC and CGC, as stated by Geweke (77), and shown in

Equation (3.26).


FY~Xz = FYZ-X FYX (3.26)

In the special case where the bivariate PGC influence of Y on X, Fryx, has been

previously conditioned out (i.e. equal to zero) using CGC, equation (3.27) is valid.


Frz~x = FZ (3.27)

This allows for the derivation of equation (3.28), which explains the empirical relationship

that allows subtraction of PGC values to retrieve the DGC value between.


FYx Xz = Fz x FYX (3.28)

A direct comparison of the efficacy of these methods for calculating DGC was achieved

using a Monte Carlo simulation of a 3 neuron chain. 100 realizations of this simulation were run

and both methods were used to recover DGC from the PGC output of the synapse at the end of

the chain. The synaptic weights were randomly generated between 15mV and 45mV for each

realization of the simulation. The results of this simulation are shown in Figure 3-6. These

results suggest that the subtraction method presented in equation (3.21)[12] can produce DGC

values similar to that of CGC as shown in equation [5].
















+ .'..
1+0 ti-
+ ,. + Y
















I I I I
+












5 20 25 30 35 0 4+
Snple weigh (mV)



comparison of Geweke's subtraction method to CGC. Each point was generated from
^ UL + P









repetitive calculation of DGC.





between synaptic weights and the recovered DGC for each synapse in the serial chain. The



















synaptic weights e1-2, C 2-3, S3-4, and o4h5 were randomly generated for each simulation using
a uniform distribution frozen within 15 mV to 45 mV linear recange of the sigmoid relationship
deri r e first exeriment. Te DGC for patways withod i s computationally simple

repetitive calculation of DGC.
Monte Carlo Simulations with Random Serial TopologiesCC


Swith random synaptic weights. This experiment was designed to describe the relationshipmV)


combetween synaptic weights and the recovered DGC for each synapse in the serial chain. Thed from

synaptica Monte Carlo simulation of the s45 werial simulation using randomly generated for each simulation usingaptic

a uniform distributis con fromsistent within 15 mV to 45 mVhe linear region expected from of the sigmoid replot shown in











derived from the first experiment. The DGC for pathways with mediated influences (F2,3, F3,4,












& F4>5) was calculated for each synapse by subtracting the conditioned F(N-I)M pathway from


FN-M and


QlngergCaj al N1 ->N2





oR

o %
06


.%
0 0


15 20 25 5 40 45
Synpto Weght Imi


Reoved Dirot C N3 >M4


Synapt Weght mV)


Reonwed Di~O tQC K2->rt


SyvnaptWenht [mV


6ynapi.Wenht(m\


Figure 3-7 DGC values can be recovered at each of the synapses in the serial chain. This was
done with randomly generated synaptic weights between 15mV and 45mV at all
synapses. The weights for each synapse were generated independently within each
simulation and the simulation was replicated 100 times. These plots demonstrate that
DGC values recovered from entangled pathways (top right, bottom right, and bottom
left) along the serial chain mirror those calculated using PGC on the first pathway
(top left). This suggests that DGC values represent the direct influence between two
neurons similar to the PGC relationship that can be calculated between neurons that
are not entangled by mediated influences.


retained. The synaptic weights S112 and Fl>2 were also retained to serve as a baseline for


comparison. According to the hypothesis, Monte Carlo distributions for the DGC pathways









should mirror the relationship between S1-2 and F1a2, a purely direct pathway. This simulation

was repeated 100 times.

Figure 3-7 a) shows that the distributions for the DGC at each synapse mirror the linear

distribution from the purely direct S192 to Fl2. The higher variance for the DGC farther down

the chain is due to additive error because of the way DGC is calculated. This empirical

relationship shows that the conditioned pathway, F(N-I ), is a representation of the mediated

component of the remaining pathway, FN&m. The empirical relationship also suggests that DGC

is an accurate representation of the DGC component between two neurons and can be used to

calculate a meaningful synaptic weight for the synapse using the relationship described in

Equation (3.25).

100 Neuron Neural Network

A larger more complex neural network simulation was used to further test the ability of

Granger Causality to recover synaptic weights from spiking activity. This simulation consists of

80 excitatory and 20 inhibitory neurons connected with 20,000 synapses. Excitatory neurons are

connected to all other neurons using randomly generated weights from a exponential distribution

where most connections are weak (1-4mV) while strong connections (up to 70mV) are sparse.

Inhibitory neurons are connected to the network using a flat distribution of weights ranging from

-1 to -10mV. Five excitatory neurons in this network were chosen and linked in a serial chain,

resembling the earlier 5 neuron chain, with synaptic weights of 40mV as shown in Figure 3-8

panel A). This network produces spontaneous spiking and bursting shown in panel B). Pairwise

Granger Causality is then used to determine the relationship between these 5 neurons shown in

panel C). Pairwise Granger Causality alone reveals that information flows directionally down

the chain with no information flow in the opposite direction. However, the structure and











A)












C)


1 2 3 4 5
from neuron


S.~ *. .I
**b j rj '. .i m.*.p




o L:t < >1..>
:timems


Figure 3-8 100 Neuron Simulation. A chain of 5 neurons is imbedded within a 100 neuron neural
network. The goal of this simulation is to extract the causal core shown in panel A
using Granger Methods. Each synaptic weight in the chain is is set at 40mV. Panel B
shows the activity of these five neurons within the full network. Panel C shows the
results from only PGC analysis. Panel D shows the remaining significant
connectivity after CGC analysis and their corresponding synaptic weights. Note that
the weights down the chain are significantly underestimated, this is likely due to the
influence of the rest of the network on recovered weights.

connectivity of the chain are not clear enough to disseminate the structure underlying this

network. Conditional Granger Causality is then used identify and remove spurious connection

revealing the structure shown in panel D). Thus, Granger Causality effectively identifies the

serial chain embedded within this 100-neuron simulation. Conditional Granger Causality and the

previously demonstrated sigmoid relationship are then used to recovery of synaptic weights also

shown in panel D). Interestingly, results from a larger neural network are not as accurate as

results shown previously from a simpler network. Further testing has shown that interaction with


* 1







[~II=









the rest of the network subtly distorts the magnitude of the recovered synaptic weights. Future

work will focus on the use Conditional Granger Causality to condition out the influences of the

other 98 neurons on each paring to improve the accuracy of recovered synaptic weights.

Application to Spike Timing Dependent Plasticity

The brain is an ever-changing, non-stationary system that belies the fact that many

measures of activity require stationarity to derive meaningful conclusions. Take for example

studies that attempt to demonstrate plasticity (4, 21, 81). One of the most common paradigms

measures activity (e.g., spike rate) during a "baseline" phase that is followed by some treatment

(e.g., learning phase, such as delivery of a tetanic pulse train). Activity is then measured again to

determine if any changes (e.g., spike rate) have occurred. The core notion is that changes in

connectivity will result in a change in the number of spikes. Unfortunately, measures like spike

rate are highly susceptible to spontaneous (random) fluctuations in brain activity. In contrast,

changes in the rate of spiking alone do not substantially influence causality estimates until the

spike rate becomes so low that either method begins to suffer. Hence, GC may provide a novel

and more reliable method to measure activity changes, while also outlining the structural

information of the network as these changes occur.

A simple neural model sensitive to spike timing dependent plasticity (STDP), a proposed

mechanism for changes in synaptic plasticity, was created using CSIM (82, 83) to investigate this

notion. This model was constructed to determine if changes in synaptic weight due to STDP

could be reliably detected with Granger Causality. The simulation was designed so that an

STDP synapse, with parameters adopted from Markram et al., between two neurons would be

exposed to a paired pulse stimulation similar to that ofBi et al. (84). This paired pulse was

designed to elicit long-term potentiation (LTP).











Input spike train
S Izh Neuron 1 Presynaptic
I4zh Neuron 2 Postsynaptic

2 .* .. ................... .. ...


0 1 2 3 4 6 6

1
i l10n 6 postsynaptic response






0 1 2 3 4 5 6
membrane potential and spikes













PGC should detect and quantify the change after the paired pulse stimulation in the post
0 02

-0 04



-0 08
0 1 2 3 4 5 6
time [sec]


Figure 3-9 A simulation to determine if changes in synaptic weight consistent with stimulation
timing dependent plasticity (STDP) at a synapse between two biologically relevant
neurons could be detected after a paired pulse stimulation was applied to the neurons.
The paired pulses were designed to produce potentiation at the synapse. The top plot
shows the action potentials generated from the pre and post synaptic neurons. The
paired pulse stimulation was delivered between seconds 2 and 4. The middle plot
shows the amplitude of the post-synaptic response in amps, note that the post synaptic
response was amplified due to the paired pulse stimulation, consistent with the effects
STDP. The bottom plot shoes the membrane voltage and action potentials fired for
the post synaptic neuron.

PGC should detect and quantify the change after the paired pulse stimulation in the post

stimulation spontaneous activity. However, changes in activity from the paired pulse stimulation

may not be great enough to detect using granger causality. The simulated neurons used were

also based on Izhikevitch's simple model of spiking neurons (3). Each simulation was six

seconds long, sampled at 1000 kHz, and repeated 10 times. The first two seconds of

spontaneous activity are the baseline, the middle two seconds are during the application of the










paired pulse stimulation to the synapse, and the final two seconds of spontaneous activity are

used as the post treatment recording.




I

OA I-



0 1 I

0-9-





Q-2 -



01
pro-telanlu post-ttanus

Figure 3-10 Statistical Analysis of Plasticity Simulation. The simulation ran in Figure 3-9 was
repeated 10 times and the resulting pre and post spike trains were used to calculate
the causal interaction between the neurons using PGC. The pre and post-tetanus
values of PGC were compared using an analysis of variance. Plotting the analysis of
variance demonstrates that the PGC values pre and post-tetanus are statistically
independent suggesting that the change in causality due to the paired pulse
stimulation can be detected and quantified using PGC.

One realization of the simulation is shown in Figure 3-9. Note that the post-synaptic

response increased after the paired pulse stimulation was applied. The spike trains for the

baseline and treatment were low-pass filtered and PGC was calculated from the pre-synaptic

neuron to the post-synaptic neuron. Ten replicates of this experiment produced 10 PGC values

for both baseline and treatment that are then pooled and compared using an analysis of variance,

shown in Figure 3-10. This analysis shows that the values are statistically different post

treatment suggesting that changes resulting from STDP can be detected using PGC.









Conclusions

Granger Causality is a powerful statistical technique that can quantify causal relationships,

ranging from interactions between different brain areas to recordings from single neurons.

However, there are a number of potential issues in interpreting synaptic efficacy that results from

PGC, including false directed influence and separation of direct and mediated influences. In this

paper, we explored several additional techniques specifically to address those issues.

Specifically, CGC is used to untangle direct and mediated influences and provide estimates of

DGC. Additionally, a less computationally intensive subtractive process can provide DGC under

special conditions. When these special conditions are not met, CGC would be required to recover

DGC values. Through network simulations, we then described how these techniques could

potentially be used to provide additional information about the underlying network structure and

to estimate the relationship between synaptic strength and DGC estimates. These methods are

useful when that strength is static or changing due to changes in plasticity.

There still remains several important weaknesses and limitations of Granger Causality that

must be considered. For example, synaptic weight estimates using DGC are dependent not only

on the variability of the system being measured and the measuring instruments, but also on the

quantity of data recorded. Thus, short recording times with little activity can substantially affect

estimates using either method. Multiple realizations of the process being examined are

recommended to produce an accurate representation of causality within a system. Perhaps one of

the most challenging problems remaining with this technique in living systems is detecting the

presence of unobserved units, where apparent causal relationships are due to unobserved

elements. This possibility must be considered when drawing conclusions about complex systems

that may contain multiple unobserved processes.









DGC was also investigated for its potential to estimate synaptic weights. The model

intentionally represents a simplified case of neural connectivity where the independent stochastic

processes that drive the neurons, along with the spike firing rates, are based on the behavior of

one type of cortical neuron. Changing these stochastic processes and neuron types in the model

may alter the sigmoid relationship between synaptic weight and GC. Different model parameters

or data from neural culture, where neuron type and synaptic weights may be unknown, will

require calibration in order to recover valid synaptic weights. However, in this study we

demonstrate that there is a linear region within the sigmoid relationship between synaptic weight

and GC, where changes in synaptic efficacy can be observed.

The 100-neuron simulation demonstrates that as the complexity of the network grows the

PGC results underestimate the synaptic weights. This may be due to the contributions of the

other 95 neurons on activity within the causal chain. While the sigmoid relationship may not

always apply, the existence of a region where synaptic weight changes linearly with PGC

suggests that DGC values from that system are comparable (i.e. a larger Granger Causality value

translates to a larger synaptic weight). This gives an investigator confidence that changes

observed during an experimental manipulation are also comparable in magnitude. This linear

relationship allows for pre-experimental and post-experimental values of DGC to be subtracted

in order to quantify plasticity. It is possible that these observations can be quantified with proper

calibration between synaptic weight and DGC values.

Conversely, we have also demonstrated that there are regions in the relationship where

changes in synaptic weight will not be reflected as changes in causality. Specifically, when the

post synaptic potential is significantly sub threshold or when it is already so large that the post

synaptic potential will almost always cause a post synaptic action potential. Changes in synaptic









weight within these regions may not be reflected in changes PGC values calculated between the

observed neurons. However, it should be noted that changes in synaptic weight in these regions

will not produce and observable changes in the spiking output of the neuron either.

Perhaps a more fundamental problem with any Granger based method is the ability to

detect the presence of inhibitory, rather than excitatory, contributions. Inhibitory synapses,

unlike excitatory synapses, do not produce a post-synaptic action potential. Thus, an inhibitory

synapse will produce GC values near zero, similar to that of null connectivity. Additionally,

inhibitory influences decrease activity in their targets and may distort serial causal influences,

even though they may be strong. Including steps to untangle inhibitory influences, perhaps

through conditional logic, may help address this problem. However, GC remains one of the most

powerful techniques to estimate causal relations or untangle complex structural topologies in

networks where that information is unknown.









CHAPTER 4
ASSESSMENT OF PLASTICITY IN-VITRO USING GRANGER CAUSALITY

Introduction

Microelectrode array (MEA) technology has been used to quantify and record

multichannel neural activity from in-vitro dissociated neural cultures for nearly three decades

(85-87). However, the underlying structure and connectivity that is responsible for producing the

neural activity observed from these cultures has remained largely unknown. Subsequently,

difficulties arise when trying to accurately relate changes in observed neural activity to

physiological changes within the network such as plasticity. Detection of plasticity across a

network is now a common theme in recent neuroscience research (20, 67, 81, 88-92). Spike

timing dependent plasticity (STDP), a process where the efficacy of a synapse is governed by the

timing of pre- and postsynaptic action potentials, is thought to be a potential mechanism by

which in-vitro plasticity occurs (83, 84, 88, 93). However, neural activity recorded using MEAs

is extracellular, whereas STDP experiments often use intercellular voltage clamp methods on

isolated neuron pairs. Ad hoc algorithms and measures are often developed to quantify possible

plasticity related changes in-vitro with varying levels of success (4, 18, 20, 81, 89). However,

these algorithms often do not provide a statistically sound measure of effective connectivity that

can be related to the actual connectivity underlying the dissociated neural network.

Jimbo et al has demonstrated that changes in the synaptic strength of pathways within

living neural networks could be identified by describing changes in elicited spike rates after

induction by high frequency stimulation or tetanization (4). Specifically, Jimbo's data suggested

that tetanization resulted in pathway specific potentiation and depression. The metric used to

identify those changes is based on changes in the spike rates recorded on each electrode before

and after a tetanic pulse train is delivered, a simple and elegant way to observe output related









changes in neural connectivity. However, observations that can be made using spike rate are

limited providing only a course quantification of neural connectivity. First, spike rate

observations are non-directional and they are also non-causal. Second, this method relies on

using a difference between pre and post treatment measurements of spike rate to detect

underlying pathways, limiting the ability to observe connectivity prior to a treatment which

modifies the underlying strength. Several methods (17) including correlation (4, 21, 94, 95),

likelihood methods (36-38), partial directed coherence (30, 50, 96), and Granger Causality (48,

49, 76) can address some of these deficiencies and have been used to identify connectivity

patterns between spiking neurons in neural simulation, in-vitro and in-vivo.

Pairwise Granger Causality (PGC) (1), a robust statistical forecasting tool borrowed from

economics, is unique among these measures because it can address all of these issues and it can

quantify the degree of causal connectivity between neurons (76). PGC makes use of

autoregressive modeling (66) to determine if the temporal activity of a pre-synaptic neuron can

be used to predict the activity of a possible post-synaptic neuron. Additionally, neural network

simulations from Chapter 3 suggest that PGC can be used to quantify synaptic connectivity from

simulated spontaneous neural activity. This chapter will explore the use of PGC as a method for

detecting pathway specific changes in plasticity in-vitro. PGC will be validated against the

pathway information derived using Jimbo's method of spike rate changes pre and post tetanus.

PGC should allow for a deeper analysis of plasticity related effects from Jimbo's protocol and

new insights will be presented after PGC has been compared against this method.

What do PGC values mean in neural networks?

Connectivity in terms of neural networks is usually expressed as a synaptic weight or

synaptic efficacy that can be related directly to the amount of voltage or current that is induced

from the pre-synaptic to the post-synaptic neuron. However, it is not intuitively clear what the









values generated by PGC mean in terms of synaptic coupling. A computer simulation, using

Izhikevitch's simple neuron model (3), was used in Chapter 3 to determine the effect of known

synaptic weights on causality estimates using PGC. These values were calculated from pre and

post-synaptic spike trains in order to better demonstrate the relationship between PGC values and

synaptic efficacy. Within this simulation the synaptic weight between two regular spiking

cortical neurons were systematically increased and the resulting spike trains from the simulation

were analyzed using PGC. The results indicated a sigmoid relationship between PGC and the

actual synaptic weight in which PGC is insensitive to variations with very low and very high

synaptic strengths. The was, however, a linear region (cf Figure 3-4) in the middle of this

sigmoid indicating a region where PGC values are linearly related to synaptic efficacy (and also

can be subject to linear arithmetical operators such as subtraction). Based on this relationship, the

PGC values recovered from the cortical cultures on the MEA will likely represent the synaptic

coupling between these two channels but may be distorted if those weights are excessively low

or high. Of course this value may not represent a specific synaptic weight as the connections

between neurons in culture may consist of thousands of synapses (97) and may be mediated

through several pathways (21). It should provide an overall estimate of the synaptic weights

coupling two neurons. Hence, this method could be used to detect changes in synaptic weights

among the pathways coupling neurons indicative of plasticity.

Methods

Dissociated Cortical Cultures

Dissociated cortical neural cultures consisted of cortical hemispheres (Brain Bits Inc.) from

E18 Wistar rat embryos that were digested using the Papain Dissociation System (Worthington

Biochemical Corporation) and mechanically triturated separating the neurons soma from the

surrounding connective tissue. After dissociation, approximately 50,000 cells were placed in the









center of a MEA that has been first coated with polyethyleneimine (96) (98) and then also treated

with laminin (Sigma) (99). The neurons were cultured in medium consisting of 90 % Dulbecco's

modified Eagle's medium (DMEM) (Gibco cat# 10569-010) and 10% Equine Serum (HyClone

cat# SH30074.03). The medium was equilibrated in an incubator to 5% carbon dioxide and

35.5C temperature before use. The medium in each culture was exchanged once per week.

Freshly dissociated neurons rapidly begin to reestablish a network when placed on the MEA,

forming synapses to neighboring cells and producing spontaneous action potentials within the

first few days (100). During the first two weeks the neurons mature and begin to express the full

complement of receptors, a process completed after one month in vitro (101-104). In this

experiment, data from six cultures were analyzed. All cultures were older than 30 days in vitro in

this experiment.

Microelectrode Arrays

The MEAs that were used for this experiment were obtained from Multichannel Systems

GmBH. The 60 electrodes of each array are arranged in an 8 x 8 grid, spaced 200 |tm apart, to

measure neural activity over a 1.6 mm2 area. These arrays consist of a glass wafer over which a

gold surface is deposited and the electrode traces are etched using photolithography. An

insulating layer of silicon nitride (SiN) is then deposited via vapor deposition over the surface

providing an electrical barrier between the gold and culture media. 30 jtm diameter holes are

then plasma etched over the circular pads and Titanium nitride (TiN) is plated onto the gold

forming a low impedance interface between the recording amplifiers and culture media. Each

electrode measures the voltage changes produced by neural soma, dendrite, and axons within an

approximate radius of 60 jtm as they fire action potentials (105). Conversely, application of bi-

phasic current or voltage pulses (e.g., 100 to 800 mV or 50 to 200 ptA) will elicit action









potentials from neurons near the stimulated channel. Neural activity was sampled and digitized at

25 kHz using Multichannel Systems MEA1060 amplifier and A/D hardware. Action potentials

were distinguished from noise using a voltage threshold 5 standard deviations above or below

estimated noise levels. Spike events were recorded into a file containing the time, channel,

height, width, and context information surrounding each event. This spike file was later

processed offline to extract information about any bursts that may have occurred. Bursts were

detected using a modification of the leaky integrator algorithm described by Gross (91) which is

applied to the spike times for each electrode. The environment both within and surrounding the

MEA during recordings was temperature and gas compositionally controlled at 35.5C and 5%

C02 duplicating the environment within the incubator in which these cells were maintained.

Media temperature within the array was controlled using Multichannel Systems TC02

temperature controller.

Stimulation Induced Plasticity Protocol

The study of plasticity is a field where much remains unexplored. However, it is one of the

most actively investigated areas of brain research. Two distinct and closely related phenomena,

long term potentiation (LTP) (106) and long term depression (LTD), have come to be accepted

as possible mechanisms by which synaptic efficacy is increased and decreased respectively.

While the timing rules thought to underlie LTD and LTP between individual neurons are easily

investigated by monitoring the magnitude of post synaptic potentials (PSP) delivered to post-

synaptic neurons using patch clamp, single neuron electrophysiological methods are problematic

when trying to determine how these mechanisms function at the network level. Neural cultures

maintained on MEAs present an opportunity to observe these phenomena on 60 electrodes

however MEAs only provide extra-cellular voltage information from these cultures. Thus, while









action potentials can be detected from the neurons grown on MEAs, subtle fluctuations like PSPs

are indistinguishable from background noise.

Jimbo et al devised a clever experiment (4) to overcome this limitation to determine the

efficacy of a tetanus to induce plasticity changes after its delivery to a single electrode in a

dissociated cortical network grown on a micro-electrode array. The protocol consists of a

baseline probe sequence, treatment induction tetanus, and a post-treatment probe sequence.

Jimbo identifies neural pathways by indicating where LTP and LTD may have occurred by

determining the change in spike rate between the probe sequences. The induction tetanus consists

of trains of 11 biphasic 600mV pulses delivered at 20 Hz repeated 20 times at 5s intervals. The

probe sequences deliver biphasic sequential probes of 600mV to each of the 60 channels

(including the ground channel) in a randomized order that is repeated 10 times for a total of 600

probes. Each of these probes evokes a burst of activity resulting from activation of the cortical

neural network across all 60 channels. The number of spikes generated from the probe within a

200 ms time window on each electrode quantifies this activity. The baseline and post-treatment

spike rate over this 200 ms window for 60 channels for each probe are then compared by

difference plot (as shown in the left panel of Figure 4-1) to determine the change in spike rate

between baseline and post-treatment probes. The red horizontal lines show instances where the

protocol enhanced the spike rate resulting from probing on that channel and the blue lines show

instances where the effect of the protocol resulted in depression of the spike rate for probes

delivered to that channel.

The use of PGC on these same spike trains should allow for more detailed description of

changes in the effective connectivity induced by the tetanus. The protocol yields spike trains for

all 60 electrodes following each of the 1200 total probes (60 channels probed x 10 probes per









channel x 2 probe series pre and post tetanus) that are delivered to the neural culture. The ten

probes per channel can be considered to be realizations of the same process, thus when

calculating PGC between the two channels the 10 source spike (pre-synaptic) trains and 10

response (post-synaptic) spike trains are pooled.

PGC can also be directionally calculated from each channel to every other channel for each

probe yielding a 60x60 matrix of directional influences for each probe, 60 times more

information than that provided by spike rate. Incidentally, directionality can be eliminated by

combining the two directional PGCs between a pair of channels to yield a measure called total

causality, similar to coherence, that is more appropriate for validation of PGC against established

non-directional spike rate methods. When PGC is calculated for each of the probed channels, this

yields a 60x60x60 matrix of PGC results for each probe sequence before and after tetanus. For

simplification these PGC matrices generated from elicited data sets will be referred to as elicited

Granger Causality (EGC) matrices. When the post-tetanus EGC matrix is subtracted from the

pre-tetanus EGC matrix, this will be referred to as the EGC difference matrix. The axes of this

EGC difference matrix are stimulus (probed) channel, source channel, and response channel and

each entry in the matrix contains the change in PGC from the first set of probes to the second set

of probes.

Results

Spike Rate versus PGC

Jimbo's experimental procedure was replicated in our lab1 on six cortical MEA cultures.

Data from neural culture 6686 is used to present results that were consistent across all of the

subjects. The timing of spikes was collected from the raw waveforms and then cleaned of noise


1 The data for this experiment was collected by Ping He as a part of a larger experiment to determine long-term
changes in plasticity in the DeMarse laboratory.










spikes by template matching. These remaining spikes were then used to create the rate based

stimulus-response plot shown in the left panel of Figure 4-1. This figure plots the changes in the

number of spikes post-tetanization by color (blue = decrease, green = no change, red = increase)

for each response channel (x-axis) for every probed channel (y-axis). The plot suggests

Prob-response pike rate pkn PGC colapwd acros source channel

a m a mm a a.
.[ -..._ .r u .. :_n:.


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m a m r ma Wmaa
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5 10 15 20 25 3035 40 45 5 55 0 5 15 20 25 30 ~5 40 45 50 55 60
1 5-15
5 10 15 20 25 30 35 40 45 50 55 00 5 10 15 20 25 30 36 40 45 50 55 80
response Cnnel response ctannel


Figure 4-1 Comparison of spike rate to Pairwise Granger Causality. The left plot shows the
change in elicited spike rate post tetanus at each channel (x-axis) for each probed
channel (y-axis). Each pixel on the plot represents the change, post-tetanization, in
elicited spike rate at the response channel from a stimulation delivered to the probed
channel. The pathways stimulated by the single channel probes on a particular
channel tend to uniformly increase (red horizontal lines) or uniformly decrease (blue
horizontal lines). The blue vertical line at response channel 52 is a result of drift
during the experiment a noisy channel. This channel would normally be removed
from analysis. The right plot is an analysis of the same data using PGC. This plot
shows the change in PGC post-tetanization for each response channel (x-axis) for
each stimulated probe channel (y-axis). The change in PGC for each pixel can be
interpreted as the change in the effective connectivity of the response channel to all
other channels. PGC similarly detects pathway specific plasticity, but does so by
looking at changes in effective connectivity between channels rather than changes in
spike rate. Notice that the analysis of channel 52 yields no effect (green represents no
change in PGC) as the noisy spikes from channel 52 have no effective relationship
with other channels. Additionally, PGC offers greater contrast of post-tetanus effects.
The enhanced contrast can especially be seen when comparing response channels 1-9
between both plots.









simultaneous enhancement and depression of spike rates similar to the data previously reported

by Jimbo et al. (4). PGC was then calculated for each of the probe sequences and reported as a

stimulus-response difference-plot shown in the right panel of Figure 4-1. This is essentially

interpreted the same as the left panel except that the colors represent a change in PGC instead of

spike rate. Note that the 3-dimensional EGC matrix is separately collapsed across the source and

response axes and the results are summed to create the 60x60 plot shown in Figure 4-1. This is

done to reduce the complexity of the EGC matrix to allow direct comparison to the spike rate

results. A simple visual comparison between the left and right panels in Figure 4-1 shows that, in

general, any changes in rate are also reflected in changes in causality. Of the 60 probe channels,

the results of more than 95% of probe channels match direction of change when comparing spike

rate and PGC stimulus-response plots.

However, there are some noticeable differences between the two plots PGC is influenced by

changes in rate but only when the rate increase on the pre-synaptic channel is complemented by

an increase on the postsynaptic activity. The data from dish 6686 was specifically chosen to help

reinforce this notion due to a unique event that occurred. Most noticeable, the spike train data

from channel 53 (the dark blue vertical stripe) in the left panel of Figure 4-1 was contaminated

with background noise and generated false spikes that were similar enough to real spikes that

they could not be filtered out. During the recording following tetanization the source of the noise

was removed resulting in a large decrease in spike rate for this channel. When PGC is calculated

for the noisy uncorrelated random spikes from this channel PGC yields values near zero, as one

would predict for temporally unrelated time series. Thus, PGC reports the changes in the

temporal relationship between channels rather than a simple change in rate. This is further

reinforced by the data from channels 1-9 that show little change in spike rate across all probe











channels in the left panel of Figure 4-1 but show changes in PGC (right panel of Figure 4-1).


Additionally, some channels show stronger changes in PGC than those shown by spike rate as


shown in examples depicted in Figure 4-2a for LTP and Figure 4-2b for LTD. This is not


surprising considering that changes in synaptic efficacy can be reflected by changes in the pre


and postsynaptic spike timings that may appear in changes in spike rate.


Spik Rda f te obe H Har e CTlhn 126


0 1 20 0 30 0 0 61
CIL Ndby migid&
PGC. for Hrff d,~ B -a i 26



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0 10 20 30 40 50 60
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pia R for oPb H.r&- CI-l 41








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Figure 4-2 A comparison of spike rate and PGC analysis for enhanced channel 26 and depressed
channel 41. This figure highlights the increased contrast provided by PGC over spike
rate by comparing spike rate (top panels) or PGC analysis (bottom panels), sorted by
initial strength (x-axis), before and after tetanization (blue square vs. green circles,
respectively). Panel A shows potentiation in terms of spike rate (top) and PGC
(bottom) for channel 26 from Figure 4-1. Panel B shows the same relationship for the
depressed channel 41. In both cases PGC provides greater contrast between baseline
and post treatment measurements for both potentiation and depression. The increased
contrast provided by PGC makes it easier to establish the statistical significance of
pathway specific changes.

Surrogate Analysis

A subtle but important difference between the two methods is that the right panel of Figure


4-1 has considerably higher contrast between effected channels and surrounding unaffected


channels i.e. less regions that are light yellow or light blue. This suggests that the signal to noise










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50 1 10 504 10


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515 M
10 -10 10 1 % M i, -10


S 10 15 2 25 30 35 40 45 S 55 60 5 10 15 20 25 90 35 40 45 50 S5 60
response channel rspse dChanniel


Figure 4-3 Surrogate analysis of the PGC analysis. Two surrogate analyses were performed on
the data from the Jimbo protocol. The first, shown in the left plot, is a result of
randomly shuffling the spike timings within the 200ms post stimulation window for
both the pre and post-tetanus prior to PGC analysis. This effectively destroys the
spike timing relationships while maintaining the spike rate over this window. The
resulting PGC analysis yields near zero PGC results for both pre and post-tetanus
probe sets. The resulting change in PGC post-tetanus shown in the left figure is near
zero (hence a uniform green field) for all pathways. This spike-shuffled surrogate
analysis also establishes a significance threshold that can be used to distinguish
actually changes shown in Figure 4-1 from noise. Note that if the spike shuffled data
were used to generate a spike rate plot, it would be identical to the left panel in Figure
4-1. The second surrogate analysis, shown in the right panel, shuffles the pre and
post-tetanus spike trains. The pathway specific patterns observed in Figure 4-1 are
obliterated. This suggests that the plasticity induction protocol is responsible for
producing these pathway specific patterns and that these patterns are not randomly
generated.

ratio for PGC is higher, allowing for a greater contrast between effected channels and

background noise. Additionally, the use of PGC also has the added benefit of having a statistical

significance level. A surrogate data set can be created from the original spike trains that

maintains the spike rate information in the spike train but scrambles the interspike intervals by

randomizing the spike timings for a given spike train. The null hypothesis for this first surrogate

is that changes in spike rate are responsible for the corresponding changes in PGC. A well

established principle of PGC is that when the two time series being regressed are made up of


PGC cdlepsed acros sources channel wiwe posm lIlanus shuffle









independent white noise PGC=0. In practice with real spike trains a surrogate data set with

randomized spike timings will produce a value that approaches zero establishing a threshold by

which the significance of the unscrambled can be determined from. GC values falling above this

threshold can be considered significant and non-random. Once the spike timings were scrambled

the same causality analysis was performed on the surrogate spike trains. The resulting analysis,

shown in the left panel of Figure 4-3, confirms that the patterns demonstrated by the original data

set are significant considering that the significance levels generated from the surrogate data set

are all in the green (i.e. close to zero change) region of the color scale. This also invalidates the

null hypothesis suggesting that the PGC analysis is not simply dependent on rate and is likely

dependent on changes in interspike timing between channels.

A second surrogate analysis was created by randomly swapping out pre-tetanus spike

trains with post tetanus spike trains from the same stimulus and source/response channels. This

surrogate, while not restricted to use with only PGC analysis, serves the purpose of determining

whether the pathway specific patterns resulting from the Jimbo protocol are randomly produced.

The null hypothesis for this surrogate is that failed probes are responsible for the patterns seen in

rate and PGC analysis. Thus, swapping probes between post and pre-tetanus analysis would test

this hypothesis. PGC analysis is then performed on this surrogate data set with results shown in

the right panel of Figure 4-3. Notice that the pathway specific patterns inherent to the non-

surrogate data set have been obliterated. This surrogate analysis randomized the pre/post-

tetanization temporal order of the spike train realizations. If the pathway specific plasticity

effects were randomly generated from failed probes, the effects should have persisted in the

surrogate analysis. Subsequently, these results suggest that the second surrogate null hypothesis

is invalid.









Discussion

The original hypothesis for this experiment was that PGC could be used to describe

pathway specific plasticity effects in dissociated cortical culture. Comparison of PGC results

against spike rate validated this hypothesis. Over 95% of the pathways described by spike rate

were found to agree with the PGC analysis. Often, those channels that differed from the spike

rate analysis were channels that showed no spike rate change but showed a change in PGC. This

was likely due to fact that PGC specifically models changes in the fine temporal dependencies

between spike trains rather than relying on course changes in rate. While spike rate has the

advantage of being simple and easy to obtain, use of PGC compliments spike rate as a more

statistically rigorous and perhaps more sensitive metric. PGC was also shown to be more

resilient to the introduction of noise spikes into the analyzed spike trains as demonstrated by the

noisy channel 52 in Figure 4-1. PGC was also shown to provide a higher contrast between pre

and post-tetanus analysis as shown in Figure 4-2. Statistical significance was more easily

established using a spike-shuffled surrogate analysis. This surrogate analysis also suggests that

pathway specific PGC plasticity results are not simply spike rate dependent but reflect true

changes in causal strength between the channels.

However, these are not the only benefits to using PGC. For comparative purposes the SGC

matrix was intentionally collapsed and the benefit of directionality was removed in order to

better validate and compare the results of PGC against the non-directional spike rate method.

The following discussion explores these directional effects providing a deeper analysis including

amplitude dependent plasticity, comparisons of spontaneous and elicited PGC analysis, and

directional plasticity that can be described by PGC.









Amplitude Dependent Plasticity

Directionality is a fundamental property of PGC that distinguishes it from other methods

such as correlation and coherence. Directionality has been intentionally removed from the

analysis so far to enable better comparison to non-directional spike rate based methodology.

However, directionality is a key component of STDP timing rules for plasticity. Directional

information about plasticity changes should allow a more complete understanding of how

induced plasticity affects the underlying structure of the network. When the EGC difference

matrix is calculated without using total causality several new options emerge for visualizing

plasticity. One use of the EGC difference matrix is that it can be used to determine how the

initial magnitude of a pathway affects magnitude and direction of change of that pathway after

tetanization. The EGC difference matrix contains 60x60x60 entries whose values can be

compared to the same number of pre-tetanus EGC matrix values.

According to amplitude dependent plasticity (ADP) shown by Bi & Poo, changes in

synaptic weight due to correlated time dependant spiking are a function of initial synaptic

strength. While potentiation of excitatory postsynaptic potentials (EPSPs) is larger for weak

synapses, those experiencing depression are reduced by a constant percentage (84). EPSPs are

commonly measured postsynaptically using single electrode voltage clamp techniques. In this

analysis, we investigate ADP among the pathways observed during Jimbo's protocol. However,

unlike in Bi & Poo's experiments, MEA electrodes are extracellular and cannot measure EPSPs.

