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Brain Dynamics, System Control and Optimization Techniques with Applications in Epilepsy

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Title:
Brain Dynamics, System Control and Optimization Techniques with Applications in Epilepsy
Creator:
Liu, Chang-Chia
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (169 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Biomedical Engineering
Committee Chair:
Pardalos, Panagote M.
Committee Members:
Uthman, Basim M.
Carney, Paul R.
Van Oostrom, Johannes H.
Roper, Steven N.
Graduation Date:
8/9/2008

Subjects

Subjects / Keywords:
Brain ( jstor )
Connectivity ( jstor )
Electrodes ( jstor )
Electroencephalography ( jstor )
Epilepsy ( jstor )
Neurons ( jstor )
Nonlinearity ( jstor )
Seizures ( jstor )
Signals ( jstor )
Time series ( jstor )
Biomedical Engineering -- Dissertations, Academic -- UF
clustering, dynamics, epilepsy, nonlinear, statistics, svm
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Biomedical Engineering thesis, Ph.D.

Notes

Abstract:
The basic mechanisms of epileptogenesis remain unclear and investigators agree that no single mechanism underlies the epileptiform activity. Different forms of epilepsy are probably initiated by different mechanisms. The quantification for preictal dynamic changes among different brain cortical regions have been shown to yield important information in understanding the spatio-temporal epileptogenic phenomena in both humans and animal models. In the first part of this study, methods developed from nonlinear dynamics are used for detecting the preictal transitions. Dynamical changes of the brain, from complex to less complex spatio-temporal states, during preictal transitions were detected in intracranial electroencephalogram (EEG) recordings acquired from patients with intractable mesial temporal lobe epilepsy (MTLE). The detection performance was further enhanced by the dynamics support vector machine (D-SVM) and a maximum clique clustering framework. These methods were developed from optimization theory and data mining techniques by utilizing dynamic features of EEG. The quantitative complexity analysis in multi-channel intracranial EEG recordings is also presented. The findings suggest that it is possible to distinguish epilepsy patients with independent bi-temporal seizure onset zones (BTSOZ) from those with unilateral seizure onset zone (ULSOZ). Furthermore, for the ULSOZ patients, it is also possible to identify the location of the seizure onset zone in the brain. Improving clinician?s certainty in identifying the epileptogenic focus will increase the chances for better outcome of epilepsy surgery in patient with intractable MTLE. Recent advances in nonlinear dynamics performed on EEG recordings have shown the ability to characterize changes in synchronization structure and nonlinear interdependence among different brain cortical regions. Although these changes in cortical networks are rapid and often subtle, they may convey new and valuable information that are related to the state of the brain and the effect of therapeutic interventions. Traditionally, clinical observations evaluating the number of seizures during a given period of time have been gold standard for estimating the efficacy of medical treatment in epilepsy. EEG recordings are only used as a supplemental tool in clinical evaluations. In the later part of this study, a connectivity support vector machine (C-SVM) is developed for differentiating patients with epilepsy that are seizure free from those that are not. To that end, a quantitative outcome measure using EEG recordings acquired before and after anti-epileptic drug treatment is introduced. Our results indicate that connectivity and synchronization between different cortical regions at higher order EEG properties change with drug therapy. These changes could provide a new insight for developing a novel surrogate outcome measure for patients with epilepsy when clinical observations could potentially fail to detect a significant difference. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2008.
Local:
Adviser: Pardalos, Panagote M.
Statement of Responsibility:
by Chang-Chia Liu.

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Copyright Liu, Chang-Chia. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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detect the effect of therapeutic interventions. This quantification would also help us

identify the regions that actively participate during epileptic seizures. In the later

chapter, I investigate possibilities for identify the regions in the brain that are actively

participate prior to epileptic seizures and the effect of therapeutic interventions on

EEG recordings. Although it is known that the fluctuations in EEG frequency and

voltage arise from spontaneous interactions between excitatory and inhibitory neurons

in circuit loops, the cause of neuron discharge remains unclear. The synchronization

of neuron activity is considered to be important for information processing in the

developing brain. There is now clear evidence that there are distinct differences between

the immature and mature brain in the pathophysiology and consequences of seizures,

abnormal synchronization activity in the developing brain can result in irreversible

alterations in neuronal connectivity [108]. In cognitive task studies, In 1980, Freeman

found "more rc, l oi spatiotemporal activities in EEG for a brief period of time when

the animal inhaled a familiar odor until the animal exhaled [109; 96]. Some earlier

studies also indicated the significant role of synchronization for physiological systems in

humans; the detectable alters in synchronization phenomena have been associated to a

number of chronic, acute diseases or the normality of brain. [110; 111]. In the field of

epilepsy research, several authors have -,i--.- -1. 1 direct relationship between alters in

synchronization phenomena and onset of the epileptic seizures using EEG recordings. For

example, Iasemidis et al., (1996) reported from intracranial EEG that the entrainment

in the largest Lyapunov exponents from critical cortical regions is a necessary condition

for onset of seizures for patients with temporal lobe epilepsy [112; 61]; Le Van Quyen

et al., showed epileptic seizure can be anticipated by nonlinear analysis of dynamical

similarity between recordings [35]. Mormann et al., showed the preictal state can be

detected based on a decrease in synchronization on intracranial EEG recordings [89; 43].

The highly complex behavior on the EEG recordings is considered to normality of brain

state, while transitions into a lower complexity brain state are regarded as a pathological












Dependency Matrix (Before Treatment PI) Dependency Matrix (After Treatment PI)


5 1.5 5 1.5

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Dependency Matrix (Before Treatment P2) Dependency Matrix (After Treatment P2)
2 2

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Dependency Matrix (Before Treatment P3) Dependency Matrix (After Treatment P3)
2 2

5 1.5 5 1.5

10 I 10

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0 0
5 10 15 5 10 15

Dependency Matrix (Before Treatment P4) Dependency Matrix (After Treatment P4)


.5 1.5 5 1.5

:- *0
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Figure 8-2. Pairwise mutual information between for all electrodes- before v.s. after
treatment









3.3 Proper Time Delay

We used the mutual information function to estimate the proper time d. 1i, between

successive components in delay vectors. In theory, the time delay used for time delay

vector reconstruction is not the subject of the embedding theorem. Since the data are

assumed to have infinite precision, from mathematical point of view delay time can be

chosen arbitrary. On the other hand, it is essential to have a good estimation for proper

time delay when dealing with none artificial data. For none artificial data, the time

delay parameter can affect the dynamical properties under studying, if time delay is very

large, the different coordinates may be almost uncorrelated. In this case, the attractor

may become very complicated, even if the underlying true attractor is simple. If delay

is too small, there is almost no difference between the different components between

delay vectors, such that all points are accumulated around the bisectrix in the embedding

space. Therefore, it is slr.--- -1. I to look for the first minimum of the time d, li1 mutual

information [75]. The concept of mutual information is given as below

Mutual information is originated from information theory and it has been used for

measuring interdependence between two series of variables. Let us denote the time series

of two observable variables as X = {x}N 1 and Y = {yj}N where N is the length of

the series and the time between consecutive observations (i.e. iii,,,,J, .irate) is fixed. The

mutual information between observations xi and yi is defined as:

Slo PY(Si, qj)


where Px,y(xi, yj) is the joint probability density of x and y evaluated at (xi, yj) and

Px(xi),Py(yj) are the marginal probability densities of x and y evaluated at xi and yj

respectively. The unit of mutual information is in bit, when based 2 logarithm is taken.

If x and y are completely independent, the joint probability density Px,y(xi, yj) equals

to the product of its two marginal probabilities and the mutual information between









[40] C.E. Elger. Future trends in epileptology. Current Opinion in Neur 4l.i ,
14(2):185-186, 2001.

[41] F. Mormann, R.G. Ai,11.. i 1. C. E. Elger, and K. Lehnertz. Seizure prediction: the
long and winding road. Brain, 130(2):314-333, 2006.

[42] S. Gigola, F. Ortiz, C. E. DAttellis, W. Silva, and S. Kochen. Prediction of epileptic
seizures using accumulated energy in a multiresolution framework. Journal of
Neuroscience Methods, 138:107-111, 2004.

[43] F. Mormann, R.G. Ail1i.. i.1: T. Kreuz, C. Rieke, P. David, C.E. Elger, and
K. Lehnertz. Automated detection of a preseizure state based on a decrease in
synchronization in intracranial electroencephalogram recordings from epilepsy
patients. Phys. Rev. E, 67(2):021912, 2003.

[44] 0. Rossler. An equation for continuous chaos. Phys. Lett., 35A:397-398, 1976.

[45] E.N. Lorenz. Deterministic nonperiodic flow. Journal of Atmospheric Sciences,
20:130-141, 1963.

[46] P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors.
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[47] A. Babloi-,-nt: and A. Destexhe. Low dimensional chaos in an instance of epilepsy.
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eeg recordings. Brain T(',' 'i,,''l i, ; 9:249-270, 1997.