Thus, the starting effective connectivity is compared to the change in effective connectivity using

pairwise Granger Causality (PGC) calculated from prior analysis. If the observed changes in

plasticity by PGC are accurate then one would expect that the magnitude and direction of

changes within the network would agree with previously reported ADP results (84).











Amplitude Dependent Plasticity In-vitro


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Figure 4-4 Amplitude dependent plasticity (ADP) can be explored in-vitro using PGC.
According to amplitude dependent plasticity (ADP), changes in synaptic weight due
to correlated time dependant spiking are a function of initial synaptic strength. This
plot shows initial magnitude of PGC prior to induction (x-axis) and the corresponding
directional change magnitude after induction (y-axis). This demonstrates that the
tetanic induction generally increases weak connections and decreases strong
connection. This plot resembles results reported by Bi & Poo 1998 where
postsynaptic potentials were observed using intercellular recordings. Data shown is a
compilation from six cortical cultures exposed to tetanization.

The entire pre-tetanization EGC matrix was used as starting weights and the EGC

difference matrix was used to report changes in those weights. These results were plotted Figure

4-4 where the initial PGC amplitude for all pathways was plotted on the x-axis and the % change

in the PGC amplitude was plotted on the y-axis. This plot contains data from 6 MEA cortical

cultures where Jimbo's protocol was used to induce plasticity. Overall Figure 4-4 suggests that

weak pathways are most often enhanced while strong pathways are weakened. At around initial

PGC values of 0.25 depression begins to be favored over potentiation as the initial strength of the










pathway increases. These results suggest that changes in connectivity post tetanus may be

amplitude dependent which is consistent with the results from Bi & Poo.

Directional Causal Influences

The EGC difference matrix is large which can make detection of network changes difficult

to visualize. Thus, when trying to understand what is happening in the EGC difference matrix it

is preferable to collapse the matrix along one of its axes for simplification. However, when


PGC collapsed across soLuce channel


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Figure 4-5 Stimulus-Response difference plot of PGC where the EGC difference matrix is
collapsed by source. While similar to the left panel of Figure 4-1, this plot makes use
of the directional properties of PGC. Essentially each pixel represents the change in
causal information coming in to each response channel. This plot shows the
directional change in PGC post-tetanization for each response channel (x-axis) for it
corresponding stimulated probe channel (y-axis). Red pixels indicate a potentiation
of inward causal flow while blue pixels indicate depressed inward causal flow. The
change in PGC for each pixel can be interpreted as the change in the effective
connectivity of the response channel to all other channels.










directionality is involved, which axis is collapsed has a profound impact on what information

that figure will present. The available axes to collapse upon are source, response, and by probe

channel. When collapsing the EGC difference matrix by source (as was done earlier to match

spike rate plots), each pixel is a summation of all of the causal influences from all source

channels into each individual response channel for a given probe. This is shown in the resulting

stimulus-response PGC plot shown in Figure 4-5. Thus, collapsing the EGC difference Matrix by

source reveals the information flow coming into every channel for each probe and produces a


PGC oollapsed across response channel
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Figure 4-6 Stimulus- Source difference plot of PGC where the EGC difference matrix is
collapsed by response. This plot makes use of the directional properties of PGC,
essentially each pixel represents the change in causal information coming out of each
source channel. This plot shows the directional change in PGC post-tetanization for
each source channel (x-axis) for it corresponding stimulated probe channel (y-axis).
Red pixels indicate a potentiation of outward causal flow while blue pixels indicate
depressed outward causal flow. The change in PGC for each pixel can be interpreted
as the change in the effective connectivity of the source channel to all other channels.









stimulus-response plot. When collapsing across the response axis, as shown in Figure 4-6, the

stimulus-source plot's pixels represent the summation of information flow from each individual

source channel to all response channels. Thus, collapsing the GC Matrix by response channel

reveals the information flow coming from every channel for each probe resulting in a stimulus-

source plot. Comparison of Figures 4-5 and 4-6 suggests that while after tetanization a pathway

may experience an increase consistent with LTP, a subset of the response channels experience an

increase in total information outflow while others experience an increase in total information

inflow. The same complementary observation can be seen with depressed probe channels that are

thought to experience LTD. Logically, this is consistent with the idea that an increase/decrease in

information outflow pre-synaptically should be matched by a corresponding increase/decrease in

information inflow post-synaptically. This reinforces the notion that the tetanus elicited changes

in the neural network consistent with STDP mechanisms.

When collapsing across the remaining probe channel axis, as shown in Figure 4-7 the source-

response plot's pixels represent the summation of information flows from a source channel to a

response channel for every stimulating probe. Thus, collapsing the EGC difference matrix by

stimulus channel reveals the directional information flow for each source-response pair for all

probes. Collapsing on this axis produces a source-response plot that is more consistent with what

could be observed in a recording of spontaneous activity from a MEA culture. The 60 source-

response slices of the EGC difference matrix of size 60x60 are also shown in Figure 4-8 at their

respective probe locations in a spatial arrangement meant to replicate the placement of the

electrodes in the MEA culture dish. These plots specifically speak to the larger amount of

information available when using PGC methods instead of spike rate. Using the spike rate

method an investigator can observe the change in spike rate at 60 channels following a probe.











PGC collapsed across probe channel


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Figure 4-7 Source-Response difference plot of PGC collapsed by probe before and after
application of tetanus. This plot shows the directional change in PGC post-
tetanization for each source channel (x-axis) for it corresponding stimulated probe
channel (y-axis). Each pixel shows the total information flow from each source to
each response channel over all probes. This plot is a sum of all of the source response
plots that are plotted later in Figure 4-8. This represents the total change in causality
from elicited activity over all probes for every pathway.

However, when using PGC, 3600 possible directional pathways can be observed for each probe.

This is largely due to the fact that rate based changes are non-directional and non-causal. So

when a probe is delivered only the changes in spike rate at distant channels can be observed.

When using PGC spike trains from any channel can be compared to any other channel. One of

Jimbo's observations was that the spike rates at distant electrodes for a specific probe channel

tend to change uniformly in one direction either depression or potentiation.





























































Figure 4-8 Source-response plots by spatial probe location. These source-response plots (each

interpreted the same as the source response plot in Figure 4-7 but for single probes

rather than all probes) are arranged topographically by the location at which the

probes were delivered to create the respective source response plots. This plot is a

difference plot showing the entirety of the 60 x 60 x 60 PGC difference matrix for

one MEA culture treated with Jimbo's protocol. Notice that while some channel were

dominated by potentiation (red) and others dominated by depression (blue), most

channels simultaneously demonstrate both potentiation and depression.


Interestingly, the plots in Figure 4-8 tell a slightly more complex story. While there are


some probes that produce changes in activity that fit this profile, in fact most probes show both


depression and potentiation within the network, information that is lost when averaging across


source or response channel axes. While potentiation or depression may win out after collapsing


to create a source-response plot, both mechanisms are most often present before collapsing.


However, PGC produces results that agree with some of Jimbo's major points. While probing at





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a channel activates the entire network, the pathways and synapses activated depend on where the

probe was delivered. This explains how a pathway can produce potentiation for one probe

channel and in turn show depression when a different probe channel activates those same

neurons. Thus, a similar conclusion as Jimbo's can be drawn from these source-response plots

that the observed changes are pathway-specific and not neuron specific.

Spontaneous/Elicited Connection

PGC can also be applied to spike trains from spontaneous rather than evoked activity. This

presents an interesting opportunity to compare source-response plots from the EGC difference

matrix to source-response plots from spontaneous activity. Recordings of spontaneous activity

were taken before and after tetanization during this experiment. Spike trains were collected from

population bursting events inherent to spontaneous activity in MEA cortical cultures, as opposed

to collection from a time window after stimulation as with elicited bursts. The spontaneous burst

events were detected using a burst detection algorithm based on several found in literature. A

window of 200ms is used to capture the activity from the burst. Unlike data collection from an

elicited time series the number of events is based on the spontaneous bursting activity of the

culture rather than based on a predetermined number of probes. The spike trains are low pass

filtered and used to calculate PGC values for every source and response pair for both pre and

post tetanus spontaneous recordings. The resulting difference matrix, or spontaneous Granger

Causality (SGC) difference matrix, is 60x60 and includes PGC values for every source to every

response channel. The resulting plot, shown in Figure 4-9, is most similar to the source-response

plot generated from the GC matrix collapsed across probe shown in Figure 4-7. Theoretically

these two plots should resemble each other as they are both source-response plots.

The connectivity in the spontaneous source-response plot is a result of the expression of

spontaneously activated pathways during population burst events. The pathways activated are










dependent on the patterns of the bursts expressed by the culture, which is dependent on many

factors possibly including spatial origin, connectivity, and environmental factors. While the

patterns are similar between elicited and spontaneous activity and are likely linked there are

many reasons why they should not completely agree. Spontaneous activity is a direct result of the

probabilistic processes that drive single unit activity in the network. Which bursts patterns occur

and those that do not are dependent on this process. In contrast, elicited bursting is not dominated

by this process. Each electrode is systematically probed and responses are observed. It is likely

PGC pfrmm spolmous ictybeforad afteetatus

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Figure 4-9 Spontaneous Source-Response plot calculated from windowed bursting events during
spontaneous (no stimulation probes) activity. This source (y-axis) response (x-axis)
plot represents the difference in PGC calculated from spontaneous bursts before and
after the delivery of the tetanus. Again the colors represent the change in causality
(blue = decrease, green = no change, and red = increase). This source response plot
should resemble the source response plot after collapsing by probe shown in figure 4-
7 if PGC analysis of spontaneous and elicited data is similar. Some similarities are
seen between these plots suggesting a connection between stimulated and
spontaneous neural activity.









that many of the pathways activated by probes are part of forced burst patterns that may be rarely

if ever generated from spontaneous activity and are thus missing from spontaneous connectivity

plots. Additionally, the elicited patterns are generated from 600 elicited burst events while the

spontaneous activity produces perhaps 20-30 burst events. Despite these differences, there is

similarities between the patterns from spontaneous and elicited source-response plots reinforcing

the linkage between spontaneous and elicited activity patterns.

Conclusions

The inverse source problem has proven a formidable obstacle for neuroscientists when

exploring network level plasticity mechanisms in vitro. While synaptic efficacy can be

monitored using glass pipette electrodes to detect and measure post-synaptic potentials, large-

scale use of these electrodes to monitor network activity is problematic. MEAs are a well-

established means of obtaining multichannel recordings from living neural networks but do so

extra-cellularly. To estimate plasticity using these arrays Jimbo et. al. devised a protocol that

reveals some pathway information for dissociated culture grown on an MEA by observing

changes in spike rate after the simultaneous induction of LTP and LTD. While spike rate is an

elegantly simple means of detecting pathways using this protocol, it is that same simplicity that

limits the amount of information that can be garnered. We have proposed the use of PGC as a

means of extracting synaptic efficacy from the spatiotemporal activity generated from elicited

and spontaneous activity. The linear relationship between PGC values and synaptic weight was

demonstrated in a simple biologically relevant neural model. The model demonstrates a linear

relationship in the region beginning where the synaptic efficacy is large enough to cause some

post-synaptic action potentials and ending where the synaptic weight is so large as to cause post-

synaptic action potentials every time the pre-synaptic neuron fires. It is only within this region

that PGC values change significantly in relation to synaptic weight suggesting that experimental









changes in PGC likely scale linearly to synaptic efficacy. Thus, there are circumstances, when

changes in synaptic efficacy are not reflected in neural spiking activity changes, and will not be

reflected in changes in PGC. However, based on this modeling it is reasonable to assume that

changes in PGC values are likely reflective of changes in synaptic efficacy.

The applicability of PGC to plasticity studies in-vitro was verified against established

experimental plasticity protocols with an established analysis method, spike rate change.

Comparison of PGC results to spike rate results shows that PGC is not only sensitive enough to

be used to detect plasticity but also provides information on the structural connectivity

underlying recorded neural activity prior to treatment.

While the two methods did not agree on all pathways, this can be easily explained by the

fact that PGC is sensitive to changes in inter-spike timing associated with LTP and LTD that

changes in spike rate may not detect. Conversely, some changes in spike rate may not be

reflected in a change in causality for similar reasons. Additionally, the creation of surrogate data

sets by shuffling the spike timings within individual realizations demonstrate that PGC possesses

a sound statistical foundation that allows discrimination between signal and noise events.

Directionality was intentionally removed to make the comparison to spike rate, however when

included it enables several new analysis methods to be used in conjunction with plasticity

experiments. With baseline connectivity available, it is possible to track changes in connectivity

after treatment in the context of these beginning values. This analysis yields results similar to

previously reported empirical observations of STDP in-vitro where whole cell patch-clamp

methods were used to monitor synaptic efficacy directly (84). Additionally, this plot suggests

that synaptic efficacies may be being driven towards equilibrium, illustrated by the transition

from potentiation to depression around values of 0.25 for PGC. Not only does this show that the









connectivity changes within the network are unlikely to be random, it also shows that there may

be some sort of homeostatic mechanism (107) at work within the network.

Changes in information outflow from a channel can be visualized when the EGC

difference matrix is collapsed across the source axes, making use of the directional property of

PGC. Conversely, changes in information inflow can be visualized when the EGC difference

matrix is collapsed along the response axis. Comparison of these two plots for individual probe

channels that experience LTP (or LTD) certain groups of neurons experience and increase

(decrease for LTD) in information outflow while other distinct groups specifically experience a

complimentary increase (decrease for LTD) of information inflow. This result suggests that the

observed changes in connectivity are specific and directional rather than uniform across the dish.

A uniform increase in connectivity could be indicative of fluctuations in environmental

conditions such as temperature rather than a result of STDP. Directionality also allows collapsing

the probe axis to create source-response plots for each channel probed, rather than just being able

to observe distant changes in spike rate in relation to the probe channel. These plots show that,

while one STDP mechanism may dominate for a probed channel, often LTP and LTD both occur

among the various pathways activated. Also, these plots reaffirm the concept that changes in

plasticity are pathway specific rather than neuron specific. This can be seen when a pathway

demonstrates potentiation in one source-response plot but shows depression when the location

probed changes. This suggests that, even though the same neurons are involved, the pathway is

different. Collapsing across these source-response plots by probe also has a logical connection to

PGC source-response plots generated from spontaneous activity. This connection may prove

useful in the design of future experiments in that it may not be necessary to perform a probe

sequence in order to observe changes in neural network structure. It is clear that PGC can









describe changes in network connectivity and not only compliments the use of spike rate but also

exceeds it in terms of the amount of information provided about the pathways underlying neural

activity. However, there are limitations of PGC that must be considered.

A causal relationship between two channels may be due to a common unobserved driving

influence. Additionally, it is possible that PGC values for a pathway will contain mediated

influences in addition to direct causal influences. These mediated influences would distort the

relationship between PGC values and synaptic weight. However, even with mediated influences

included, the changes observed are indicative of changes in the underlying structure of the

network for that pathway, not necessarily directly between two individual neurons. The

complexity of the synaptic connectivity within any given pathway prohibits a complete

description of all the synaptic weights within that pathway, given the finite fixed electrode

placement of MEAs. However, PGC makes the most of the data available by enabling a deeper

analysis of these pathways, than previously implemented methods. Results from simulated

pathways suggest that PGC can relate changes in "synaptic weights" within a pathway to

changes in the spiking output of postsynaptic targets. This suggests that changes in PGC

observed in pathways in-vitro represent changes in synaptic weights somewhere within these

pathways. This conclusion is further reinforced by in-vitro PGC results that suggest that

amplitude dependent plasticity occurs within pathways. This phenomenon has been well

described at the cellular level but never at the network level. These results suggest that PGC can

describe changes within in-vitro pathways that are known to occur at synapses between

individual neurons. Additionally, directional analysis of the EGC difference matrix suggests

that some areas experience specific changes in output while others experience corresponding

specific changes in input. This further suggests that pathway specific changes are real and









directional rather than random and undirected byproducts of natural variation. While much work

needs to be done to fully capture individual changes between neurons, these results suggest that

PGC is a useful tool for uncovering effective relationships in dissociated in-vitro cortical neural

networks grown on microelectrode arrays.









CHAPTER 5
DIFFERENTIAL TETANAZATION OF SINKS AND SOURCES IN DISSOCIATED
CORTICAL CULTURE

Introduction

Granger Causality was used in the previous chapter to identify changes network within in-

vitro cortical networks grown on microelectrode arrays. Granger Causality was also shown to

provide a better representation of the effective connectivity between channels before and after

the application of a tetanus. Moreover, unlike spike rate based methods, Granger Causality could

measure connectivity without the necessity of tetanization (i.e., it does not require a difference

between pre and post measurements). In this chapter, information about effective connectivity

will be used to predict apriori changes in plasticity prior to experimental manipulation. The

central hypothesis of this chapter is that if information about connectivity can be known before

hand and it is accurate, then this information can be used to predict the direction in which

plasticity will change when tetanizing a specific channel on the array.

Granger Causality provides information about which channels are causality related to other

channels and the direction of that relationship. In essence, the experimenter knows who is

connected to whom and how they are connected (i.e., direction). Between single neurons where a

presynaptic neuron is synaptically connected to a post synaptic neuron, it has been shown that

stimulating the presynaptic neuron followed -1 to 100 ms later by a stimulation of the

postsynaptic neuron will increase the efficacy of that synapse. Conversely, stimulating the post

synaptic neuron before the presynaptic neuron results in the depression of that synapse (a

decrease in efficacy) (84, 108). This is known as spike timing dependant plasticity (STDP) (83,

109) where the timing of activity or stimulation determines the strength of the synapses in a

network. If information about the structural connectivity between neurons recorded with our

MEAS were known then it should be possible to select a causally connected pair of neurons and









stimulate that pair to either potentiate or depress the synapses between them dependant on the

timing of the stimulation (pre vs post or post vs pre).

However, each channel on the MEA records from several neurons simultaneously. In

addition, a single stimulation pulse (e.g., a single biphasic 600 mV 200 us pulse) may activate

neural activity in a wide area surrounding the stimulation electrode, but this is of course

dependant on the voltage that is applied. Instead of attempting to identify a particular pair of

neurons perhaps identifying channels whose primary relationship to others was to drive activity

in those channels could be a solution (i.e., an analog of a pre-post synaptic relationship).

Stimulating that driving channel would essentially be similar to a presynaptic-stimulation

/postsynaptic-activation. In other words, spike timing dependant potentiation of that synapse and

perhaps result in an increase in causal strength between that channel and others. Conversely,

stimulating a channel, whose primary relationship with others is as a target (i.e., a sink where

other channels converge to drive activity on this channel), should result in the depression of that

pathway. In other words, a postsynaptic neuron stimulated before its presynaptic counterparts.

Of course, identification of causal sinks and sources within the cortical culture would be

critical for the execution of this stimulation paradigm. For this study, a sink is defined as a node

(channel) within the culture where the balance of causal information flow is inward to the node.

In contrast, a source is defined as a node where the balance of causal information flow is outward

from the node. When PGC is calculated between two channels A and B, PGC analysis yields

results from A to B as well as from B to A. Sinks and sources could then be identified when PGC

analysis is carried out amongst all channel parings. Summing PGC values for all pathways

leading out of a channel, EFout, represents total information flow coming out of that channel.

Summing PGC values for all pathways coming into the same channel, XFin, represents total









information flow coming in to that channel. Positive values of the difference, XFout YFin, would

indicate a source within the neural network (i.e. dominated by information outflow) while

negative values would indicate a sink (i.e. dominated by information inflow). See Seth et al for a

similar identification technique that has been tested in artificial networks (55).

In this chapter, an experiment is described that tests whether differential tetanization of

sink channels versus source channels produces differing outcomes. Activity from in-vitro

cortical networks will be measured to identify the location of potential sinks and sources among

the channels on the array. A source channel will then be tetanized and changes in plasticity will

be measured. The prediction is that the causal flow out of this source channel should increase. A

sink channel will then be tetanized and the prediction is that causal flow into this sink channel

should decrease.

Background

Identification of Neural Sinks and Sources

Identification of sources and sinks of neural activity in multichannel data has proven useful

in many theoretical and experimental neuroscience paradigms. For example, identification of

sinks, sources, and the corresponding current flow within hippocampal slice is useful for

mapping neural circuits within the hippocampus (110, 111). Causality in the form of the Directed

Transfer Function has been used in epilepsy to detect the source of seizure generation also

known as the seizure focus (112) from EEG. This is done so that the seizure focus can be

surgically removed, effectively ending seizures in a patient. Essentially, the seizure focus is

located by finding a source of causation within the epileptic brain. This method has also been

used with ECoG signals to determine the ictal source (113, 114). Ding et al. refined the use of









the Directed Transfer Function so as to be used with EEG rather than ECoG because the use of

ECoG electrodes is invasive requiring placement under the dura.

Plasticity by Tetanization

Plasticity has been studied in-vitro using MEAs in several published experimental

protocols (4, 18, 21, 115). Most often plasticity is induced at an active channel (where activity

has been recorded) using a tetanic stimulation. The frequency, amplitude, and duration of these

tetanic stimulations have varied between investigators. For example, Jimbo (4) applied a

tetanization sequence of 100ms (20 trains of 10 bipolar 100lts 0.6V pulses at 20Hz every 5

seconds) while Shahaf (18) employed a constant train at 1, 0.5, or 0.3Hz (each pulse is biphasic

50tjA 500|ts) until the desired plasticity was achieved. Tateno et al (21) applied tetanic pulse

trains (similar to those used by Jimbo) to two channels simultaneously in the same culture.

Where the tetanus is delivered in the culture varies between investigators as well. Often

the tetanus is simply delivered at a highly active channel that elicits a network burst response

when stimulated. Interestingly, it has been shown that channels whose activity is strongly

correlated with the channel during tetanus are most often potentiated while those that are weakly

correlated are most often depressed (4). This sort of reasoning is merely an observation however

and requires the experiment to be conducted first before the direction can be determined. In

contrast, we are able to determine what the effective connectivity is for a channel prior to

tetanization based merely on spontaneous neural activity recorded before any stimuli are

delivered. The correlated changes described by Jimbo et al are however consistent with our

notion that when these channels are tetanized (they would appear as sources in our analysis) the

activity produced by that channel should be enhanced (i.e., increase).








Experimental Design

The experiment used 10 mature (all subjects were more than 35 days old) cortical neural

networks grown on MEAs. Each subject was exposed to 2 tetanic pulse trains (20 0.2Hz trains


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Figure 5-1 Topographical map of sinks and sources spatially laid out and superimposed on a
picture of the microelectrode array. Sinks are blue while sources are red (green
indicates neutral) all calculated from spontaneous bursting activity using Pairwise
Granger Causality. Row 4, Column 4 represents a strong sink and Row 7, Column 5
represents a strong source. Each of these would be candidates for tetanization to
differentially induce depression or potentiation, respectively..

of 10 bipolar 100[ts 0.5V pulses at 20Hz) 1 per day delivered to either a sink or source. Sinks

and sources will be identified immediately prior to tetanization from baseline recordings of

spontaneous activity for 10 minutes. We will then identify burst events in these spike trains

using an algorithm that uses a thresholded global spike rate to detect the onset of each burst. The

end of the burst is detected using the quiescent time after the burst where no spiking occurs

(116). A 200ms window of spiking activity is then extracted following the onset of each burst.









These spike trains windows were then low pass filtered at 50 Hz so that Pairwise Granger

Causality can be used to determine causal relationships between all combinations of channels.

Since each burst is treated as a realization of the same process over 100 realizations can typically

be used to calculate PGC from the 10-minute recordings.

The total inflow of causation into an electrode is calculated by summing the PGC values that

include the specified electrode as the destination of causation. Similarly the total outflow of

causality is calculated by summing the PGC values that include the specified electrodes as the

source of causation. The difference between total outflow and total inflow will be used to

determine whether a particular electrode is a sink (negative values of this difference) or a source

(positive values of this difference). This value will be calculated for every electrode in the MEA

and displayed spatially to allow selection of the electrode for tetanization. An example of a map

of sinks and sources within a cortical culture is shown in Figure 5-1. This plot shows the

difference of total inflow and total outflow represented by color (red = source, green = neutral,

and blue = sink) for each electrode overlaid over an image of a microelectrode array. This type of

arrangement allows for easy spatial identification of where sinks and sources are in the cortical

culture. The electrode selected for tetanization will be limited to electrodes that are not on the

edge of the culture to better allow observation of effects from tetanization. Each dish will be

tetanized once per day on either a sink or a source depending on the requirements of the

experimental design lattice. The first day of this protocol will consist of 5 randomly selected

cultures tetanized on sinks and 5 cultures tetanized on sources. The second day will reverse the

tetanization of sinks and sources for each culture. Thus, over the course of this two day

experiment each dish will be tetanized on a sink and a source.









Spontaneous vs. Elicited Activity as a Measure of Plasticity

Spontaneous and elicited bursting are both based on the structure of the same underlying

network and logically reflect similar patterns of connectivity. However, spontaneous bursting

activity is initiated with firing probabilities and patterns that are generated dynamically within

the neural network. Elicited bursting patterns, while dependent on the same underlying structure,

is evoked artificially and can generate patterns that spontaneous activity may not (also a possible

reason the test probe sequences can induce plasticity). Changes in spontaneous bursting activity

revealed using correlation matrix based methods have been used to identify plasticity related

changes post tetanus (20). In this experiment, the difference between pre-tetanization

connectivity estimated by PGC from spontaneous baseline activity prior to tetanization will be

subtracted from PGC estimates post-tetanization to determine the changes in connectivity that

result from tetanization protocol.

This experiment will use two primary means of detecting plasticity after tetanus. Both of

these methods were developed in chapter 4. First, PGC will be used on spontaneous activity (pre

and post tetanization) to create a source response difference-plot to determine whether pathway

specific potentiation and depression occurs as a result of the tetanization. Essentially this will

mean looking for potentiated and depressed source and response pathways. Blue (depressed

pathway) or red (potentiated pathway) horizontal and vertical lines in source response plots

would be an indicator that plasticity may have occurred. Additionally, an amplitude dependent

plasticity effect (ADP) (84) was also found from the data analysis in chapter 4. Bi and Poo

found that the post synaptic potentials (PSPs) (the magnitude of the voltage will be referred to as

"weight") in the post-synaptic neuron in a simple neuron pair exposed to negative spike timing

(post-synaptic neuron fires before pre-synaptic neuron) would elicit depression. These depressed

pathways would decrease by a constant percentage independent of starting "weight". However,










when exposed to a positive spike timing (pre before post) potentiation would occur that was

dependent on the starting "weight". Specifically, smaller PSP were increased by a larger

percentage than larger ones. In chapter 4, this relationship was found in pathways analyzed

using PGC from elicited spiking activity, rather than measing PSPs directly. This analysis is

made possible by PGC providing a measure of effective connectivity in the pre and post





10.1 + .-- --

-W
Uo < W
I"



20 L





O4
R


30 67
2 s --i





4 .; h m

0 m


10 20 30 40 50 60
response electrode


Figure 5-2 Source response plot for culture 9059 tetanized on a sink. This plot shows the change
in PGC post-tetanization. The y-axis is the source electrode while the x-axis is the
response electrode. Changes range from depression shown in blue to potentiation
shown in red. Green indicated no change. Notice that channel 40 experienced a
increase in causal output (red horizontal stripe at source electrode 40) and
simultaneously experienced a decrease in causal input (blue vertical stripe at response
electrode 40).

tetanization results. Thus, it is possible to use the pre-tetanization PGC analysis as starting

"weights" for the observed pathways and use the resulting post-tetanization PGC analysis to









calculate a change in these "weights". This ADP analysis will also be used to determine whether

plasticity occurs during this experiment. ADP results will also be statistically analyzed to

determine the direction of any change in plasticity is consistent with out predictions.

Results and Discussion

Results

The experiment was run over the course of two days as described earlier. During the

experiment it was easy to notice that cultures grown on certain MEAs showed a larger amount of

causality between electrodes than those that were likely experiencing failures in the insulation of

the electrode tracing. Cracks in the insulation cause leakage of current from the electrode tracing

(117). These cracks can decrease the signal to noise ratio of the electrodes making it

increasingly difficult to distinguish actual spiking events from noise events. Based on this

observation, 2 of the cultures were removed from the statistical analysis (dish 9299 and 9058).

The are included in the tables, however. Several of the MEAs, where insulation damage was

likely, were not removed because the effect was not as severe. PGC was calculated for before

and after the tetanization for all subjects in the experiment. The measures mentioned earlier we

used to first determine whether plasticity occurred during the course of this experiment.

First, a source-response difference plot was constructed for each trial to qualitatively

determine what type of plasticity effects were seen post-tetanization. An example of these plots

are seen in Figure 5-2 for a culture tetanized on a sink and Figure 5-3 for a culture tetanized on a

source, demonstrating that both potentiation (indicated by red) and depression (indicated by

blue) occurred when tetanizing on sinks and sources. Notice that there are horizontal lines

representing changes in the output of a channel and vertical lines representing changes in the

input to a channel. These results are consistent with plasticity shown by Jimbo, showing














10 I I II aI I I
i- I I I I I








-,-A *N urn 1 -MI m



2 30


0
U m J U E















40 -
A2 E H








10 20 30 40 50 60
response electrode

Figure 5-3 Source-response plot for culture 9449, which was tetanized on a source. This plot
shows the change in PGC post-tetanization. The y-axis is the source electrode while
the x-axis is the response electrode. Changes range from depression shown in blue to
potentiation shown in red. Green indicated no change. Some channels were
depressed while others were potentiated.

simultaneous potentiation and depression for pathways in the culture. Second, the PGC analysis

was examined to look for any ADP trends. Plasticity effects similar to those reported in Chapter

4 were found. This analysis will be elaborated on in the ADP analysis section. However, both

of these key indicators suggest that plasticity consistent with the experiment in Chapter 4

occurred. The remaining analysis will determine if these effects were consistent with the

predictions made earlier.
predictions made earlier.









Table 5-1 Global culture causality changes post-tetnanization. The change in causality was
assessed for sinks and sources by summing total causality for all of the pathways in
the culture pre and post tetanus and taking the difference between these values. No
consistent differential is seen between sinks and sources. Greyed out values were not
used in statistical analysis.
Culture Change in total causality after Change in total causality after
Tetanization at a Sink Tetanization at a source
9447 42.22 118.86
9449 -229.07 -97.04
9448 -89.39 -90.88
9450 -22.82 -155.91

9291 51.9 -17.26
9272 16.2 62.67
9059 9.34 11.25

9053 -13.85 -39.65
Mean -29.43 -25.99

Qualitative inspection of the stimulus response plots for each subject shows no

observable difference between tetanization of a sink versus a source. A variant of this metric

used was quantitatively comparing the difference in a sum of all causality for all pathways pre

and post-tetanization. This was done for the entire array as a global measure and also for the

tetanized channels alone. A statistical analysis of the resulting difference in causality finds that

there is no difference between tetanization on a sink versus a source (F(7,7)=0.001, the raw

results are shown in Table 5-1). Cultures experience both global increases and global decreases

for tetanization both on sinks and sources. This result is not unexpected since changes described

by Jimbo involve correlations with the tetanized channel. When looking at the tetanized channels

alone there is a noticeable change in the magnitude of the causality for the pathways leading out

of sources and into sinks. Statistical analysis shows that while the direction of change

potentiationn versus depression) is not consistent across all subjects sinks or sources, the

magnitude of the changes are statistically significant (F(7,7)=6.40 for sinks and F(7,7) = 3.23










Table 5-2 Tetanized channel total causality changes post-tetanization. The change in causality
was assessed for sinks and sources by summing total causality for only the tetanized
pathway pre and post tetanus and taking the difference between these values. No
consistent differential is seen between sinks and sources. Grayed out values were not
included in statistical analysis.
Culture Tetanized Sink Tetanized Sink Tetanized Source Tetanized Source
Change in Outward Change in Inward Change in Change in
Pathways Pathways Outward Inward Pathways
Pathways


9447
9449
9448
9450

9291
9272
9059

9053
Mean


1.84
-1.78
-0.44
-0.04

-0.07
-0.29
-0.78

-3.16
-0.59


10.44
-10.12
-8.18
-11.01

-1.56
-.012
0.80

-4.12
-2.97


-0.78
-15.44
-3.87
-1.38

1.04
0.58
1.72

-3.91
-2.75


-1.24
0.42
-0.09
-0.92

-0.59
-0.01
0.22

0.38
-0.22


02 0.4 06 08
baseline causality


0 02 0.4 06
baseline causality


Figure 5-4 Culture wide Amplitude Dependent plasticity. Each pathway's initial causality is
plotted on the x-axis while it's corresponding change after tetanization is the y-axis.
This is a compilation of the effects seen for all 10 cultures. The effect for sinks is
seen in panel A while the effect for sources is seen in panel B. There is no observable
or statistical difference between these two plots and no differential effect between
sinks and sources.


0.8 1


\











sources) for pathways leading out of sources and into sinks when compared to the magnitude of

changes for pathways in the opposite direction.

ADP is the final method used to determine if plasticity related effects followed the

primary hypothesis. This was not only used to compare sinks versus sources, but to also get an

idea of whether a pathways magnitude could be used to predict changes post tetanization. Again,

the results are compared globally across the entire dish and on the tetanized channels alone.

Global ADP is plotted for all pathways in Figure 5-4 panel A) for all sinks and panel B) for all



&







1 1
S0o5 0.5

-0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8
baielin causality basel.n causal.y


1 1

| 0.5 + .5













Figure 5-5 Tetanized channel amplitude dependent plasticity. The initial causality of pathways
involving the tetanized channel's is plotted on the x-axis while it's corresponding
change after tetanization is the y-axis. Panel A are the pathways coming out of the
tetanized sinks. Panel B are the pathways coming into tetanized sinks. Panel C are the

pathways coming out of the tetanized sources. Panel D are the pathways coming into
tetanized sources. There is no observable or statistical difference between these two
plots showing no differential effect between sinks and sources.









sources. Visual inspection of these graphs shows them to be very similar. A linear model for

ADP (initial causality versus change in causality) was calculated for each of these graphs to

allow for a quantitative comparison. Both models are standard linear models of the form

y=mx+b. The slope, m for the sink plot is -0.20 while the slope for the source plot is -0.14.

These slopes suggest that larger weights in both cases typically are depressed. This is true for all

models in this study. It is worth noting that the intercepts of all models in this study are small and

positive, suggesting a small region where small weights are typically potentiated. It is of

immediate interest that the norm of the residuals is high for both models at 13.0 and 13.8

respectively. The variability in higher magnitude starting causalities is likely the source of the

poor fit. Essentially, these models cannot be considered different, thus there was no global

differential effect. This finding is in line with previous global observations. The same metric

and methods are then applied to only the tetanized channels segregated by sink and source as

well as by pathways coming into and pathways coming out of that channel, shown in Figure 5-5.

The parameters for the models fit to the plots are shown in Table 5-2. These plots and models fit

to them may explain why large significant deviations were found on channels coming out of

sinks and into sources from earlier sink source analysis. On first inspection the slope of

tetanized sink pathways average -0.23 while tetanized source pathways average -0.11, suggesting

that sink pathways may be depressed more. However, the norm or the residuals for pathways

coming into sinks and pathways out of sources remain large at 3.22 and 3.94 respectively,

suggesting high variability especially in the higher starting causality region. The norms of the

residuals pathways coming out of sinks and into sources is much better at 1.06 and 0.70

respectively. However, it's easy to notice that the goodness of fit for these pathways is likely

improved because of the virtual lack of strong initial causalities for these pathways. This









suggests that the statistical effect shown in the earlier sink source analysis may be a result of the

way sinks and sources were chosen rather than any differential effects due to tetanization.

Overall, this suggests that no significant differential effects were detected.

Conclusions

This experiment did produce plasticity similar to the results reported earlier in Chapter 4. This

can be seen in the comparison of the ADP trends from this experiment and the replicate of the

Jimbo experiment. These results suggest that the tetanic stimulations in this experiment

produced similar ADP related changes seen in Chapter 4. The cultures also showed

simultaneous potentiation and depression another hallmark of the Jimbo protocol. This result

suggests that changes that occurred post tetanization are not simply dependent on global shifts in

spike rate. These effects were seen in Figure 5-2 and 5-3 for both sinks and sources. This

suggests that it does not matter whether spontaneous or elicited activity is used to reveal the

effects of tetanization using PGC.

The predicted direction of that changes based on stimulating a sink versus a source were

not supported. The prediction made in this experiment was that a differential effect would be

seen when sink and sources, identified using connectivity derived from spontaneous activity,

were tetanized. It is possible that the effective connectivity revealed from spontaneous activity is

not a good predictor of what pathways will correlate with a pathway when it is tetanized.