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area in temporal lobe epilepsy characterized by neuronal complexity loss. Electroen-
,, ,,l,, i. Clin. N,-ur.ph .l 95:108-117, 1995.

[50] U.R. Acharya, O. Fausta, N. Kannathala, T. Chuaa, and S. Laxminarayan.
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volume 898, pages 366-381. Springer-Verlag, 1981.

[53] J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev.
Mod. Phys., 57:617-656, 1985.











yiai =0 (5-16)
i 1
0 < ai < C, i = 1,...n (5-17)

The solution of the primal problem is given by w = u* '':' ,: where w is the vector

that is perpendicular to the separating hyperplane. The free coefficient b can be found

from ai(yi(w xi + b) 1) = 0, for any i such that ci is not zero. D-SVM map a given EEG

data set of binary labeled training data into a high dimensional feature space and separate

the two classes of data linearly with a maximum margin hyperplane in the dynamical

feature space. In the case of nonlinear separability, each data point x in the input space

is mapped into a different space using some nonlinear mapping function p. A nonlinear

kernel function, k(x, f), can be used to substitute the dot product < p(x), p(') >. This

kernel function allows the D-SVM to operate efficiently in a nonlinear high-dimensional

feature space without being adversely affected by dimensionality of that space.

5.6 Connectivity Support Vector Machine

In this subsection, we describe the framework of C-SVM. Instead of modeling the

deterministic evolution of the physiological state from time, we now model the evolution of

an ensemble of possible states by implementing EEG representation as a "information path

, :r;1 or "Brain C .,,,... /.;.-,'.I The brain connectivity can be formulated as follows.

Let G be an undirected graph with vertices V1,... V, where Vi represents electrode i.

There is an edge (link) with the weight i, for every pair of nodes Vi and Vj corresponding

to the connectivity of the brain dynamics between these two electrodes. The connectivity

or synchronization can be viewed as been activated by interactions between neurons in the

local circuitry underlying the recording electrodes. Figure 5-4 represents a hypothetical

brain graph in which each connected path denotes the underlying connectivity. With this

graph model, the attributes of C-SVM inputs are the pair-wised relation between two

time series profiles rather than time stamps of a time series profile. In this context, the










known that the orbitofrontal areas communicate with each other more than other parts

of the brain. This has led us to the conclusion that the brain areas that are selected to

be in the maximum clique are the vulnerable brain areas, rather than the epileptogenic

areas. In other words, the brain areas) that are highly synchronized could be governed

or manipulated by the epileptogenic areas so that they continuously show strong neuronal

communication through the synchronization of EEG signals (measured by cross-mutual

information).


RL2 -
RE I I II I1
kP3 I.I
RLI

I 1 IIIII ii ini
LF23 Seizure Onset-
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RT3 -
RT2 -
RTI
LT4 I s ai II oIll b -
LT3 -
LT2
LTI


RD2- J-


.D3
0 20 4- 60 SO 100 120 140 160 180


Figure 7-7. Electrode selection using the maximum clique algorithm for Case 1


7.4.4 Implications of the Results

In normal brain functions, the orbitofrontal areas (both left and right) of the

brain are highly synchronized active most of the time as it is considered to be the

brain's executive function, and the temporal lobe areas are separated into left and

right cortical hemispheres that work independently from each other. We hypothesized

that this operation in the brain should be applied to the epileptic brains, even in the

pre-seizure period. As we predicted, from the spectral partitioning results, both left and

right orbitofrontal areas were also highly synchronized and active as well as right and left

temporal lobe areas during the pre-seizure state. This -ir-'-i -;that the epileptogenetic













Fpl Fp2

F F3 F4 F8



T3 C3 C4 T4



P3 P4
T5 T6


Oz r18 Channels




Figure 5-1. Scalp electrode placement


5.2 EEG Data Information

In this study, the dataset consists of continuous short-term (about 60 minutes)

multi-channel scalp EEG recordings of 10 epileptic patients, 5 with medically intractable

epilepsy (TLE) and 5 were seizure after treatment. The 19-32 channels scalp EEG

recordings were obtained using standard 10-20 system, Nicolet BMSI 6000. Figure 5-1

shows the location of the electrodes on the scalp.

Table 8-1 shows the EEG description from 10 subjects, EEG signals were recorded at

sampling rate 250Hz. For consistency, we analyze and investigate EEG time series using

bipolar electrodes only from 18 standard channels for every patient.EEG recordings from

each subjects were inspected by certificated electroencephalographers. We randomly and

uniformly sample two 30-second EEG epochs from each subject. Since EEG recordings

were digitized at the sampling rate of 250 Hz, the length of each EEG epoch 7,500 points.









where m is given as an integer and ry is a positive real number. The value of N is the

length of compared subsequences in S, and rd specifies a tolerance level.


d(xi,xj) max Si+k Sj+k (2-32)
O
d(xi, xj) represents the maximum distance between vectors xi and xj in their respective

scalar components.
n-m+l C7 rf
(rf)= In (2-33)
n-m-
i=1
Finally the approximate entropy is given by:


ApEn(m, rf, N) m(rf)- )m+l(rf). (2-34)


The parameter rf corresponds to an a priori fixed distance between neighboring

trajectory and r, is chosen according to the standard deviation estimated from data.

Hence, rf can be viewed as a filtering level and the parameter m is the embedding

dimension determining the dimension of the phase space. Heuristically, ApEn quantifies

the likelihood that subsequences in S of patterns that are close and will remain close

on the next increment. The lower ApEn value indicates that the given time series is

more regular and correlated, and larger ApEn value means that it is more complex and

independent.

2.8 Dynamical Support Vector Machine (D-SVM)

The underlying dynamics of preictal transitions is changing from case to case, this

requires analytical tools which is capable for identifying the changes in brain dynamics

when preictal transitions take place. The detection performance is further improved by the

dynamics support vector machine (D-SVM), a method developed from optimization theory

and data mining techniques by utilizing dynamic features of EEG.

D-SVM performs classification by constructing an N-dimensional hyper plane that

separates the data into two different classes. The maximal margin classifier rule is used to

construct the D-SVM. The objective of maximal margin D-SVM is to minimize the bond









types of connection between two or more sources, and quantifying the synchronization

between different brain areas (measured by different electrodes) is crucial to a greater

understanding of the brain connectivity network. The synchronization may be attributable

to the brain's anatomical, functional, or dynamical connectivity. In this study, the

synchronization patterns are postulated to reflect the seizure evolution (epileptogenic

process), and we shall use electrode synchronization as a similarity measure of EEG signals

from different brain areas. This is fine in theory, however there are a few complexity issues

in calculation of multivariate measures. First, in spite of the theoretical capability of

multi-variate methods to calculate common patterns from several sources simultaneously,

the calculation complexity increases exponentially with the number of sources.

Therefore, we use multivariate measures for quantifying the synchronization from

only 2 electrodes at a time. Specifically, a simple signal processing used to calculate the

synchronization between electrode pairs is employ, -1 in this study. Then we apply a data

mining technique based on network-theoretical methods to the multivariate analysis of

EEG data.

7.3.2 Brain Synchronization

In general, statistical similarity measures can be categorized into two groups: linear

and nonlinear dependence measures. The linear measure is mainly used for measuring a

linear relationship between two or more time series. For example, the most commonly used

measure is cross-correlation function, which is a standard method of estimating the degree

of correlation in time domain between two time series. The result of a cross correlation

function can be calculated at different time lags of two time series to show the level of

redundancy at different time points. Frequency coherence is another linear similarity

measure, which calculates the synchrony of activities at each frequency [121]. Although

the information from cross-correlation function and frequency coherence has been shown

to be identical [122], the similarity between two EEG signals in different frequency bands

such as delta, theta, beta, alpha and gamma, is still commonly used to investigate EEG









2.9 Statistical Distance ............... ........... .. 56
2.10 Cross-Validation .............. . . .. 57
2.11 Performance Evaluation of D-SVM .............. ...... 57
2.12 Patient Information and EEG Description ............. .. .. 58
2.13 Results ..................... ............ .... 58
2.14 Conclusions ............... .............. .. 59

3 QUANTITATIVE COMPLEXITY ANALYSIS IN MULTI-CHANNEL INTRACRANIAL
EEG RECORDINGS FROM EPILEPSY BRAIN ................ .. 60

3.1 Introduction ............... . . .. 60
3.2 Patient and EEG Data Information .............. . .. 61
3.3 Proper Time Delay ............... . ..... 63
3.4 The Minimum Embedding Dimension ............. .. .. 65
3.5 Data Analysis ............... ........... .. 66
3.6 Conclusions ............... .............. .. 67

4 DISTINGUISHING INDEPENDENT BI-TEMPORAL FROM UNILATERAL
ONSET IN EPILEPTIC PATIENTS BY THE ANALYSIS OF NONLINEAR
CHARACTERISTICS OF EEG SIGNALS .................. ... 72