However, this only explains why an ADP effect may not be seen. This does not account for the

predictions made based on STDP rules. The likely reason why these predictions did not work is

because STDP rules are based on changes between a single pre synaptic neuron and a single

postsynaptic neuron. Predictions in such a simple network would be easy to predict. The

cortical culture used in these experiments is created from 50,000 dissociated embryonic neurons

that randomly connect to form a neural network. The actual connectivity in these cultures is









much more complex than the controlled conditions in the experiments by Bi and Poo (118). The

MEA can simultaneously observe the activity of nearly 100 neurons. These neurons may have

thousands of synapses with other neurons in the culture (97) and many of these connections may

be bidirectional. Additionally, it is difficult in this complex network to control what pathways

experience positive and negative spike timings. Reverberating connections and feedback loops

may present many spike timings to the same synapse. Additionally, the effects of the spike

timings from spontaneous bursting during and after the tetanus may have an effect. Perhaps the

STDP based predictions, which were based on results from two isolated mono-directionally

coupled neurons, are too simplistic given the complexity of the connectivity in these dissociated

cultures.

This experiment was devised to be a proof of concept for being able to measure the

effective connectivity in the dish and then be able to manipulate that connectivity based on this

knowledge. While the predictions for this experiment may have been too simplistic for this

system, future studies could benefit from the use of PGC to reveal connectivity before

experimental manipulation. Future studies may also benefit from a reduction of the complexity

of the cortical culture by reducing the density of the culture. Another possible strategy may be to

use a paired pulse paradigm to better control the spike timings in the targeted pathways.









CHAPTER 6
ANALYSIS OF TEMPORAL LOBE SEIZURES USING GRANGER CAUSALITY

Introduction

Prior research in this body of work have focused on using Granger Causality metrics to

reveal effective connectivity relationships in neural simulations and in in-vitro dissociated

cortical culture. These experiments provided new insights into the connectivity underlying these

networks and new metrics of plasticity. However, one ultimate goal for this research is to

provide an analytical technique to better understand on how the intact brain functions. Data from

the Evolution into Epilepsy Project at the University of Florida presents a unique opportunity to

explore the in-vivo dynamics and effective connectivity relationships of seizures from an animal

model of temporal lobe epilepsy (TLE). High quality 32 channel data from microwires was

continuously recorded in vivo from rats at a high sampling rate (25 KHz per channel). The data

from these animals include multiple seizure events from several animals. The 32 electrodes were

bilaterally located in the hippocampus and afforded the opportunity to look at 992

( n! f
n( k)- ; n=32 channels, k=2 for bivariate model different directional hippocampal
(n k)! )

interactions on multiple time scales using Granger Causality.

In this chapter, the first goal is to first determine if Granger analysis can provide

anatomically relevant descriptions of effective connectivity in that anatomical structure. In

contrast to the dissociated cortical cultures used in the previous two experiments, the anatomy

and the physiology of the hippocampus is well established (119). However, descriptions of the

effective connectivity of the hippocampus from implanted microwire arrays are rare in general

and nonexistent for seizure behavior. The second goal is to provide a better description of the

dynamics and causal relationships with and between hemispheres underlying seizure activity and









how they differ from those observed during baseline activity. This chapter will also provide a

brief description of TLE, animal models of TLE, and the Evolution into Epilepsy Project.

Temporal Lobe Epilepsy

Epilepsy is a disorder that effects roughly 2% of the population. The most common type

of epilepsy among adult patients is Temporal Lobe Epilepsy (TLE) (120). The symptoms of

TLE are well defined and classified by the International League Against Epilepsy (ILAE).

Simple Partial Seizures involve small areas of the temporal lobe and typically do not affect

consciousness. They are often characterized by auras or periods where sensations such as sounds

or smells are felt that are not actually present. Complex partial seizures can affect

consciousness and the ability to interact with others. Often, the symptoms include staring,

automatic movements of the hands and mouth, and other abnormal behaviors. Seizures

associated with TLE can also spread to the rest of the brain causing Secondarily Generalized

Tonic-Clonic seizures. The tonic phase of this type of seizure is characterized by stiffening of the

limbs and body in either a flexed or extended position. The following clonic phase is

characterized by uncontrolled jerking of the limbs and body. Electrophysiologically, TLE is

characterized by seizure activity that can be recorded from the hippocampus. Physiologically,

TLE is also characterized by hippocampal sclerosis consisting of pyramidal cell loss, granule cell

dispersion, and gliosis within the hippocampus. Also, patients with TLE can often be cured of

seizure activity (80-90% success) by the removal of the hippocampus, suggesting that the

hippocampus implicating it as a potential source of TLE seizures.

There are several animal models used to study TLE. These include models where TLE is

induced pharmacologically and also those that use electrical stimulation. Of those that use

stimulation some require stimulation to elicit a seizure while others involve spontaneous seizure

generation. For example, kindling is produced by repeated subthreshold focal electrical









stimulation of the brain. After these stimulations have been delivered for sufficient time the

targeted area reaches a "kindled state" where a single stimulation will produce a secondary

generalized seizure. However, these seizures remain stimulus locked requiring a stimulus pulse

to generate seizure events. In contrast, Lothman (121) has developed an animal model with

spontaneous seizures in which the rat is stimulated once with a high frequency superthreshold

stimulation designed to immediately induce status epilepticus. The resulting limbic seizures

damage the hippocampus and result in recurrent spontaneous seizures that develop and increase

in severity over time. This model is used by researchers at the University of Florida in the

Evolution into Epilepsy (EIE) Project (40) to study the electrophysiological and physiological

changes that occur between the time of stimulation and the expression of spontaneous seizures.

Animal Model of Chronic Limb Epilepsy

The TLE animal model used in the Evolution into Epilepsy Project has been in use for the

last two decades (121). In the weeks that follow this stimulation, the rat progressively

demonstrates recurrent spontaneous hippocampal seizures. The temporal and electrographic

progression of rats treated with this model closely resembles that of human patients that have

chronic mesial temporal seizures. The surgical and electrophysiological techniques used by the

Evolution Into Epilepsy (EIE) project team at the University of Florida are described below.

The rats were prepared according to procedures approved by the University of Florida

IACUC. The rats are adult male Sprague-Daly rats that are 50 days old. The surgical procedure

consists of a craniotomy from 1.7 mm lateral to 3.5mm lateral of the bregma so that electrodes

could be implanted into areas due to their prominent role in epileptic discharges (122). A 2x8

rectangular array of electrodes was implanted bilaterally in both hemispheres so that they cover

the CA1/CA2 area to the DG. The electrode placement is shown in Figure 6-1 with the surgery

shown in the left panel of Figure 6-2. The spacing between electrodes is 400 |tm on the short





















S.V TV Recording Richi side
Left side
SRecordin f array 2
Array 1 I

St1imulaling ,
/'' \ / /^eectrode

ii \ J I 8.0/
I I

2X 4 Ba ~ ~a \k
// /,, \\

/,f' I 24,2-3,2.21.21, I9. 1,,s,

SIS 14 I S i ,: I [I Z .1627.2S.210, ,9J1 J:
S 2. 3, 4, 5. 6. 7. 8
lhectrode configuration showing lxoationi of channels.

Figure 6-1 Diagram of microelectrode placement in the rats brain. The upper left is a
representation of the skull of a rat showing the bregma and lamda locations. The
cartoon in the main caption shows the orientation of the electrode strips and the
corresponding location of individual electrodes. (Used with permission from Dr. Paul
Carney).



































Figure 6-2 Placement of the electrodes during surgery. This image shows a single 16 electrode
array being lowered into position into the rat's hippocampus. The electrodes vary in
depth so that they are lowered to a depth where targeted structures lie in the
hippocampus. The location of the electrode tips was later confirmed using MRI after
the rat has been sacrificed.(Used with permission of Dr. Paul Carney).

axis and 200 jtm on the long axis at a depth of 4 mm from the surface of the brain. The length

for each electrode is designed such that the tip of each electrode is within the brain region of

interest. The location of the end of each array closest to the centerline of the brain is 4 mm

caudal to the bregma and 1.7 mm lateral. A single Teflon coated bipolar twist electrode was

placed in the posterior ventral hippocampus in the right hemisphere of the animal, posterior to

the electrode array for stimulation into status epilepticus. After the experiment was completed

the rats were sacrificed and high-resolution magnetic resonance imaging (33) was used to verify

the position of the electrodes, as shown in the lower right panel of Figure 6-2.

































Figure 6-3 Example of the rat environment and recording chamber. These cages are designed so
that the electrode cables can be constantly attached to the rat's head while also
allowing the rat to freely explore its environment.

The rats were kept in an 30 cm enclosure designed to permit the electrodes to be connected

to the data acquisition system while the rats engaged in exploratory activity. This enclosure is

shown in Figure 6-3. The rats were allowed to heal for one-week post surgery before stimulation.

A 10 second train of 50Hz 1 ms bipolar 240[tA pulses was delivered every 12 seconds for 60

minutes to induce status epilepticus in the rats. 32 channels of simultaneous recordings were

taken from the rats 24 hours a day for 76 days while the rat was engaged in sleep and normal

exploratory behavior. Data was acquired using a Tucker-Davis (Alachua, FL) Pentusa at a

sampling frequency of 24.4141 kHz, band pass filtered from 0.5 Hz to 12kHz, and stored to disk.


~I











Time Domain Pawi grage Causaliy Betwee Electrodes Durig Seiaue E in Rt004

4.5
28

424


20 3

16 2.5
-2
12

8

40.5

1 0
1 4 8 12 16 20 24 28 32
Response Electrod


Figure 6-4 Initial Pairwise Granger Causality analysis of four 60-second seizures for a single
animal. These plots are used to display the causal interactions between all electrode
combinations. The plots are organized in terms of the source electrode (y-axis) and
the response electrode (x-axis) for the Granger analysis. The magnitude of the causal
interaction between source and response electrodes is represented by the color scale
in the left of the figure, with blue being zero and red being strong interaction. This
plot clearly shows two regions, once consisting of the first sixteen electrodes (bottom
left quadrant) and the other consisting of the last sixteen electrodes (top right
quadrant). These areas correspond to the electrode arrays in each hemisphere. The
average within-hemisphere interactions are strong while the hemisphere to
hemisphere (top left and bottom right quadrants) interactions appear weak across the
four seizures.

Seizure Analysis with Pairwise Granger Causality

Seizures detection was conducted manually by visual detection of the raw traces and

verification against video records. The seizures were catalogued and the start time of the seizure

was noted. A 10-minute window surrounding each seizure was then extracted for all 32 channels

from the raw recording and low pass filtered to 1kHz. This frequency was chosen because it

maintains most of the features in the data while allowing for a reasonable model order for

autoregressive modeling required for parametric PGC. A 60 second window was then chosen

within the 10-minute period of filtered data that encompassed the seizure.









Spectral Pairwise Granger Causality was then calculated for all possible electrode pairings

(992 directional combinations) for the entire 60-second window for 4 seizures to see if PGC was

able to describe causal interactions for epileptic seizures. The stationarity of the time series was

addressed by normalizing the mean and variance. The Bayesian information criterion (BIC) was

used to determine the correct model order, which was found to be m=25. The resulting spectral

analysis was then integrated across the spectral domain to produce a single time domain value

for each electrode pairing for each window resulting in a 32x32 matrix. Granger analysis was

then plotted as 32x32 image, shown in Figure 6-4. These plots are organized so that each pixel

represents the causal influence of a driving channel (y-axis) and a driven channel (x-axis). The

magnitude of this causal relationship is represented by color with blue being zero ranging

through light blue, green, yellow, and red with red being the strongest interactions. This plot

clearly distinguishes two distinct regions, a lower left and a top right quadrant, that have strong

within region causality but weak cross-hemisphere interactions.

Based on promising results from this initial analysis, a higher resolution analysis was

conducted where PGC was calculated over a series of 1 second moving windows for each seizure

independently. The overlap between these 1 second moving windows could be modified to

create a real time movie of the interactions. A 50% overlap was used for each seizure resulting in

129 data windows for PGC. This data for each window was then plotted as 32x32 image and

combined into a movie at 2 frames per second. This allows the dynamics of 32 interacting

channels for each seizure to be better visualized for qualitative analysis.

Calculation of average PGC values by the eight electrodes in each area (L-DG, R-DG, L-

CA1, R-CA1) allowed creation of an additional PGC index to condense the spatial and temporal

interactions quantitatively. Each seizure was visually analyzed to identify and window specific









stages. A single value for the interaction between each indexed area was calculated that

represents the average for that stage and was independent of the length of the stage. This

provides 4x4 matrix for each stage for each seizure. Analysis of variance was calculated

between different directional interactions over the pool of 15 seizures from the three animals

used in this analysis. Additionally, baseline interictal activity was analyzed using this method

for comparison.

Finally, a higher time resolution movie was made for one of the seizures. This movie is 30

frames per second and includes behavioral video, a trace of the recorded waveforms, and the

corresponding PGC analysis. The same one-second data window was used to calculate PGC

interaction, however this analysis window was moved by 1/30th of a second for each

corresponding video frame. Frames from this movie are displayed in the following figures to

better describe the seizure stages that will be identified and described in the results.

Results

Qualitative visual analysis of the raw waveform, behavioral videos, and PGC movies

reveal a common progression of events during all 15 seizures into five major stages. Figures 6-4

through 6-9 highlight examples of interictal activity 60 seconds prior to seizure and also from

each of these stages. These plots are frames from the high-resolution movie compiled for one

animal. Each figure includes a snapshot of surveillance footage of the rat in the bottom left

corner, four of 32 electrode voltage traces (one from each major region L-DG, R-DG, L-CA1, R-

CA1) on the right, and a 32 x 32 Granger Causality analysis in the upper left quadrant. The

electrode traces are normalized against each other and are time locked to the surveillance video.

A one second window highlighted between the red bars in on each voltage trace represents the

current analysis window whose results are depicted at the top left.









Each Granger analysis plot represents the interactions by electrodes in the source area (y-

axis) versus electrodes in the destination area (x-axis) for each electrode combination. The

magnitude of the Granger Causality analysis for each pairing is represented by color with blue

being near zero ranging to red representing the strongest interactions. For example, in Figure 6-5

the bright red area in the lower left are interactions from electrodes in the L-CA1 (y-axis) to

electrodes in the L-CA1 (x-axis). This example would be interpreted as interactions within the

L-CA1 area during the pre-ictal period roughly 30 seconds before seizure.

The seizures begin in stage 1, shown in Figure 6-6, in which a mixture of intrahemisphere

activity was dominated by causal activity within the CA1 region and from the CA1 region to the

DG. This activity gradually builds up over time. Behaviorally, Stage 1 often includes the

beginning of the tonic phase of the seizure, where the animal remains motionless and blinks its

eyes. As this causal activity becomes stronger, it begins to spread across hemispheres at the

beginning of stage 2.

Stage 2 is the most striking, marked by high-magnitude mono-directional causal

reverberations across hemispheres as well as substantially increased intra-hemisphere causal

interactions. These transfers often reverberate sequentially from one hemisphere to the other.

An example of a directional transfer between hemispheres is shown in Figure 6-7 from the left to

the right hemisphere. After several directional transfers the seizure moves into stage 3, shown in

Figure 6-8, marked by mainly intra-hemisphere activity. However, unlike the intra-hemisphere

waveforms of stage 1, the intra-hemisphere Granger causality interaction patterns are both more

dynamic, of larger magnitude, and less confined to a single area or pathway. Only week

occasional cross hemisphere directed transfers occur in stage 3.














R-DG


O R-CAI





]4.-j


L-CAL L-DG R-CAI
I- fl r I I. F : DII FI


R-DG _______





R-CA1








L-CAG
L .. c>li. ., .L^ *^ ...





L

I- I I
45 50
TIME [seconds]


55 60


Figure 6-5 Example of data and Granger analysis during the interictal (pre-ictal) stage. This plot
is of one frame approximately 30 seconds prior to seizure. The right panel includes
sample raw data traces from each of the four main areas recorded from including the
right and left DG and the left and right CA1. The segment of data analyzed is
between the vertical red lines in the sample electrode traces. The top left panel is a
source (y-axis) response (x-axis) representation of the PGC analysis with the
electrodes organized by area. An interictal spike is included in the analysis for this
frame. Most of this activity is within CA1 and is also directed from the CA1 to the
DG. The bottom left panel is a frame from surveillance footage of the rat's behavior.
In this frame the rat is either resting or sleeping.



























RESPONSE ELECTRODE












1851 60 665 70 75
L-.CA L-DG R-CA1 R-DR
RESPONSE ELECTRODE












1851 6a 65 70 75
i1 1 i I




Figure 6-6 Beginning of the seizure in the high resolution movie classified as Stage 1. Intra-
hemisphere activity is the primary interaction unique to this seizure stage. The right
panel includes a sample raw data trace from each of the main areas recorded from
including the right and left DG and the left and right CA1. The segment of data
analyzed is between the vertical red lines in the sample electrode traces. The top left
is a source (y-axis) response (x-axis) representation of the PGC analysis with the
electrodes organized by area. The PGC analysis shows directional activity within the
left CA1 as well as a directional influence of the left CA1 on the left DG. The bottom
left panel is surveillance footage of the rat's behavior. In this frame the rat has just
woken up and may be exhibiting tonic behavior.

















R-ODG1
SR-CAkI









I-i rI AII I

L-DG





L-CA1




1 ,c 60 65 70) 75
TIME [seconds)




Figure 6-7 Example of a directional transfer during Stage 2. The right figure includes a sample
raw data trace from each of the main areas recorded from including the right and left
DG and the left and right CA1. The segment of data analyzed is between the vertical
red lines in the sample electrode traces. The top left is a source (y-axis) response (x-
axis) representation of the PGC analysis with the electrodes organized by area. This
frame shows an example of a directional transfer from the left hemisphere to the right
hemisphere. In following frames this transfer is reciprocated from right to left. The
bottom left panel is surveillance footage of the rat's behavior. In this frame the rat is
exhibiting tonic behavior where the rat is awake and blinking its eyes but not moving.

















SF-



if L :


L 1. I- -. -
RESPONSE ELECTRODE


2253


R-DG


'1 I"I


.1 .1


70 75 80
TIME (seconds)


Figure 6-8 Example from Stage 3 Late intra-hemisphere. The right figure includes a sample
raw data trace from each of the main areas recorded from including the right and left
DG and the left and right CA1. The segment of data analyzed is between the vertical
red lines in the sample electrode traces. The top left is a source (y-axis) versus
response (x-axis) representation of the PGC analysis with the electrodes organized by
area. Most of the causality is isolated within hemisphere with occasional weak cross
hemisphere interactions. The bottom left panel is surveillance footage of the rat's
behavior. In this frame, the rat is still exhibiting tonic behavior.


..... iLWLa .
















R-DG




R-CA1




L-UG


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2821


II


. I I I II


gFlCAI

Lii
SL-DG
CC~

Iss


95 100 105 11
TIME (seconds)


Figure 6-9 Stage 4 Transition. The right figure includes a sample raw data trace from each of
the main areas recorded from including the right and left DG and the left and right
CA1. The segment of data analyzed is between the vertical red lines in the sample
electrode traces. The top left is a source (y-axis) response (x-axis) representation of
the PGC analysis with the electrodes organized by area. The causality here seems to
be bidirectional for nearly all electrodes. The bottom left panel is surveillance
footage of the rat's behavior. In this frame, the rat is beginning to transition from
tonic to clonic behavior.


1. ..i.~ ,,ll. Ll~I.LLkI jIA~ia.L LahLt IL 6 L.~ liUd


I


0 77"


I

















SR-CAI -D






LL-DG-G









L-CA1




298 00 105 110 115
TIME (seconds)




Figure 6-10 Stage 5 Post transition. The right figure includes a sample raw data trace from each
of the main areas recorded from including the right and left DG and the left and right
CA1. The segment of data analyzed is between the vertical red lines in the sample
electrode traces. The top left is a source (y-axis) response (x-axis) representation of
the PGC analysis with the electrodes organized by area. The dominant driving
pattern is from the DG to the CA1 in this stage. The bottom left panel is surveillance
footage of the rat's behavior. In this frame the rat is showing clonic behavior
including rearing and rhythmic shaking of the forelimbs.









Stage 4 is usually a brief and highly causal epoch where all areas intra and cross

hemisphere are bi-directionally causal, shown in Figure 6-9. In the PGC movies, this stage

consists of near saturation of causal effect with nearly the entire 32x32 frame shows strong

causal interactions. Behaviorally, stage 4 marks the transition from

the tonic phase where the rat is motionless and appears tense to the clonic phase of the seizure

where the rats appendages jerk rapidly and rhythmically. Immediately after stage 4, the pattern

smoothly transitions into stage 5, which is made up of heavy spiking activity with a causality

pattern similar to that observed during interictal activity.

Quantitative analysis between anatomical regions in hippocampus was conducted by

creating an index from multiple electrodes common to that region, as described earlier. The 32

electrodes split between hemispheres were divided into those in the DG and those within the

CA1/2 region. This results in four areas to compare using quantitative analysis. This index was

calculated for each seizure at each Stage of the seizure (1-5) for all possible interactions

including a scaled within area index. This allowed statistical comparison of the seizure stages

across multiple seizures (n=15). The interactions between and within the 4 indexed areas for

each seizure stage (16 interactions total) were compared sequentially with neighboring stages as

well as to inter-ictal indexes.

All Stages were significantly different from inter-ictal baseline activity, Fs(14,14) > 6.9 p

< 0.01. Figure 6-11 highlights the statistical comparisons between stages. Starting with stage 2

in panel C each of the 16 area indexes is compared with its analog in the previous stage using

dots to indicate significance levels. One dot indicates a 80% confidence interval, Fs(14,14) >

1.58, two dots a 90% confidence interval, Fs(14,14) > 2.02, three dots a 95% confidence interval,

Fs(14,14) > 2.48, while four dots indicates a 99% confidence interval, Fs(14,14) > 3.70.









The statistical analysis revealed gross changes between areas when transitions were made

from one seizure stage to another as well as differentiated all seizure stages from inter-ictal

baseline activity patterns. While the differences between seizure and baseline events are not

unexpected, some interesting discoveries emerged from the analysis of the seizure progression.

For instance, the transition between stages, such as from Stage 1 to Stage 2, is statistically

distinct and discreet, as shown in Figure 6-11 panel C). These results also suggest that Stage 4

has only between-hemisphere activity in common with Stage three (no dots in top left and

bottom right of Panel E) and within-hemisphere activity in common with Stage 5 (no dots in

bottom left and top right of Panel F). This suggests that Stage 4 has elements of Stage 3 and

Stage 5 within it, suggesting Stage 4 is a transition between 4 and 5. When comparing Stage 3

and Stage 5 there are no significant differences between within-hemisphere interactions (7 out of

8 Fs(14,14) < 1.58, p=0.20) while between-hemisphere interactions where significantly different

(7 out of 8 F(14,14) > 2.48 p=0.05) indicating a decrease in between hemisphere interactions.

However, this analysis does little to capture the effects of short intense events such as the

directional transfers shown in Stage 2. These transfers, while intense and obvious to discern in

the causality movies used for quantification, are poorly reflected because of averaging over the

analysis window. An example of one of these directional transfers was shown in Figure 6-7.

Thus, Stage 2 and Stage 3 cross hemisphere indexes are not significantly different when

compared to each other, see Figure 6-11 Panel D) (all Fs(14,14) < 1.58, p=0.20). However, CAl

to DG intra-hemispherical relationships between Stage 2 and Stage 3 were significantly different,

Fs(14,14) > 3.70, p=0.01, an observation that was not made during qualitative analysis.

This analysis suggests that there is a large decrease in CA1 to DG intra-hemisphere

interactions from Stage 2 to Stage 3. Qualitatively, a transition is observed in Stage 4 where the









dominant information flow from the CA1/2 to DG areas for both hemispheres reverses so that

the dominant information flow becomes from the DG to CA1/2. This effect is highlighted by a

comparison of Stage 2 (the highest magnitude instance of CAl to DG interactions) to Stage 5

(the highest magnitude instance of CAl to DG interactions) right and left CA1 to DG influences,

F(14,14) = 21.2 and 13.3, respectively. This suggests that CA1 to DG influences are effectively

eliminated by the start of Stage 5.

Discussion

Currently, synchronization is thought to be the dominant dynamic occurring across the

limbic system during an epileptic seizure. Synchronization is defined as adjustment of rhythms

of oscillating objects due to their interaction (123). Synchronization across a diffuse excitable

network is thought to be a possible mechanism that initiates seizure in the limbic system (124).

In contrast others suggest that a large amount of asynchrony is necessary to maintain seizure

events once they have begun (125). The results from the Granger analysis do indeed find

synchronization within the hippocampus prior to, during, and after the ictal event. However, the

detailed interactions that occur during these seizures are largely unknown. Hence, the detailed

interactions illustrated in these results using Granger Causality analysis have never previously

been described in the study of TLE. These results provide a new picture of highly dynamic

directional interactions among specific brain areas during a seizure that follow a common

progression. For example, one of the distinct features in Stage 1 of nearly all seizures is an

increase in local interactions within the CA1 area whose causal strength builds over time

eventually influencing activity in DG. This is followed by a series of directional cross-

hemisphere reverberations of Stage 2 in which the seizure appears to transfer back and forth

from one hemisphere to the other. Together, these results suggest a hierarchical progression of









causation within the CA1 that spreads to the DG and finally crosses over into the neighboring

hemisphere.

In contrast, if a coherence analysis were conducted over these data (a tool commonly used

in this literature), it would suggest that these directional transfers were actually bidirectional

phenomena because this method is non-directional. Non-directional measures such as coherence

would also not be able to detect the reversal in directional connectivity between the dentate and

CA1 during the Stage 4 transition. According to these measures the entire seizure would seem to

be dominated by bidirectional synchronization across the entire network. However, Granger

Causality analysis tells a very different story. The dominant directional influences in the

beginning stage of the seizure are abnormal with the CA1 areas showing directional causation on

the DG. This is the opposite of baseline interictal patterns that often show directional influences

from the dentate to the CA1. Additionally, this abnormal pattern is time locked behaviorally

with the tonic portion of the seizure. Interestingly, the flip of the directional connectivity to

connectivity resembling baseline connectivity occurs during the transition from the tonic to

clonic behavior. The implications of this directional flip are not completely clear, however.

One of the possibilities explaining the driving influence of CA1 to DG during the early

seizure may be related to one of the weaknesses of the Granger Causality based methods. In this

case the entorhinal cortex may be differentially driving both of these areas creating a false

directed influence from CA1 to DG, instead of the influence of CA1 to DG being mediated

through the entorhinal cortex. The entorhinal cortex has been implicated as a possible initiator of

ictal events where disinhibited neurons in the entorhinal cortex may directly fire pyramidal cells

in CA1 at least in a pilocarpine model (119). The entorhinal cortex is also anatomically

connected to the DG. Hence, both of these situations are supported by the literature and may









even be happening simultaneously. It is unlikely based on the anatomical organization of the

hippocampus that the effects of CAl are mediated through CA2 and CA3 to the DG. Further

experimental analysis using conditional granger causality with electrode coverage within the

entorhinal cortex will be necessary to determine which of these possibilities is correct.

Nevertheless, these results provide strong support for the use of the Granger Causality

metric as an important tool for seizure detection. The stage 4 transition from abnormal driving

(CA1 to DG) to a normal driving pattern (DG to CA1) suggests that at some point prior to

seizure the driving pattern became abnormal. The abnormal driving influence has been seen in

the pre-ictal analysis at least a minute before seizure. An natural question to ask would be "When

does this driving influence become abnormal?" Once identified, the transition from normal to

abnormal driving influences could potentially be used as an indicator of an impending seizure.

However, the role of the entorhinal cortex in this connectivity should be determined before this

relationship is used for seizure detection. For example, if the entorhinal cortex turns out to be

the source of the abnormal driving influence it may better aid detection if electrodes were placed

in this area. Additionally, detection of this abnormal driving influence may increase the

sensitivity of current seizure detection methods. This could be useful for detecting epochs

during epileptigenesis that are not easily detected using behavioral or current electrographic

seizure detection methods.

In conclusion, using Granger Causality to reveal effective connectivity has led to a new

understanding of the interactions within the hippocampus during seizure. This understanding

provides the opportunity to make new predictions about seizure detection and suggests new

experiments that should expand our understanding of TLE. However, we have only used the









most basic of GC based tools in this analysis. Future work could make use of conditional and

spectral GC analysis to paint a clearer picture of effective connectivity during seizure.











A) Interictal B) Stage 1 Early Intrahemisphere



12 12
R-D O L. R














C) Stage 2 Directional Transfers D) Stage 3 Late Intrahemisphere




-2.O 00 @ LL Ro
.R-c ** **





a OD Q R
w &I* ** B


** ** ** *








Figure 6-11 Quantitative analysis of the 5 seizure stages. Analysis was over 15 seizures from 3
subjects. A Granger Causality index was created for each of four areas: Left and
Right DG, and Right and Left CAl. The effects for the electrodes in each of the four
areas were summed and normalized to create these indices. The y-axis is the driving
area and the x-axis is the receiving area, the magnitude of the interaction is described
by the color ranging from blue (low interaction) to red (high interaction). Panel A)
shows the pre-seizure interactions while the remaining five are during seizure. The
black dots in panels C)-F) are used to indicate statistical difference for each
interaction index from its corresponding index in the previous stage. The number of
dots represent the significance level of that interaction (p < 0.2, p < 0.10, p < 0.05, p
< 0.01, for 1-4 dots respectively).(Figure continued on next page)











E) Stage 4 Transition






R-Ii
H-t A ^


F) Stage 5 Post Transition


0


LTDG


R-CAWI


Figure 6-11. Continued









CHAPTER 7
SUMMARY OF RESULTS AND CONCLUSIONS

Summary of Results

This research explored the use of Granger Causality and associated methods to reveal

effective connectivity in a wide variety of neural systems. These include a neural network

simulation in chapter 3, in-vitro dissociated cortical cultures in chapters 4 and 5, and finally in-

vivo microelectrodes in the hippocampus of epileptic rats in Chapter 6. In each case, the goal

was to reveal information about the actual connectivity underlying the system by using Granger

Causality to calculate a measure of effective connectivity from the output of that system. The

difficulty in recovering the actual connectivity increases as the complexity of the system

increases. For example, increasing the number of modeling neurons in the simulated neural

networks demonstrated this relationship. However, even in the extremely complex in-vitro and

in-vivo systems Granger Causality was able to provide novel and useful insights into the

workings of these neural systems.

The neural networks in Chapter 3 used a simple neural network model modified such that

the input synaptic weights were known prior to simulation. The outputs from these simulations

were used to recover the synaptic weights used to drive the network. In a simple network of two

neurons the relationship between pre and post synaptic is a sigmoid with a linear section between

15mV and 45mV where Granger Causality increases from near zero to a little greater than 1.

Granger Causality does not have an upper bound like that of coherence, thus the high end of the

sigmoid being around 1 is merely coincidence. Increasing the complexity of this network to 100

neurons was shown to distort this simple relationship and requiring another tool, conditional

granger causality, to recover the initial synaptic weights driving the network. CGC was needed

to remove not only false direct influences but also to untangle mediated influences from direct









influences. Usually, untangling a mediated influence from a direct influence requires the use of

conditional granger causality. However, it was found that under the special circumstance where

a false direct influence was conditioned out (i.e. shown to be equal to zero and entirely mediated

through the confounded pathway), the magnitude of the false connection could be simply

subtracted from the confounded pathway. This shortcut relationship is from previous work by

Geweke (126). The ability of both CGC and Geweke's shortcut to recover the direct influence

from a confounded pathway were both evaluated and found to be effectively equivalent.

However, as the complexity and length of the causal chain grew the ability of these methods

decreased.

When the size of the network was increased to 100 neurons these methods consistently

underestimated the original synaptic weights. This was likely due to the unknown effects of the

network complexity on these pathways. These suggest that more work is needed to accurately

recover synaptic weights from these networks. However, while the sigmoid curve equation

shown in Chapter 3 does not always hold numerically, synaptic weight does scale linearly within

the region where a presynaptic neuron begins to affect the firing rate of a postsynaptic neuron up

the the point where it always causes the post synaptic neuron to fire. This relationship suggests

that changes in effective connectivity calculated using Granger Causality observed in complex

networks are likely linear changes that could relate to synaptic efficacy if the sigmoid

relationship were know. This suggests that Granger Causality methods could be useful in

revealing relationships in living neural networks.

In-vitro dissociated cortical neural networks grown on microelectrode arrays were used for

the experiments in Chapters 4 and 5. Jimbo (4) used spike rate to describe simultaneous pathway

specific potentiation and depression in these cultures. This was done with accomplished by









probing on each of the 60 channels 10 times and measuring the spike rates at all electrodes in the

ensuing burst. The probe sequence and corresponding spike rates were used to establish a

baseline spike rate response for every electrode and then used post treatment to assess possible

changes. The treatment was a high frequency high amplitude pulse train or tetanus delivered to a

single electrode that consistently responded to probing. This tetanus was designed to induce

plasticity related changes in the culture and shown by Jimbo et al to potentiate or depress neural

pathways. These results of Chapter 4 were consistent with Jimbo's finding identifying potential

pathways within the dish using a comparison of changes in spike rate. PGC was then applied to

the baseline and post-treatment probe data. The results were shown to agree with spike rate

results, but PGC showed several marked improvements over Jimbo's analysis.

Admittedly spike rate is a computationally simpler method when compared to PGC.

However, PGC showed many advantages over using spike rate for describing plasticity. First, it

showed a higher contrast between baseline and post-treatment results. One of the difficulties in

the use of spike rate as a measure is establishing the statistical significance of changes. The

higher contrast PGC provides makes this task easier. Second, PGC has a significance threshold

that can be obtained using a surrogate where the spike timings within the data are shuffled. This

effectively destroys the spike timing relationships used to calculate PGC while maintaining the

spike rate characteristics of this data. This analysis of this surrogate data demonstrated that the

observed changes in the original data set were not simply due to changes in rate for if they were,

the results would have been equivalent. Third, PGC gives a statistically relevant baseline

measurement of effective connectivity establishing a baseline connectivity map. Spike rates

from the baseline alone do not have any meaning without being used in conjunction with the









post-treatment spike rates. Thus, baseline spike rates provide no information about baseline

effective connectivity.

Finally, PGC is a directional measure while spike rate is a non-directional non-causal

measure. Results calculating using spike rate result in a 60x60 matrix with axes of probe channel

and response channel. The use of PGC on the same data results in a 60x60x60 matrix with axes

of probe channel, response channel, and source channel. Thus, in addition to giving similar

results as spike rate, PGC provides 60 times more information. This extra information was

exploited to learn more about the cultures. Comparison of probe/source plots to probe/response

plots for potentiated pathways showed that some channel experienced an increase in output while

others experienced and increase in input post-tetanus. This same relationship was shown in

depressed channels except with decreases in output and input. This suggests that changes were

not just global changes in rate and were likely reflected in their postsynaptic targets as well. The

pre-tetanus effective connectivity calculated from PGC was also used to measure amplitude

dependent plasticity by comparing the initial connectivity to changes in connectivity post-

tetanus. Analysis of this data across 6 subjects showed that the amplitude dependent plasticity

trends matched trends seen in work done by Bi & Poo (118), where potentiated pathways

increase a lot if they are small and depressed pathways change by a constant percentage

independent of initial strength.

A second experiment was developed based on the results shown from the analysis of the

Jimbo protocol. This experiment made use of the ability of PGC to measure connectivity prior to

tetanization. Once PGC was calculated from the pre-tetanus data sinks and sources of causal

flow were identified within the culture. A sink was defined as a channel where the balance of

causal information flow was inward, while a source was defined as a channel where the balance









of causal information flow was outward. The central hypothesis in this experiment was that

tetanization on a sink would elicit depression at the sink while tetanization at a source would

elicit potentiation. The experiment was carried out on 10 cultures. The results suggested that

changes in plasticity did occur. However the direction of results did not match the predictions

made prior to the experiment. Perhaps one reason this experiment was not successful was that

the design and predictions underestimated the complexity of connectivity in the neural culture.

One solution might be to reduce the complexity of the network by reducing the density of the

neurons. Additionally, a paired pulse paradigm at pre and postsynaptic targets would likely

increase the probability of plasticity induction only on targeted pathways.

In-vivo, PGC was used to explore effective connectivity of seizures in rats from the

Evolution into Epilepsy project. While this analysis was mostly empirical, comparison of the

effective connectivity during seizure and during interictal and baseline was conducted.

Qualitative analysis of the results from PGC on 15 seizures from 3 subjects showed that there

were 5 distinct stages of the seizure. Stage 1 consisted of the buildup of activity within the CA1

area and also of a directed influence from the CA1 to the dentate gyms. This connectivity is

already abnormal when compared to interictal and baseline results. Baseline connectivity often

highlights a directed influence from the dentate gyms to CA1, which is also the expected

anatomical influence. Stage 2 consists of large reverberating directional transfers from one

hemisphere to the other accompanied by large within hemisphere interactions mainly from CAl

to the dentate gyms. Stage 3 shows a large decrease in the magnitude of directional transfers

however the within and between hemisphere interactions continue. This stage often contains the

largest observable waveforms in the ictal event. Stage 4 or the transition stage marks the time

when the connectivity flips to a pattern more resembling that of baseline activity where the









dentate gyms drives the activity of CA1. This stage is marked by a brief period of strong

bidirectional causality between most electrodes in both hemispheres. Behaviorally, the

surveillance video of the rat shows that Stage 4 also marks the transition from tonic to clonic

behavior in the rat. Stage 5 or the post transition stage shows a strong causal influence from the

dentate gyms to the CA1 as the rat clonically seizes. Shortly after Stage 5, the seizure usually

ends and the rat engages in normal exploratory behavior.

Quantitative analysis was also carried out by analysis of variance between granger

causality indexes created for combinations of each of the four regions left and right CA1 and left

and right dentate gyms for each stage of the seizure. It served as no surprise that stages that were

qualitatively different from each other were also quantitatively different. However, quantitative

analysis confirmed that all seizure stages were different than interictal activity. Also,

quantitative analysis confirmed that Stage 4 was likely a transition stage. It was shown that Stage

3 and Stage 5 had causal interactions in common with Stage 4 that they did not have with each

other (i.e. Stage 4 contains aspects of Stage 3 and 5). Overall, the results from this analysis were

quite instructive and suggest that a great deal of information can be gleaned using Granger

Causality not only about causal bindings during seizure, but also about changes in effective

connectivity during the period in which these seizures develop (epileptigenesis).

Overall Conclusions

The aims of this body of work were to exploit the ability of Granger Causality to reveal

effective connectivity. Granger Causality tools were used to explore effective connectivity in

simulated, in-vitro, and in-vivo neural systems. The simulated neural network models provided a

means of evaluating the ability of granger causality to relate back to the initial structure of the

neural network. The results from this model support the notion that linear changes in effective

connectivity observed using PGC in a living system likely reflect actual linear changes in the









underlying connectivity. Further the models provided a change to test conditional granger

causality's ability to identify false directed influences and to detangle direct and mediated

influences in confounded pathways.

The in-vitro work with cortical cultures suggests that PGC is a useful tool in describing

changes in plasticity. The advantage of using a statistically sound, directional, and causal

method that yields a measure of effective connectivity allowed much more information to be

extracted from Jimbo's protocol. This allowed phenomena such as amplitude dependent

plasticity to be measured from spike timings. These results contribute more evidence that

suggests the plasticity is being observed using Jimbo's protocol. These new measures also

allowed new predictions to be made about the response of the culture to a differential

tetanization on sink and sources. While the prediction that sinks would be depressed and sources

would be enhanced proved to be too simplistic, the PGC analysis suggested that plasticity

occurred as a result of tetanization. Future predictions that take this complexity into

consideration may be successful by using Granger Causality to estimate effective connectivity

prior to experimental treatment.

Finally, PGC was able to show previously unreported causal relationships during seizure

that may be useful for efforts to prevent seizures from occurring. At some point prior to the ictal

event the effective connectivity in the hippocampus appears to become abnormal with the CA1

driving activity in the dentate gyms. If this flip to abnormality can be detected it may provide an

opportunity to intervene at an opportune moment to prevent the immanent ictal event by

correcting this abnormal activity pattern. Additionally, the use of PGC to analyze epileptigenesis

could provide new insight that may lead to treatments for a condition that is notoriously hard to

treat pharmacologically.









In each of these cases, using Granger Causality to reveal effective connectivity led to a

new understanding of these systems. This understanding provided the opportunity to make new

predictions about these systems and suggest new experiments that were previously not feasible.

Overall, Granger Causality has proven to be a useful tool in exploring effective connectivity in a

wide variety of neural systems and in the future, may become one of the traditional tools used in

neuroscience.









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BIOGRAPHICAL SKETCH

Alex Cadotte was born in Glens Falls, New York in 1976. He grew up in Franklin,

Virginia where he graduated with honors from Franklin High School in 1994. He received his

B.S. in chemical engineering from Virginia Tech in 1999. He worked in industry from 1999 to

2002 at companies such as General Electric and Coming Optical Fiber in Wilmington, NC. Alex

moved to Gainesville, FL, in August of 2002 and completed a Master of Science in August of

2004 at the University of Florida in biomedical engineering specializing in tissue and neural

engineering. Alex began his PhD at the University of Florida in August of 2005, advanced to

candidacy in April of 2006, and completed a Doctorate in Philosophy on November 20, 2007

specializing in neural computation and neural engineering. This dissertation is a summary of the

work done by Alex under the mentorship of his doctoral committee consisting of Dr. Thomas

DeMarse (chair), Dr. Mingzhou Ding, Dr. Paul Camey, Dr. William Ditto, and Dr. Jason Frazier

(external) in this field of study.





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1 ASSESSING CONNECTIVTY IN SIMULATED, IN-VITRO, AND IN-VIVO NEURAL NETWORKS USING GRANGER CAUSALITY By ALEX JOSEPH CADOTTE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Alex Joseph Cadotte

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3 To my Aunt Claire & Uncle Bill Parker.

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4 ACKNOWLEDGMENTS I thank Tom DeMarse for his guidance and patien ce for the past five years. I thank the other graduate students I have worked with in Neural Hybrid Com putation Laboratory including, Ping He, Vinh Trin, Karl Docke ndorf, Nicky Grimes, Pamela Douglas, Hany Elmariah, Crystal Stephens, Memming Park, and Amber Scarlatos for the many interesting conversations and collaborations. I specifically thank Ping He fo r the data provided from the Jimbo replicate experiment. Also, I specifically thank Amber S carlatos, for the extensive collaborations we had during the past 2 years. I thank Mingzhou Ding for his valuable teachings, time, and conversations and also members of his laboratory including Yonghong Chen, Anil Bollimunta, Yan Zhang, and Raj Rajagovindan. I thank Paul Carney, William Ditto, and all those associated with the Evolution into Epilepsy project for the work behind the data that was provided me for analysis. I thank all of the staff and faculty within the Biomedical Engineering department. I thank Yanni Constantinidis for his influence on my teaching st yle and for his career advice. His presence in the Biomedical Engineering department ha s been missed, may he rest in peace. I thank my family and friends for their support over the past four years of graduate school. I thank my parents and my brothers for believi ng in me. Thanks to Shane Calfee and Courtney Calfee for suggesting UF for grad school, GO Gators Id also like to thank the many friends I have made here at the University of Flor ida including Cameron Siggs, John Azeke, Neel Bhatavadekar, Andres Argibay, Ning Jiang Ar gibay, John Quinlan, Amber Pouncy, and Lance Gardner. Finally, Id like to thank Theresa Mc Guirk for her patience, love, and support while I was hard at work producing this document.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 ABSTRACT...................................................................................................................................11 CHAP TER 1 INTRODUCTION AND BACKGROUND OF CAUSAL METHODS IN NEUROSCIENCE.................................................................................................................. 13 Introduction................................................................................................................... ..........13 Review of Causal Methods in Neuroscience.......................................................................... 15 Non-Parametric Methods.................................................................................................16 Cross Correlation...................................................................................................... 16 Coherence.................................................................................................................17 Mutual Information..................................................................................................19 Parametric Approaches....................................................................................................20 Likelihood Methods................................................................................................. 20 Methods using Autoregressive Modeling................................................................20 Partial Directed Coherence, the Dire cted Transfer Function, and Granger Causality ...............................................................................................................21 2 MATHMATICAL BASIS FOR GRANGER CAUSALITY ................................................. 28 Mathematical Basis for Tim e Domain Measures...................................................................28 Granger Causality............................................................................................................28 Autoregressive Modeling................................................................................................28 Time Domain Granger Causality..................................................................................... 31 Spectral and Conditional Granger Causality..........................................................................32 Spectral Granger Causality..............................................................................................33 Conditional Granger Causality........................................................................................ 35 Limitations of Causal Methods....................................................................................... 37 3 RECOVERY OF SYNAPTIC WEIGHTS F ROM SPIKING ACTIVITY USING GRANGER CAUSALITY..................................................................................................... 41 Introduction................................................................................................................... ..........41 Conditional Granger Causality and Me diated vs. Direct Relationships ................................. 44 Simulation and Analysis.........................................................................................................45 Mono-directional Simulation........................................................................................... 47 Relationship between Synaptic Weight and Granger Causality...................................... 49

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6 Mediated vs. Direct Causal Rela tionships under Serial Connectivity ............................. 51 Monte Carlo Simulations with Random Serial Topologies ..................................... 55 100 Neuron Neural Network....................................................................................57 Application to Spike Timi ng Dependent Plasticity ..................................................59 Conclusions.............................................................................................................................62 4 ASSESMENT OF PLASTICITY IN-VITRO USING GRANGER CAUSALITY ............... 65 Introduction................................................................................................................... ..........65 What do PGC values mean in neural networks?.................................................................... 66 Methods..................................................................................................................................67 Dissociated Cortical Cultures.......................................................................................... 67 Microelectrode Arrays.....................................................................................................68 Stimulation Induced Plasticity Protocol.......................................................................... 69 Results.....................................................................................................................................71 Spike Rate versus PGC....................................................................................................71 Surrogate Analysis...........................................................................................................74 Discussion...............................................................................................................................77 Amplitude Dependent Plasticity...................................................................................... 78 Directional Causal Influences.......................................................................................... 80 Spontaneous/Elicited Connection.................................................................................... 85 Conclusions.............................................................................................................................87 5 DIFFERENTIAL TETANAZATION OF SINKS AND SOURCEs IN DISSOCIATED CORTICAL CULTURE .........................................................................................................92 Introduction................................................................................................................... ..........92 Plasticity by Tetanization..................................................................................................... ..95 Experimental Design............................................................................................................ ..96 Spontaneous vs. Elicited Activity as a Measure of Plasticity ................................................. 98 Results and Discussion......................................................................................................... 100 Results........................................................................................................................ ...100 Conclusions...................................................................................................................106 6 ANALYSIS OF TEMPORAL LOBE SEIZUR E S USING GRANGER CAUSALITY...... 108 Introduction................................................................................................................... ........108 Temporal Lobe Epilepsy............................................................................................... 109 Animal Model of Chronic Limb Epilepsy..................................................................... 110 Seizure Analysis with Pairwise Granger Causality..............................................................114 Results...................................................................................................................................116 Discussion.............................................................................................................................126 7 SUMMARY OF RESULTS AND CONCLUSSIONS........................................................ 132 Summary of Results............................................................................................................. .132 Overall Conclusions............................................................................................................ ..137

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7 LIST OF REFERENCES.............................................................................................................140 BIOGRAPHICAL SKETCH.......................................................................................................150

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8 LIST OF TABLES Table page 3-1 A table of the Pairwise Granger Causal influences calculated between the 5 neurons in the sim ulation displayed in Figure 3-5.......................................................................... 53 3-2 Pairwise Granger Causal influences calcu lated between the 5 neurons in the simulation displayed in Figure 3-5 afte r the use of significance thresholds, conditional granger causality analysis, a nd removal of mediated influences.................... 53 5-1 Global culture causality changes post-tetnanization. .......................................................102 5-2 Tetanized channel total causa lity changes post-tetanization. ..........................................103

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9 LIST OF FIGURES Figure page 2-1 Conditional Pairwise Granger Causality Example............................................................ 36 3-1 Topology diagram of a simple five neuron network.......................................................... 46 3-2 Power spectra and cross-spectra calcul ated from the 5 neuron model simulation............ 48 3-3 Granger Causality Spectra................................................................................................. 49 3-4 The PGC results from a mono-directiona l sim ulation with the weight from neuron 4 to 5 varied from 0-75mV in increments of 5mV............................................................... 50 3-5 Five neuron simulation.................................................................................................... ..52 3-6 Direct influences between N2 and N3 were recovered by both m ethods to allow a comparison of Gewekes subtraction method to CGC. ................................................... 55 3-7 DGC values can be recovered at each of the synapses in the serial chain. ........................ 56 3-8 100 Neuron Simulation..................................................................................................... .58 3-9 A simulation to determine if changes in synaptic weight consistent with stim ulation timing dependent plasticity (STDP) at a synapse between two bi ologically relevant neurons could be detected after a pair ed pulse stimulation was applied to the neurons...............................................................................................................................60 3-10 Statistical Analysis of Plasticity Simulation......................................................................61 4-1 Comparison of spike rate to Pairwise Granger C ausality.................................................. 72 4-2 A comparison of spike rate and PGC analysis for enhanced channel 26 and depressed channel 41. .................................................................................................................... .....74 4-3 Surrogate analysis of the PGC analysis. ............................................................................ 75 4-4 Amplitude dependent plasticity (ADP) can be explored in-vitro using PGC. According to am plitude dependent plasticity (ADP), changes in synaptic weight due to correlated time dependant spiking are a function of initial synaptic strength...............79 4-5 Stimulus-Response difference plot of PGC where the EGC difference matrix is collapsed by source............................................................................................................80 4-6 StimulusSource difference plot of PGC where the EGC difference m atrix is collapsed by response........................................................................................................81

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10 4-7 Source-Response difference plot of P GC collapsed by probe before and after application of tetanus......................................................................................................... 83 4-8 Source-response plots by spatial probe location................................................................ 84 4-9 Spontaneous Source-Res ponse plot calculated from windowe d bursting events during spontaneous (no stimul ation probes) activity.........................................................86 5-1 Topographical map of sinks and sources spatially laid out and superim posed on a picture of the microelectrode array.................................................................................... 96 5-2 Source response plot for culture 9059 tetanized on a sink................................................. 99 5-3 Source-response plot for culture 9449, which was tetanized on a source. ...................... 101 5-4 Culture wide Amplitude Dependent plasticity. ................................................................ 103 5-5 Tetanized channel amplitude dependent plasticity.......................................................... 104 6-1 Diagram of microelectrode placem ent in the rats brain................................................... 111 6-2 Placement of the electrodes during surgery..................................................................... 112 6-3 Example of the rat enviro nm ent and recording chamber................................................. 113 6-4 Initial Pairwise Granger Causality analysis of four 60-second seizures for a single anim al......................................................................................................................... ......114 6-5 Example of data and Granger analysis during the interictal (pre-ictal) stage. .................118 6-6 Beginning of the seizure in the high resolution movie classified as Stage 1. .................. 119 6-7 Example of a directiona l transfer during Stage 2............................................................. 120 6-8 Example from Stage 3 Late intra-hemisphere.............................................................. 121 6-9 Stage 4 Transition...................................................................................................... ....122 6-10 Stage 5 Post transition................................................................................................ ...123 6-11 Quantitative analysis of the 5 seizure stages.................................................................... 130

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11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ASSESSING CONNECTIVTY IN SIMULATED, IN-VITRO, AND IN-VIVO NEURAL NETWORKS USING GRANGER CAUSALITY By Alex Joseph Cadotte December 2007 Chair: Thomas DeMarse Major: Biomedical Engineering Since the early 1980s, interest in synchronizatio n between regions in the brain has led to the adaptation of forecasting measures such as Gr anger Causality from fields like economics to neuroscience. The basic principle behind Gra nger Causality is that a directional causal relationship can be inferred from a time series X to a time series Y when the introduction of previous measurements of X reduces the variance of the error time series in the linear modeling of Y. Granger Causality methods are used in the Neural Hybrid Co mputing Laboratory to investigate functional relationships between neural units in simulation, in-vitro, and in-vivo. Computer simulations using biol ogically plausible neural-netwo rk models have shown that the direct component of pairwise Granger Causa lity (PGC) scales linearly with synaptic efficacy within biologically relevant ranges. This s uggests that the PGC values obtained from living neural networks may be useful in calculati ng the underlying connectivity of living networks. Thus, PGC was used in-vitro to identify connectivity within neural pathways in a living neural network of 50,000 neurons grown on a microelectrode array (MEA). An MEA is an 8x8 grid of electrodes embedded in a cell cultur e dish. Data recorded from two plastcity experiments using these MEA cultures was analyzed using PGC. The resulting difference in PGC connectivity

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12 before and after the induction stage of this expe riment reveals that PGC was able to describe plasticity within neural pathways in-vitro. PGC was also applied in-vivo to spontaneous epileptic seizures recorded from a 32electrode array implanted in a rats hippocampus. Th ese rats were electrica lly stimulated causing limbic status epilepticus and later began to de velop spontaneous seizures consistent with Lothmans animal model of chronic limbic epileps y. Results from the analysis of spontaneous limbic seizures reveal the dynamics of the di rectional causal influences during seizure. Interestingly, they indicate a sudden change in direction of causal in fluence between the CA1 and dentate gyrus regions of the hippocam pus near the end of status epilepticus.

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13 CHAPTER 1 INTRODUCTION AND BACKGROUND OF C AUS AL METHODS IN NEUROSCIENCE Introduction Multiv ariate time series analysis is incr easingly used during the last 20 years in neuroscience to determine the relationships be tween simultaneously recorded signals. Granger Causality, a predictive forecasting tool originating fr om the field of economics (1), is one of the latest examples of multivariate time series analysis that has been adapted for use with neural systems (2). According to Granger Causality, a causal relationship is said to exist when, by introducing previous measurements of the first time series into the linear regression of the second time series, the error of the linear regression of the second time series is reduced. In essence, this says that the second time series is better pred icted with information a bout the first time series because the second time series contains time dela yed elements of the first time series when a causal relationship exists. This is the basic prin ciple behind the calculation of Granger Causality. Expanding on this basic principle, the relations hip between higher order multivariate systems can also be explored. However, reducing the multivariate system to pairwise relationships to determine causal relationships is often insuffi cient to sort out causal relations when those relations are mediated through another element rath er than direct. While it is necessary to use Granger Causality to determine the interdepende ncy and directionality of relationships (e.g., neural connections), Conditional Granger Causality is required to resolve whether relationships (e.g., neural pathways) are direct or mediat ed. Conditional Granger Causality does this by comparing error terms from bivariate and trivaria te linear regressions. If the addition of a third time series to the linear regres sion that is already known to infl uence a first time series does not add to the prediction of a first, it can be infe rred that its influence is already accounted for by mediation through the second time series.

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14 This body of work focuses on the use of Granger Causality based methods as a tool to determine the effective c onnectivity in neural systems where the anatomical connectivity may be unknown or unclear. Included is a review of cu rrent causal methods, an explanation of the mathematical basis for Granger Causality, and four experiments where outputs from neural networks, both simulated and living, are analyzed using Granger Causality to determine effective connectivity. These include modeled, in-vitro, a nd in-vivo paradigms. First, a simple neural network model (3) will be used to determine the relationship between Granger Causality and synaptic weights used to drive the model. A modeled system is ideal because the synaptic weights are known and can be compared with thos e weights recovered usi ng Granger Causality. Understanding this relationship is a prerequisite to experiments in living systems because it will define what kind of relations hip between Granger Causality and the underlying anatomical synaptic efficacy underlying th e living neural network. Next, data from cortical cultures grown on microelectrode arrays will be the focus of two in-vitro experiments in Chapters 4 and 5. Th e first experiment based on the results from the modeling section determines the ability of us ing Granger methods to recover the effective connectivity of a dissociated neural network. Granger Causality was used to measure changes in effective connectivity observed af ter treatment with a protocol de signed to induce plasticity (4). Grangers results are compared to estimates base d on changes in firing rate. The second in-vitro experiment investigated Granger Causalitys ability to provide information of effective connectivity prior to experimental manipulations. Using this e ffective connectivity sinks, where the balance of effective connect ivity is directed inward, and sources, where the balance of effective connectivity is directed outward will be identified. Sinks and sources will be differentially subject to the same high frequenc y plasticity induction s timulation protocol, or tetanization, with the prediction that tetanization of sinks would yield depression while

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15 tetanization of sources would yield potentiation. This prediction is based on the differential orientation of the connectivity of these pa thways with the rest of the culture. Finally, Chapter 6 examines the ability of Granger Causality to measure progressive changes in effective connectivity during epileptic seizures measured in a chronic limbic epileptic rat model. This data analysis will be conducted on in vivo 32-channel microelectrode arrays data in rat hippocampus originally r ecorded by the Evolution into Ep ilepsy project team to study the mechanisms of epileptigenesis. The goal of this work is to determine if Granger Causality methods can successfully be used to describe th e changes in effective connectivity before and during seizures to learn mo re about in effective conn ectivity during seizure. Review of Causal Methods in Neuroscience Many experim ental platforms are used in the neuroscience community to answer scientific and practical questions about the beha vior and function of the nervous system. As these platforms have evolved over the last half a century, so to has the quality, content, and availability of data sets from these systems. These platforms have been adapted to better answer questions of causality and bindi ng between various elements requiring multichannel data to be acquired. Subsequently, the methods used to analyze multichannel data have evolved alongside or even been borrowed from other disciplines, providing meaningful analysis of this data. Some examples of these methods are cross correlation, c oherence, mutual information, partial directed coherence, the directed transfer function, and Granger causality. These methods fall into two broad categories, those that are non-parametric (calculated directly from data) and those that are parametric (the data is fit to a model and parameters from the model are used to calculate a result). The following sections will provide a br ief background and examples using each of these methods.

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16 Non-Parametric Methods Cross Correlation Cross correlation is one of the m ost basic and longest used algor ithms that has been used to determine the interdependence between two time se ries. The cross correla tion function measures the linear dependence between time series as a function of their time delay, A significant value of the cross correlati on function at a time delay may indicate a causal relationship. Consider two time series x ( t ) and y ( t ) that have been normalized to have a zero mean and unit variance. The cross correlat ion function of these two time series is shown in equation (1.1), where N is the length of the tim e series and is the time delay. Cxy() 1 N x ( k ) y (k 1 N k ) (1.1) The output of the cross correlation function ranges from -1 to 1 for each value of where -1 is complete inverse linear correlation and 1 is complete linear direct correlation. Values of where the cross correlation function is equal to zeros represent circumstances where there is no linear correlation. In practice, a statistical threshold is calc ulated around zero perhaps using a surrogate data set to determine when significant correlations exist. The cross correlation function has been used in neuroscience for over 50 years, since it was used to measure correlation between EEG signals (5). Use of correlation based methods is widespread in neuroscience and a full elaborati on of it use is beyond the scope of this research (see (6-9) for an excellent review). This meas ure has more recently been used to describe correlations among neurons (10-17) and specifically in studies of plasticity (4, 18-20). For example, Jimbo (4) found that neural activity that was highly corre lated with a tetanic stimulation pulse train tended to be potentiated (an increase in effective connectivity). In contrast, neural activity that was weakly correlated with tetanic s timulation typically resulted in

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17 depression (a decrease in effectiv e connectivity). Jimbo suggests that the st rength of correlation between these channels during tetanization may be representative of the total number of synapses in the pathway between the electrodes. Tateno and Jimbo (21) use a nor malized cross correlation analysis for all neuron pairs during a similar pl asticity experiment. Th e difference in cross correlation (post-tetanization corre lation pre-tetanus correlation) between all neuron pairs after tetanic stimulation highlighted ch anges in the correlation that was indicative of plasticity within a living neural network. Segev et al (14) has used cross correlation to describe connectivity patterns in multichannel data from similar living neural networks during synchronized spiking events known as bursts. Segev demonstrated that bursting activity contains statistically separable subgroups that he suggests may be a template for storage and retrieval of information. Overall, cross correlation has been one of th e primary methods used in neuroscience to demonstrate relationships especially in multichannel data. It has the advantage of being a widely understood, straightforward, and simple method. Ho wever, it does have some limitations. The results from cross correlation can often be subject to interpretation and be ambiguous (7). Additionally, for results to be sta tistically significant often a larg e amount of data is necessary. Directional information can be ascertained using cross correlation but this can become problematic and ambiguous over large data sets because manual interpretation is often required. Additionally, cross correlati on, as presented here, is strictly a time domain measure. To examine correlations in the spectral domain known as coherence additional manipulations are required. Coherence Coherence is the linear correlation between tw o time series as a function of frequency. Also known as squared coherence or the coherence spectrum, this measure is simply the normalized Fourier transform of equation (1.1). Skipping several step s of this derivation, the resulting equation for squared coherence is show n in equation (1.2).

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18 C ( f ) Sxy( f ) 2Sxx( f ) Syy( f ) (1.2) The spectral matrix S, used to calculate coherence, is non-parametric calculating using Welchs method to create a periodogram. An esti mate of the average spectra can be estimated over several trials of event rela ted data by averaging the resultan t periodograms. However, this method only provides an estimate of coherence and significance would need to be established by calculating confidence intervals. The resulting co herence values for each frequency vary from 0 to 1, where 0 would indicate no co rrelation at that frequency and 1 would indicate maximum linear correlation at that frequency. Like correlation, coherence is also widely us ed in neuroscience. Recent reviews detail the use of coherence in neuroscience (9) and in hum an epilepsy (22) over the past 40 years. While coherence shares many of the weaknesses of correla tion, a critical weakness of this method is the lack of directionality. This sp ecific weakness spurred the deve lopment (23) of a method called directed coherence also known as Gersch causality (24) that was initially used identify epileptic foci using three electrodes. Gersch stated one ch annel is said to drive th e other channels if the first channel explains or accounts for the linear relation between the two. This method has also found various neural applications (25) including EEG (26) (27, 28) fMRI (29) and spike trains (27, 30). Unfortunately this method has been show n to be sensitive to noise contamination (25), which can potentially lead to misidentification of directional influences While coherence is often calculated directly from data (non parametrically) it can also be calculated using parameters from autoregressive models fit to the data (31). These parametric approaches will be elaborated later in this re view. Mutual information, a non-parametric method that uses probabilistic estimates, is discussed next.

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19 Mutual Information This measure falls under the auspices of information theory and probability. The primary measure used in information theory is entropy. Entropy is defined for a variable X as the sum over all possible outcomes M of pi (probability that an outcome occurs) multiplied by the base 2 logarithm of pi as shown in equation (1.3) as Shannon entropy. Consequently, the Shannon entropy will be high for a uniform system where a ll states have an equal probability and lower for a system with a Poisson distribution where certain states have much higher probability. Ix pi i1 Mlog2pi (1.3) It is this concept that is applied to the calculation of Mutual Info rmation (MI) between two variables. MI is defined as: MIxy pijlog pijpipj (1.4) where pij is the joint probability that the outcomes fo r x and y occur. Esse ntially, this measure describes how much extra information is known about one signal by knowing the outcomes of the other one. MI will e qual zero if the two variab les are independent because pij would be equal to pi pj. Mutual Information has been used to estimate th e relationship between dynamic stimuli and the response in corresponding sensory areas in-vivo for various animal s (32), including the response motion sensitive neurons in the visual cortex of prim ates to visual stimuli (33). It has also been used to estimate synaptic efficacy in vitro between pyramidal neurons in-vitro (34). Joint probability density can also be esti mated between spike trains using Mutual Information (17, 35). Like cross correlation, Mutual In formation also requires large amounts of data (9) in order to

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20 produce a reliable representation of the probability distribution. Thus, the use of Mutual Information on experimental time-scales can sometimes be problematic. Parametric Approaches Likelihood Methods Maximum Likelihood is a parametric approach th at creates statistical models of a system for analysis. A parametric probability model is assumed for a process under study. Methods associated with Likelihood met hods are very case dependent of the system being analyzed and the statistical model chosen (17) and cannot be el aborated on completely w ithin this review. The likelihood is represented as joint probability density of the expe rimental data as a function of the unknown or free parameters in the model. Thus, the models parameters are estimated from the experimental data. If the experime ntal data fits the chosen model well, this is thought to be an optimal way of representing the experimental pr ocess (17). This method has been used to analyze the interactions of simulta neously recorded spike trains in the cats auditory system (36, 37) and place cells within the hippocampus of ra ts (38). While this methods shows promise, extreme care must be taken in choosing an a ppropriate model to estimate the probability distribution. Similar to Mutual Information, this process ofte n requires large amounts of data that can make analysis of experimental systems problematic. Methods using Autoregressive Modeling Several parametric methods make use of autore gressive modeling (AR) in order to assess a systems causality. These methods, including Pa rtial Directed Coherence (PDC), the Directed Transfer Function (DTF), and Gran ger Causality (GC) each use different parameters from an AR model fit to the system in ques tion. Additionally, it should be not ed that the parameters of an AR model have been used on their own to assess changes in neural systems over time (39, 40). For example, Sanchez et. al. specifically highlig hts the changes in AR coefficients calculated

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21 from a spiking time series over the developmen t of spontaneous limbic seizures in-vivo. A general form for the AR m odel is shown in equation (1.5), where X is a vector consisting of p time series, A is a p x p coefficient matrix and E is the uncorrelated noise vector with a covariance matrix of X ( t ) A ( m ) X ( t m ) E ( t )m1p (1.5) AR modeling makes use of the autocorrelation function to model the current value of a variable from past instances of that same variable ( p =1). A multivariate AR (MVAR) model ( p >1) uses this same framework to model the curre nt value of a variable using not only past instances of that same variable, but past instan ces of the other variables in that multivariate system. It is easy to see why autoregressive modeling has become such a popular tool in neuroscience where multichannel recordings are common. A more comprehensive mathematical review of AR modeling will be expanded upon in Chapter 2. Several more advanced methods collect in formation from the AR model and perform operations on them to determine a measure for causality or direction. Often these methods employ two or more AR models, which are calc ulated including and excluding a variable of interest. This methodology calcula tes a particular variables e ffect on AR model coefficients and linear regression error that is used to calculate a measure of causality. Partial Directed Coherence, the Directed Transfer Function, and Granger Causality Three methods have emerged that use AR models to determine dire ctional connectivity. Partial Directed Coherence (PDC), the Directed Transfer Function (DTF), and Granger Causality all use aspects of an AR model including and excluding a time series hyp othesized to have a driving relationship with a sec ond time series. This basic principle is common to all three methods. While these methods are often considered to be the same or similar, there are subtle

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22 differences between these methods. For exampl e, PDC and DTF are exclusively frequency domain measures that rely on the Fourier transf orm of the AR model resulting in new frequency domain coefficients. In contrast, Granger Causality has formulati ons in the time domain and the frequency domain. While PDC and DTF are closely related, both being derived from this new coefficient matrix, DTF differs from PDC in that the coefficient matrix is inversed, Hij( f ) Aij 1( f ), and the DTF is squared when compared to the PDC. The equations for PDC and the DTF are shown below in equations (1.6) and (1.7) respectively. Notice the similar for mulation for both equations. ji( f ) Aij( f ) kAkj( f ) 2 (1.6) j i( f ) Hij( f ) 2tHil( f ) 2 (1.7) Another major difference between Granger Caus ality and the other tw o methods is that Granger Causality is a statistically relevant qua ntity. For example, the time domain version of Granger Causality, shown in equation (1.8) is the ratio of the varian ce fro m two error time series. The DTF and PDC are ratios of spectral coefficients from the AR model, which dont have any meaningful statistical interpreta tion. Granger Causality is simply a more statistically rigorous measure. This does not mean that the DTF and PD C are not useful tools. In fact, these methods have produced interesting result s in the analysis of neural time series including EEG (41-43), neural simulation (43, 44) and livi ng neural networks (45-47). Fans elow et al. (46) uses PDC to reveal causal influences from the sensory cortex to the thalamus during rhythmic activity associated with whisker twitching behavior in rats. This sugges ts that sensory cortex driven thalamic bursting, associated with whisker tw itching behavior during ex ploratory behavior, may

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23 prime the thalamo-cortical loop for sensory signal detection. Schelter et al (43) calculated PDC between EEG and EMG signals and determined that the motor cortex likely participates in the generation of essential tremor. Ginter et al uses the DTF to determine if the signature of imagined movements of the index finger can be dete cted reliably in the cortex using EEG. This information is relevant in the creation of a br ain machine interface (BMI ). This research area seeks to help amputees and those suffering from va rious forms of paralysis (due to spinal cord injury for example) control pros thetic devices by imagining move ment of the affected limb. These are just a few examples of the use PFC and the DTF. These methods were often developed to improve spectral information conten t but are only empirical measures that dont have a firm statistical foundati on. Thus, they often are used in parallel with the Granger Causality metric. FY X ln var( X( t )) var(X( t )) (1.8) Granger Causality has found similar applicatio ns within neuroscience. The original concept of Granger causality has played a significant role in the field of economics since the 1960s. Within the last 15 years, it is now incr easingly used in neuros cience to understand the dynamics of brain circuitry. In addition to the time domain version of Granger Causality, shown in equation (1.8), Granger Causality also has formulat ions in the frequency dom ain and can be used to separate mediated and direct influences (31). The math behind these concepts will be explored further in Chapter 2. There are a wide variety of systems that Gr anger Causality and asso ciated methods have been applied to with success. For example, Zhu et al (48, 49) explores causal interactions between neurons during learning in the primary motor cortex of rhesus monkeys. Specifically, Zhu monitored the interactions between 8 electrode channels in the primary motor cortex. The

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24 monkeys used in this experiment were trained in a reaching task prior to surgery implanting the electrode array into motor cortex. A string was then attached to the arm of the monkey that interfered with the completion of previously learned motor task. The monkey adapts or essentially learns to complete the same task w ith this perturbing string attached. Recordings from the 8 electrodes during this re-learning phase were associated with changes in the strength and direction of causal intera ctions over time. Coupling st rength between the electrodes increased during the learning pha se, but was attenuated to pre-learning levels once adaptation had been achieved. In this experiment the netw ork appears to be essent ially rewired before and after learning and may reflect plasticity a ssociated with the re-learning process. Functional Magnetic Resonance Imaging (fMR I) allows for three-dimensional brain activity mappings to be collected during a behavioral task over time. The dynamics observed during fMRI recordings can be used to determine functional connectivity between brain areas using Granger Causality (50, 51). An fMRI time series can be considered a filtered and sampled version of a local field potential (LFP) genera ted from the observed neuronal population. The fMRI time series for each area is functionally a measure of the blood oxygen levels in that area. This can be used as an indicator for brai n activity based on consumption of blood oxygen. Casual analysis of a simulated fMRI time series between two brain areas shows a nearly identical relationship to that of the original LFPs genera ted in the simulation (50). A cognitive task where various stimuli were delivered requiring a left or right response wa s observed using fMRI (50) and analyzed using Granger Causality. Frontal areas were found to drive the parietal areas during these tasks. These results were interpre ted as neural correlates of executive control and working memory. A similar study (51) showed th at pre-motor areas could be used to predict activity in the motor cortex.

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25 A variety of neural systems are addressed in Kaminski et al (52), including simulated local field potentials, simulated spike trains, human sleep analysis, and vis uomotor integration in macaque monkeys using LFPs (local field potentials). Analysis of human EEG patterns during sleep reveals a causal influence from anterior P3 and P4 and the posterior region P3 electrode (the international 10-20 system was used for electrode placement). These patterns were found to be consistent with anatomical projections from the pre-frontal cortex. Additionally, these effects were observed to fade during the transition from sleep to wakefulness. In-vivo LFP data was analyzed from macaques that were trained to pe rform a visuomotor pattern discrimination task. The data was collected from 15 bipolar twist electrodes over 888 trials of a go no-go task. Using these trials to reduce the length of the data sample necessary, a 50 ms moving window of Granger Causality values was used to reveal cau sal interaction through the 500 ms trials. This analysis reveals that two cortical areas, a striate and prestriate area, shown to be coherent during the task, show a directional transfer from the striate to a prestriate which is followed by feedback in the opposite direction 50 ms later. Note that these directional interactions were unattainable using coherence. A more detailed analysis of this experimental paradigm can be found in Brovelli et al (53) and Chen et al (54). In Chen et al (54) a Spectral (15-30Hz) Conditional Granger Causality analysis was used to show th at some cortical areas were mediating effects from other areas while other relationships were direct. Note that a conditional analysis is necessary to ascertain when mediation occurs in certain circumstances. Kaminski et al also introduces a method to address the statistical significance of results using surrogate data analysis, a method that will be used in all studies in th is research. Additionally, Kaminskis methods for preprocessing point process data (A first order Butterworth set at 10% of the sampling frequency) for causal analysis between simulate d spike-trains are used in this research.