4.1 Introduction .................. ................ .. 72
4.2 Materials and Methods .................. .......... .. 75
4.2.1 EEG Description .................. ......... .. 75
4.2.2 Non-Stationarity. .................. ......... .. 76
4.2.3 Surrogate Data Technique .................. .. 77
4.2.4 Estimation of Maximum Lyapunov Exponent . . 78
4.2.5 Paired t-Test ............... ......... .. 79
4.3 Results ................... ......... .. ...... 80
4.4 Discussion ............... ............... .. 84

5 OPTIMIZATION AND DATA MINING TECHNIQUES FOR THE SCREENING
OF EPILEPTIC PATIENTS. .................. ......... .. 91

5.1 Introduction ............... ................ .. 91
5.2 EEG Data Information ............. . . ... 93
5.3 Independent Component Analysis ................ .. 94
5.4 Dynamical Features Extraction ............... ... .. 95
5.4.1 Estimation of Maximum Lyapunov Exponent . . ... 95
5.4.2 Phase/Angular Frequency ..... ........... . .. 96
5.4.3 Approximate Entropy .................. ..... .. 97
5.5 Dynamical Support Vector Machine ................ . .. 98
5.6 Connectivity Support Vector Machine ............... . .. 100
5.7 Training and Testing: Cross Validation ............. . 102
5.8 Results and Discussions .................. ......... 102









the electroencephalographers can search for specific EEG configurations and link it to

particular physiological states or neurological disorders. However, performing the visual

inspection on long term EEG recordings is time consuming and requires continuously

cautions from examiner. Inaccurate diagnosis could lead to severe consequences, especially

in life-threatening conditions such as in emergency room (ER)or intensive care unit (ICU).

There is currently no reliable tool for rapid EEG screening that can quickly detect and

identified the abnormal configurations in EEG recordings. There is a need for developing a

reliable technique which would serve as an initial medical diagnosis and prognosis tool.

SVM has been successfully implemented for biomedical research on analyzing very

large data sets. Moreover SVM has been recently applied for the use of epileptic seizure

prediction and it has been shown to achieved 71' sensitivity and 7'- specificity for

EEG recordings from 3 patients [82]. Nurettin Acir and Cuneyt Guzelis introduced a

two-stage procedure SVM for the automatic epileptic spikes detection in a multi-channel

EEG recordings [83]. Bruno Gonzalez-Velldnet et al., reported it is possible to detect the

epileptic seizures using three features of the electroencephalogram (EEG), namely, energy,

decay (damping) of the dominant frequency, and cyclostationarity of the signals [84].

Along with this directions, the abnormal EEG identification problem can be modeled as

binary classification problem i.i, i or abnormal ". Embedded with neuron network

and connectivity concepts we first proposed and described an application of connectivity

support vector machine C-SVM, C-SVM is based on network modeling concepts and

connectivity measures to compare the EEG signals recorded from different brain regions.

A detail flow chart of the proposed C-SVM framework is given in Figure ??. We also

uses three dynamical features of EEG 1. Angular frequency 2. Approximate entropy 3.

Short-term largest lyapunov exponent to conduct the dynamical SVM in the second part

of this study.









[110] M.C. Mackey and L. Glass. Oscillation and chaos in physiological control systems.
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[120] S. Ken, G. Di Gennaro, G. Giulietti, F. Sebastiano, D. De Carli, G. Garreffa,
C. Colonnese, R. Passariello, J. Lotterie, and B. Maraviglia. Quantitative evaluation
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[121] W.H. Miltner, C. Braun, M. Arnold, H. Witte, and E. Taub. Coherence of
gamma-band eeg activity as a basis for associative learning. Nature, 397:434-436,
1999.

























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CHAPTER 6
SPATIO-TEMPORAL EEG TIME SERIES ANALYSIS

6.1 Introduction

The degree of synchronization is an important indication of how information

is processing in the brain. The quantification of neuronal synchronization has been

investigated using different approaches, from linear cross-correlation to phase synchronization

or advanced dynamical interdependence analysis. These synchronization tools have also

been applied to EEG recordings and have been shown to be able to detect increased in

synchronization measures prior to the seizures [89]. Synchronization can be quantified in

both space and time domain. For a multi-variate system, understanding the interactions

among its various variables, whose behavior can be represented along time as time-sequences,

presents many challenges. One of the key aspects of highly synchronized systems

with spatial extent is their ability to interact both across space and time, which

complicates the ain i,~-i-; greatly. In biological systems such as the central nervous

system, this difficulty is compounded by the fact that the components of interest have

nonlinear complicated dynamics that can dictate overall changes in the system behavior.

The exact figure of how to quantify the information exchanges in a system remains

ambiguous. Studies on multi-variate time series analysis have resulted in development of

a wide range of signal-processing tools for quantification of synchronization in systems.

However, the general consensus on how to quantify this phenomenon is largely uncertain.

In the literature, synchronization between variables can be categorized as identical

synchronization, phase synchronization and generalized synchronization. In the following

chapters, I undertake an in-depth analysis of preictal and interictal synchronization

behavior, focusing on EEG recordings from patients with temporal lobe and generalized

seizures.



















































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CHAPTER 2
EPILEPSY AND NONLINEAR DYNAMICS

2.1 Introduction

The beginning and termination of epileptic seizures reflect intrinsic, but poorly

understood properties of the epileptic brain. One of the most challenging tasks in the

field of epilepsy research has been remained the search for basic mechanism that underlies

seizures. Traditional studies into the seizure activity have focus upon neuronal apparatuses

such as neurotransmitters, receptors or specific ionophores. However, a seizure involves

large portions of the cerebral cortex, therefore, it is likely that investigation into the

epileptic brain as a system will elucidate important greater information than traditional

approaches.

The development of preictal transitions can be considered as a sudden increase of

synchronous neuronal firing in the cerebral cortex that may begin locally in a portion

of one cerebral hemisphere or begin simultaneously in both cerebral hemisphere. By

observing the occurrence of epileptic seizures, it is reasonable to believe that there

are multiple states exist in a epileptic brain and the sequences of the states are not

deterministic. The preictal transitions are detectable EEG dynamical changes by applying

methods developed from nonlinear dynamics. Several groups have reported that seizures

are not sudden transitions in and out of the abnormal ictal state; instead, seizures follow

a certain dynamical transition that develops over time [32-39] see [40; 41] for review. In

an study of Pijn et al. in 1991, authors were able to demonstrate decrease in the value

of correlation dimension at seizure onset in the rat model. In early 1990s, Iasemidis et

al., first estimated the largest Lyapunov exponent and reported seizure was initiated

detectable transition period by analyzing spatiotemporal dynamics of the EEG recordings;

this transition process is characterized by: (1) progressive convergence of dynamical

measures among specific anatomical areas dynamicall entrainment "and (2) following

the overshot brain resetting mechanism during post ictal state. Martinerie et al., (1998)









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

BRAIN DYNAMICS, SYSTEM CONTROL AND OPTIMIZATION TECHNIQUES
WITH APPLICATIONS IN EPILEPSY

By

C('!h ig-Chia Liu

August 2008

('C! ,i: Panagote M. Pardalos
Major: Biomedical Engineering

The basic mechanisms of epileptogenesis remain unclear and investigators agree

that no single mechanism underlies the epileptiform activity. Different forms of epilepsy

are probably initiated by different mechanisms. The quantification for preictal dynamic

changes among different brain cortical regions have been shown to yield important

information in understanding the spatio-temporal epileptogenic phenomena in both

humans and animal models.

In the first part of this study, methods developed from nonlinear dynamics are used

for detecting the preictal transitions. Dynamical changes of the brain, from complex

to less complex spatio-temporal states, during preictal transitions were detected in

intracranial electroencephalogram (EEG) recordings acquired from patients with

intractable mesial temporal lobe epilepsy (\ TLE). The detection performance was

further enhanced by the dynamics support vector machine (D-SVM) and a maximum

clique clustering framework. These methods were developed from optimization theory

and data mining techniques by utilizing dynamic features of EEG. The quantitative

complexity analysis in multi-channel intracranial EEG recordings is also presented. The

findings -,.;; -1 that it is possible to distinguish epilepsy patients with independent

bi-temporal seizure onset zones (BTSOZ) from those with unilateral seizure onset zone

(ULSOZ). Furthermore, for the ULSOZ patients, it is also possible to identify the location

of the seizure onset zone in the brain. Improving clinician's certainty in identifying the
