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26 Seth and Edelman (55, 56) employed grange r causality to disti nguish patterns of connectivity and causality within co mplex neural network simulations. Seth (56) defines a set of tools to analyze the output of causal analysis of a multivariate system. While calculation of pairwise granger causality in a la rge neural system can be strai ght forward, making sense of the large amount of data resulting from these data se ts can be challenging. This paper defines several concepts that can help reduce the complexity of results. These include causal density, causal flow, causal reciprocity, causal disequilibrium, casual s ources, and causal sinks. Causal density is defined as the fraction of interac tions among nodes that are causally significant. Causal flow is a measure of the difference be tween casual inflow and causal outflow from a node. Causal reciprocity is a measure of feedb ack between nodes. Causal disequilibrium refers to an imbalance in reciprocity and flow between nodes. Causal sinks refer to nodes where the balance of information flows into the node, while causal sources refer to nodes where the balance of information flows out of the node. These basic de finitions help lead to tools that can be used to understand the effects of causal flow within a network. Seth de monstrates the use of the tools in a 32-neuron neural network model of target fi xation using a head/eye system. Results from this model and analysis using these tools allow new hypothesis in cluding being able to predict the effects of lesions within simulated networ ks. Seths further work with Edelman (55) includes the idea of a causal core within large neural populations during behavior. A causal core is defined as neural interactions within the netw ork present when a reference neuron, such as an output motor neuron, fires. This idea allows th e linkage of neural inte ractions with specific outputs from the network. Seth and Edelman use the Darwin X system, an autonomous robot with a simulated neural network brain based on cortical and hippocampal anatomy. The robot performed a dry version of the water maze test for rates where the robots goal was to find a hidden platform in a static environment with spat ial clues. Over time the robot learned to find

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27 the platform more quickly similarly to rats that have performed the water maze. The behavioral output from the Darwin X system was analyzed using the causal core idea and Granger Causality. Consistent causal co res could be identified for specif ic outputs of reference neurons. Changes in the causal core during learning sequen ces were also observed. The causal core was observed to reduce in size and become more refine d as Darwin X learned to move more directly to the hidden platform. These examples show that Granger Causality has proven to be a useful tool for uncovering effective and functional relationshi ps in various neural systems. These tools and associated methods will be used in several e xperiments in simulated, in-vitro, and in-vivo neural networks. We will begin with a more comprehensive review of the autoregressive modeling and the properties of Granger Causal tools presented in the next chapter.

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28 CHAPTER 2 MATHMATICAL BASIS FOR GRANGER CAUSALITY Mathematical Basis for Time Domain Measures Granger Causality The goal of Granger Causality (GC) is to prov ide a strong mathematical basis to determine causal influence and direction of neural interaction. The basic id ea can be traced back to Wiener (57). He proposed that, for two simultaneously measured time series, one series can be called causal to the other if we can better predict the second series by incorporating past knowledge of the first one. This concept was later adopted and formalized by Granger in 1969 (1) for linear regression models of stochastic processes. Specifically, if the va riance of the prediction error for the second time series at the present time is reduced by including past measurements from the first time series in a linear regression model, then the first time series can be said to have a causal (directional or driving) influe nce on the second time series. Th e advantage over cross correlation metrics is that GC is more robust with respect to changes in overall rate of activity, provides a measure of the causal influence, is directiona l, and often requires less data for analysis. A description of this technique is provided in the following sections de scribing autoregressive modeling as well as time-domain, spectral, and conditional methods. A more detailed review by Ding et al in a recently published handbook on causal methods can be found here (2). The purpose of this chapter is to de scribe what Granger Causality doe s and does not do. This is accomplished by describing how GC produces a measure of effective connectivity and discussion of several of its inherent weaknesses. Autoregressive Modeling Autoregressive (AR) modeling of neural time series is th e foundation of parametric Granger Causality methods. Time series in multivar iate neural data are typically recorded from

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29 multiple electrodes or channels. While AR models are often presented in a theoretical context, the examples reported here will be provided in the context of a multichannel recording from a neural process. Consider th e multivariate random process, ( t ), consisting of p independent channels of neural data. ( t ) X1( t ) X2( t ) ... Xp( t ) (2.1) A recording of time series from these channels would be considered a single real ization of this neural process. Multiple realizations of the neural process are advantageous to creating a more accurate model describing this process. ( t )is described using an AR model of model order m assuming that ( t )is a stationary process. ( t ) A (1) ( t 1) ... A ( m ) ( t m ) E ( t ) (2.2) A ( i )are p by p coefficient matrices where i 1,2,..., m and E ( t )is a zero mean uncorrelated noise vector with a covariance matrix of A ( i )and are unknown matrices that can be estimated from the realizations of ( t ). This is done by right multiplying Equation (2.2) by ( t k ), where T indicates a transposed matrix The expectation is then taken on the resulting equation yielding th e Yule-Walker equations for k 1,2,..., m containing a total of mp2model coefficients to be solved for. R ( k ) A (1) R ( k 1) .... A ( m ) R ( k m ) 0 (2.3)

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30 R ( n ) X ( t ) XT( t n ) is the auto covariance function of ( t ). Note that lag n is n k 1 ranging from 0 to m-1. The auto-covariance function is then calculated from each realization, x( t ),of ( t )of length N. R ( n ) 1 N n x ( i ) xT( i n )i1 Nn (2.4) If multiple realizations of the data are availa ble the auto covariance functions are averaged across realizations. Once the auto co variance function has been calculated the mp2model coefficients of the A ( i )coefficient matrix can be solved for because there are mp2equations within equation (2.3). There are several methods that can be used to solve th is including sim ply solving for each of these coefficients or us ing methods such as the Levinson, Wiggins, and Robertson method or Morfs method, used for this analysis. Once A ( i ) is known, the noise covariance matrix is obtained using the following relationship. R (0) A ( i ) R ( i )i 1 m (2.5) This preceding section describes the es timation of a MVAR for any model order m However, it is necessary determine the correct model order for use with each system being studied. The correct model order can be determined using several methods including the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The BIC is more often used for neural applicati ons as it compensates for the large N (number of data points) common to neural data sets. Thus, the BIC will be the primary method used to determine model order in these studies. AIC ( m ) 2log[det( )] 2 mp2N (2.6)

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31 BIC ( m ) 2log[det( )] 2 mp2log N N (2.7) The BIC is plotted as a function of the model order m The correct model order usually corresponds to a minimization of BIC. However, when the model order becomes too large this may result in excessive computation time. Of ten, a smaller model order with a similarly minimized value of BIC is used rather than the absolute minimum BIC for this reason. The resulting time domain MVAR model is th e basis for the calculation of both time and spectral domain granger causality, which w ill be discussed in the following section. Time Domain Granger Causality With the core AR modeling tools now fully described, Parametric Granger Causality can be calculated from the resulting models. GC methods make use of the variance of prediction errors from various combinations of AR models. Pairwise Granger Causality (PGC) is used to investigate the directional interactions between two time series. Two AR models are required to determine for each direction of influence. Consider two time series X(t) and Y(t) where separate AR models are created to predict cu rrent value of the X series from m (a previously determined model order) previous X values and: X ( t ) bXX( j ) X ( t j )j 1 m( t ), 1 var(( t )) Y ( t ) bYY( j ) Y ( t j )j 1 m( t ), 1 var(( t )) (2.8) The variance of the error series ( t ), 1, is a gauge of the linear prediction accuracy of X(t) as 1 is for Y(t). Now consider a bi-variate multivariate autoregressive (MVAR) model, W ( t ), where both X(t) and Y(t) are calculated fr om the previous values of the X and a Y time series:

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32 W( t )X ( t )aXX( j ) X ( tj )j 1maXY( j ) Y ( tj )j 1m( t ) Y ( t )aYX( j ) X ( tj )j 1maYY( j ) Y ( tj )j 1m( t ) (2.9) The variance of the new error series is a gauge of the prediction accuracy of the new expanded predictor. W 2222 var(( t ))cov(( t ),( t )) cov(( t ),( t ))var(( t )) (2.10) Based on Wieners idea, Grange r (1) formulated that if th e X prediction is improved by incorporating past knowledge of the Y series, the Y time series can then be said to have a Granger-causal influence on the X time series. This is the basis for the time domain version of Granger Causality where the variance of the linear prediction error of X alone, 1, is compared to the variance of linear pred iction error of X including Y, 2: FYX ln 12 (2.11) Note that when 12(i.e. the linear prediction error is not improved by including Y) that this relationship will yield a PGC value of zero. Driving in the opposite direction is addressed by simply reversing the roles of the two time series. It is clear from this definition that timing plays an essential role in directional causal influences. Spectral and Conditional Granger Causality Two additional developments of Granger's causality are important for the proposed research. First, natural time se ries, including ones from neurobiol ogy, contain oscillatory aspects in specific frequency bands. A spectral repres entation of Granger Caus ality was developed by Geweke (58), allowing casual interactions to be quantified at specifi c frequencies. Second,

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33 pairwise Granger Causality can often yield false direct influences due to parallel driving by or mediation through elements not included in the AR model. When data from at least 3 or more time series is available, such as from neurobiol ogical multi-electrode arrays, conditional Granger causality (54, 59) can be used to address these limitations. Th ese methods are described in the following section. Spectral Granger Causality Geweke (58) published his groundbreaking pape r in 1982 where he identified a novel way to decompose the time domain causality measure in to its spectral content. His method compares the intrinsic power (defined as the power of a ti me series generated by the stochastic processes intrinsic to that time series) of one time series and the total power of the same time series, which may contain a causal influence from a second time series, in the frequency domain. When there is no causality between the two time series (i.e. the total power is equal to the intrinsic power) the log ratio of the total to intrinsic power is equal to zero. The frequency domain function W ( f ) is obtained by taking the Fourier transf orm of the previously described MVAR W ( t ): A ( f ) W ( f ) E ( f ) (2.12) A ( f ) is the spectral coefficient matrix and E ( f ) is the spectral noise vector resulting from the Fourier Transform performed on W ( t ). Rearrangement of (2.12) by setting the matrix inversion A ( f )1equal to H ( f ) (commonly known as the tr ansfer function) yields: W ( f ) H ( f ) E ( f ) (2.13) The spectral matrix S ( f )can now be calculated that contai ns the auto spectra and cross spectra of W ( t ).

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34 S ( f )H ( f )3H*( f ) (2.14) H*( f )is the transpose and conjugate of H ( f ). 3 is the covariance matrix of the noise terms from W ( t ). This spectral matrix, containing the total power of the X time series, is retained. Spectral Granger Causality is based on decomposing the total power (variance) at a given frequency f for one time series into the sum of an intrinsic part and a causal part from the other series. A new spectral matrix is calculated in the same way except that both sides of equation (2.14) are left m ultiplied by Gewekes normalization transform: P10 22 1 10cov(( t ),( t )) var(( t )) 1 (2.15) Gewekes normalization transform isolates the intrinsic power of X by allowing the intrinsic power component of the total power to be calculated separately from the causal power component of X(59). The X component of the new transfer function, shown in equation (2.16) is then used to calcu late the intrin sic power of the x time series. Hxx( f )Hxx( f ) 22 Hxy( f ) (2.16) Taking the logarithm of the ratio of the to tal power over the intrinsic power gives the Granger causality spectrum. Fy x( f ) ln Sxx( f ) % Hxx( f ) 2% H* xx( f ) (2.17)

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35 From the value of the causality spectrum, one can evaluate how much of one signals variance at f is explained by the second signal exerting causal influence. Geweke showed that the integration of this spectral quantity over fr equency is the time domain Granger causality. Conditional Granger Causality Highly m ultivariate data can be analyzed by PGC, reducing the problem to a bivariate case. This approach is straightforward and often e ffective for simple systems. However, this methodology can yield distorted resu lts in more complex systems. Consider the following two situations. In one case, causal influence from Z to X is mediated through Y. For the second case, Z influences X through Y, but Z also has a direct influence on X. These two cases are indistinguishable using a bi-variate GC analysis; in fact the anal ysis will identify both cases as the second case. Conditional GC (CGC) is an al gorithm that can be used in both in the time domain and in the frequency domain for three time series (31, 60) to overcome this shortcoming. We will only describe the time domain version of CGC, as the core idea of spectral CGC is very similar to the previously descri bed Spectral Granger Causality. U ( t ) X ( t ) Axx( j ) X ( t j )j 1 p Axy( j ) Y ( t j )j 1 p Axz( j ) Z ( t j )j 1 p( t ) Y ( t ) Ayx( j ) X ( t j )j 1 p Ayy( j ) Y ( t j )j 1 p Ayz( j ) Z ( t j )j 1 p( t ) Z ( t ) Azx( j ) X ( t j )j 1 p Azy( j ) Y ( t j )j 1 p Azz( j ) Z ( t j )j 1 p( t ) (2.18) Time domain CGC compares the prediction of X including Y in a bi-variate MVAR model (from the previously described W ( t )), with the prediction of X including Y and Z in the trivariate MVAR model U ( t ) shown in figure 2-1.. U xxxyxzxyyyyzxzyzzz var(( t ))cov(( t ),( t ))cov(( t ),( t )) cov(( t ),( t ))var(( t ))cov(( t ),( t )) cov(( t ),( t ))cov(( t ),( t ))var(( t )) (2.19)

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36 Figure 2-1 Conditional Pairwise Granger Causality Example. If Pairwise Granger Causality were applied to determine the connectivity of both of these network configurations the results for both would resemble panel a). Using Pairwise Granger Causality alone, it is not possible to differentiate between these network configurations. Conditional Granger Causality is required to determine if the connection from Y to X is real. Conditional Granger Causality doe s this by determining if Y affects X directly or if the effect of Y mediated through Z by dete rmining how well X is predicted by Z with and without the influence of Y. The influence of Z on X is entirely mediated through Y when the prediction of X is not improved in the trivariate MVAR model over th e bivariate MVAR model. Specifically, CGC is calculated by taking the log of the ratio of th e variance of the prediction error of X from W ( t ) to the variance of the pred iction error of X from U ( t ). This is formalized in the equation for CGC: FZ X Yln 2xx (2.20) Conditional Granger Causality ca lculated using this equation is greater than zero when some of the power from Z is dir ectly causal on X. The result would be equal to zero when the influence of Z on X is entirely mediated through Y. Thus, this method allows the two examples stated earlier to be easily differentiable. While Conditional Granger Causality addresses specific deficiencies in the original formulation of Granger Causality, there remain several further weaknesses that need to be considered when drawing conclusion from a causal analysis.

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37 Limitations of Causal Methods Since the tim e of Aristotle, philosophers have reached little consensus on a common definition for the concept of causality (61). Ho wever, causation is a c oncept critical to many disciplines including law, medicine and the sciences. A functional definition of this concept is necessary to develop tests of causality that are useful in the context of these disciplines. Interestingly, the debate over the definition of causality has provided insights into several weaknesses of causal methods including Granger Causality that must be considered when designing sound experiments. Clive Granger best describes th e inherent limitations of his or iginal method in his accounts of the origins of Granger Causality (62) (61). He describes how several of his peers noted that his method is not real causality, it is only Granger causality. Granger also describes several axioms that must be assumed to be valid. The first among these is that the past and present may cause the future, but the future cannot cause the pa st (61). Second, everyt hing in the universe is described and this set of information includes no redundant information. In this hypothetical universe, it would be easy to talk about the relationship between a variable X and a variable Y. However, these assumptions are difficult to satisfy in real systems. In reality, causality algorithms can be effected by many factors in cluding sampling rate, time scale selection, seasonal influences, signal to no ise ratio, variable selection, variable availability, and even chance. Time scale and sampling rate are two variables that can have intertwined yet diametrically opposed effects on a causal analysis. Selection of time scale and sampling rate is dependent on the processes being observed. If the sampling rate is too low, it can be difficult to observe time delayed influences in the system because the syst em may be interacting at time scales of a higher frequency than the sampling rate. Conversely, if the sampling rate is too high it can be

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38 computationally intractable to model far enough into the future (i.e. high model order m ) to capture interactions. Additionally, it is possible that two time series may interact differently on two largely different time scales that may require separate analysis. These examples demonstrate that proper consideration must be given to se lection of time scale a nd sampling rate depending on the system being analyzed. The data in these experiments are often recorded at very high (up to 25kHz) sampling rates that must be low passed filtered to provide a proper sampling rate and time scale to study causal interactions. All data is down sampled to 1kHz prior to analysis. Spike train point processes are further low pass filtered to 50Hz (10% of nyquist for 1kHz) to create a continuous time series so that AR models can be used. It is also not possible to desc ribe all variables in an experi mental system. Consequently, unobserved time series are a factor that must al ways be considered wh en talking about causal influences. Implementation and subsequent inte rpretation of Granger Causality inherently depends entirely on the availability and selecti on of variables for anal ysis. Specifically, the causal relationship between two neurons may be influenced by a third and likely many other neurons that are not recorded from (e.g., a neuron too distant from an electrode to be measured). The likelihood of this situation increases with the complexity of the system being observed. Consequently, the contribution of these unobserve d neurons would not be accounted for. This can have several effects on the relationships be tween observed time series. One possibility is that a common driving input at differing time delays could create a false causal influence where no real connectivity even exists. A priori knowledge of possible an atomical structure can help in this type of situation. So called seasonal influences also present an interesting possibility. Two time series may be causal because of a similar common driv ing by an unobserved seasonal process.

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39 Subsequently, the causality betw een these time series may wax and wane dependent of this seasonal influence or possibly di ssapear altogether. Another possible scenario is that the influence between two time series could include mediated influences that cannon be conditioned out using previously mentioned conditional methods This would tend to distort the magnitude of the influences reported from analysis. This is a likely possibility in complex neural systems that cannot be easily remedied. This problem imp acts not only GC analysis but also every other multivariate statistic used in neuroscience today. Thus, it is clear that results from this type of analysis could potentially suffer from poor experimental designs where the important variables are unobserved or uncontrolled. Additionally, there is another specific weakne ss of GC relating to the analysis of point processes. Often, the output from an observed ne uron is described in terms of when that neuron fires action potentials. Point pr ocesses are not a continuous func tion over time. AR models can only describe continuous processes. In order to compensate for this, spik e files from individual neurons must be low pass filtered prior to causal analysis. Subsequently, high frequency information above the cutoff for the low pass filter is not included in analysis. However, unpublished results from a non-parametric version of Granger Causality for point processes being developed by Mingzhou Ding and colleagues are very similar to parametric results. Specifically, the non-parametric method does not show any significant in teractions over the cutoff of the low pass filter used for the parametric method. These strengths and weaknesses of Granger Caus ality must be considered carefully before application to neural systems. In the following chapters, these methods will be used to describe the effective connectivity using the observable ou tputs of these systems to better understand the

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40 underlying systems. These weaknesses will be carefully considered during experimental design, analysis, and subsequent interpretation.

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41 CHAPTER 3 RECOVERY OF SYNAPTIC WEIGHTS FR OM SPIKING ACTIVI TY USING GR ANGER CAUSALITY Introduction Recent adva nces in multichannel recording t echniques enable simultaneous access to the electrophysiological acti vity of hundreds to thousands of neurons. These advances allow investigators to address one of the primary challenges in neuroscience, lin king the structure of a network to its potential functions. For example, in cognitive neuroscience, by recording the activations of brain structures during various behavioral perfor mances, investigators now have the technology for understanding how different brai n structures interact to produce behavior. This experimental advance has been supported by a simultaneous advance in the quality of analytical tools that quantify neur onal interactions, such as the clas sical cross-correlation (14, 18, 21, 63-65) and coherence functions (66). At th eir core, however, these methods do not provide directional or causal relations hips between different channels whether those channels are recorded from individual neurons or from differen t brain regions. In cont rast, Granger Causality (GC) (1) has emerged in recent years as a techni que of choice to provide a mathematically sound method for estimating causal relations. Pairwise Granger causality (PGC) was initia lly developed in the economics community to describe and quantify the causal relationship be tween data from two different economic time series. The foundation for this technique can be traced back to Wiener (57), who proposed that for any two simultaneously recorded time series, one series could be called causal to another if incorporating past knowledge of the first time se ries permits more accurate prediction of the second series. Granger formalized this idea in the context of linear regression models of stochastic processes (1). Specifically, consid er two simultaneously recorded time series x1, x2, x3 xn and y1, y2, y3 yn. Suppose one would like to construc t a linear predictor of the current

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42 value of the x series based on m prior values: xn = a1xn-1 + a2xn-2 + amxn-m + n. This is nothing more than a single variable autoregressive model in which standard procedures can be applied to yield the model order and m odel coefficients, where the variance of error series, n, is a gauge of the prediction accuracy. Now consider a predictor of the current values of the x series by including both the prev ious values of the x series and the prev ious values of the y series, namely: xn = b1xn-1 + b2xn-2 + bmxn-m + c1yn-1 + c2yn-2 + cmyn-m + n. The variance of the error series, n, is then a gauge of the prediction accuracy of the new expanded predictor that includes information about the y series. Based on Wieners idea, Granger formulated that if the variance (n)/variance (n) is less than one in some suitable stat istical sense (i.e. the prediction of x is improved by incorporating past knowledge of y) then we say that the y series has a causal influence on the x series. The concept of Granger causality has played a significant role in the field of economics since 1960s. Only recently has its potential for un derstanding the dynamics of brain circuitry been realized in neuroscience. For example, Zhu et al (48, 49) explores causal interactions between neurons during learning in the primary motor cortex of rhesus monkeys. Fanselow et al. (46) reveals causal influences fr om the sensory cortex to the th alamus during rhythmic activity associated with whisker twitching behavior in rats It is expected that Granger causality based methods will experience significant growth in applic ations to a wide variety of neural systems where information transfers exist among individu al neurons or different brain areas, including data from neural simulations, in vivo electroencephalogram (EEG), and in vivo microwire arrays (27, 30, 37, 43-47, 49, 50, 52, 56, 67-74). These studies have mainly focused on revealing structures of interac tion in the recorded circuits. The physiological meaning underlying th e statistical values of Granger causality

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43 remains unexplored. The goal of this study is to examine whether values of Granger causality can be used to recover synaptic weights betw een interacting neurons in a complex network. While intracellular recording met hods are able to discern the syna ptic weight from postsynaptic potentials, they cannot rev eal network level activity in vitro and are of very limited use in vivo Thus, to understand the synaptic relationships th at regulate network activity, a technique to reconstruct the weights from only spik e train outputs must be devised. For a simple two-neuron network in which neur on A is synaptically connected to C, our first result is to show that the aforementi oned PGC can easily discriminate a direct causal relationship from A to C when the strength of th at coupling is even moderately high and, more importantly, the values of PGC can be related to synaptic efficacy. For more complex networks the pairwise approach encounter s significant methodological limita tions when direct, mediated, and serial influences exist between neurons. For example, consider a three-neuron case in which neuron A is synaptically coupled to a mediating neuron, B, who is then connected to neuron C. First, PGC would identify an erroneous causal connection between A and C, even though this pair is not directly coupled. The reason is that the activity of A, does in fact, causally influence the activity of C, but does so only through the mediating relationship involving B. Second, when trying to recover the synaptic efficacy from B to C, due to the serial influence from A, the values of PGC from B to C are no longer a true reflection of the synaptic weight between the two neurons. These problems can become even more acute in more complex network with many interacting ne urons or brain areas. The second result of this paper is to describe a technique to recover the Direct Gr anger Causality (DGC) and the synaptic efficacy between neurons within networks with more real istic topologies, including direct, mediated, and serial influences.

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44 The organization of this chapter is as follows. Existing methods such as Pairwise Granger Causality and Conditional Granger Causality are described in terms of their application to spiking neural networks. A comput ational shortcut that allows fa ster recovery of DGC is also explored. These methods are then systema tically expanded to accommodate increasingly complex network conditions including direct, medi ated, and serial influences. Each solution is illustrated with biologically plausible, simulate d networks. Finally, we examine how DGC could be used to estimate both synaptic weights with in a network and how it might provide a useful measure for describing changes in network wide plasticity. Conditional Granger Causality and Mediated vs. Direct R elationships In Chapter 2, the mathematical basis for both Pairwise and Conditional Granger causality was described. One of the limitations of PGC was th at when it is used in a multivariate system, often these results do not reflect the true connectivity. False dire cted influences and cases where direct and mediated influences are confounded within the same pathway are the two primary problems. In the following section we will describe how these relationships can be untangled using Granger Methods. Chapter 2 also shows th at CGC can be used to identify and remove erroneous direct connections that are actually mediated though othe r elements. However, at this point, we are left with a second problem. The effect of Y on X is embedded in the relationship reported between Z and X. This results in a stronger PGC value from Z to X when compared to the PGC value from Y to Z, even though the actual synaptic weights between these pairs may be identical. The direct influences need to be separa ted from mediated influences in order to more accurately quantify synaptic interactions within this network. Direct influences, after being recovered from an entangled pathway, will be refe rred to as Direct Granger Causality (DGC) to avoid confusion with values recovered using onl y PGC. Note that in neuron pairs that do not contain mediated influences, PGC already represents the DGC between these neurons.

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45 One way to recover DGC from an entangled pathway would be to make the relationship between X and Z conditional on Y using CGC. This was the solution described by Geweke (59) shown in equation (3.21). However, this method is computationally expensive in small networks and possibly intractable in large netw orks due to the exponentia l number of potential combinations that must be addressed. Interestingl y, Geweke also noted th at it might be possible, when the influence from Y to X is entirely me diated through Z, to dire ctly subtract the PGC calculated from Y to X from the PGC value from Z to X. This would recover the same quantity as calculating CGC from X to Y conditiona l on Z, without having to perform the computationally intensive CGC analysis. This con cept will be evaluated empirically later in the paper. FZX | Yln var( xx ( Y X )( t )) var(xx ( X Y Z )( t )) (3.21) Simulation and Analysis In this sec tion, we illustrate the analytical solutions described earlier within a simple, biologically plausible neural netw ork model. We have chosen Iz hikevitchs simple neuron model (3, 75), a stochastically driven neural network with modifiable synaptic weighting that provides biologically relevant spikes that can then be an alyzed using GC(76, 77). Briefly, the model is a reduction of the Hodgkin-Huxley model (78) to a simple two-dimensional system of differential equations. One equation models membrane voltage, v, equation (3.22), while the other models and a recov ery variable, u shown in equation (3.23). v 0.04 v2 5 v 140 u I (3.22) u a ( bv u ) (3.23) The auxiliary after spike resetting of the neuron is mediated by equation (3.24).

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46 if v30 mV then vc uud (3.24) The variable u is a membrane recovery variable that accounts for K+ activation and Na+ inactivation in the neuron. The variables a, b, c, and d are dimensionless parameters that effect the temporal characteristics of the action potential, while v and u are the first time derivatives of v and u. These variables remain constant an d are the default values used by Izhikevich for regular spiking cortical neurons (3) Each time step of the model represents 1 ms, equivalent to a Figure 3-1 Topology diagram of a simple five neuron network. In panel a) the synaptic weights from N1 to N2 and N4 to N5 have been syna ptically coupled. N3 is left uncoupled to demonstrate that PGC will indicate null c onnectivity. Equivalent but independent random processes drive each of the five ne urons. 100 realizations of this network using Izhikevichs simple neuron model with these weights yields the results shown in panel b) using Pairwise Granger Caus ality. These results demonstrate that Pairwise Granger Causality can, not only resolve the difference between null and actual connectivity, but also determine th e directionality of those influences.

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47 1kHz sampling rate of a realistic neural system During simulation, neurons can spontaneously fire based on spontaneous super-threshold fluctu ations of membrane noise injected into each neuron and from synaptic inputs from other neurons in the network. Mono-directional Simulation The first m odel, illustrated in Figure 3-1, c onsisted of a five neuron network composed of mono-directional connectivity (i.e., one way sing ular direction for causa l relations) from Neuron 4 (N4) to 5 (N5) and from Neuron 2 (N2) to 1 (N1), with neighboring Neuron 3 (N3) uncoupled (zero synaptic weight). Hence, the activity of N2 and N4 will have a causal influence on firing of N1 and N5, respectively. In contrast, only the equivale nt independent random process intrinsic to N3 drives N3. It is important to note th at the intrinsic random proce sses that drive these neurons are equivalent, but independent The cross-power spectra for all neurons during simulated activity are shown in Figure 3-2. During the simulation, all five neurons were active, indicated by the auto-spectra (along the diagonal) for each neuron. As expected from the configuration of the synaptic weights, the cross-power spectra for N1 to N2 and N4 to N5 is larger than that of N3 with its neighbors. The spectra resulting from Granger causality is shown in Figure 3-3. This figure shows causality by frequency from 0 to 50 Hz between N4 to N5 (top panel) and the converse relation, N5 to N4 (bottom panel). Peak causality between N4 to N5 occurred at frequencies near 20 Hz. In contrast, spectral causality within the reverse relationship, N5 to N4, was very low. This verifies the absence of a bi-directional relationship between the pair. Similar results using other modeling systems can be found elsewhere (48, 49, 76).

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48 Figure 3-2 Power spectra and cross-spectra calcu lated from the 5 neuron model simulation. These plots are organized in matrix format mirroring that of the spectral matrix so that that row 1 column 1 represents the intrinsic power spectra for N1, while row 1 column 2 represents the cross-spectra between N1 and N2. Thus, the diagonal contains the individual ne urons spectral power while the upper triangular graphs represent Cross-spectra between neurons. No tice that Neuron 3 (Row 3, Column 3) has intrinsic power but demons trates no significant cross-sp ectra (Other plots in both Row 3 and other plots in Co lumn 3) with its neighbors as it is independent and uncoupled to the other neurons. Neuron pairs 1,2 and 4,5 both demonstrate significant cross-spectra (Row 1 Column 2 and Row 4 Column 5) consistent with initial synaptic weights. However, it is not possible to determine directionality from this information.

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49 Figure 3-3 Granger Causality Spectr a. The causality from N4 to N5 is apparent, with a maximum value of approximately 0.92 around 20Hz, while the reverse direction of N5 to N4 approaches zero at all frequencie s and is not statistically sign ificant. This verifies that PGC provides directional information that cannot be discerned from power spectra, shown in Figure 3-2. The Pairwise Gra nger Causality for this directional neuron coupling would be reported as 0.92. Relationship between Synaptic Weight and Granger Causality At this point, we have shown how PGC can provide an estim ate of the relationships between elements. One important consideration, however, is how PGC values correspond to actual synaptic weights, especially when those wei ghts might be in constant flux. In other words, how is an estimate of causality using PGC related to the actual underlying synaptic weight especially when those weights may be changing?

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50 Figure 3-4 The PGC results from a mono-directio nal simulation with the weight from neuron 4 to 5 varied from 0-75mV in increments of 5mV. Synaptic weight is plotted on the xaxis while the resulting PGC values calcula ted from the spike timing is plotted on the y-axis. The relationship between synaptic weight and PGC is sigmoid described by equation 12. Notice the region approxim ately between 15mV and 45mV where the relationship between synaptic weight and PGC is linear. Each increment in the graph was simulated 100 times to provide the mean and variance shown in the plot. The simulation was modified such that the connectivity between N1 and N2 was systematically incremented from 0 mV to 75 mV to observe the effects on our estimate of causality. Each increment was simulated 100 ti mes to determine both the average estimate by PGC and variance of that estimate. The re sults of those simulati ons indicate a sigmoid relationship between the s ynaptic weight from N1 to N2 (S1 2) and resultant GC, F1 2 shown in Figure 3-4. There is a regi on between 15mV and 45mV wher e the relationship between S1 2

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51 and F1 2 is linear with a slope of 0.033 causal units /mV. Synaptic weights above this range do not produce significant changes on F1 2, suggesting that 45mV is the maximum effective synaptic weight. Similarly, weights below 15 mV produce only slight chan ges in the estimate of causality. The data from this experiment were fit with a sigmoid equation, shown in Equation (3.25), to quantify the functional relationship between Granger estim ates and actual synaptic weights. An example of estimated synaptic weight s using this equation is presented in panel d of Figure 3-5. FY X 1 1 384 e 0.2124 S (3.25) Mediated vs. Direct Causal Relati onships under Serial Connectivity PGC works well for recovering synaptic weights between two neurons using Equation (3.25). However, there are situations where PGC values will n ot adhere to this relationship due to added complexity within the network. Of ten PGC values include not only the direct information between nodes but also include mediat e influences that could distort the relationship between PGC and synaptic weights. This become s more common as the number of elements and connectivity increase within a network. This issue is addressed in the second simulation, where a serial topology was created to demonstrate discrimination between direct and mediated connectivity as well as separation and recovery of individual syna ptic weights within a chain of serially connected neurons. The five-neuron topology, shown in Figure 3-5 a), consisted of causal

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52 Figure 3-5 Five neuron simulation. This was carri ed out where a) shows the synaptic weights before simulation. Calculation of PGC fo r all possible connecti ons for the 5 neurons yields the plot shown in b) Notice that using PGC alone many new false connections are shown. When CGC is used to eliminate the false connection the plot is reduced to what is shown in c). After CGC plot c) begins to resemble the connectivity pattern as shown in a), however, the values associated with c) do not scal e with the synaptic weights shown in a). A further step usi ng CGC a second time or using Gewekes subtraction method is required to detangle direct and mediated influences. The results after the use of Gewekes subtrac tion method to calculate DGC are shown in d) along with corresponding approx imations of synaptic weight. synaptic weights of 45 mV from N1 through N5 similar to a synfire chai n (79, 80). The original PGC estimate of causality, shown in panel (b), in dicates erroneous connec tions among the entire pool of neurons reflecting the complex causal re lationships between each element. Table 3-1 shows the PGC results for each relationship. Note also that the PGC values, shown along the elements of the serial chain, steadily increase from N1 to N5 (diagonal in Table 3-1), reflecting the embedded mediated causal influences cascading throughout the chain.

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53 Table 3-1 A table of the Pairwise Granger Causal influences calculated between the 5 neurons in the simulation displayed in Figure 3-5. The right column indicates what the from neuron is and the top row indi cates what the to neuron. For example, to locate the influence of neuron 4 on neuron 1 a value of 0.01 is reported in From N4, To N1. To N1 To N2 To N3 To N4 To N5 From N1 0.92 0.55 0.38 0.48 From N2 0.01 1.38 0.94 0.78 From N3 0.01 0.01 2.02 1.25 From N4 0.01 0.01 0.02 2.11 From N5 0.00 0.01 0.02 0.01 CGC was applied to discriminate between false direct from mediated connections in order to determine the networks true connectivity. Panel c of Figure 3-5 shows the results of CGC, where the topology has been correctly estimated but includes the systematic errors in estimating the strength of coupling inherent within this method due to the causal cascade inherent in serial chains. The problem arises because values remaining after calculating CGC represent a combination of direct and mediated values that correspond to an increase in total information flow down the chain. This is due to the accumula tion of mediated influences towards the end of the chain. Thus, the direct components betw een any two neurons must be separated from mediated influences to relate the direct component to a m eaningful synaptic weight. Recovery of direct synaptic weights could be achieved using CGC for a second time on the mediated pathway. However, a simpler solution may exist. If PGC simply sums the effects Table 3-2 Pairwise Granger Causal influences calculated between the 5 neurons in the simulation displayed in Figure 3-5 after th e use of significance th resholds, conditional granger causality analysis, and removal of me diated influences. This table contains only direct causal influences. All other va lues reported in Table 3-1 were found to be false or insignificant. The right column indicates what the from neuron is and the top row indicates what the to neuron. For exam ple, to locate the influence of neuron 4 on neuron 1 a value of 0.01 is repo rten in From N4, To N1. To N1 To N2 To N3 To N4 To N5 From N1 0.92 From N2 0.82 From N3 1.08 From N4 0.86 From N5

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54 of previous elements along a serially connected ch ain, then it may be possible to simply subtract the previously conditioned out PGC values that represent the mediat ed component of the distorted PGC value. For example, by subtracting N1 3 (0.55) from N2 3 (1.37), a value of 0.82 is obtained, which better estimates the actual synaptic weights incorp orated into the model. This process was repeated for each pair and the resul ting estimates are shown in panel (d) of Figure 35 and Table 3-2. The relationship between CGC and the simple subtraction of N1 3 to N2 3 is derived from the properties of both PGC and CG C, as stated by Geweke (77), and shown in Equation (3.26). FY XZ FYZ X FY X (3.26) In the special case where the bivariate PGC influence of Y on X, FY X, has been previously conditioned out (i.e. equal to zero) using CGC, equation (3.27) is valid. FYZ X FZ X (3.27) This allows for the derivation of equation (3.28), which explains the empirical relationship that a llows subtraction of PGC values to retrieve the DGC value between FY XZ FZ X FY X (3.28) A direct comparison of the efficacy of th ese methods for calculating DGC was achieved using a Monte Carlo simulation of a 3 neuron chain. 100 realizations of this simulation were run and both methods were used to recover DGC from the PGC output of the synapse at the end of the chain. The synaptic weights were random ly generated between 15mV and 45mV for each realization of the simulation. The results of this simulation are shown in Figure 3-6. These results suggest that the subtraction method presented in equation (3.21)[12] can produce DGC values s imilar to that of CG C as shown in equation [5].