Z -


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z-Hz-


-


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z-Hz-H


.z *z
z-Hz-H


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z
ZHH


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z-Hz-H


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oocu
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- 0 0-00 0 0-] 0 -0 00 0 0] 0 0 0 --
.zH0^ *z *z z z z z z z z * *
~~~~~~~00~~ ~ ~ ~ ~ HH~HHHH~~HH~0HH


.z *z *z *z *
~HHZHH3HHUHH~HH


So
* zc
ZHHd


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r1
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45


40


35


30


25


20


15

10


5


0 L
0 5 10 15 20
time (s)


Figure 2-8. Z component of Lorenz system


25 30 35















4-

Sp-value = 0.9955

0
0-

i-- 5, ',







1 2 3






Patients
,, A














Figure 4-10. Nonlinearities across recording areas during interictal state for BTSOZ
1 2 3


patientsPatients
Figure 4-10. Nonlinearities across recording areas during interictal state for BTSOZ
patients


in patients with ULSOZ. Further studies on a larger sample of patients to validate these

results are warranted. Success of this study will provide more much-needed information to

guide electroencephalographer and clinician to improve the likelihood of successful surgery.









Table 3-1. Patients and EEG data statistics for complexity analysis
Patient # Gender Age Focus (RH/LH) Length of EEG (hr.) Number of seizure
P1 M 19 RH 20h 37m 05s 3
P2 M 33 LH 09h 43m 57s 3


transverse and B lateral views of the brain, illustrating the depth and subdural electrode

placement for EEG recordings are depicted. Subdural electrode srips are placed over the

left orbitfrontal (LOF), right orbitofrontal (ROF), left subtemporal (LST), and right

subtemporal (RST) cortex see Figure 3-1. The EEG recording data for epilepsy patients

were obtained as part of pre-surgical clinical evaluation. They had been obtained using

a Nicolet BMSI 4000 and 5000 recording system, using s 0.1 Hz high-pass and a 70 hz

low pass filter. Each recording contains a total number of 28 to 32 intracranial electrodes

(8 subdural and 6 hippocampal depth electrodes for each cerebral hemisphere). Prior to

storage, the signals were sampled at 200Hz using an analog to digital converter with 10

bits quantization. The recordings were stored digitally onto high fidelity video type. Two

epilepsy subjects (see Table 3-1) were included in this study.



(A) (B)


ROF 3 2 3 LOF LTD

RST LST


LOF
LST

RTD LTD


Figure 3-1. Electrode placement









5. Infections: Infections of the nervous system may result in seizure activity. These

include infection of the covering of the brain and the spinal fluid (meningitis),

infection of the brain (encephalitis), and human immunodeficiency virus (HIV) and

related infections.

6. Tumors: Cancerous (malignant) and benign brain tumors may be associated with

seizures. The location of the lesion influences the risk.

7. Cerebral palsy: Epilepsy is often a symptom of cerebral palsy, which results from

lack of oxygen, infection, or trauma during birth or infancy.

8. Febrile seizures: Febrile seizures occur in small children and are caused by high

fever.

1.3 Classification of Epileptic Seizure (ICES 1981 revision)

The ICES (Commission on Classification and Terminology of the ILAE, 1981), The

ICES recognizes 18 subclasses of simple focal seizures belonging to four groups, four

modalities of complex focal seizures, and three types of secondarily generalized seizures.

1.3.1 Partial Onset Seizures

Partial onset seizures are those in which, in general, the first clinical and EEG

changes indicate initial activation of a limited group of neurons. A partial seizure is

classified primarily on the basis of whether or not consciousness is impaired during the

attack. When consciousness is not impaired, the seizure is classified as a simple focal

seizure. When consciousness is impaired, the seizure is classified as complex focal seizures.

In patients with impaired consciousness, aberrations of behavior (automatisms) may

occur. A partial onset seizure may not terminate, but instead progress to a generalized

motor seizure. Impaired consciousness is defined as the inability to respond normally

to exogenous stimuli by virtue of altered awareness and/or responsiveness. There is

considerable evidence that partial onset seizures usually have unilateral hemispheric

involvement and only rarely have bilateral hemispheric involvement; complex partial onset









series acquired from electrode x is presented by electrode y and vice versa. Let X be the

set of data points where its possible realizations are x, x2, x3, ..., x~ with probabilities

P(xI), P(2), P(x3),...P(x,). The Shannon entropy H(X) of X is defined as Mutual
information has been applied for measuring the interdependency between two time series.

Many previous studies have shown its superior performance over the traditional linear

measures [99-104].

Kraskov et al., 2004 introduced two classes of improved estimators for mutual

information M(X, Y) from samples of random points distributed according to some joint

probability density p(x, y). In contrast to conventional estimators based on histogram

approach, they are based on entropy estimates from k nearest neighbour distances. Let

us denote the time series of two observable variables as X = {xi} and Y ={yj}N1,

where N is the length of the series and the time between consecutive observations (i.e.,
.',,1i].:,u. period) is fixed. Then the mutual information is given by:


( P((xi) ,yj) (66)
i j

where px(i) = -dx, p,(i) = -dy and


p(i,j) / p(x, y)dxdy (6-17)

"J "denotes the integral over bin i. If nx(i) and ny(j) are the number of data points in
the ith bin of X and jth bin of Y; n(i,j) is the number of data points in the intersection

bin (i,j). The probabilities are estimated as px(i) n (i)/N, px(j) n(j)/N and

p(i,j) a nx(j)/N. Rather then bin approach the mutual information can be estimated
from k-nearest neighbor statistics.

We first estimate H(X) from X by


H(X) t N P(X xi). (6-18)
i= 1


























Figure 5-3. Support vector machines


input of C-SVM is the degree of connectivity between different brain regions. Given n

time series data points, each with m time stamps, the proposed framework will decrease

the number attributes by 2(n 1)/m times. Let I be the total number of data points, the

dimensionality can be reduced from A E Rlxnxm to A E cRx -).

The connectivity among the 18 EEG channels is calculated for each sample as shown

in Figure 5-4. The connectivity are Euclidean Distance-based between channel i and

channel j, i / j, for i,j = 1,..., 18. For example: let Ci and Cj denote EEG time

series from channel i and channel j, respectively. Each epoch of time series has length

30 seconds, which is equal to 7,500 points. So the size of vector C, (or Cj) is 7,500. The

connectivity between C, and Cj using Euclidean distance we obtain:

7500


E Uij k 1
7500

,and 18 x 17 connectivity profiles for each sample. Thus, the C-SVM transforms each EEG

time series sample into this with (18 x 17) number of attributes. C-SVM largely reduces

number of attributes from m = (18 x 7500) to m = (18 x 17) and also saves memory

resources and computational time.









t = 0 t = oo), the structure of the trajectory (path) will shrink for a dissipative

system. For a dissipative system, after a sufficient long time,the number of variables d

used to describe state space reduced to a small set of A. This set of state variable is called

an Ii ,1 II i .i ". An attractor can be classified into one of the following four different

categories:

1. Saddle point: For any given initial conditions, after a sufficient long time, the

solution may converge to the same final state (fixed point). An example for this

attractor is a constant series.


x(t) = x(0), t oo (2-1)


2. Limit cycle: Instead of converging to a fixed point the dynamical system may

converge to a set of states, which are visited periodically. A limited cycle attractor is

a closed trajectory through state space.


x(t) x(t + T), (2-2)


where T denotes the period of this cycle.

3. Limit tori: A limit tori attractor is a limit cycle attractor with multiple period.

This attractor will no longer be closed and limited cycle becomes a limit torus.

4. Strange attractor: The existence of this type of attractor was unknown until the

development of nonlinear dynamics. A strange attractor is defined as an attractor

that shows sensitivity to initial conditions (exponential divergence of neighboring

trajectories), it may appear to be stochastic in time domain. A strange attractor

exhibits regular structure in the phase space (See Figure 2-1, 2-5 for R6ssler and

Lorenz attractor).

Recall an attractor is a set of state variables; geometrically an attractor can be a

point, a curve, a manifold, or even a complicated set with a fractal structure known as the

-li i i,.- attractor". Describing these attractors has been one of the achievements of chaos









7.4.3 Maximum Clique Algorithm

In this section, we discuss the results from analyzing the structural property of

the brain network using the maximum clique approach. As mentioned earlier, the idea

of applying the maximum clique technique is different from the one using the spectral

partitioning approach as we are only interested in the most highly synchronized group of

electrodes in the brain network. We adopted the algorithm to find a maximum clique in

the brain connectivity graph after deleting the insignificant arcs in the original complete

graph as follows: Let G = G(V, E) be a simple, undirected graph where V = {1,..., n}

is the set of vertices (nodes), and E denotes the set of arcs. Assume that there is no

parallel arcs (and no self-loops joining the same vertex) in G. Denote an arc joining vertex

i and j by (i, j). We define a clique of G as a subset C of vertices with the property that

every pair of vertices in C is connected by an arc; that is, C is a clique if the subgraph

G(C) induced by C is complete. Then, the maximum clique problem is to find a clique C

with maximum cardinality (size) IC|. The maximum clique problem can be represented

in many equivalent formulations (e.g., an integer programming problem, a continuous

global optimization problem, and an indefinite quadratic programming). In this paper, we

represent it in a simple integer programming form given by

We analyzed 3 epochs of 3-hour EEG recordings, 2 hours before and 1 hour after

a seizure, from Patient 2 who had the epileptogenic areas on both right and left mesial

temporal lobes. Figures 7-7 and 7-8 demonstrate the electrode selection of the maximum

clique group during two hours before and one hour after the seizure onset. During the

period before the seizure onset, both figures manifested a pattern where all the LD

electrodes were consistently selected to be in the maximum clique. During the seizure

onset, the size of the maximum clique increases drastically. This is very intuitive because,

in temporal lobe epilepsy, all of the brain areas are highly synchronized. We visually

inspected the raw EEG recordings before and during the seizure onsets and found a

similar semiological pattern of the seizure onset electrodes from the L(T)D areas











z (to


x(I n)
end of
data set


z (I)"
z1Q


"fiduciry"
trajectory


Figure 2-9. Estimation of Lyapunov exponent (Lmax)


and duration T. If Dt is the sampling period, then


T = (N- 1)- Dt At (p- 1). -. (2-26)


If the evolution time At is given in second, then the unit of L is bit/sec. The selection of

p is based from Takens' embedding theorem and was estimated from epoches during ictal

EEG recordings. Takens' embedding theorem is defined:


f(t) A (x(t), x(t + ),...x(t + 2n T))T, (2-27)


using the above defined fx, even if one only observes one variable x(t) for t oo, one can

construct an embedding of the system into a p = 2m + 1 dimensional state space.

The dimension of the ictal EEG attractor is found between 2 to 3 in the state

space. Therefore according to Takens' the embedding dimension would be at least

p = 2 3 + 1 7. The selection of '- is chosen as small as possible to capture the

highest frequency component in the data.

2.6 Phase/Angular Frequency

Phase/ angular frequency ,max estimates the rate of change of the stability of a

dynamical system. Thus, it complements the Lyapunov exponent, which measures the

local stability of the system. The difference in phase between two evolved states X(ti) and






























2










-1


-2


-3


-4


-5



0 5 10 15 20
time (s)


Figure 2-3. Y component of Rbssler system


25 30 35

















































Figure 8-1. Nonlinear interdependences for electrode FP1


before and after treatment. Although the results indicate that the mutual information

and nonlinear interdependencies measures could be useful in determining the treatment

effects for patients with ULD. To prove the usefulness of the proposed study, a larger

patient population is needed. The approaches in this study are a bivariate measures, since

a multivariate measure is not easy to model and has not been resolved. The decoupling

between frontal and occipital cortical regions may be caused by decreased driving force

deep inside the brain. In other words, the effect of the treatment may reduce the couple

strength between thalamus and cortex in ULD subjects. Nevertheless, the limitations

must be mentioned, it has been reported that it is necessary to take into account the


P3
Before After


Before After


Before


After









above results -ii--.- -1 the existing the treatment effects on the coupling strength and

directionality of information transport between different brain cortical regions.


Table 8-2. Topographical distribution
(DE))
Electrode DE for P1
(1) Fpl F3, C4, P4, F7
T4, T5, 01
(2) Fp2 C3, C4, F8, T4
T5, Pz
(3) F3 Fpl, C4, P4, 02
(4) F4 C4, P4 02
(5) C3 Fp2, C4, P3, 01
(6) C4 Fpl, Fp2, F3, F4, C3
Al N/A
A2 N/A
(7) P3 C3, 02
(8) P4 Fpl, F3, F4
(9) 01 Fpl, C3
(10) 02 F3, F4, P3
(11) F7 Fpl, Fz, Pz
(12) F8 Fp2
(13) T3 NONE
(14) T4 Fpl, Fp2, Cz
(15) T5 Fpl, Fp2
(16) T6 NONE
(17) Fz F7
(18) Pz Fp2, F7, Cz
(19) Cz T4, Pz


for treatment decoupling effect (DE: Decouple Electrode


DE for P2
Fp2, F3, F8, T5

Fpl, F4, T6, 02

Fpl, C3, P3, Pz
Fp2, P4, 02, Fz
F3, P3, 01
P4, T6, 02
N/A
N/A
F3, C3, T3, Pz
F4, C4
C3, T5, T3, F7
Fp2, F4, C4
C3, T5, 01
Fpl, C4, P4
P3, 01
Fp2, 01
Fp2, C4
C4
Fpl, F4
F3, P3, Cz
Pz


DE for P3
F3, F7

F8

Fpl
Cz
P3, 01
P4, 02
N/A
N/A
01
C4
P3, Pz
C4
Fpl
Fp2
NONE
NONE
NONE
NONE
Cz
02
F4. Fz


DE for P4
F3, P3, Fz, T5
F8, T4, Fz
C3, C4, T5, P4
Cz
Fpl, F7
C4, Fz
Fp2, P3, 01
Fpl, Fp2, F3
N/A
N/A
Fpl, C3, Cz
Fp2,
C3
Pz
F3
Fp2,
NONE
Fp2,
Fp2, Cz
NONE
Fpl, Fp2, F3
02
Fp2, T5, P3


8.5 Conclusion and Discussion

The effectiveness of the new AEDs is currently accessed using myoclonus severity with

the UMRS. As mentioned above, it is not easy to perform such evaluation scheme precisely

especially in the later stages of the disease. Furthermore, the UMRS is a skewed measure

that may not detect functional changes in a patient when these changes may be clinically

important. The outcome from UMRS in this study did not evaluate the severity of the

patients accurately, as the P1 was with less severity of ULD determined by the clinical

experienced neurologist. The present study measure the synchronization behaviors and

nonlinear interdependences, in a straightforward manner, in the cortical network during









similarity patterns [123; 121]. For example, [124] used frequency coherence measures to

investigate the interactions between medial limbic structures and the neocortex during

ictal periods (seizure onsets). In another study by [125], the coherence pattern of cortical

areas from epileptic brain was investigated to identify a cortical epileptic system during

interictal (normal) and ictal (seizure) periods.

Although linear measures are very useful and commonly used, they are insensitive

to nonlinear coupling between signals, and non-linearities are quite common in neural

contexts. To be able to investigate more of the interdependence between EEG electrodes,

nonlinear measures should be applied. Nonlinear measures have been widely used to

determine the interdependence among EEG signals from different brain areas. For

example, [106] and [35] studied the similarity between EEG signals using nonlinear

dynamical system approaches. They applied a time-delay embedding technique to

reconstruct a trajectory of EEG in phase space and used the idea of generalized

synchronization proposed by [126] to calculate the interdependence and causal relationships

of EEG signals.

We propose an approach to investigate and quantify the synchronization of the brain

network, specifically tailored to study the propagation of epileptogenic processes. [127]

investigated this propagation, where the average amount of mutual information during

the ictal period (seizure onset) was used to identify the focal site and study the spread of

epileptic seizure activity. Subsequently, [128] applied the information-theoretic approach

to measure synchronization and identify causal relationships between areas in the brain

to localize an epileptogenic region. Here, we apply an information-theoretic approach,

called cross-mutual information, which can capture both linear and nonlinear dependence

between EEG signals, to quantify the synchronization between nodes in the brain network.

In order to globally model the brain network, we represent the brain synchronization

network as a graph.
































Figure 2-11. Phase/Angular frequency of Lyapunov exponent (max)


in the human neonate and in epileptic activity in electrocardiograms (Diambra, 1999)

[63]. Mathematically, as part of a general theoretical framework, ApEn has been shown

to be the rate of approximating a Markov chain process [62]. Most importantly, compared

ApEn with Kolmogrov-Sinai (K-S) Entropy (Kolmogrov, 1958), ApEn is generally finite

and has been shown to classify the complexity of a system via fewer data points via

theoretical analysis of both stochastic and deterministic chaotic processes and clinical

applications [62; 64-66]. Here I give brief description about ApEn calculation for a time

series measured equally in time with length n. Suppose S = s, a2, ..., s is given and use

the method of delay we obtain the delay vector xl, x, ..x, xn-m+l in R':


Xi Si, Si+l, ., Si+m-l, (2-30)

C,(r) number ofxjsuch thatd(x xj) < rf
c (r) (2-31)
NV- m+1









epileptogenic focus will increase the chances for better outcome of epilepsy surgery in

patient with intractable MTLE.

Recent advances in nonlinear dynamics performed on EEG recordings have shown the

ability to characterize changes in synchronization structure and nonlinear interdependence

among different brain cortical regions. Although these changes in cortical networks are

rapid and often subtle, they may convey new and valuable information that are related

to the state of the brain and the effect of therapeutic interventions. Traditionally, clinical

observations evaluating the number of seizures during a given period of time have been

gold standard for estimating the efficacy of medical treatment in epilepsy. EEG recordings

are only used as a supplemental tool in clinical evaluations. In the later part of this study,

a connectivity support vector machine (C-SVM) is developed for differentiating patients

with epilepsy that are seizure free from those that are not. To that end, a quantitative