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55 Figure 3-6 Direct influences be tween N2 and N3 were recovered by both methods to allow a comparison of Gewekes subtraction method to CGC. Each point was generated from a Monte Carlo simulation of the serial simu lation using randomly generated synaptic weights between 15mV and 45mV. This plot suggests that bo th methods provide results consistent with the linear region e xpected from the sigmoid plot shown in Figure 3-4. However, Gewekes subtra ction method is computationally simple providing the researcher with a clear advantage in large networks that may require repetitive calculation of DGC. Monte Carlo Simulations with Random Serial Topologies A Monte Carlo simulation was conducted to furt her test DGC in a 5 neuron serial chain with random synaptic weights. This experiment was designed to desc ribe the relationship between synaptic weights and the recovered DGC for each synapse in the serial chain. The synaptic weights S1 2, S2 3, S3 4, and S4 5 were randomly generated for each simulation using a uniform distribution from within 15 mV to 45 mV linear range of th e sigmoid relationship derived from the first experiment. The DGC for pathways with mediated influences (F2 3, F3 4,

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56 & F4 5) was calculated for each synapse by subtracting the conditioned F(N-1)M pathway from FN M and Figure 3-7 DGC values can be reco vered at each of the synapses in the serial chain. This was done with randomly generated synaptic weights between 15mV and 45mV at all synapses. The weights for each synapse were generated independently within each simulation and the simulation was replicated 100 times. These plots demonstrate that DGC values recovered from entangled path ways (top right, bottom right, and bottom left) along the serial chain mirror those calculated using PGC on the first pathway (top left). This suggests that DGC values represent the direct influence between two neurons similar to the PGC relationship that can be calculated between neurons that are not entangled by mediated influences. retained. The synaptic weights S1 2 and F1 2 were also retained to serve as a baseline for comparison. According to the hypothesis, Mont e Carlo distributions for the DGC pathways

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57 should mirror the relationship between S1 2 and F1 2, a purely direct pathway. This simulation was repeated 100 times. Figure 3-7 a) shows that the distributions for the DGC at each synapse mirror the linear distribution from the purely direct S1 2 to F1 2. The higher variance for the DGC farther down the chain is due to additive error because of the way DGC is calculated. This empirical relationship shows that the conditioned pathway, F(N-1)M is a representation of the mediated component of the remaining pathway, FN M. The empirical relationshi p also suggests that DGC is an accurate representation of the DGC com ponent between two neurons and can be used to calculate a meaningful synaptic weight for the synapse using the relationship described in Equation (3.25). 100 Neuron Neural Network A larger more complex neural network simulati on was used to further test the ability of Granger Causality to recover syna ptic weights from spiking activity. This simulation consists of 80 excitatory and 20 inhibitory neurons connected with 20,000 synapses. Excitatory neurons are connected to all other neurons using randomly generated weights from a exponential distribution where most connections are weak (1-4mV) while strong connections (up to 70mV) are sparse. Inhibitory neurons are connected to the network using a flat dist ribution of wei ghts ranging from -1 to -10mV. Five excitatory neurons in this network were chosen and linked in a serial chain, resembling the earlier 5 neuron chai n, with synaptic weights of 40mV as shown in Figure 3-8 panel A). This network produces spontaneous sp iking and bursting shown in panel B). Pairwise Granger Causality is then used to determine the relationship between these 5 neurons shown in panel C). Pairwise Granger Ca usality alone reveals that info rmation flows directionally down the chain with no information flow in the oppos ite direction. However, the structure and

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58 Figure 3-8 100 Neuron Simulation. A chain of 5 neurons is imbedded within a 100 neuron neural network. The goal of this simulation is to extract the causal core shown in panel A using Granger Methods. Each synaptic weight in the chain is is set at 40mV. Panel B shows the activity of these five neurons w ithin the full network. Panel C shows the results from only PGC analysis. Pa nel D shows the remaining significant connectivity after CGC analysis and their corresponding synaptic we ights. Note that the weights down the chain are significantly und erestimated, this is likely due to the influence of the rest of the network on recovered weights. connectivity of the chain are not clear enough to disseminate the structure underlying this network. Conditional Granger Causality is then used identify and remove spurious connection revealing the structure shown in panel D). Thus, Granger Causal ity effectively identifies the serial chain embedded within this 100-neuron si mulation. Conditional Granger Causality and the previously demonstrated sigmoid relationship are th en used to recovery of synaptic weights also shown in panel D). Interestingly, results from a larger neural network are not as accurate as results shown previously from a simpler network. Further testing has show n that interaction with

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59 the rest of the network subtly distorts the magnitude of the rec overed synaptic weights. Future work will focus on the use Conditional Granger Caus ality to condition out the influences of the other 98 neurons on each paring to improve the accuracy of recovered synaptic weights. Application to Spike Timing Dependent Plasticity The brain is an ever-changing, non-stationary system that belies the fact that many measures of activity require stationarity to de rive meaningful conclusi ons. Take for example studies that attempt to demonstrate plasticity (4, 21, 81). One of the most common paradigms measures activity (e.g., spike rate) during a basel ine phase that is followed by some treatment (e.g., learning phase, such as delivery of a tetanic pulse train). Activity is then measured again to determine if any changes (e.g., spike rate) have occurred. The core notion is that changes in connectivity will result in a change in the numbe r of spikes. Unfortunately, measures like spike rate are highly susceptible to spontaneous (random) fluctuations in brain activity. In contrast, changes in the rate of spiking alone do not subs tantially influence causality estimates until the spike rate becomes so low that either method begins to suffer. Hence, GC may provide a novel and more reliable method to measure activity ch anges, while also out lining the structural information of the network as these changes occur. A simple neural model sensitive to spike ti ming dependent plasticity (STDP), a proposed mechanism for changes in synaptic plasticity, was created using CSIM (82, 83) to investigate this notion. This model was construc ted to determine if changes in synaptic weight due to STDP could be reliably detected with Granger Causal ity. The simulation was designed so that an STDP synapse, with parameters adopted from Markram et al., between two neurons would be exposed to a paired pulse stimulation similar to th at of Bi et al. (84). This paired pulse was designed to elicit long-te rm potentiation (LTP).

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60 Figure 3-9 A simulation to determine if changes in synaptic weight consistent with stimulation timing dependent plasticity (STDP) at a synapse between two biologically relevant neurons could be detected after a paired pul se stimulation was applied to the neurons. The paired pulses were designed to produce potentiation at the synapse. The top plot shows the action potentials generated from the pre and post synaptic neurons. The paired pulse stimulation was delivered be tween seconds 2 and 4. The middle plot shows the amplitude of the post-synaptic res ponse in amps, note that the post synaptic response was amplified due to the paired puls e stimulation, consistent with the effects STDP. The bottom plot shoes the membrane voltage and action potentials fired for the post synaptic neuron. PGC should detect and quantify the change afte r the paired pulse stim ulation in the post stimulation spontaneous activity. However, changes in activity from the paired pulse stimulation may not be great enough to dete ct using granger causality. Th e simulated neurons used were also based on Izhikevitchs simple model of spiking neurons (3). Each simulation was six seconds long, sampled at 1000 kHz, and repeated 10 times. The first two seconds of spontaneous activity are the base line, the middle two seconds are during the appli cation of the

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61 paired pulse stimulation to the synapse, and th e final two seconds of spontaneous activity are used as the post treatment recording. Figure 3-10 Statistical Analysis of Plasticity Simulation. The simulation ran in Figure 3-9 was repeated 10 times and the resulting pre and post spike trains were used to calculate the causal interaction between the neurons using PGC. The pre and post-tetanus values of PGC were compared using an analysis of variance. Plotting the analysis of variance demonstrates that the PGC values pre and post-tetanus are statistically independent suggesting that the change in causality due to the paired pulse stimulation can be detected and quantified using PGC. One realization of the simulation is shown in Figure 3-9. Note th at the post-synaptic response increased after the paired pulse stimul ation was applied. The spike trains for the baseline and treatment were low-pass filtered and PGC was calculated from the pre-synaptic neuron to the post-synaptic neuron. Ten replicates of this ex periment produced 10 PGC values for both baseline and treatment that are then poole d and compared using an analysis of variance, shown in Figure 3-10. This analysis shows that the values are statistically different post treatment suggesting that changes resulting from STDP can be detected using PGC.

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62 Conclusions Granger Causality is a powerful statistical technique that can quantify causal relationships, ranging from interactions between different br ain areas to recordi ngs from single neurons. However, there are a number of potential issues in interpreting synaptic efficacy that results from PGC, including false directed influence and separati on of direct and mediated influences. In this paper, we explored several additional techniques specifically to address those issues. Specifically, CGC is used to untangle direct and mediated influences and provide estimates of DGC. Additionally, a less computationally inte nsive subtractive process can provide DGC under special conditions. When these special conditions are not met, CGC would be required to recover DGC values. Through network simulations, we then described how these techniques could potentially be used to provide additional inform ation about the underlying network structure and to estimate the relationship between synaptic strength and DGC estimates. These methods are useful when that strength is static or changing due to changes in plasticity. There still remains several important weaknesse s and limitations of Granger Causality that must be considered. For example, synaptic we ight estimates using DGC are dependent not only on the variability of the system being measured and the measuring instruments, but also on the quantity of data recorded. Thus, short recording tim es with little activity can substantially affect estimates using either method. Multiple real izations of the process being examined are recommended to produce an accurate representation of causality within a system. Perhaps one of the most challenging problems remaining with this technique in living systems is detecting the presence of unobserved units, where apparent causal relationships are due to unobserved elements. This possibility must be considered when drawing conclusions about complex systems that may contain multiple unobserved processes.

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63 DGC was also investigated for its potential to estimate synaptic weights. The model intentionally represents a simplified case of neur al connectivity where the independent stochastic processes that drive the neurons, along with the spike firing rates, are based on the behavior of one type of cortical neuron. Changing these st ochastic processes and neuron types in the model may alter the sigmoid relationship between synap tic weight and GC. Different model parameters or data from neural culture, where neuron t ype and synaptic weights may be unknown, will require calibration in order to recover valid sy naptic weights. However, in this study we demonstrate that there is a linear region within the sigmoid relationship between synaptic weight and GC, where changes in synaptic efficacy can be observed. The 100-neuron simulation demonstrates that as the complexity of the network grows the PGC results underestimate the synaptic weights. This may be due to th e contributions of the other 95 neurons on activity within the causal ch ain. While the sigmoid relationship may not always apply, the existence of a region where synaptic weight changes linearly with PGC suggests that DGC values from that system are co mparable (i.e. a larger Granger Causality value translates to a larger synaptic weight). This gives an investigator c onfidence that changes observed during an experimental manipulation ar e also comparable in magnitude. This linear relationship allows for pre-experi mental and post-experimental valu es of DGC to be subtracted in order to quantify plasticity. It is possible that these observations can be quantified with proper calibration between synaptic weight and DGC values. Conversely, we have also demonstrated that there are regions in the relationship where changes in synaptic weight will not be reflected as changes in causality. Specifically, when the post synaptic potential is signifi cantly sub threshold or when it is already so large that the post synaptic potential will almost always cause a pos t synaptic action potential. Changes in synaptic

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64 weight within these regions may not be reflecte d in changes PGC values calculated between the observed neurons. However, it should be noted that changes in synaptic we ight in these regions will not produce and observable changes in the spiking output of the neuron either. Perhaps a more fundamental problem with a ny Granger based method is the ability to detect the presence of inhibitory, rather than excitatory, contributions Inhibitory synapses, unlike excitatory synapses, do not produce a post-s ynaptic action potential. Thus, an inhibitory synapse will produce GC values near zero, simila r to that of null connectivity. Additionally, inhibitory influences decrease act ivity in their targets and may distort serial causal influences, even though they may be strong. Including step s to untangle inhibitory influences, perhaps through conditional logic, may help address this problem. However, GC remains one of the most powerful techniques to estimate causal relations or untangle complex structural topologies in networks where that information is unknown.

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65 CHAPTER 4 ASSESMENT OF PLASTICITY IN-VITRO USING GRANGER CAUSALITY Introduction Microelectrode array (MEA) technology has been used to quantify and record multichannel neural activity from in-vitro dissociated neural cultures for nearly three decades (85-87). However, the underlying st ructure and connectivity that is responsible for producing the neural activity observed from these cultures has remained largely unknown. Subsequently, difficulties arise when trying to accurately relate changes in observed neural activity to physiological changes within the network such as plasticity. Detection of plasticity across a network is now a common theme in recent neuros cience research (20, 67, 81, 88-92). Spike timing dependent plasticity (STDP), a process wh ere the efficacy of a synapse is governed by the timing of preand postsynaptic action potential s, is thought to be a potential mechanism by which in-vitro plasticity occurs ( 83, 84, 88, 93). However, neur al activity recorded using MEAs is extracellular, whereas STDP experiments of ten use intercellular voltage clamp methods on isolated neuron pairs. Ad hoc algorithms and m easures are often developed to quantify possible plasticity related changes in-vitro with varying levels of success (4, 18, 20, 81, 89). However, these algorithms often do not provide a statistically sound measure of eff ective connec tivity that can be related to the actual connectivity underlying the dissociated neural network. Jimbo et al has demonstrated that changes in the synaptic strength of pathways within living neural networks could be identified by describing changes in elicited spike rates after induction by high frequency stimul ation or tetanization (4). Specifically, Jimbos data suggested that tetanization resulted in pathway specific po tentiation and depression. The metric used to identify those changes is based on changes in the spike rates reco rded on each electrode before and after a tetanic pulse train is delivered, a si mple and elegant way to observe output related

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66 changes in neural connectivity. However, observations that can be made using spike rate are limited providing only a course quantification of neural conn ectivity. First, spike rate observations are non-directional and they are also non-causal. Second, this method relies on using a difference between pre and post treatme nt measurements of spike rate to detect underlying pathways, limiting the ability to obser ve connectivity prior to a treatment which modifies the underlying strength. Several methods (17) including correlation (4, 21, 94, 95), likelihood methods (36-38), partial directed cohe rence (30, 50, 96), and Granger Causality (48, 49, 76) can address some of these deficiencies and have been used to identify connectivity patterns between spiking neur ons in neural simulation, in-vitro and in-vivo. Pairwise Granger Causality (PGC) (1), a robus t statistical forecasting tool borrowed from economics, is unique among these measures because it can address all of th ese issues and it can quantify the degree of causal connectivity between neurons (76). PGC makes use of autoregressive modeling (66) to determine if th e temporal activity of a pre-synaptic neuron can be used to predict the activity of a possible po st-synaptic neuron. Additionally, neural network simulations from Chapter 3 suggest that PGC can be used to quantify synaptic connectivity from simulated spontaneous neural activity. This chap ter will explore the use of PGC as a method for detecting pathway specific changes in plasticity in-vitro. PGC will be validated against the pathway information derived using Jimbos method of spike rate changes pre and post tetanus. PGC should allow for a deeper analysis of plasti city related effects from Jimbos protocol and new insights will be presented after PGC ha s been compared against this method. What do PGC values mean in neural networks? Connectivity in terms of neural networks is usually expressed as a synaptic weight or synaptic efficacy that can be rela ted directly to the am ount of voltage or current that is induced from the pre-synaptic to the post-synaptic neuron However, it is not intuitively clear what the

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67 values generated by PGC mean in terms of syna ptic coupling. A comput er simulation, using Izhikevitchs simple neuron model (3), was used in Chapter 3 to determ ine the effect of known synaptic weights on causality estimates using PGC. These values were calculated from pre and post-synaptic spike trains in or der to better demonstrate the rela tionship between PGC values and synaptic efficacy. Within this simulation the synaptic weight between two regular spiking cortical neurons were systematically increased an d the resulting spike trains from the simulation were analyzed using PGC. The results indica ted a sigmoid relationship between PGC and the actual synaptic weight in which PGC is insensitive to variations with very low and very high synaptic strengths. The was, however, a linear region (cf. Figure 3-4) in the middle of this sigmoid indicating a region where PGC values are li nearly related to synaptic efficacy (and also can be subject to linear arithmetical operators such as subtraction). Based on this relationship, the PGC values recovered from the cortical cultures on the MEA will likely represent the synaptic coupling between these two channels but may be distorted if those wei ghts are excessively low or high. Of course this value may not represen t a specific synaptic weight as the connections between neurons in culture may consist of thous ands of synapses (97) and may be mediated through several pathways (21). It should provide an overall estimate of the synaptic weights coupling two neurons. Hence, this method could be used to detect changes in synaptic weights among the pathways coupling neur ons indicative of plasticity. Methods Dissociated Cortical Cultures Dissociated cortical neural cultur es consisted of cortical hemis pheres (Brain Bits Inc.) from E18 Wistar rat embryos that were digested us ing the Papain Dissociation System (Worthington Biochemical Corporation) and mechanically triturated separating the neurons soma from the surrounding connective tissue. After dissociation, approximately 50,000 cells were placed in the

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68 center of a MEA that has been firs t coated with polyethyleneimine ( 96) (98) and then also treated with laminin (Sigma) (99). The neurons were cultured in medium consisting of 90 % Dulbeccos modified Eagles medium (DMEM) (Gibco ca t# 10569-010) and 10% Equi ne Serum (HyClone cat# SH30074.03). The medium was equilibrated in an incubator to 5% carbon dioxide and 35.5C temperature before use. The medium in each culture was exchanged once per week. Freshly dissociated neurons rapidly begin to reestablish a network when placed on the MEA, forming synapses to neighboring cells and produ cing spontaneous action potentials within the first few days (100). During the first two weeks the neurons mature and begin to express the full complement of receptors, a process completed after one month in vitro (101-104). In this experiment, data from six cultures were analyzed. A ll cultures were older than 30 days in vitro in this experiment. Microelectrode Arrays The MEAs that were used for this experiment were obtained from Multichannel Systems GmBH. The 60 electrodes of each array are arranged in an 8 x 8 grid, spaced 200 m apart, to measure neural activity over a 1.6 mm2 area. These arrays consist of a glass wafer over which a gold surface is deposited and the electrode traces are etched using photolithography. An insulating layer of silicon nitr ide (SiN) is then deposited via vapor deposition over the surface providing an electrical barrier between the gold and culture media. 30 m diameter holes are then plasma etched over the circ ular pads and Titanium nitrid e (TiN) is plated onto the gold forming a low impedance interface between the r ecording amplifiers and culture media. Each electrode measures the voltage changes produced by neural soma, dendrite, and axons within an approximate radius of 60 m as they fire action potentials (1 05). Conversely, application of biphasic current or voltage pulses (e.g., 100 to 800 mV or 50 to 200 A) will elicit action

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69 potentials from neurons near the s timulated channel. Neural activity was sampled and digitized at 25 kHz using Multichannel Systems MEA1060 amp lifier and A/D hardware. Action potentials were distinguished from noise using a voltage th reshold 5 standard deviations above or below estimated noise levels. Spike events were record ed into a file containing the time, channel, height, width, and context information surrounding each event. This spike file was later processed offline to extract information about an y bursts that may have occurred. Bursts were detected using a modification of the leaky integr ator algorithm described by Gross (91) which is applied to the spike times for each electrode. The environment both within and surrounding the MEA during recordings was temp erature and gas compositionally controlled at 35.5C and 5% CO2 duplicating the environment within the incu bator in which these cells were maintained. Media temperature within the array was c ontrolled using Multichannel Systems TC02 temperature controller. Stimulation Induced Pl asticity Protocol The study of plasticity is a field where much remains unexplored. However, it is one of the most actively investigated areas of brain resear ch. Two distinct and closely related phenomena, long term potentiation (LTP) (106) and long term depression (LTD), have come to be accepted as possible mechanisms by which synaptic effi cacy is increased and decreased respectively. While the timing rules thought to underlie LTD and LTP between individual neurons are easily investigated by monitoring the magnitude of pos t synaptic potentials (PSP) delivered to postsynaptic neurons using patch cl amp, single neuron electrophysiol ogical methods are problematic when trying to determine how th ese mechanisms function at the network level. Neural cultures maintained on MEAs present an opportunity to observe these phenomena on 60 electrodes however MEAs only provide extracellular voltage information from these cultures. Thus, while

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70 action potentials can be detected from the neurons grown on MEAs, subtle fluctuations like PSPs are indistinguishable from background noise. Jimbo et al devised a clever experiment (4 ) to overcome this limitation to determine the efficacy of a tetanus to induce plasticity change s after its delivery to a single electrode in a dissociated cortical network gr own on a micro-electrode array. The protocol consists of a baseline probe sequence, treatment induction te tanus, and a post-treatment probe sequence. Jimbo identifies neural pathways by indicat ing where LTP and LTD may have occurred by determining the change in spike rate between th e probe sequences. The in duction tetanus consists of trains of 11 biphasic 600mV pul ses delivered at 20 Hz repeated 20 times at 5s intervals. The probe sequences deliver biphasic sequential pr obes of 600mV to each of the 60 channels (including the ground channel) in a randomized order that is repe ated 10 times for a total of 600 probes. Each of these probes evokes a burst of ac tivity resulting from activation of the cortical neural network across all 60 channels. The number of spikes generated fro m the probe within a 200 ms time window on each electrode quantifies this activity. The baseline and post-treatment spike rate over this 200 ms window for 60 ch annels for each probe are then compared by difference plot (as shown in the left panel of Figu re 4-1) to determine the change in spike rate between baseline and post-treatment probes. The red horizontal lines show instances where the protocol enhanced the spike rate resulting from probing on that channel and the blue lines show instances where the effect of the protocol resu lted in depression of the spike rate for probes delivered to that channel. The use of PGC on these same spike trains should allow for more detailed description of changes in the effective connectiv ity induced by the tetanus. The protocol yields spike trains for all 60 electrodes following each of the 1200 total probes (60 channels probed x 10 probes per

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71 channel x 2 probe series pre and post tetanus) that are delivered to the neural culture. The ten probes per channel can be considered to be re alizations of the same process, thus when calculating PGC between the two channels the 10 source spike (pre-s ynaptic) trains and 10 response (post-synaptic) spike trains are pooled. PGC can also be directionally calculated from each channel to every other channel for each probe yielding a 60x60 matrix of directional in fluences for each probe, 60 times more information than that provided by spike rate. Incidentally, directionality can be eliminated by combining the two directional PGCs between a pair of channels to yield a measure called total causality, similar to coherence, that is more appropriate for validation of PGC against established non-directional spike rate methods. When PGC is calculated for each of the probed channels, this yields a 60x60x60 matrix of PGC results for each probe sequence before and after tetanus. For simplification these PGC matrices generated from elicit ed data sets will be referred to as elicited Granger Causality (EGC) matrices. When the posttetanus EGC matrix is subtracted from the pre-tetanus EGC matrix, this will be referred to as the EGC difference matrix. The axes of this EGC difference matrix are stimulus (probed) cha nnel, source channel, and response channel and each entry in the matrix contains the change in PGC from the first set of probes to the second set of probes. Results Spike Rate versus PGC Jimbos experimental procedure was replicated in our lab1 on six cortical MEA cultures. Data from neural culture 6686 is used to present results that were consis tent across all of the subjects. The timing of spikes was collected from the raw waveforms and then cleaned of noise 1 The data for this experiment was collected by Ping He as a part of a larger experiment to determine long-term changes in plasticity in the DeMarse laboratory.

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72 spikes by template matching. These remaining spikes were then used to create the rate based stimulus-response plot shown in th e left panel of Figure 4-1. This figure plots the changes in the number of spikes post-tetanization by color (blu e = decrease, green = no change, red = increase) for each response channel (x-axis) for every probed channel (y-axis). The plot suggests Figure 4-1 Comparison of spike rate to Pairwise Granger Causality. Th e left plot shows the change in elicited spike rate post tetanus at each channel (x-a xis) for each probed channel (y-axis). Each pixel on the plot re presents the change, post-tetanization, in elicited spike rate at the re sponse channel from a stimulation delivered to the probed channel. The pathways stimulated by the single channel probes on a particular channel tend to uniformly increase (red horizontal lines) or uniformly decrease (blue horizontal lines). The blue vertical line at response channel 52 is a result of drift during the experiment a noisy channel. This channel would normally be removed from analysis. The right plot is an analysis of the same data using PGC. This plot shows the change in PGC post-tetanization for each response channel (x-axis) for each stimulated probe channel (y-axis). The change in PGC for each pixel can be interpreted as the change in the effective connectivity of the response channel to all other channels. PGC similarly detects pa thway specific plasticity, but does so by looking at changes in effective connectivity between channels rather than changes in spike rate. Notice that the analysis of channe l 52 yields no effect (green represents no change in PGC) as the noisy spikes from channel 52 have no effective relationship with other channels. Additionally, PGC offers greater contrast of post-tetanus effects. The enhanced contrast can especially be seen when comparing response channels 1-9 between both plots.

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73 simultaneous enhancement and depression of spike rates similar to the da ta previously reported by Jimbo et al. (4). PGC was then calculated for each of the probe sequences and reported as a stimulus-response difference-plot shown in the ri ght panel of Figure 4-1. This is essentially interpreted the same as the left panel except that the colors represent a change in PGC instead of spike rate. Note that the 3-dimensional EGC matr ix is separately collapsed across the source and response axes and the results ar e summed to create the 60x60 plot shown in Figure 4-1. This is done to reduce the complexity of the EGC matrix to allow direct comparison to the spike rate results. A simple visual comparison between the le ft and right panels in Figure 4-1 shows that, in general, any changes in rate are also reflected in changes in causality. Of the 60 probe channels, the results of more than 95% of probe channels match direction of change when comparing spike rate and PGC stimulus-response plots. However, there are some noticeable differences between the two plots PGC is influenced by changes in rate but only when the rate increase on the pre-synaptic channel is complemented by an increase on the postsynaptic activity. The data from dish 6686 was specifically chosen to help reinforce this notion due to a unique event that occurred. Most noticeable, the spike train data from channel 53 (the dark blue vertical stripe) in the left panel of Fi gure 4-1 was contaminated with background noise and generated false spikes th at were similar enough to real spikes that they could not be filtered out. During the recording following tetanization the source of the noise was removed resulting in a large decrease in spike rate for this channel. When PGC is calculated for the noisy uncorrelated random spikes from this channel PGC yields values near zero, as one would predict for temporally unrelated time se ries. Thus, PGC reports the changes in the temporal relationship between channels rather th an a simple change in rate. This is further reinforced by the data from channels 1-9 that show little change in spike rate across all probe

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74 channels in the left panel of Figure 4-1 but s how changes in PGC (right panel of Figure 4-1). Additionally, some channels show stronger changes in PGC than those shown by spike rate as shown in examples depicted in Figure 4-2a for LTP and Figure 4-2b for LTD. This is not surprising considering that change s in synaptic efficacy can be reflected by changes in the pre and postsynaptic spike timings that may appear in changes in spike rate. Figure 4-2 A comparison of spike rate and PGC analysis for enhanced channel 26 and depressed channel 41. This figure highlights the incr eased contrast provided by PGC over spike rate by comparing spike rate (top panels) or PGC analysis (bottom panels), sorted by initial strength (x-axis), before and after tetanization (blue square vs. green circles, respectively). Panel A shows potentiation in terms of spike rate (top) and PGC (bottom) for channel 26 from Figure 4-1. Panel B shows the same relationship for the depressed channel 41. In both cases PGC provides greater contrast between baseline and post treatment measurements for both pot entiation and depression. The increased contrast provided by PGC makes it easier to establish the statis tical significance of pathway specific changes. Surrogate Analysis A subtle but important difference between the tw o methods is that the right panel of Figure 4-1 has considerably higher contrast between effected channels and surrounding unaffected channels i.e. less regions that are light yellow or light blue. This s uggests that the signal to noise

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75 Figure 4-3 Surrogate analysis of the PGC analys is. Two surrogate analyses were performed on the data from the Jimbo protocol. The firs t, shown in the left plot, is a result of randomly shuffling the spike timings with in the 200ms post stimulation window for both the pre and post-tetanus pr ior to PGC analysis. This effectively destroys the spike timing relationships while maintaining the spike rate over this window. The resulting PGC analysis yields near zero PGC results for both pre and post-tetanus probe sets. The resulting change in PGC pos t-tetanus shown in the left figure is near zero (hence a uniform green field) for all pa thways. This spike-shuffled surrogate analysis also establishes a significance th reshold that can be used to distinguish actually changes shown in Figure 4-1 from noise Note that if the spike shuffled data were used to generate a spike rate plot, it would be identical to the left panel in Figure 4-1. The second surrogate analysis, shown in the right panel, shuffles the pre and post-tetanus spike trains. The pathway sp ecific patterns observed in Figure 4-1 are obliterated. This suggests that the plastic ity induction protocol is responsible for producing these pathway specific patterns and that these patte rns are not randomly generated. ratio for PGC is higher, allowing for a greate r contrast between effected channels and background noise. Additionally, the use of PGC also has the added benefit of having a statistical significance level. A surrogate data set can be created from the original spike trains that maintains the spike rate information in the spike train but scrambles the interspike intervals by randomizing the spike timings for a given spike tr ain. The null hypothesis for this first surrogate is that changes in spike rate are responsibl e for the corresponding changes in PGC. A well established principle of PGC is that when the two time series being regressed are made up of

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76 independent white noise PGC=0. In practice with real spike trai ns a surrogate data set with randomized spike timings will produce a value that approaches zero establishing a threshold by which the significance of the unscrambled can be de termined from. GC valu es falling above this threshold can be considered significant and nonrandom. Once the spike timings were scrambled the same causality analysis was performed on the surrogate spike trains. The resulting analysis, shown in the left panel of Figure 4-3, confirms th at the patterns demonstrat ed by the original data set are significant considering that the significanc e levels generated from the surrogate data set are all in the green (i.e. close to zero change) region of the color s cale. This also invalidates the null hypothesis suggesting that the PGC analysis is not simply de pendent on rate and is likely dependent on changes in inters pike timing between channels. A second surrogate analysis was created by randomly swapping out pre-tetanus spike trains with post tetanus spike trains from the same stimulus and source/response channels. This surrogate, while not restricted to use with only PGC analysis, serves the purpose of determining whether the pathway specific patterns resulti ng from the Jimbo protocol are randomly produced. The null hypothesis for this surrogate is that fail ed probes are responsible for the patterns seen in rate and PGC analysis. Thus, swapping probes between post and pr e-tetanus analysis would test this hypothesis. PGC analysis is then performed on this surrogate data se t with results shown in the right panel of Figure 4-3. Notice that th e pathway specific patterns inherent to the nonsurrogate data set have been obliterated. This surrogate an alysis randomized the pre/posttetanization temporal order of th e spike train realizations. If the pathway specific plasticity effects were randomly generated from failed probes, the effects should have persisted in the surrogate analysis. Subsequently, these results sugge st that the second surrogate null hypothesis is invalid.

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77 Discussion The original hypothesis for this experiment was that PGC could be used to describe pathway specific plasticity effects in dissociat ed cortical culture. Comparison of PGC results against spike rate validated this hypothesis. Over 95% of the pa thways described by spike rate were found to agree with the PGC analysis. Ofte n, those channels that differed from the spike rate analysis were channels that showed no spike rate change but showed a change in PGC. This was likely due to fact that PGC specifically mode ls changes in the fine temporal dependencies between spike trains rather than relying on cour se changes in rate. While spike rate has the advantage of being simple and easy to obtain, use of PGC compliments spike rate as a more statistically rigorous and perhap s more sensitive metric. PGC was also shown to be more resilient to the introduction of noise spikes into the analyzed spike trains as demonstrated by the noisy channel 52 in Figure 4-1. PGC was also shown to provide a highe r contrast between pre and post-tetanus analysis as shown in Figure 42. Statistical significance was more easily established using a spike-shuffled surrogate analys is. This surrogate analysis also suggests that pathway specific PGC plasticity results are not simply spike rate dependent but reflect true changes in causal strength between the channels. However, these are not the only benefits to using PGC. For comparative purposes the SGC matrix was intentionally collapsed and the benefit of directionality was removed in order to better validate and compare the results of PGC against the nondirectional spike rate method. The following discussion explores these directional effects provi ding a deeper analysis including amplitude dependent plasticity, comparisons of spontaneous and elicit ed PGC analysis, and directional plasticity that can be described by PGC.

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78 Amplitude Dependent Plasticity Directionality is a fundamental property of PG C that distinguishes it from other methods such as correlation and coherence. Directional ity has been intentiona lly removed from the analysis so far to enable be tter comparison to nondirectional spike rate based methodology. However, directionality is a key component of STDP timing rules for plasticity. Directional information about plasticity changes should allow a more complete understanding of how induced plasticity affects the underlying structur e of the network. When the EGC difference matrix is calculated without using total causali ty several new options emerge for visualizing plasticity. One use of the EGC difference matrix is that it can be used to determine how the initial magnitude of a pathway affects magnitude and direction of change of that pathway after tetanization. The EGC difference matrix co ntains 60x60x60 entries w hose values can be compared to the same number of pre-tetanus EGC matrix values. According to amplitude dependent plasticity (ADP) shown by Bi & Poo, changes in synaptic weight due to correlated time dependa nt spiking are a function of initial synaptic strength. While potentia tion of excitatory postsynaptic pote ntials (EPSPs) is larger for weak synapses, those experiencing depression are redu ced by a constant percentage (84). EPSPs are commonly measured postsynaptically using single electrode voltage clamp techniques. In this analysis, we investigate ADP among the pathwa ys observed during Jimbo s protocol. However, unlike in Bi & Poos experiments, MEA electro des are extracellular a nd cannot measure EPSPs. Thus, the starting effective connect ivity is compared to the change in effective connectivity using pairwise Granger Causality (PGC) calculated from prior analysis. If the observed changes in plasticity by PGC are accurate then one would expect that the magnitude and direction of changes within the network would agree with previously reported ADP results (84).

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79 Figure 4-4 Amplitude dependent plasticity (ADP) can be explored in-vitro using PGC. According to amplitude dependent plasticity (ADP), changes in synaptic weight due to correlated time dependant spiking are a function of initial synaptic strength. This plot shows initial magnitude of PGC prior to induction (x-axis) and the corresponding directional change magnitude after induction (y-axis). This demonstrates that the tetanic induction generally increases weak connections a nd decreases strong connection. This plot resembles resu lts reported by Bi & Poo 1998 where postsynaptic potentials were observed using in tercellular recordings. Data shown is a compilation from six cortical cultures exposed to tetanization. The entire pre-tetanization EGC matrix wa s used as starting weights and the EGC difference matrix was used to repo rt changes in those weights. Th ese results were plotted Figure 4-4 where the initial PGC amplitude for all pathwa ys was plotted on the x-axis and the % change in the PGC amplitude was plotted on the y-axis. This plot contains data from 6 MEA cortical cultures where Jimbos protocol was used to indu ce plasticity. Overall Figure 4-4 suggests that weak pathways are most often enhanced while strong pathways are weakened. At around initial PGC values of 0.25 depression begins to be favor ed over potentiation as the initial strength of the

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80 pathway increases. These results suggest that changes in connectivity post tetanus may be amplitude dependent which is consistent with the results from Bi & Poo. Directional Causal Influences The EGC difference matrix is large which can make detection of network changes difficult to visualize. Thus, when trying to understand what is happening in the EGC difference matrix it is preferable to collapse the matrix along one of its axes for simplification. However, when Figure 4-5 Stimulus-Response difference plot of PGC where the EGC difference matrix is collapsed by source. While similar to the left panel of Figure 4-1, this plot makes use of the directional properties of PGC. Essentially each pixel represents the change in causal information coming in to each response channel. This plot shows the directional change in PGC post-tetanization for each response cha nnel (x-axis) for it corresponding stimulated probe channel (y-axi s). Red pixels indicate a potentiation of inward causal flow while blue pixels indicate depressed inward causal flow. The change in PGC for each pixel can be inte rpreted as the change in the effective connectivity of the response ch annel to all other channels.