outcome measure using EEG recordings acquired before and after anti-epileptic drug

treatment is introduced. Our results indicate that connectivity and synchronization

between different cortical regions at higher order EEG properties change with drug

therapy. These changes could provide a new insight for developing a novel surrogate

outcome measure for patients with epilepsy when clinical observations could potentially

fail to detect a significant difference.









False positive (FP): False positive answers denoting incorrect classifications of

negative cases into the positive cases;

A classification result is considered to be true positive if the D-SVM classify a

interictal EEG epoch as a preictal EEG sample.

False negative (FN): False negative answers denoting incorrect classifications of

positive cases into the negative cases;

A classification result is considered to be true positive if the D-SVM classify a

preictal EEG epoch as a interictal EEG sample.

The performance of the D-SVM is evaluate using sensitivity and specificity:

Sensitivity (TPFN)

Specificity TNP

The sensitivity can be interpreted as the probability of accurately classifying EEG epochs

in the positive case. Specificity can be consider as the probability of accurately classifying

EEG epochs in the negative class. In general, one alv-wb wants to increase the sensitivity

of classifiers by attempting to increase the correct classifications of positive cases (TP). On

the other hand, false positive rate can be considered as (1 specificity) which one wants

to minimize.

2.12 Patient Information and EEG Description

The information of the patients and the EEG recording are summarized in the table

below. For each patient, we randomly selected 200 epochs from interictal state, each epoch

is 10.24 seconds long in duration as input to D-SVM classification scheme. The interictal

and ictal state is defined as: 1. interictal state: 1 hour away from ictal state 2. preictal

state: 5 minutes data length prior to ictal state

2.13 Results

The results of this study indicate that D-SVM can correctly the detect preictal state

with high sensitivity and specificity. For the patients with bi-lateral seizure onset zone

the performance of D-SVM is better than those with uni-lateral seizure onset zone. The









activated simultaneously. Low EEG frequency indicates less responses of the brain, such

as sleep, whereas higher EEG frequency implies the increased alertness. Given the above

descriptions, an acquired EEG time series can be defined as a record of the fluctuating

brain activity measured at different times and spaces. The high degree of synchronicity for

two different brain regions implies strong connectivity among them and vice versa. We will

interchangeably use the terms synchronicity and connectivity for rest of the chapter.

Although the brain may have originally emerged as an organ with functionally

dedicated regions, recent evidence -.-. -: --I that the brain evolved by preserving,

extending, and re-combining existing network components, rather than by generating

complex structures de novo [114; 115]. This is significant because it -ii:--. -I- (1)

the brain network is arranged such that the functional neural complexes supporting

different cognitive functions share many low-level neural components, and (2) the

specific connection topology of the brain network may pl i, a significant role in seizure

development. This line of thinking is also supported by [70], which demonstrates

that specific connected structures are either significantly abundant or rare in cortical

networks. If seizures evolve in this fashion, then we should be able to make some specific

empirical hypotheses regarding the evolution of seizures, that might be borne out by

investigating the synchronization between the activity in different brain areas, as revealed

by quantitative analysis of EEG recordings. The goal of this study is to test the following

two hypotheses. First, we should expect the brain activity in the orbitofrontal areas are

highly correlated while the activity in the temporal lobe and subtemporal lobe areas

are highly correlated with their own side (left only or right only) during the pre-seizure

period. The high correlation can be viewed as a recruitment operation initiated by an

epileptogenic area through a regular communication channel in the brain. Note that

the connection of these brain areas has been a long-standing principle in normal brain

functions and we believe that the same principle should hold in the case of epilepsy as

well. Second, we should expect some brain regions to be consistently active, manifested









the first local minimum of I(Q, S), rather than some subsequent minimum, should

probably be chosen for the sampling interval T,.

3.4 The Minimum Embedding Dimension

Dynamical systems processing d degree of freedom which may choose to proceed on a

manifold of much lower dimension, so that only small portions of the degrees of freedom

are actually active. In such case it is useful to estimate the behaviors of degrees of freedom

over a period of time, and it is obvious that this information can be obtained from that

dimension of attractor from the corresponding system. If one chooses the embedding

dimension too low this results in points that are far apart in the original phase space

being moved closer together in the reconstruction space. Takens delay embedding theorem

states that a pseudo-state space can be reconstructed from infinite noiseless time series

(when one choose d > 2dA) is often been used when reconstructing the delay vector [52].

There are several classical algorithms used to obtain the minimum embedding dimension

[76; 74; 77]. The classical approaches usually require huge computation power and vast

among of data. Another limitation of these algorithms is that they usually subjective to

different types of data.We evaluated the minimum embedding dimension of the attractors

from the EEG by using Caos method. The notions here followed "Practical method for

determining the minimum embedding dimension of a scalar time series". Suppose that we

have a time series (x, x2, X3, ..., N). Applying the method of delay we obtain the time

delay vector as follows:


yi(d) = (xi,Xi+ ,..., Xi+(d+l)-), i = 1,2,...,N (d 1); (3-6)

where d is the embedding dimension and T is the time-delay and yi(d) means the ith

reconstructed vector with embedding dimension d. Similar to the idea of the false nearest

neighbor method, defining

II y(d + 1) n(i,d)(d + 1) ||
a(, d)=) id)(d i ,2,...,N dr (3-7)
II yi(d) Yn(i,d)(d) 11
















1111


R.A
RLu
R1.
E.2 -
RLI
RF
RF3
RF2
RFI
LFN
LF3
LF2
LF -I
RtT
RT3
RT2
RTI -
LT -


RDI -
RD -
R D
R D2
LDII
LDW
LTi1
LD3 -
LDI I


Minutes


Figure 7-8. Electrode selection using the maximum clique algorithm for Case 2


Figure 7-9. Electrode selection using the maximum clique algorithm for Case 3



processes slowly develop themselves through a regular communication channel in the brain

network, rather than abruptly disrupt, collapse, or change the way brains communicate.

From this observation, we postulate that this phenomenon may be a reflection of neuronal

recruitment in seizure evolution. This observation confirms our first hypothesis. In


Seizure Onset- r



A I Il III
..1m iiI


I I I I I I I









addition, we have found that nodes in the brain network are clustered during the seizure

evolution. Most brain areas seem to be communicating with their physiological neighbors

during the process. The key process of seizure evolution could be the step where the

epileptogenic areas) govern or manipulate the other vulnerable, or easily synchronized,

brain areas to communicate with their neighbors. This can be viewed as a recruitment

of other brain areas done by the epileptogenic areass. In most cases, the recruitment of

seizure development should start with a weaker group, which in our case is represented

by a vulnerable brain area. After enough neurons have been recruited, the disorders of

epileptic brains spread out abnormal functions from than localized areas of cortex or

other vulnerable areas throughout the cortical networks and the entire brain network.

This phenomenon was shown by the results of our maximum clique approach, which

confirms our second hypothesis. In addition, a different type of maximum clique patterns

may be useful in the identification of incoming seizures. This study .i i.-- -1 that, in the

future, this framework may be used as a tool to provide practical seizure interventions.

For example, one can locate and stimulate the brain areas that seem to be vulnerable to

the seizure evolution by electrical pulses through the monitoring process of the maximum

clique. This will drastically reduce the risk of seizure to epilepsy patients.

7.5 Discussion and Future Work

In this study, we attempted to study seizure evolution by investigating some neuronal

interactions among different brain areas. Analyzing multidimensional time series data

like multichannel EEG recordings is a very complex process. The study of the brain

network needs to involve the neuronal activities from not only a single source or a small

group of sources, but also the entire brain network. Here we applied the cross-mutual

information technique, a measure widely used in the information theory, to capture the

neuronal interactions through the brain's synchronization patterns. Then we modeled the

global interactions using network/graph-theoretic approaches, spectral partitioning and

maximum clique. These approaches are used to generalize the brain network investigation









3-2 Average minimum embedding dimension profiles for Patient 1 (seizure 1) . 67

3-3 Average minimum embedding dimension profiles for Patient 1 (seizure 2) . 68

3-4 Average minimum embedding dimension profiles for Patient 1 (seizure 3) . 69

3-5 Average minimum embedding dimension profiles for Patient 2 (seizure 4) . 70

3-6 Average minimum embedding dimension profiles for Patient 2 (seizure 5, 6) 71

4-1 32-channel depth electrode placement ............... .... 76

4-2 Degree of nonlinearity during preictal state .................. .. 81

4-3 Degree of Nonlinearity during postictal state ................ 82

4-4 STLmax and T-index profiles during interictal state ............... ..83

4-5 STLma and T-index profiles during preictal state ................ ..84

4-6 STLmax and T-index profiles during postictal state ............... ..85

4-7 Nonlinearities across recording areas during interictal state for ULSOZ patients 86

4-8 Nonlinearities across recording areas during perictal state for ULSOZ patients 86

4-9 Nonlinearities across recording areas during postictal state for ULSOZ patients. 87

4-10 Nonlinearities across recording areas during interictal state for BTSOZ patients 88

4-11 Nonlinearities across recording areas during preictal state for BTSOZ patients 89

4-12 Nonlinearities across recording areas during postictal state for BTSOZ patients 90

5-1 Scalp electrode placement .................. ........... .. 93

5-2 EEG dynamics feature classification .................. ...... .. 99

5-3 Support vector machines .................. ............ .. 101

5-4 Connectivity support vector machine .................. ..... 102

7-1 EEG epochs for RTD2, RTD4 and RTD6 (10 seconds) . . ..... 126

7-2 Scatter plot for EEG epoch (10 seconds) of RTD2 vs. RTD4 and RTD4 vs.
RTD 6 ......................... ...... .. ... . 126

7-3 Cross-mutual information for RTD4 vs. RTD6 and RTD2 vs. RTD4 ..... ..127

7-4 Complete connectivity graph (a); after removing the arcs with insignificant connectivity
(b) ... .... ..... ... .................. ..... 128

7-5 Spectral partitioning .................. .............. .. 129





























* Temporal depth (LTD & RTD))

* Subtemporal (LST & RST)

D Orbltofrontal (LOF & ROF)






p-value = 0.9975


4








3








2








1-







n ---


'C.?

V)''.~"


Patients


Figure 4-12. Nonlinearities across recording areas during postictal state for BTSOZ

patients


;: "A.


S *'d







',wV
,A
,


V.9








<.v


At
A*









4.~




A e/
.:I


~ -''' -~"" -' -'''''


'~'~'









CHAPTER 3
QUANTITATIVE COMPLEXITY ANALYSIS IN MULTI-CHANNEL INTRACRANIAL
EEG RECORDINGS FROM EPILEPSY BRAIN

3.1 Introduction

Epilepsy is a brain disorder characterized clinically by temporary but recurrent

disturbances of brain function that may or may not be associated with destruction or

loss of consciousness and abnormal behavior. Human brain is composed of more than

10 to the power 10 neurons, each of which receives electrical impulses (known as action

potentials) from others neurons via synapses and sends electrical impulses via a sing

output line to a similar (the axon) number of neurons (Shatz 1981). When neuronal

networks are active, they produce a change in voltage potential, which can be captured by

an electroencephalogram (EEG).

The EEG recordings represent the time series that match up to neurological activity

as a function of time. The structure of EEG recordings represent the inter activities

among the groups of neurons. Many investigators have applied nonlinear dynamical

methods to a broad range of medical applications. Recent developments in nonlinear

dynamics have shown the abilities to explain some underlining mechanisms of brain

behavior [67-71]. It is known that a dynamical system with d degree of freedom may

evolve on a manifold with a lower dimension, so that only portions of the total number

of degree of freedom are actually active. For a simple system with limit cycles, it is

obvious that time-delay embedding produce an equivalent reconstruction of the true

state. According to embedding theorem from Whitney (1936), an arbitrary D-dimension

curved space can be mapped into a Cartesian (rectangular) space of 2d + 1 dimensions

without having any self intersections, hence satisfying the uniqueness condition for

an embedding [51]. Sauer et al. (1991) generalized Whitneys and Takens' theorem to

fractural attractors with dimension Df and showed the embedding space only need to have

a dimension greater than 2Df [72]. Although it is possible for a fractal to be embedded in

another fractal, we only consider the integer embedding. Takens delay embedding theorem































15



10














-15



-15
I I I I
0 5 10 15 20
time (s)


Figure 2-6. X component of Lorenz system


25 30 35









reported decrease in complexity quantify by the correlation density. Le Van Quyen et

al. (1999) showed drop in dynamical similarity before seizures using a measure called

dynamicall similarity index "; Lehnertz and Elger (1998) demonstrated seizure prediction

by dynamicall complexity "time series analysis. Litt et al., (2001) showed increase in

accumulated -S,1 I! energy "prior to seizure onset. See also [42]. Mormann et al., (2003)

detected the preictal state based on decrease in "synchronization "measures [43].

The basic text of nonlinear dynamics and nonlinear dynamical models are presented

in the following sections. Nonlinear dynamical measures namely (the largest Lyapunov

exponent (Lmax), Phase/ Angular frequency (+), Approximate entropy (ApEn)) were

used for detecting the preictal transitions in intracranial EEG recordings acquired from

patients with intractable MTLE. Since the underlying dynamics of preictal transitions

is changing from case to case, this demands sophisticate analytical tools which have the

ability for identifying the changes of brain dynamics when preictal transitions occur. The

preictal detection performance was further improved by proposed dynamics support vector

machine (D-SVM), a classification method developed from optimization theory and data

mining techniques. The detection performances were summarized in the later part of this

chapter.

2.2 Dynamical Systems and State Space

In this section, the basic theory about nonlinear dynamical systems will be given. A

dynamical system consists of a set of d state variables, such that each state of the system

map to a point p E M. Thus M is d dimensional manifold. A system is said to be a

dynamical system if state of the system changes with time. Let us denote p(t) be the state

of a system at time T, as time evolve (e.g., t = 0 t = 10, 000), the evolution of the

state of the system through state space will form a path. This is path is call "trajectory

". If the current state p(t) uniquely determines all the future state in time, the system is

said to be a deterministic dynamical system. If the mapping is not unique, the system is

called a stochastic dynamical system. As p(t) evolve for a sufficient amount of time (e.g.,









[136] D.V. Moretti, C. Miniussi, G.B. Frisoni, C. Geroldi, O. Zanetti, G. Binetti, and
P.M. Rossini. Hippocampal atrophy and eeg markers in subjects with mild cognitive
impairment. Clinical N ,.,. il'l.;.: ,; 118(12):2716-2729, 2007.

[137] R. Quian Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger. Performance
of different synchronization measures in real data: A case study on
electroencephalographic signals. Phys. Rev. E, 65:041903, 2002.

[138] H. Unverricht. Die MI;, I.' .-.:, Franz Deutick, L. il. i.- 1891.

[139] H.B. Lundborg. Die progressive /l;,. '..ai ,-Epilepsie (Unverrichts -I;,.1. /, '.:,).
Almqvist and Wiksell, Uppsala, 1903.

[140] N.K. C!i. --, P. Mir, M.J. Edwards, C. Cordivari, D.M., S.A. Schneider, H.-T. Kim,
N.P. Quinn, and K.P. Bhatia. The natural history of unverricht-lundborg disease: A
report of eight genetically proven cases. Movement Disorders, Vol. 23, No. 1:107-113,
2007.

[141] E. Ferlazzoa, A. Magauddaa, P. Strianob, N. Vi-Hongc, S. Serraa, and P. Gentonc.
Long-term evolution of eeg in unverricht-lundborg disease. E1,.:I, I/"; Research,
73:219-227, 2007.

[142] M.C. Salinsky, B.S. Oken, and L. Morehead. Intraindividual analysis of antiepileptic
drug effects on eeg background rhythms. Electr ,' ,''., i',;,',i' 1/1,1; and Clinical
N(. ;-I,,i;.: ...,I ; 90(3):186-193, 1994.

[143] T.M. Cover and J.A. Thomas. Elements of Information The ..;, Wiley, New York,
1991.

[144] M.R. C'!I ~i i 1: Bootstrap Methods: A Practitioner's Guide. Wiley-Interscience,
1999.

[145] M. Steriade and F. Amzica. Dynamic coupling among neocortical neurons during
evoked and spontaneous spike-wave seizure activity. Journal of N *,*, (i, ,'.:li. .i~;
72:2051-2069, 1994.

[146] E. Sitnikova and G. van Luijtelaar. Cortical and thalamic coherence during
spikewave seizures in wag/rij rats. F/'.:/* I"-; Research, 71:159-180, 2006.

[147] E. Sitnikova, T. Dikanev, D. Smirnov, B. Bezruchko, and G. van Luijtelaar. Granger
causality: Cortico-thalamic interdependencies during absence seizures in wag/rij rats.
Journal of Neuroscience Methods, 170(2):245-254, 2008.









study, prior to calculate Lma, the EEG recordings were first divided into non-overlapping

window with 10.24 second in duration. The same segmentation procedure was also used by

lasemidis et al., (1991) such segmentation technique is often applied especially for medical

time series [55].

4.2.3 Surrogate Data Technique

The degree of nonlinearity of a signal can be examined by testing the null hypothesis:

"The signal results from a Gaussian linear stochastic process ". One way to test this

hypothesis is to estimate the difference in a discriminating statistic between the original

EEG and its surrogate [81]. There are three different procedures for surrogate data.

1. Surrogates are realizations of independent identically distributed (iid) random