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81 directionality is involved, which axis is co llapsed has a profound impact on what information that figure will present. The available axes to collapse upon are source, response, and by probe channel. When collapsing the EGC difference ma trix by source (as was done earlier to match spike rate plots), each pixel is a summation of all of the causal influences from all source channels into each individual response channel fo r a given probe. This is shown in the resulting stimulus-response PGC plot shown in Figure 4-5. Thus, collapsing the EGC difference Matrix by source reveals the information flow coming into every channel for each probe and produces a Figure 4-6 StimulusSource difference plot of PGC where the EGC difference matrix is collapsed by response. This plot makes us e of the directional properties of PGC, essentially each pixel represents th e change in causal information coming out of each source channel. This plot shows the direc tional change in PGC post-tetanization for each source channel (x-axis) for it correspond ing stimulated probe channel (y-axis). Red pixels indicate a potentiation of outward causal flow wh ile blue pixels indicate depressed outward causal flow. The change in PGC for each pixel can be interpreted as the change in the effectiv e connectivity of the source ch annel to all other channels.

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82 stimulus-response plot. When collapsing across th e response axis, as shown in Figure 4-6, the stimulus-source plots pixels represent the summation of information flow from each individual source channel to all re sponse channels. Thus, collapsing th e GC Matrix by response channel reveals the information flow coming from every channel for each probe resulting in a stimulussource plot. Comparison of Figures 4-5 and 4-6 suggests that while after tetanization a pathway may experience an increase consistent with LTP, a subset of the response channels experience an increase in total information outflow while othe rs experience an increase in total information inflow. The same complementary observation can be seen with depressed probe channels that are thought to experience LTD. Logically, this is consistent with the id ea that an increase/decrease in information outflow pre-synaptically should be matched by a corresponding increase/decrease in information inflow post-synaptically. This reinfor ces the notion that the tetanus elicited changes in the neural network consistent with STDP mechanisms. When collapsing across the remaining probe cha nnel axis, as shown in Figure 4-7 the sourceresponse plots pixels represent the summation of information fl ows from a source channel to a response channel for every stimulating probe. Thus, collapsing the EGC difference matrix by stimulus channel reveals the di rectional information flow for each source-response pair for all probes. Collapsing on this axis produces a source-response plot that is more consistent with what could be observed in a record ing of spontaneous activity fr om a MEA culture. The 60 sourceresponse slices of the EGC difference matrix of size 60x60 are also shown in Figure 4-8 at their respective probe locations in a spatial arrange ment meant to replicate the placement of the electrodes in the MEA culture dish. These plots specifically speak to th e larger amount of information available when using PGC methods instead of spike rate. Using the spike rate method an investigator can obser ve the change in spike rate at 60 channels following a probe.

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83 Figure 4-7 Source-Response difference plot of PGC collapsed by probe before and after application of tetanus. This plot show s the directional change in PGC posttetanization for each source channel (x-axis) for it corresponding stimulated probe channel (y-axis). Each pixel shows the to tal information flow from each source to each response channel over all probes. This plot is a sum of all of the source response plots that are plotted later in Figure 4-8. This represents the total change in causality from elicited activity over all probes for every pathway. However, when using PGC, 3600 possible directional pathways can be observed for each probe. This is largely due to the fact that rate based changes are non-directional and non-causal. So when a probe is delivered only the changes in sp ike rate at distant channels can be observed. When using PGC spike trains from any channel can be compared to any other channel. One of Jimbos observations was that the spike rates at distant electrodes for a specific probe channel tend to change uniformly in one direc tion either depressi on or potentiation.

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84 Figure 4-8 Source-response plots by spatial probe location. These source-response plots (each interpreted the same as the source response plot in Figure 4-7 but for single probes rather than all probes) are arranged t opographically by the lo cation at which the probes were delivered to crea te the respective source respons e plots. This plot is a difference plot showing the entirety of the 60 x 60 x 60 PGC difference matrix for one MEA culture treated with Jimbos protoc ol. Notice that while some channel were dominated by potentiation (re d) and others dominated by depression (blue), most channels simultaneously demonstrate both potentiation and depression. Interestingly, the plots in Fi gure 4-8 tell a slightly more complex story. While there are some probes that produce changes in activity that fit this profile, in fact most probes show both depression and potentiation within the network, information that is lost when averaging across source or response channel axes. While potentiation or depression may win out after collapsing to create a source-response plot, both mechanis ms are most often present before collapsing. However, PGC produces results that agree with some of Jimbos major points. While probing at

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85 a channel activates the entire network, the path ways and synapses activated depend on where the probe was delivered. This explains how a pathway can produce potentiation for one probe channel and in turn show depression when a different probe channel activates those same neurons. Thus, a similar conclusion as Jimbos can be drawn from these source-response plots that the observed changes are pathwa y-specific and not neuron specific. Spontaneous/Elicited Connection PGC can also be applied to spike trains from spontaneous rather than evoked activity. This presents an interesting opportunity to compare source-response plots from the EGC difference matrix to source-response plots from spontaneous activity. Recordings of spontaneous activity were taken before and after tetanization during this experiment. Spike trains were collected from population bursting events inherent to spontaneous activity in ME A cortical cultures, as opposed to collection from a time window after stimulation as with elicited bursts. The spontaneous burst events were detected using a burst detection algorithm based on several found in literature. A window of 200ms is used to captu re the activity from the burst. Unlike data collection from an elicited time series the number of events is based on the spontaneous bursting activity of the culture rather than based on a predetermined nu mber of probes. The spike trains are low pass filtered and used to calculate PGC values for every source and response pair for both pre and post tetanus spontaneous recordi ngs. The resulting difference ma trix, or spontaneous Granger Causality (SGC) difference matrix, is 60x60 and in cludes PGC values for every source to every response channel. The resulting plot, shown in Figur e 4-9, is most similar to the source-response plot generated from the GC matrix collapsed across probe shown in Figure 4-7. Theoretically these two plots should resemble each other as they are both source-response plots. The connectivity in the spontaneous source-respon se plot is a result of the expression of spontaneously activated pathwa ys during population burst events The pathways activated are

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86 dependent on the patterns of the bursts expre ssed by the culture, which is dependent on many factors possibly including spatial origin, conne ctivity, and environmenta l factors. While the patterns are similar between elicited and spont aneous activity and are likely linked there are many reasons why they should not co mpletely agree. Spontaneous activ ity is a direct result of the probabilistic processe s that drive single unit activity in th e network. Which bursts patterns occur and those that do not are dependent on this process. In contrast, el icited bursting is not dominated by this process. Each electrode is systematically probed and responses are observed. It is likely Figure 4-9 Spontaneous Source-Re sponse plot calculated from wi ndowed bursting events during spontaneous (no stimulation probes) activity This source (y-axis) response (x-axis) plot represents the difference in PGC calcu lated from spontaneous bursts before and after the delivery of the tetanus. Again the colors represent the change in causality (blue = decrease, green = no change, and red = increase). This source response plot should resemble the source response plot af ter collapsing by probe shown in figure 47 if PGC analysis of spontaneous and elicited data is similar. Some similarities are seen between these plots suggesting a connection between stimulated and spontaneous neural activity.

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87 that many of the pathways activated by probes are part of forced burst patterns that may be rarely if ever generated from spontaneous activity a nd are thus missing from spontaneous connectivity plots. Additionally, the e licited patterns are generated from 600 elicited burst events while the spontaneous activity produces perhaps 20-30 burst events. Despit e these differences, there is similarities between the patterns from spontaneo us and elicited source-response plots reinforcing the linkage between spontaneous and elicited activity patterns. Conclusions The inverse source problem has proven a form idable obstacle for neuroscientists when exploring network level plastici ty mechanisms in vitro. While synaptic efficacy can be monitored using glass pipette elec trodes to detect and measure post-synaptic potentials, largescale use of these electrodes to monitor network activity is problematic. MEAs are a wellestablished means of obtaining multichannel recordings from living neural networks but do so extra-cellularly. To estimate plasticity using thes e arrays Jimbo et. al. devised a protocol that reveals some pathway information for disso ciated culture grown on an MEA by observing changes in spike rate after the simultaneous indu ction of LTP and LTD. While spike rate is an elegantly simple means of detecti ng pathways using this protocol, it is that same simplicity that limits the amount of information that can be ga rnered. We have proposed the use of PGC as a means of extracting synaptic efficacy from the spatiotemporal activity generated from elicited and spontaneous activity. The lin ear relationship between PGC valu es and synaptic weight was demonstrated in a simple biologically relevant neural model. The model demonstrates a linear relationship in the region beginning where the sy naptic efficacy is large enough to cause some post-synaptic action potentials and ending where the synaptic weight is so large as to cause postsynaptic action potentials every time the pre-synaptic neuron fires. It is only within this region that PGC values change significantly in relation to synaptic weight sugges ting that experimental

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88 changes in PGC likely scale linearly to synaptic efficacy. Thus, there are circumstances, when changes in synaptic efficacy are not reflected in neural spiking activity changes, and will not be reflected in changes in PGC. However, based on this modeling it is reasonable to assume that changes in PGC values are likely reflective of changes in synaptic efficacy. The applicability of PGC to plasticity studi es in-vitro was verified against established experimental plasticity protocols with an esta blished analysis method, spike rate change. Comparison of PGC results to sp ike rate results shows that PGC is not only sensitive enough to be used to detect plasticity but also provi des information on the structural connectivity underlying recorded neural ac tivity prior to treatment. While the two methods did not agree on all path ways, this can be easily explained by the fact that PGC is sensitive to changes in inter-spike timing associated with LTP and LTD that changes in spike rate may not detect. Conversel y, some changes in spike rate may not be reflected in a change in causal ity for similar reasons. Additionally, the creation of surrogate data sets by shuffling the spike timings within indivi dual realizations demonstrate that PGC possesses a sound statistical foundation that allows discrimination between signal and noise events. Directionality was intentionally removed to ma ke the comparison to spike rate, however when included it enables several new analysis methods to be used in conjunc tion with plasticity experiments. With baseline connectivity available, it is possible to track changes in connectivity after treatment in the context of these beginning values. This analysis yields results similar to previously reported empirical observations of STDP in-vitro where whole cell patch-clamp methods were used to monitor synaptic efficacy directly (84). Additionall y, this plot suggests that synaptic efficacies may be being driven towards equilibrium, illustrated by the transition from potentiation to depression around values of 0.25 for PGC. No t only does this show that the

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89 connectivity changes within the network are unlikel y to be random, it also shows that there may be some sort of homeostatic mechanis m (107) at work within the network. Changes in information outflow from a ch annel can be visualized when the EGC difference matrix is collapsed across the source ax es, making use of the directional property of PGC. Conversely, changes in information inflow can be visualized when the EGC difference matrix is collapsed along the response axis. Co mparison of these two plots for individual probe channels that experience LTP (or LTD) certain groups of neurons experience and increase (decrease for LTD) in information outflow while other distinct groups specifically experience a complimentary increase (decrease for LTD) of inform ation inflow. This result suggests that the observed changes in connectivity are specific and di rectional rather than uniform across the dish. A uniform increase in connectivity could be in dicative of fluctuations in environmental conditions such as temperature rather than a resu lt of STDP. Directionality also allows collapsing the probe axis to create source-response plots for each channel probed, rather than just being able to observe distant changes in spike rate in relation to the probe channel. These plots show that, while one STDP mechanism may dominate for a probed channel, often LTP and LTD both occur among the various pathways activated. Also, these plots reaffirm the concept that changes in plasticity are pathway specific rather than neuron specific. This can be seen when a pathway demonstrates potentiation in one source-respons e plot but shows depression when the location probed changes. This suggests th at, even though the same neurons are involved, the pathway is different. Collapsing across these source-response plots by probe also has a logical connection to PGC source-response plots generated from spont aneous activity. This connection may prove useful in the design of future experiments in th at it may not be necessary to perform a probe sequence in order to observe changes in neural network structure. It is clear that PGC can

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90 describe changes in network connectivity and not only compliments the use of spike rate but also exceeds it in terms of the amount of informati on provided about the pathways underlying neural activity. However, there are limitations of PGC that must be considered. A causal relationship between two channels may be due to a common unobserved driving influence. Additionally, it is possible that PG C values for a pathway will contain mediated influences in addition to direct causal influences. These mediated influences would distort the relationship between PGC values and synaptic wei ght. However, even with mediated influences included, the changes observed are indicative of changes in the underlying structure of the network for that pathway, not necessarily di rectly between two individual neurons. The complexity of the synaptic connectivity with in any given pathway prohibits a complete description of all the synaptic weights within that pathway, given the finite fixed electrode placement of MEAs. However, PGC makes the most of the data available by enabling a deeper analysis of these pathways, than previously implemented methods. Results from simulated pathways suggest that PGC can relate changes in synaptic weights within a pathway to changes in the spiking output of postsynaptic targets. This suggests that changes in PGC observed in pathways in-vitro represent change s in synaptic weights somewhere within these pathways. This conclusion is further reinfor ced by in-vitro PGC results that suggest that amplitude dependent plasticity occurs within pathways. This phenomenon has been well described at the cellular level but never at the network level. Th ese results suggest that PGC can describe changes within in-vitro pathways that are known to occur at synapses between individual neurons. Additionally, directional analysis of the EGC difference matrix suggests that some areas experience specific changes in output while others experience corresponding specific changes in input. This further sugge sts that pathway specifi c changes are real and

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91 directional rather than random a nd undirected byproducts of natural variation. While much work needs to be done to fully capture individual changes be tween neurons, these re sults suggest that PGC is a useful tool for uncovering effective relationships in dissoci ated in-vitro cortical neural networks grown on microelectrode arrays.

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92 CHAPTER 5 DIFFERENTIAL TETANAZATION OF SINKS AND SOURCES IN DISSOCIATED CORTICAL CULTURE Introduction Granger Causality was used in the previous ch apter to identify changes network within invitro cortical networks grown on microelectrode arrays. Grange r Causality was also shown to provide a better representation of the effective connectivity between channels before and after the application of a tetanus. Moreover, unlike sp ike rate based methods, Granger Causality could measure connectivity without the necessity of tetanization (i.e., it does not require a difference between pre and post measurements). In this chapter, information about effective connectivity will be used to predict a priori changes in plasticity prior to experimental manipulation. The central hypothesis of this chapter is that if information a bout connectivity can be known before hand and it is accurate, then this information can be used to predict the direction in which plasticity will change when tetanizi ng a specific channel on the array. Granger Causality provides information about wh ich channels are causality related to other channels and the direction of that relationship. In essence, the experimenter knows who is connected to whom and how they are connected (i.e., direction). Between single neurons where a presynaptic neuron is synaptically connected to a post synaptic neuron, it has been shown that stimulating the presynaptic neuron followed ~1 to 100 ms later by a stimulation of the postsynaptic neuron will increase the efficacy of that synapse. Conversely, stimulating the post synaptic neuron before the pres ynaptic neuron results in the de pression of that synapse (a decrease in efficacy) (84, 108). Th is is known as spike timing depe ndant plasticity (STDP) (83, 109) where the timing of activity or stimulation determines the strength of the synapses in a network. If information about the structural connectivity between neurons recorded with our MEAS were known then it should be possible to select a causally connected pair of neurons and

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93 stimulate that pair to either potentiate or de press the synapses between them dependant on the timing of the stimulation (pre vs post or post vs pre). However, each channel on the MEA records from several neurons simultaneously. In addition, a single stimulation pulse (e.g., a si ngle biphasic 600 mV 200 us pulse) may activate neural activity in a wide area surrounding the stimulation electrode, but this is of course dependant on the voltage that is applied. Instead of attempting to identify a particular pair of neurons perhaps identifying channels whose primar y relationship to others was to drive activity in those channels could be a solution (i.e., an analog of a pre-post synaptic relationship). Stimulating that driving channe l would essentially be similar to a presynaptic-stimulation /postsynaptic-activation. In other words, spike timing dependant pot entiation of that synapse and perhaps result in an increase in causal streng th between that channel and others. Conversely, stimulating a channel, whose primary relationship w ith others is as a target (i.e., a sink where other channels converge to drive activity on this channel), should result in the depression of that pathway. In other words, a postsynaptic neuron stimulated before its presynaptic counterparts. Of course, identification of causal sinks and s ources within the cortical culture would be critical for the execution of this stimulation para digm. For this study, a sink is defined as a node (channel) within the culture wher e the balance of causal informati on flow is inward to the node. In contrast, a source is defined as a node where the balance of causal information flow is outward from the node. When PGC is calculated between two channels A and B, PGC analysis yields results from A to B as well as from B to A. Sinks and sources could then be identified when PGC analysis is carried out amongst all channel pa rings. Summing PGC values for all pathways leading out of a channel, Fout, represents total information fl ow coming out of that channel. Summing PGC values for all pathways coming into the same channel, Fin, represents total

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94 information flow coming in to that channe l. Positive values of the difference, Fout Fin, would indicate a source within the ne ural network (i.e. dominated by information outflow) while negative values would indicate a si nk (i.e. dominated by information inflow). See Seth et al for a similar identification technique that has been tested in artificial networks (55). In this chapter, an experiment is described that tests whether differential tetanization of sink channels versus source channels produces differing outcomes. Activity from in-vitro cortical networks will be measured to identify the location of potential sinks and sources among the channels on the array. A source channel will then be tetanized and changes in plasticity will be measured. The prediction is th at the causal flow out of this source channel should increase. A sink channel will then be tetanized and the prediction is that causal flow into this sink channel should decrease. Background Identification of Neur al Sinks and Sources Identification of sources and sinks of neural activity in multichannel data has proven useful in many theoretical and experime ntal neuroscience paradigms. Fo r example, identification of sinks, sources, and the corresponding current fl ow within hippocampal slice is useful for mapping neural circuits within th e hippocampus (110, 111). Causality in the form of the Directed Transfer Function has been used in epilepsy to detect the source of seizure generation also known as the seizure focus (112) from EEG. This is done so that the seizure focus can be surgically removed, effectively ending seizures in a patient. Essentially, the seizure focus is located by finding a source of causa tion within the epileptic brain. This method has also been used with ECoG signals to determine the ictal source (113, 114). Ding et al. refined the use of

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95 the Directed Transfer Function so as to be used with EEG rather than ECoG because the use of ECoG electrodes is invasive requiring placement under the dura. Plasticity by Tetanization Plasticity has been studied in-vitro using MEAs in several published experimental protocols (4, 18, 21, 115). Most of ten plasticity is induced at an active channel (where activity has been recorded) using a tetanic stimulation. The frequency, amplitude, and duration of these tetanic stimulations have varied between inve stigators. For example, Jimbo (4) applied a tetanization sequence of 100ms (20 trains of 10 bipolar s 0.6V pulses at 20Hz every 5 seconds) while Shahaf (18) employed a constant trai n at 1, 0.5, or 0.3Hz (each pulse is biphasic 50 A 500 s) until the desired plasticity was achieved. Tateno et al (21) applied tetanic pulse trains (similar to those used by Jimbo) to two channels simultaneously in the same culture. Where the tetanus is delivered in the culture varies between investigators as well. Often the tetanus is simply delivered at a highly activ e channel that elicits a network burst response when stimulated. Interestingly, it has been shown that channels w hose activity is strongly correlated with the channel during tetanus are most often potentiated while those that are weakly correlated are most often depressed (4). This sort of reasoning is merely an observation however and requires the experiment to be conducted firs t before the direction can be determined. In contrast, we are able to determine what the effective connectivity is for a channel prior to tetanization based merely on spontaneous neural activity recorded before any stimuli are delivered. The correlated changes described by Ji mbo et al are however consistent with our notion that when these channels are tetanized (the y would appear as sources in our analysis) the activity produced by that channel should be enhanced (i.e., increase).

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96 Experimental Design The experiment used 10 mature (all subjects were more than 35 days old) cortical neural networks grown on MEAs. Each subject was exposed to 2 tetanic pulse trains (20 0.2Hz trains Figure 5-1 Topographical map of sinks and source s spatially laid out and superimposed on a picture of the microelectrode array. Sinks are blue while sources are red (green indicates neutral) a ll calculated from spontaneous bursting activity using Pairwise Granger Causality. Row 4, Column 4 repres ents a strong sink and Row 7, Column 5 represents a strong source. Each of thes e would be candidates for tetanization to differentially induce depression or potentiation, respectively.. of 10 bipolar s 0.5V pulses at 20Hz) 1 per day delivered to either a sink or source. Sinks and sources will be identified immediately prio r to tetanization from baseline recordings of spontaneous activity for 10 minutes. We will then identify burst events in these spike trains using an algorithm that uses a th resholded global spike rate to dete ct the onset of each burst. The end of the burst is detected using the quiescent time after the burst where no spiking occurs (116). A 200ms window of spiking activity is then extracted following the onset of each burst.

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97 These spike trains windows were then low pass filtered at 50 Hz so that Pairwise Granger Causality can be used to determine causal relatio nships between all combinations of channels. Since each burst is treated as a realization of th e same process over 100 realizations can typically be used to calculate PGC from the 10-minute recordings. The total inflow of causation into an electrode is calculated by summing the PGC values that include the specified electrode as the destinati on of causation. Similarly the total outflow of causality is calculated by summing the PGC values that include the specified electrodes as the source of causation. The difference between to tal outflow and total inflow will be used to determine whether a particular electrode is a sink (negative values of this difference) or a source (positive values of this difference). This value will be calculated for every electrode in the MEA and displayed spatially to allow selection of the electrode for tetanization. An example of a map of sinks and sources within a co rtical culture is shown in Fi gure 5-1. This plot shows the difference of total inflow and total outflow repr esented by color (red = source, green = neutral, and blue = sink) for each electrode overlaid over an image of a microelectrode array. This type of arrangement allows for easy spatial identification of where sinks and sources are in the cortical culture. The electrode selected for tetanization will be limited to electrodes that are not on the edge of the culture to better allow observation of effects from tetanization. Each dish will be tetanized once per day on either a sink or a source depending on the requirements of the experimental design lattice. The first day of this protocol will consist of 5 randomly selected cultures tetanized on sinks and 5 cultures tetanized on sources. The second day will reverse the tetanization of sinks and sources for each cult ure. Thus, over the course of this two day experiment each dish will be te tanized on a sink and a source.

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98 Spontaneous vs. Elicited Activity as a Measure of Plasticity Spontaneous and elicited burs ting are both based on the stru cture of the same underlying network and logically reflect si milar patterns of conn ectivity. However, spontaneous bursting activity is initiated wi th firing probabilities a nd patterns that are gene rated dynamically within the neural network. Elicited bursting patterns, while dependent on the sa me underlying structure, is evoked artificially and can generate patterns that spontaneous activity may not (also a possible reason the test probe sequences can induce plasticity). Changes in spontaneous bursting activity revealed using correlation matrix based methods have been used to iden tify plasticity related changes post tetanus (20). In this experime nt, the difference between pre-tetanization connectivity estimated by PGC from spontaneous baseline activity prior to tetanization will be subtracted from PGC estimates pos t-tetanization to determine the changes in connectivity that result from tetanization protocol. This experiment will use two primary means of de tecting plasticity after tetanus. Both of these methods were developed in chapter 4. Firs t, PGC will be used on spontaneous activity (pre and post tetanization) to create a source response difference-plot to determine whether pathway specific potentiation and depression occurs as a re sult of the tetanization. Essentially this will mean looking for potentiated and depressed sour ce and response pathways. Blue (depressed pathway) or red (potentiated pathway) horizontal and vertical lines in source response plots would be an indicator that plasticity may have occurred. Additionally, an amplitude dependent plasticity effect (ADP) (84) was also found from the data analysis in chapter 4. Bi and Poo found that the post synaptic potentials (PSPs) (the magnitude of the voltage will be referred to as weight) in the post-synaptic neuron in a simp le neuron pair exposed to negative spike timing (post-synaptic neuron fires before pre-synaptic neuron) would elicit depression. These depressed pathways would decrease by a constant percentage independent of startin g weight. However,

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99 when exposed to a positive spike timing (pre be fore post) potentiation would occur that was dependent on the starting w eight. Specifically, smaller PSP were increased by a larger percentage than larger ones. In chapter 4, this relationship was found in pathways analyzed using PGC from elicited spiking activity, rather than measing PSPs directly. This analysis is made possible by PGC providing a measure of effective connectivity in the pre and post Figure 5-2 Source response plot for culture 9059 teta nized on a sink. This plot shows the change in PGC post-tetanization. The y-axis is the source electrode while the x-axis is the response electrode. Changes range from de pression shown in blue to potentiation shown in red. Green indicated no change. Notice that channel 40 experienced a increase in causal output (red horizontal stripe at source electrode 40) and simultaneously experienced a decrease in causal input (blue vertical stripe at response electrode 40). tetanization results. Thus, it is possible to us e the pre-tetanization PGC analysis as starting weights for the observed pathways and use the resulting post-tetanization PGC analysis to

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100 calculate a change in these weights. This ADP analysis will also be used to determine whether plasticity occurs during this experiment. ADP results will also be statistically analyzed to determine the direction of any change in plasticity is consistent with out predictions. Results and Discussion Results The experiment was run over the course of two days as describe d earlier. During the experiment it was easy to notice that cultures gr own on certain MEAs showed a larger amount of causality between electrodes than those that were likely experiencing failures in the insulation of the electrode tracing. Cracks in the insulation ca use leakage of current from the electrode tracing (117). These cracks can decrea se the signal to noise ratio of the electrodes making it increasingly difficult to distinguish actual spiking events from noise events. Based on this observation, 2 of the cultures were removed from the statisti cal analysis (dish 9299 and 9058). The are included in the tables, however. Several of the MEAs, where insulation damage was likely, were not removed because the effect was not as severe. PGC was calculated for before and after the tetanization for all subjects in the experiment. The measures mentioned earlier we used to first determine whether plasticity occu rred during the course of this experiment. First, a source-response difference plot was constructed for each trial to qualitatively determine what type of plasticity effects were se en post-tetanization. An example of these plots are seen in Figure 5-2 for a culture tetanized on a sink and Figure 5-3 for a culture tetanized on a source, demonstrating that bot h potentiation (indi cated by red) and depression (indicated by blue) occurred when tetanizing on sinks and s ources. Notice that there are horizontal lines representing changes in the output of a channel and vertical lin es representing changes in the input to a channel. These results are consiste nt with plasticity shown by Jimbo, showing

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101 Figure 5-3 Source-response plot for culture 9449, which was tetanized on a source. This plot shows the change in PGC post-tetanization. The y-axis is the source electrode while the x-axis is the response electrode. Cha nges range from depression shown in blue to potentiation shown in red. Green indi cated no change. Some channels were depressed while others were potentiated. simultaneous potentiation and depression for pathways in the culture. Second, the PGC analysis was examined to look for any ADP trends. Plastici ty effects similar to those reported in Chapter 4 were found. This analysis will be elaborated on in the ADP analysis section. However, both of these key indicators suggest that plasticity consistent with the experiment in Chapter 4 occurred. The remaining analysis will determin e if these effects were consistent with the predictions made earlier.

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102 Table 5-1 Global culture causality changes post-tetnanization. The change in causality was assessed for sinks and sources by summing tota l causality for all of the pathways in the culture pre and post tetanus and taking the difference between these values. No consistent differential is seen between sinks and sources. Greyed out values were not used in statistical analysis. Culture Change in total causality after Tetanization at a Sink Change in total causality after Tetanization at a source 9447 42.22 118.86 9449 -229.07 -97.04 9448 -89.39 -90.88 9450 -22.82 -155.91 9299 12.16 14.77 9291 51.9 -17.26 9272 16.2 62.67 9059 9.34 11.25 9058 21.95 3.06 9053 -13.85 -39.65 Mean -29.43 -25.99 Qualitative inspection of the stimulus response plots for each subject shows no observable difference between tetanization of a si nk versus a source. A va riant of this metric used was quantitatively comparing the difference in a sum of all causality for all pathways pre and post-tetanization. This was done for the en tire array as a global measure and also for the tetanized channels alone. A statistical analysis of the resulting difference in causality finds that there is no difference between tetanization on a sink versus a source (F(7,7)=0.001, the raw results are shown in Table 5-1). Cultures expe rience both global increases and global decreases for tetanization both on sinks and so urces. This result is not unexpected since changes described by Jimbo involve correlations with the tetanized channel. When looking at the tetanized channels alone there is a noticeable change in the magnitude of the causality for the pathways leading out of sources and into sinks. Statistical analysis shows that wh ile the direction of change (potentiation versus depression) is not consistent across all su bjects sinks or sources, the magnitude of the changes are st atistically significant (F(7,7)=6. 40 for sinks and F(7,7) = 3.23

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103 Table 5-2 Tetanized channel total causality chan ges post-tetanization. Th e change in causality was assessed for sinks and sources by summi ng total causality for only the tetanized pathway pre and post tetanus and taking th e difference between these values. No consistent differential is seen between sinks and sources. Grayed out values were not included in statistical analysis. Culture Tetanized Sink Change in Outward Pathways Tetanized Sink Change in Inward Pathways Tetanized Source Change in Outward Pathways Tetanized Source Change in Inward Pathways 9447 1.84 10.44 -0.78 -1.24 9449 -1.78 -10.12 -15.44 0.42 9448 -0.44 -8.18 -3.87 -0.09 9450 -0.04 -11.01 -1.38 -0.92 9299 0.12 -1.68 0.10 0.60 9291 -0.07 -1.56 1.04 -0.59 9272 -0.29 -.012 0.58 -0.01 9059 -0.78 0.80 1.72 0.22 9058 1.88 0.12 1.63 -0.06 9053 -3.16 -4.12 -3.91 0.38 Mean -0.59 -2.97 -2.75 -0.22 Figure 5-4 Culture wide Amplitude Dependent plasticity. Each pathways initial causality is plotted on the x-axis while its corresponding change after tetanization is the y-axis. This is a compilation of the effects seen for all 10 cultures. The effect for sinks is seen in panel A while the effect for sources is seen in panel B. There is no observable or statistical difference between these tw o plots and no differen tial effect between sinks and sources.

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104 sources) for pathways leading out of sources and into sinks when compared to the magnitude of changes for pathways in the opposite direction. ADP is the final method used to determin e if plasticity related effects followed the primary hypothesis. This was not only used to co mpare sinks versus source s, but to also get an idea of whether a pathways magnitude could be used to predict changes post tetanization. Again, the results are compared globally across the entire dish and on the tetanized channels alone. Global ADP is plotted for all pathways in Figure 5-4 panel A) for all sinks and panel B) for all Figure 5-5 Tetanized channel amplitude dependent plasticity. The initial causality of pathways involving the tetanized channe ls is plotted on the x-ax is while its corresponding change after tetanization is the y-axis. Panel A are the pathways coming out of the tetanized sinks. Panel B are the pathways coming into tetanized sinks. Panel C are the pathways coming out of the tetanized sources. Panel D ar e the pathways coming into tetanized sources. There is no observable or statistical difference between these two plots showing no differential eff ect between sinks and sources.

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105 sources. Visual inspection of thes e graphs shows them to be very similar. A linear model for ADP (initial causality versus change in causali ty) was calculated for each of these graphs to allow for a quantitative comparison. Both models are standard linear models of the form y=mx+b. The slope, m for the sink plot is -0.20 while the slope for the source plot is -0.14. These slopes suggest that larger we ights in both cases typically are de pressed. This is true for all models in this study. It is worth noting that the intercep ts of all models in this study are small and positive, suggesting a small region where small we ights are typically potentiated. It is of immediate interest that the norm of the residu als is high for both models at 13.0 and 13.8 respectively. The variability in higher magnitude starting causalities is likely the source of the poor fit. Essentially, these models cannot be considered different, thus there was no global differential effect. This finding is in line with previous global observation s. The same metric and methods are then applied to only the tetanized channels segregated by sink and source as well as by pathways coming into and pathways comi ng out of that channel, shown in Figure 5-5. The parameters for the models fit to the plots are shown in Table 5-2. These plots and models fit to them may explain why large significant devi ations were found on channels coming out of sinks and into sources from earlier sink source an alysis. On first inspection the slope of tetanized sink pathways average -0.23 while teta nized source pathways average -0.11, suggesting that sink pathways may be depressed more. However, the norm or the residuals for pathways coming into sinks and pathways out of sources remain large at 3.22 and 3.94 respectively, suggesting high variability especially in the higher starting caus ality region. The norms of the residuals pathways coming out of sinks and in to sources is much better at 1.06 and 0.70 respectively. However, its easy to notice that the goodness of f it for these pathways is likely improved because of the virtual lack of strong initia l causalities for these pathways. This

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106 suggests that the statistical effect shown in the earlier sink source analysis may be a result of the way sinks and sources were chosen rather than any differential effects due to tetanization. Overall, this suggests that no significant differential effects were detected. Conclusions This experiment did produce plasticity similar to the results reported earlier in Chapter 4. This can be seen in the comparison of the ADP trends from this experiment and the replicate of the Jimbo experiment. These results suggest that the tetanic stimulations in this experiment produced similar ADP related changes seen in Chapter 4. The cultures also showed simultaneous potentiation and depression another ha llmark of the Jimbo protocol. This result suggests that changes that occurre d post tetanization are not simply dependent on global shifts in spike rate. These effects were seen in Figure 5-2 and 5-3 for both sinks and sources. This suggests that it does not matter wh ether spontaneous or elicited act ivity is used to reveal the effects of tetanization using PGC. The predicted direction of that changes base d on stimulating a sink versus a source were not supported. The prediction made in this experi ment was that a differential effect would be seen when sink and sources, identified using connectivity derived from spontaneous activity, were tetanized. It is possible that the effective conn ectivity revealed from spontaneous activity is not a good predictor of what path ways will correlate with a pa thway when it is tetanized. However, this only explains why an ADP effect ma y not be seen. This does not account for the predictions made based on STDP rules. The likel y reason why these predictions did not work is because STDP rules are based on changes betwee n a single pre synaptic neuron and a single postsynaptic neuron. Predictions in such a simple network would be easy to predict. The cortical culture used in these experiments is created from 50,000 dissociated embryonic neurons that randomly connect to form a neural network. The actual connectivity in th ese cultures is

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107 much more complex than the controlled conditions in the experiments by Bi and Poo (118). The MEA can simultaneously observe the activity of nearly 100 neurons. These neurons may have thousands of synapses with other neurons in th e culture (97) and many of these connections may be bidirectional. Additionally, it is difficult in this complex network to control what pathways experience positive and negative spike timings. Reverberating connections and feedback loops may present many spike timings to the same syna pse. Additionally, the effects of the spike timings from spontaneous bursting during and after the tetanus may have an effect. Perhaps the STDP based predictions, which were based on re sults from two isolated mono-directionally coupled neurons, are too simplistic given the complexity of the c onnectivity in these dissociated cultures. This experiment was devised to be a proof of concept for being able to measure the effective connectivity in the dish and then be able to ma nipulate that connectivity based on this knowledge. While the predictions for this experiment may have been too simplistic for this system, future studies could benefit from th e use of PGC to reveal connectivity before experimental manipulation. Future studies may also benefit from a reduction of the complexity of the cortical culture by reduci ng the density of the culture. A nother possible strategy may be to use a paired pulse paradigm to better control the spike timings in the targeted pathways.

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108 CHAPTER 6 ANALYSIS OF TEMPORAL LOBE SEIZURES USING GRANGER CAUSALITY Introduction Prior research in this body of work have focused on using Granger Causality metrics to reveal effective connectivity relationships in neural simulations and in in-vitro dissociated cortical culture. These experiments provided ne w insights into the conn ectivity underlying these networks and new metrics of plasticity. Howeve r, one ultimate goal for this research is to provide an analytical technique to better understand on how the intact brain functions. Data from the Evolution into Epilepsy Project at the Univer sity of Florida presents a unique opportunity to explore the in-vivo dynamics and ef fective connectivity relationships of seizures from an animal model of temporal lobe epilepsy (TLE). High quality 32 channel data from microwires was continuously recorded in vivo from rats at a hi gh sampling rate (25 KHz per channel). The data from these animals include multiple seizure events from several animals. The 32 electrodes were bilaterally located in the hippocampus and afforded the opportunity to look at 992 n n k! ; n=32 channels, k=2 for bivariate model different directional hippocampal interactions on multiple time s cales using Granger Causality. In this chapter, the first goal is to first determine if Granger analysis can provide anatomically relevant descriptions of effective connectivity in that anatomical structure. In contrast to the dissociated cortical cultures used in the previous two experiments, the anatomy and the physiology of the hippocampus is well establ ished (119). However, descriptions of the effective connectivity of the hippocampus from im planted microwire arrays are rare in general and nonexistent for seizure behavior. The second goal is to provide a be tter description of the dynamics and causal relationships with and betw een hemispheres underlying seizure activity and

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109 how they differ from those observed during baseline activity. This chapter will also provide a brief description of TLE, animal models of TLE, and the Evolution into Epilepsy Project. Temporal Lobe Epilepsy Epilepsy is a disorder that effects roughly 2% of the population. The most common type of epilepsy among adult patients is Temporal L obe Epilepsy (TLE) (120). The symptoms of TLE are well defined and classi fied by the International Leag ue Against Epilepsy (ILAE). Simple Partial Seizures involve small areas of the temporal lobe and typically do not affect consciousness. They are often characterized by au ras or periods where sensations such as sounds or smells are felt that are not actually presen t. Complex partial seizures can affect consciousness and the ability to interact with ot hers. Often, the symptoms include staring, automatic movements of the hands and mouth, and other abnormal behaviors. Seizures associated with TLE can also spread to the rest of the brain causing Secondarily Generalized Tonic-Clonic seizures. The tonic phase of this type of seizure is characterized by stiffening of the limbs and body in either a flexed or extended position. The following clonic phase is characterized by uncontrolled jerk ing of the limbs and body. Electrophysiologically, TLE is characterized by seizure activity that can be recorded from th e hippocampus. Physiologically, TLE is also characterized by hippocampal sclerosis consisting of pyramidal cell loss, granule cell dispersion, and gliosis within the hippocampus. Also, patients with TLE can often be cured of seizure activity (80-90% success) by the rem oval of the hippocampus, suggesting that the hippocampus implicating it as a pot ential source of TLE seizures. There are several animal models used to study TLE. These include models where TLE is induced pharmacologically and also those that use electrical stimulation. Of those that use stimulation some require stimulation to elicit a seizure while others invol ve spontaneous seizure generation. For example, kindling is produced by repeated subthres hold focal electrical

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110 stimulation of the brain. After these stimula tions have been delivered for sufficient time the targeted area reaches a kindled state wher e a single stimulation will produce a secondary generalized seizure. However, these seizures re main stimulus locked requiring a stimulus pulse to generate seizure events. In contrast, Loth man (121) has developed an animal model with spontaneous seizures in which the rat is stimulated once with a high frequenc y superthreshold stimulation designed to immediat ely induce status epilepticus. The resulting limbic seizures damage the hippocampus and result in recurrent s pontaneous seizures that develop and increase in severity over time. This model is used by researchers at the University of Florida in the Evolution into Epilepsy (EIE) Project (40) to study the electrophysiolo gical and physiological changes that occur between the ti me of stimulation and the expr ession of spontaneous seizures. Animal Model of Chronic Limb Epilepsy The TLE animal model used in the Evolution in to Epilepsy Project has been in use for the last two decades (121). In the weeks that follow this stimulation, the rat progressively demonstrates recurrent spontaneous hippocampal seizures. The temporal and electrographic progression of rats treated with this model closely resembles that of human patients that have chronic mesial temporal seizures. The surgic al and electrophysiological techniques used by the Evolution Into Epilepsy (EIE) proj ect team at the University of Florida are described below. The rats were prepared according to proce dures approved by the University of Florida IACUC. The rats are adult male Sprague-Daly rats that are 50 days old. The surgical procedure consists of a craniotomy from 1.7 mm lateral to 3.5mm lateral of the bregma so that electrodes could be implanted into areas due to their prominent role in ep ileptic discharges (122). A 2x8 rectangular array of electrodes wa s implanted bilaterally in both hemispheres so that they cover the CA1/CA2 area to the DG. The electrode plac ement is shown in Figure 6-1 with the surgery shown in the left panel of Figure 6-2. The spacing between electrodes is 400 m on the short

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111 Figure 6-1 Diagram of microelectrode placement in the rats brain. The upper left is a representation of the skull of a rat showing the bregma and lamda locations. The cartoon in the main caption shows the orient ation of the electrode strips and the corresponding location of indi vidual electrodes. (Used with permission from Dr. Paul Carney).

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112 Figure 6-2 Placement of the electrodes during surger y. This image shows a single 16 electrode array being lowered into position into the ra ts hippocampus. The electrodes vary in depth so that they are lowered to a dept h where targeted stru ctures lie in the hippocampus. The location of the electrode tips was later confirmed using MRI after the rat has been sacrificed.(Used with permission of Dr. Paul Carney). axis and 200 m on the long axis at a depth of 4 mm from the surface of the brain. The length for each electrode is designed such that the tip of each electrode is with in the brain region of interest. The location of the end of each array cl osest to the centerline of the brain is 4 mm caudal to the bregma and 1.7 mm lateral. A si ngle Teflon coated bipol ar twist electrode was placed in the posterior ventral hippocampus in the right hemisphere of the animal, posterior to the electrode array for stimulation into status epilepticus. After the experiment was completed the rats were sacrificed and high-resolution magnetic resonance imag ing (33) was used to verify the position of the electrode s, as shown in the lower right panel of Figure 6-2.

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113 Figure 6-3 Example of the rat environment and r ecording chamber. These cages are designed so that the electrode cables can be constantly attached to the rats head while also allowing the rat to freely e xplore its environment. The rats were kept in an 30 cm enclosure de signed to permit the electrodes to be connected to the data acquisition system while the rats enga ged in exploratory activit y. This enclosure is shown in Figure 6-3. The rats were allowed to he al for one-week post surgery before stimulation. A 10 second train of 50Hz 1 ms bipolar 240 A pulses was delivered every 12 seconds for 60 minutes to induce status epilepticus in the rats. 32 channels of simultaneous recordings were taken from the rats 24 hours a day for 76 days while the rat was engaged in sleep and normal exploratory behavior. Data wa s acquired using a Tucker-Davis (Alachua, FL) Pentusa at a sampling frequency of 24.4141 kHz, band pass filtered fr om 0.5 Hz to 12kHz, and stored to disk.

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114 Figure 6-4 Initial Pairwise Gra nger Causality analysis of four 60-second seizures for a single animal. These plots are used to display th e causal interactions between all electrode combinations. The plots are organized in te rms of the source electrode (y-axis) and the response electrode (x-axis) for the Gra nger analysis. The magnitude of the causal interaction between source and response elec trodes is represented by the color scale in the left of the figure, with blue being zero and red being strong interaction. This plot clearly shows two regions once consisting of the first sixteen electrodes (bottom left quadrant) and the other consisting of the last sixteen electrodes (top right quadrant). These areas correspond to the el ectrode arrays in each hemisphere. The average within-hemisphere interactions are strong while the hemisphere to hemisphere (top left and bottom right quadran ts) interactions app ear weak across the four seizures. Seizure Analysis with Pairwise Granger Causality Seizures detection was conducted manually by visual detection of the raw traces and verification against video records. The seizures were catalogued and the st art time of the seizure was noted. A 10-minute window surrounding each seiz ure was then extracted for all 32 channels from the raw recording and low pass filtered to 1kHz. This frequency was chosen because it maintains most of the features in the data while allowing for a reas onable model order for autoregressive modeling required for parametr ic PGC. A 60 second window was then chosen within the 10-minute period of filtered data that encompassed the seizure.

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115 Spectral Pairwise Granger Causality was then ca lculated for all possible electrode pairings (992 directional combinations) for the entire 60-second window for 4 seizures to see if PGC was able to describe causal interactions for epileptic seizures. The stationarity of the time series was addressed by normalizing the mean and variance. The Bayesian information criterion (BIC) was used to determine the correct model order, which was found to be m =25. The resulting spectral analysis was then integrated across the spectral domain to produce a single time domain value for each electrode pairing for each window resu lting in a 32x32 matrix. Granger analysis was then plotted as 32x32 image, shown in Figure 6-4. These plots are organized so that each pixel represents the causal influence of a driving channel (y-axis) and a driven channel (x-axis). The magnitude of this causal relationship is repr esented by color with blue being zero ranging through light blue, green, yellow, an d red with red being the stronge st interactions. This plot clearly distinguishes two distinct regions, a lower left and a top right quadrant, that have strong within region causality but weak cross-hemisphere interactions. Based on promising results from this initial analysis, a higher resolution analysis was conducted where PGC was calculate d over a series of 1 second moving windows for each seizure independently. The overlap between these 1 second moving windows could be modified to create a real time movie of the in teractions. A 50% overlap was used for each seizure resulting in 129 data windows for PGC. This data for each window was then plotted as 32x32 image and combined into a movie at 2 frames per second. This allows the dynamics of 32 interacting channels for each seizure to be better visualized for qualitative analysis. Calculation of average PGC values by the eight electrodes in each area (L-DG, R-DG, LCA1, R-CA1) allowed creation of an additional PGC index to condense the spatial and temporal interactions quantitatively. Each seizure was visually analyzed to identify and window specific

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116 stages. A single value for the interaction between each indexed area was calculated that represents the average for that stage and was i ndependent of the length of the stage. This provides 4x4 matrix for each stage for each seiz ure. Analysis of variance was calculated between different directional in teractions over the pool of 15 se izures from the three animals used in this analysis. Additionally, baseline in terictal activity was an alyzed using this method for comparison. Finally, a higher time resolution movie was made for one of the seizures. This movie is 30 frames per second and includes behavioral video, a trace of the record ed waveforms, and the corresponding PGC analysis. The same one-sec ond data window was used to calculate PGC interaction, however this an alysis window was moved by 1/30th of a second for each corresponding video frame. Frames from this movie are displayed in the following figures to better describe the seizure stages that will be identified and described in the results. Results Qualitative visual analysis of the raw wave form, behavioral videos, and PGC movies reveal a common progression of even ts during all 15 seizures into five major stages. Figures 6-4 through 6-9 highlight examples of interictal activity 60 seconds pr ior to seizure and also from each of these stages. These plots are frames from the high-resolution movie compiled for one animal. Each figure includes a snapshot of surv eillance footage of the rat in the bottom left corner, four of 32 electrode voltage traces (one from each major region L-DG, R-DG, L-CA1, RCA1) on the right, and a 32 x 32 Granger Causa lity analysis in the uppe r left quadrant. The electrode traces are norma lized against each other and are time locked to the surveillance video. A one second window highlighted be tween the red bars in on each voltage trace represents the current analysis window whose results are depicted at the top left.

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117 Each Granger analysis plot represents the in teractions by electrodes in the source area (yaxis) versus electrodes in the destination area (x-axis) for each electrode combination. The magnitude of the Granger Causality analysis for each pairing is represented by color with blue being near zero ranging to red representing the st rongest interactions. For example, in Figure 6-5 the bright red area in the lower left are interac tions from electrodes in the L-CA1 (y-axis) to electrodes in the L-CA1 (x-axis). This example w ould be interpreted as interactions within the L-CA1 area during the pre-ictal period roughly 30 seconds before seizure. The seizures begin in stage 1, shown in Figure 6-6, in which a mixture of intrahemisphere activity was dominated by causal activity within the CA1 region and from the CA1 region to the DG. This activity gradually builds up over ti me. Behaviorally, Stage 1 often includes the beginning of the tonic phase of the seizure, wher e the animal remains motionless and blinks its eyes. As this causal activity beco mes stronger, it begins to spread across hemispheres at the beginning of stage 2. Stage 2 is the most striking, marked by high-magnitude mono-directional causal reverberations across hemispheres as well as substantially increased intra-hemisphere causal interactions. These transfers of ten reverberate sequentially from one hemisphere to the other. An example of a directional transf er between hemispheres is shown in Figure 6-7 from the left to the right hemisphere. After severa l directional transfers the seizure moves into stage 3, shown in Figure 6-8, marked by mainly intra-hemisphere activity. However, unlike the intra-hemisphere waveforms of stage 1, the intra-hemisphere Gra nger causality interaction patterns are both more dynamic, of larger magnitude, and less confined to a single area or pathway. Only week occasional cross hemisphere directed transfers occur in stage 3.

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118 Figure 6-5 Example of data and Granger analysis duri ng the interictal (pre-ictal) stage. This plot is of one frame approximately 30 seconds prior to seizure. The right panel includes sample raw data traces from each of the f our main areas recorded from including the right and left DG and the left and right CA 1. The segment of data analyzed is between the vertical red lines in the sample electrode traces. The top left panel is a source (y-axis) response (x-axis) repres entation of the PGC analysis with the electrodes organized by area. An interictal spike is included in the analysis for this frame. Most of this activity is within CA 1 and is also directed from the CA1 to the DG. The bottom left panel is a frame from surveillance footage of the rats behavior. In this frame the rat is either resting or sleeping.

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119 Figure 6-6 Beginning of the seizure in the high re solution movie classified as Stage 1. Intrahemisphere activity is the primary interaction unique to this seizure stage. The right panel includes a sample raw data trace from each of the main areas recorded from including the right and left DG and the left and right CA1. The segment of data analyzed is between the verti cal red lines in the sample electrode traces. The top left is a source (y-axis) response (x-axis) repr esentation of the PGC analysis with the electrodes organized by area. The PGC analys is shows directional activity within the left CA1 as well as a directional influence of the left CA1 on the left DG. The bottom left panel is surveillance footage of the rat s behavior. In this frame the rat has just woken up and may be exhi biting tonic behavior.

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120 Figure 6-7 Example of a directiona l transfer during Stage 2. The right figure includes a sample raw data trace from each of the main areas recorded from including the right and left DG and the left and right CA1. The segment of data analyzed is between the vertical red lines in the sample electrode traces. Th e top left is a source (y-axis) response (xaxis) representation of the PGC analysis w ith the electrodes organized by area. This frame shows an example of a directional transf er from the left hemisphere to the right hemisphere. In following frames this transf er is reciprocated from right to left. The bottom left panel is surveillance footage of the rats behavior. In this frame the rat is exhibiting tonic behavior where the rat is aw ake and blinking its eyes but not moving.

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121 Figure 6-8 Example from Stage 3 Late intra-he misphere. The right figure includes a sample raw data trace from each of the main areas recorded from including the right and left DG and the left and right CA1. The segment of data analyzed is between the vertical red lines in the sample electrode traces. The top left is a source (y-axis) versus response (x-axis) representation of the PGC analysis with the electrodes organized by area. Most of the causality is isolated within hemisphere with occasional weak cross hemisphere interactions. The bottom left pa nel is surveillance footage of the rats behavior. In this frame, the rat is still exhibiting tonic behavior.

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122 Figure 6-9 Stage 4 Transition. The right figure includes a sample raw data trace from each of the main areas recorded from including the right and left DG and the left and right CA1. The segment of data analyzed is be tween the vertical red lines in the sample electrode traces. The top left is a source (y-axis) respons e (x-axis) representation of the PGC analysis with the electrodes organize d by area. The causality here seems to be bidirectional for nearly all electrodes. The bottom left panel is surveillance footage of the rats behavior. In this fram e, the rat is beginning to transition from tonic to clonic behavior.

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123 Figure 6-10 Stage 5 Post transition. The right fi gure includes a sample raw data trace from each of the main areas recorded from including th e right and left DG and the left and right CA1. The segment of data analyzed is be tween the vertical red lines in the sample electrode traces. The top left is a source (y-axis) respons e (x-axis) representation of the PGC analysis with the electrodes organized by area. The dominant driving pattern is from the DG to the CA1 in this st age. The bottom left panel is surveillance footage of the rats behavior. In this frame the rat is showing clonic behavior including rearing and rhythmic shaking of the forelimbs.

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124 Stage 4 is usually a brief and highly caus al epoch where all areas intra and cross hemisphere are bi-directionally causal, shown in Figure 6-9. In the PG C movies, this stage consists of near saturation of causal effect with nearly th e entire 32x32 frame shows strong causal interactions. Behaviorally, stage 4 marks the transition from the tonic phase where the rat is motionless and a ppears tense to the clonic phase of the seizure where the rats appendages jerk ra pidly and rhythmically. Immediately after stage 4, the pattern smoothly transitions into stage 5, which is made up of heavy spiking activity with a causality pattern similar to that observed during interictal activity. Quantitative analysis between anatomical regions in hippocampus was conducted by creating an index from multiple electrodes common to that region, as described earlier. The 32 electrodes split between hemispheres were divide d into those in the DG and those within the CA1/2 region. This results in four areas to co mpare using quantitative analysis. This index was calculated for each seizure at each Stage of th e seizure (1-5) for all possible interactions including a scaled within area index. This al lowed statistical comparison of the seizure stages across multiple seizures (n=15). The interacti ons between and within the 4 indexed areas for each seizure stage (16 interactions total) were compared sequentially with neighboring stages as well as to inter-ictal indexes. All Stages were significantly different from inter-ictal baseline activity, Fs(14,14) > 6.9 p < 0.01. Figure 6-11 highlights the statistical comparisons between stages. Starting with stage 2 in panel C each of the 16 area indexes is compar ed with its analog in th e previous stage using dots to indicate significance levels. One dot i ndicates a 80% confidence interval, Fs(14,14) > 1.58, two dots a 90% confidence in terval, Fs(14,14) > 2.02, three dots a 95% confidence interval, Fs(14,14) > 2.48, while four dots indicates a 99% confidence interval, Fs(14,14) > 3.70.

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125 The statistical analysis revealed gross change s between areas when transitions were made from one seizure stage to another as well as di fferentiated all seizure stages from inter-ictal baseline activity patterns. While the differen ces between seizure and baseline events are not unexpected, some interesting discoveries emerged fr om the analysis of the seizure progression. For instance, the transition between stages, such as from Stage 1 to Stage 2, is statistically distinct and discreet, as shown in Figure 6-11 panel C). These results also suggest that Stage 4 has only between-hemisphere activity in common with Stage three (no dots in top left and bottom right of Panel E) and w ithin-hemisphere activity in common with Stage 5 (no dots in bottom left and top right of Panel F). This s uggests that Stage 4 has elements of Stage 3 and Stage 5 within it, suggesting Stage 4 is a tran sition between 4 and 5. When comparing Stage 3 and Stage 5 there are no significant differences betw een within-hemisphere interactions (7 out of 8 Fs(14,14) < 1.58, p=0.20) while between-hemisphe re interactions where significantly different (7 out of 8 F(14,14) > 2.48 p=0.05) indicating a d ecrease in between hemisphere interactions. However, this analysis does little to capture the effects of short intense events such as the directional transfers shown in Stage 2. These transfers, while intense and obvious to discern in the causality movies used for qua ntification, are poorly reflecte d because of averaging over the analysis window. An example of one of these directional transfers was shown in Figure 6-7. Thus, Stage 2 and Stage 3 cross hemisphere indexes are not significantly different when compared to each other, see Figure 6-11 Panel D) (all Fs(14,14) < 1.58, p=0.20). However, CA1 to DG intra-hemispherical relations hips between Stage 2 and Stage 3 were significantly different, Fs(14,14) > 3.70, p=0.01, an observation that was not made during qualitative analysis. This analysis suggests that there is a la rge decrease in CA1 to DG intra-hemisphere interactions from Stage 2 to Stage 3. Qualitativ ely, a transition is observed in Stage 4 where the

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126 dominant information flow from the CA1/2 to DG areas for both hemispheres reverses so that the dominant information flow becomes from the DG to CA1/2. This effect is highlighted by a comparison of Stage 2 (the highest magnitude in stance of CA1 to DG inte ractions) to Stage 5 (the highest magnitude instance of CA1 to DG inte ractions) right and left CA1 to DG influences, F(14,14) = 21.2 and 13.3, respectively. This sugg ests that CA1 to DG influences are effectively eliminated by the start of Stage 5. Discussion Currently, synchronization is thought to be the dominant dynamic occurring across the limbic system during an epileptic seizure. Synchronization is defi ned as adjustment of rhythms of oscillating objects due to their interaction (123). Synchronization across a diffuse excitable network is thought to be a possible mechanism that initiates seiz ure in the limbic system (124). In contrast others suggest that a large amount of asynchrony is necessary to maintain seizure events once they have begun (125). The resu lts from the Granger analysis do indeed find synchronization within the hippocampus prior to, dur ing, and after the ictal event. However, the detailed interactions th at occur during these seiz ures are largely unknown. Hence, the detailed interactions illustrated in these results using Granger Causality analysis have never previously been described in the study of TLE. These results provide a new picture of highly dynamic directional interactions among specific brain areas during a seizure that follow a common progression. For example, one of the distinct feat ures in Stage 1 of nearly all seizures is an increase in local interactions within the CA1 area whose causal strength builds over time eventually influencing activity in DG. This is followed by a series of directional crosshemisphere reverberations of Stage 2 in which the seizure appears to transfer back and forth from one hemisphere to the other. Together, th ese results suggest a hierarchical progression of

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127 causation within the CA1 that spreads to the DG and finally crosses over into the neighboring hemisphere. In contrast, if a coherence analysis were c onducted over these data (a tool commonly used in this literature), it would suggest that these directional transfers were actually bidirectional phenomena because this method is non-directional. Non-directional measures such as coherence would also not be able to detect the reversal in directional connectivity between the dentate and CA1 during the Stage 4 transition. According to th ese measures the entire seizure would seem to be dominated by bidirectional synchronization ac ross the entire network. However, Granger Causality analysis tells a very different story. The dominant directional influences in the beginning stage of the seizure ar e abnormal with the CA1 areas s howing directional causation on the DG. This is the opposite of baseline interict al patterns that often show directional influences from the dentate to the CA1. Additionally, this abnormal pattern is time locked behaviorally with the tonic portion of the seizur e. Interestingly, the flip of the directional connectivity to connectivity resembling baseline connectivity occu rs during the transition from the tonic to clonic behavior. The implications of this directional flip are not completely clear, however. One of the possibilities explai ning the driving influence of CA1 to DG during the early seizure may be related to one of the weaknesses of the Granger Causality based methods. In this case the entorhinal cortex may be differentially driving both of these areas creating a false directed influence from CA1 to DG, instea d of the influence of CA1 to DG being mediated through the entorhinal cortex. The en torhinal cortex has been impli cated as a possible initiator of ictal events where disinhibited neurons in the ento rhinal cortex may directly fire pyramidal cells in CA1 at least in a pilocarpine model (119). The entorhinal cortex is also anatomically connected to the DG. Hence, both of these situ ations are supported by the literature and may

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128 even be happening simultaneously. It is unlik ely based on the anatomical organization of the hippocampus that the effects of CA1 are mediat ed through CA2 and CA3 to the DG. Further experimental analysis using conditional granger causality with electrode coverage within the entorhinal cortex will be necessary to determ ine which of these possibilities is correct. Nevertheless, these results provide strong s upport for the use of the Granger Causality metric as an important tool for seizure detecti on. The stage 4 transiti on from abnormal driving (CA1 to DG) to a normal driving pattern (DG to CA1) suggests that at some point prior to seizure the driving pattern became abnormal. The abnormal driving influence has been seen in the pre-ictal analysis at least a minute before seizure. An natura l question to ask would be When does this driving influence become abnormal? Once identified, the transition from normal to abnormal driving influences could potentially be us ed as an indicator of an impending seizure. However, the role of the entorhin al cortex in this connectivity s hould be determined before this relationship is used for seizure detection. For ex ample, if the entorhinal cortex turns out to be the source of the abnormal driving influence it ma y better aid detection if electrodes were placed in this area. Additionally, detection of this abnormal driv ing influence may increase the sensitivity of current seizure de tection methods. This could be useful for detecting epochs during epileptigenesis that are not easily detect ed using behavioral or current electrographic seizure detection methods. In conclusion, using Granger Causality to re veal effective connectivity has led to a new understanding of the interactions within the hippocampus duri ng seizure. This understanding provides the opportunity to make new predicti ons about seizure det ection and suggests new experiments that should expand our understanding of TLE. Howe ver, we have only used the

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129 most basic of GC based tools in this analysis. Future work could make use of conditional and spectral GC analysis to paint a clearer pictur e of effective connectiv ity during seizure.

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130 Figure 6-11 Quantitative analysis of the 5 seizure stages. Analysis was over 15 seizures from 3 subjects. A Granger Causality index was created for each of four areas: Left and Right DG, and Right and Left CA 1. The effects for the elec trodes in each of the four areas were summed and normalized to create these indices. The y-axis is the driving area and the x-axis is the rece iving area, the magnitude of the interaction is described by the color ranging from blue (low interaction) to red (h igh interaction). Panel A) shows the pre-seizure interactions while th e remaining five are during seizure. The black dots in panels C)-F) are used to indicate statistical difference for each interaction index from its co rresponding index in the previ ous stage. The number of dots represent the significance level of that in teraction (p < 0.2, p < 0.10, p < 0.05, p < 0.01, for 1-4 dots respectively).( Figure continued on next page)

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131 Figure 6-11. Continued

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132 CHAPTER 7 SUMMARY OF RESULTS AND CONCLUSSIONS Summary of Results This research explored the use of Granger Causality and associated methods to reveal effective connectivity in a wide variety of neural systems. These include a neural network simulation in chapter 3, in-vitro dissociated corti cal cultures in chapters 4 and 5, and finally invivo microelectrodes in the hippocampus of epileptic rats in Chapter 6. In each case, the goal was to reveal information about the actual c onnectivity underlying the system by using Granger Causality to calculate a measure of effective connectivity from the output of that system. The difficulty in recovering the actual connectivity increases as the complexity of the system increases. For example, increasing the number of modeling neurons in the simulated neural networks demonstrated this rela tionship. However, even in the extremely complex in-vitro and in-vivo systems Granger Causality was able to provide novel and useful insights into the workings of these neural systems. The neural networks in Chapter 3 used a simp le neural network model modified such that the input synaptic weights were known prior to simulation. The outputs from these simulations were used to recover the synaptic weights used to drive the network. In a simple network of two neurons the relationship between pre and post synap tic is a sigmoid with a linear section between 15mV and 45mV where Granger Causal ity increases from near zero to a little greater than 1. Granger Causality does not have an upper bound like that of coherenc e, thus the high end of the sigmoid being around 1 is merely coincidence. In creasing the complexity of this network to 100 neurons was shown to distort this simple re lationship and requiring another tool, conditional granger causality, to recover the initial synaptic weights driving the network. CGC was needed to remove not only false direct influences but al so to untangle mediated influences from direct

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133 influences. Usually, untangling a mediated influen ce from a direct influence requires the use of conditional granger causality. Ho wever, it was found that under the special circumstance where a false direct influence was conditioned out (i.e. s hown to be equal to zero and entirely mediated through the confounded pathway), the magnitude of the false connection could be simply subtracted from the confounded pathway. This s hortcut relationship is from previous work by Geweke (126). The ability of both CGC and Geweke s shortcut to recove r the direct influence from a confounded pathway were both evaluated and found to be effectively equivalent. However, as the complexity and length of the causal chain grew the ability of these methods decreased. When the size of the network was increased to 100 neurons these methods consistently underestimated the original synaptic weights. This was likely due to the unknown effects of the network complexity on these pathways. These sugge st that more work is needed to accurately recover synaptic weights from these networks However, while the sigmoid curve equation shown in Chapter 3 does not always hold numeri cally, synaptic weight does scale linearly within the region where a presynaptic neuron begins to affect the firing rate of a postsynaptic neuron up the the point where it always causes the post synap tic neuron to fire. This relationship suggests that changes in effective connectivity calculate d using Granger Causality observed in complex networks are likely linear changes that could relate to synaptic efficacy if the sigmoid relationship were know. This suggests that Gr anger Causality methods could be useful in revealing relationships in living neural networks. In-vitro dissociated cortical neural networks grown on microelectrode arrays were used for the experiments in Chapters 4 and 5. Jimbo (4) used spike rate to describe simultaneous pathway specific potentiation and depression in these cultures. This was done with accomplished by

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134 probing on each of the 60 channels 10 times and m easuring the spike rates at all electrodes in the ensuing burst. The probe sequence and correspon ding spike rates were used to establish a baseline spike rate response for every electrode and then used post treatme nt to assess possible changes. The treatment was a high frequency high amplitude pulse train or tetanus delivered to a single electrode that consisten tly responded to probing. This tetanus was designed to induce plasticity related changes in the culture and show n by Jimbo et al to poten tiate or depress neural pathways. These results of Chapter 4 were cons istent with Jimbos findi ng identifying potential pathways within the dish using a comparison of changes in spike rate. PGC was then applied to the baseline and post-treatment probe data. The results were shown to agree with spike rate results, but PGC showed several marked improvements over Jimbos analysis. Admittedly spike rate is a computationally simpler method when compared to PGC. However, PGC showed many advantages over using sp ike rate for describing pl asticity. First, it showed a higher contrast between baseline and post -treatment results. One of the difficulties in the use of spike rate as a meas ure is establishing the statistical significance of changes. The higher contrast PGC provides makes this task ea sier. Second, PGC has a significance threshold that can be obtained using a surrogate where the spike timings within the data are shuffled. This effectively destroys the spike timing relationships used to calculate PGC while maintaining the spike rate characteristics of this data. This anal ysis of this surrogate data demonstrated that the observed changes in the original da ta set were not simply due to changes in rate for if they were, the results would have been equivalent. Third, PGC gives a statistica lly relevant baseline measurement of effective connectivity establishi ng a baseline connectivity map. Spike rates from the baseline alone do not have any meani ng without being used in conjunction with the

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135 post-treatment spike rates. Thus, baseline sp ike rates provide no information about baseline effective connectivity. Finally, PGC is a directional measure while spike rate is a nondirectional non-causal measure. Results calculating using spike rate resu lt in a 60x60 matrix with axes of probe channel and response channel. The use of PGC on the sa me data results in a 60x60x60 matrix with axes of probe channel, response channel, and sour ce channel. Thus, in addition to giving similar results as spike rate, PGC provides 60 times more information. This extra information was exploited to learn more about th e cultures. Comparison of probe/ source plots to probe/response plots for potentiated pathways showed that some channel experienced an increase in output while others experienced and increase in input post-te tanus. This same relationship was shown in depressed channels except with decreases in output and input. Th is suggests that changes were not just global changes in rate a nd were likely reflected in their postsynaptic targets as well. The pre-tetanus effective connectivity calculated fr om PGC was also used to measure amplitude dependent plasticity by comparing the initial connectivity to changes in connectivity posttetanus. Analysis of this data across 6 subjects showed that the amplitude dependent plasticity trends matched trends seen in work done by Bi & Poo (118), where potentiated pathways increase a lot if they are small and depresse d pathways change by a constant percentage independent of initial strength. A second experiment was developed based on th e results shown from the analysis of the Jimbo protocol. This experiment made use of th e ability of PGC to measure connectivity prior to tetanization. Once PGC was calculated from the pre-tetanus data sinks and sources of causal flow were identified within the culture. A sink was defined as a channel where the balance of causal information flow was inward, while a source was defined as a channel where the balance

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136 of causal information flow was outward. The ce ntral hypothesis in this experiment was that tetanization on a sink would elicit depression at the sink while te tanization at a source would elicit potentiation. The experiment was carried out on 10 cultures. The results suggested that changes in plasticity did occur. However the di rection of results did no t match the predictions made prior to the experiment. Perhaps one reas on this experiment was not successful was that the design and predictions underestimated the comple xity of connectivity in the neural culture. One solution might be to reduce the complexity of the network by reducing the density of the neurons. Additionally, a paired pulse paradigm at pre and postsynaptic targets would likely increase the probability of plasticity induction only on targeted pathways. In-vivo, PGC was used to explore effective connectivity of seizures in rats from the Evolution into Epilepsy project. While this an alysis was mostly empirical, comparison of the effective connectivity during se izure and during interictal and baseline was conducted. Qualitative analysis of the results from PGC on 15 seizures from 3 subjects showed that there were 5 distinct stages of the seizure. Stage 1 consisted of the buildup of activity within the CA1 area and also of a directed influence from the CA 1 to the dentate gyrus. This connectivity is already abnormal when compared to interictal and baseline results. Base line connectivity often highlights a directed influence from the dentat e gyrus to CA1, which is also the expected anatomical influence. Stage 2 consists of la rge reverberating directi onal transfers from one hemisphere to the other accompanied by large with in hemisphere interactions mainly from CA1 to the dentate gyrus. Stage 3 shows a large decr ease in the magnitude of directional transfers however the within and between hemisphere interactions continue. This stage often contains the largest observable waveforms in the ictal event. Stage 4 or the transition stage marks the time when the connectivity flips to a pattern more resembling that of baseline activity where the

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137 dentate gyrus drives the activity of CA1. This stage is marked by a brief period of strong bidirectional causality between most electrodes in both hemi spheres. Behaviorally, the surveillance video of the rat shows that Stage 4 also marks the transition from tonic to clonic behavior in the rat. Stage 5 or the post transition st age shows a strong causal influence from the dentate gyrus to the CA1 as the rat clonically seiz es. Shortly after Stage 5, the seizure usually ends and the rat engages in normal exploratory behavior. Quantitative analysis was also carried out by analysis of variance between granger causality indexes created for combinations of each of the four regions left and right CA1 and left and right dentate gyrus for each stage of the seizure. It served as no surprise that stages that were qualitatively different from each other were also quantitatively different. However, quantitative analysis confirmed that all seizure stages were different than interictal activity. Also, quantitative analysis confirmed that Stage 4 was lik ely a transition stage. It was shown that Stage 3 and Stage 5 had causal interactions in common w ith Stage 4 that they did not have with each other (i.e. Stage 4 contains aspects of Stage 3 and 5) Overall, the results from this analysis were quite instructive and suggest th at a great deal of informati on can be gleaned using Granger Causality not only about causal bindings during seizure, but also about changes in effective connectivity during the period in which these seizures develop (epileptigenesis). Overall Conclusions The aims of this body of work were to exploi t the ability of Granger Causality to reveal effective connectivity. Granger Ca usality tools were used to e xplore effective connectivity in simulated, in-vitro, and in-vivo neural systems. The simulated neural ne twork models provided a means of evaluating the ability of granger causality to relate back to the initial structure of the neural network. The results from this model su pport the notion that linear changes in effective connectivity observed using PGC in a living system likely reflect actual linear changes in the

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138 underlying connectivity. Further the models prov ided a change to test conditional granger causalitys ability to identify fa lse directed influences and to detangle direct and mediated influences in confounded pathways. The in-vitro work with cortical cultures suggests that PGC is a useful tool in describing changes in plasticity. The a dvantage of using a st atistically sound, dire ctional, and causal method that yields a measure of effective conn ectivity allowed much more information to be extracted from Jimbos protocol. This allo wed phenomena such as amplitude dependent plasticity to be measured from spike timings. These results contribute more evidence that suggests the plasticity is bei ng observed using Jimbos protocol. These new measures also allowed new predictions to be made about th e response of the cultu re to a differential tetanization on sink and sources. While the prediction that sinks would be depressed and sources would be enhanced proved to be too simplistic, the PGC analysis suggested that plasticity occurred as a result of tetani zation. Future predictions that take this complexity into consideration may be successful by using Granger Causality to estimate effective connectivity prior to experimental treatment. Finally, PGC was able to show previously unr eported causal relati onships during seizure that may be useful for efforts to prevent seizures from occurring. At some point prior to the ictal event the effective conne ctivity in the hippocampus appears to become abnormal with the CA1 driving activity in the dent ate gyrus. If this flip to abnorma lity can be detected it may provide an opportunity to intervene at an opportune mome nt to prevent the im manent ictal event by correcting this abnormal ac tivity pattern. Additionally, the us e of PGC to analyze epileptigenesis could provide new insight that may lead to treatme nts for a condition that is notoriously hard to treat pharmacologically.

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139 In each of these cases, using Granger Causalit y to reveal effective connectivity led to a new understanding of these systems. This unders tanding provided the opportunity to make new predictions about these systems and suggest new e xperiments that were previously not feasible. Overall, Granger Causality has proven to be a usef ul tool in exploring ef fective connectivity in a wide variety of neural systems and in the future, may become one of the traditional tools used in neuroscience.

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152 BIOGRAPHICAL SKETCH Alex Cadotte was born in Glens Falls, New York in 1976. He grew up in Franklin, Virginia where he g raduated with honors from Franklin High School in 1994. He received his B.S. in chemical engineering from Virginia Te ch in 1999. He worked in industry from 1999 to 2002 at companies such as General Electric and Co rning Optical Fiber in Wilmington, NC. Alex moved to Gainesville, FL, in August of 2002 and completed a Master of Science in August of 2004 at the University of Florida in biomedical engineering specializi ng in tissue and neural engineering. Alex began his PhD at the Univer sity of Florida in August of 2005, advanced to candidacy in April of 2006, and completed a Doctorate in Philosophy on November 20, 2007 specializing in neural co mputation and neural engi neering. This disserta tion is a summary of the work done by Alex under the mentorship of his doctoral committee consisting of Dr. Thomas DeMarse (chair), Dr. Mingzhou Ding, Dr. Paul Carney, Dr. William Ditto, and Dr. Jason Frazier (external) in this field of study.


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