variables with the same mean, variance, and probability density function as the

original data. The iid surrogates were generated by randomly permuting in temporal

order the samples of the original series. This shuffling process will destroy the

temporal information and thus generated surrogates are mainly random observation

drawn (without replacement) from the same probability distribution as original data.

2. Fourier transform (FT) surrogates are constrained realizations of linear stochastic

processes with the same power spectra as the original data. FT surrogate series were

constructed by computing the FT of the original series, by substituting the phase of

the Fourier coefficients with random numbers in the range while keeping unchanged

their modulus, and by applying the inverse FT to return to the time domain. To

render completely uncoupled the surrogate pairs, two independent white noises

where used to randomize the Fourier phases.

3. Auto regressive (AR) surrogates are typical realizations of linear stochastic processes

with the same power spectra as the original series. By generating a Gaussian time

series with the same length as the data, and reordered it to have the same rank

distribution. Take the Fourier transform of this and randomize the phases (FT).

Finally, the surrogate is obtained by reordering the original data to have the same









Arnhold et al., (1999) introduced another nonlinear interdependence measure

H(k) (X Y) as

H(k)(X Y) =log R() (6-33)
N 1 R}(X Y)
H(k)(XIY) = 0 if X and Y are completely independent, while it is possible if closest in

Y implies also closest in X for equal time indexes. H(k)(XIY) would be negative if close

pairs in Y would correspond mainly to distant pairs in X. H(k)(XIY) is linear measures

thus is more sensitive to weak dependencies compare to mutual information. Arnhold et

al., (1999) also showed H was more robust against noise and easier to interpret than S.

Since H is not normalized Quiroga et al., (2002) introduced another N(XIY):

1 R, (X) ~?R(X|Y)
N(k)(XY) R ) (6-34)
N R. ( X)

which is normalized between 0 and 1. The opposite interdependencies S(YIX), H(YIX),

and N(YIX) are defined in complete analogy and they are in general not equal to

S(XIY), H(XIY), and N(XIY), respectively. Using nonlinear interdependencies on

several chaotic model (Lorenz, Roessler, and Heenon models) Quiroga et al., (2000)
showed the measure H is more robust than S.

The .-i-vilii. I ry of above nonlinear interdependencies is the main advantage over

other nonlinear measures such as the mutual information or the phase synchronization.

This .,-vmmetry can give information about "driver-response "relationships but can also

reflect different properties of dynamical systems when it is importance to detect causal

relationships. It should be clear that the above nonlinear interdependencies measures are

bivariate measures. Although it quantified the "driver-response "for given input-the whole

input space under study might be driven by other unobserved systemss.

6.6 Discussions

It is believed that synchronization occurs due to both local and global discharges of

the neurons. From the epilepsy perspective, quantifying the changes in spatiotemporal

interactions could potentially lead to the development of seizure-warning systems and









For X and Y time series we define d( = I xjYd = ,I yj I as the distances

for xi and yi between every other point in matrix spaces X and Y. One can rank these

distances and find the knn for every xi and yi. In the space spanned by X, Y, similar

distance rank method can be applied for Z = (X, Y) and for every zi = (xi, yi) one can

also compute the distances d) = zj and determine the knn according to some

distance measure. The maximum norm is used in this study:

d() -max{ | xj, i, yj }, d) = xi xj\. (6-19)

Next let '() be the distance between zi and its kh" neighbor. In order to estimate the

joint probability density function (p.d.f.), we consider the probability Pk () which is the

probability that for each zi the kth nearest neighbor has distance dc from zi. This

probability means that k 1 points have distance less than the kth nearest neighbor and

N k 1 points have distance greater than and k 1 points have distance less than
-. Pk () is obtained using the multinomial distribution:


Pk() k dc -(1 p)N-k-1, (6-20)

where pi is the mass of the e-ball. Then the expected value of logpi will be:

E(logpi) = (k)- Q(N), (6-21)

where Q(-) is the 1:.it,,,,na, function:


(t) F(t)-d ) (6-22)
dt

where F(-) is the gamma function. It holds that b(1) =C where C is the Euler -

Mascheroni constant (C 0.57721). The mass of the e-ball can be approximated (if we

consider the probability density function inside the ball is the same)by:


(e) cdP(X= x), (6-23)











from ai(yi(w xi + b) 1) = 0, for any i such that ai is not zero. D-SVM map a given EEG

data set of binary labeled training data into a high dimensional feature space and separate

the two classes of data linearly with a maximum margin hyperplane in the dynamical

feature space. In the case of nonlinear separability, each data point x in the input space is

mapped into a different dynamical feature space using some nonlinear mapping function

p. Figure 2-12 and 2-13 show the 3D plot for for entropy, angular frequency, and Lmax

during interictal (100 data points dynamical features 2 hours prior to seizure onset)and

preictal state (100 data points dynamical features sampled 2mins prior to seizure onset).


a


I





5$5



4.

4"




5
L00
4D0
1 ***


t$



4 3
*
*


p


W *


3900


S#tU


3700


360o


Ap V% 3500 3


Figure 2-12. Three dimension plot for entropy, angular frequency and Lmax during
interictal state









[14] J.S. Lockard, W.C. Congdon, and L.L. DuC'l! ii.:,. Feasibility and safety of vagal
stimulation in monkey model. Epilepsia, 31(S2):S20-S26, 1990.

[15] B.M. Uthman, B.J. Wilder, J.K. Penry, C. Dean, R.E. R-i-m-i, S.A. Reid, E.J.
Hammond, W.B. Tarver, and J.F. Wernicke. Treatment of epilepsy by stimulation of
the vagus nerve. N ,. ,,..../,; 43(7):13338-13345, 1993.

[16] G.L. MorrisIII and W.M. Mueller. Long-term treatment with vagus nerve
stimulation in patients with refractory epilepsy. the vagus nerve stimulation study
group e01-e05. N, ,,,. l..,. 54(8):1712, 2000.

[17] D. Ko, C. Heck, S. Grafton, M.L.J. Apuzzo, W.T. Couldwell, T. C', i'. J.D. Day,
V. Zelman, T. Smith, and C.M. DeGiorgio. Vagus nerve stimulation activates
central nervous system structure in epileptic patients during pet blood flow imaging.
Neurosurgery, 39(2):426-431, 1996.

[18] T.R. Henry, R.A.E. B 1: i,-, J.R. Votaw, P.B. Pennell, C.M. Epstein, T.L. Faber, S.T.
Grafton, and J.M. Hoffman. Brain blood flow alterations induced by therapeutic
vagus nerve stimulation in partial epilepsy: I. acute effects at high and low levels of
stimulation. Epilepsia, 39(9):983-990, 1998.

[19] E. Ben-Menachem, A. Hamberger, T. Hedner, E.J. Hammond, B.M. Uthman,
J. Slater, T. Treig, H. Stefan, R.E. R ,i- lic, J.F. Wernicke, and B.J. Wilder. Effects
of vagus nerve stimulation on amino acids and other metabolites in the csf of
patients with partial seizures. F1I.:I I/"-; Research, 20(3):221-227, 1995.

[20] S.E. Krahl, K.B. Clark, D.C. Smith, and R.A. Browning. Locus coeruleus lesions
suppress the seizure-attenuating effects of vagus nerve stimulation. Epilepsia,
39(7):709-714, 1998.

[21] J. Malmivu and R. Plonse. Bioelectromagnetism-Principles and Applications of
Bioelectric and Biomagnetic Fields. Oxford University Press, 1995.

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Nervenkr, 87:527-570, 1929.

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[25] D.A. Princea and B.W. Connors. Mechanisms of interictal epileptogenesis.
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[26] M.E. Weinand, L.P. Carter, W.F. El-Saadany, P.J. Sioutos, D.M. Labiner, and
K.J. Oommen. Cerebral blood flow and temporal lobe epileptogenicity. Journal of
Neuosurgery, 86:226-232, 1997.










algorithm:
begin






end
procedure:
begin
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
end


Figure 7-6. Maximum clique algorithm


initiated a highly organized rhythmic patterns and the patterns started to propagate

throughout all the brain areas. We initially speculated that the epileptogenic areas could

be the ones that are highly synchronized long before a seizure onset. In the previous case,

we observed that the L(T)D electrodes are the one that started the seizure evolution.

However, in a further investigation of EEG recordings from the same patient, we found

some contrast results. In Figure 7-9, the electrode selection pattern of the maximum

clique demonstrates a very highly synchronized group of electrodes in both left and right

orbitofrontal areas during the 2-hour period preceding the seizure. After visual inspection

on the raw EEG recordings, this seizure was initiated by the R(T)D area. Generally, it is


maximum clique

sort all nodes based on vertex ordering
LIST = ordered nodes
cbc = 0 current best clique size
depth = 0 current depth level
enter-next-depth(LIST,depth)

enter-next-depth(LIST,depth)

m = the number of nodes in the LIST
depth=depth+l
for a node in position i in the LIST
if depth+(m-i)< cbc then
return prune the search
else
mark node i
if no adjacent node then
cbc=depth (maximum clique found)
else
enter to next depth (adjacent node of i, depth)
end
end
unmark node i
if depth=1
delete node i from LIST
end
end







(A) Focus Ara (RmJ A-B e-----4------ (B)
A-C -._- -
I Subtemporalarea on fcal ide (RST} dA-0 --.--
o E- - -


pat uienCt 15 S BC or ......
6 _B-O
B A-F -----.-----

|C-D ------------------------
v> ,C -D t 1

0 DE r -- --...-
D-F -+- -m
2E-F ----. ---t

I I I I I I I I I I
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 2 3 4 5 response variable: Mean Tindex vJlu
Patients simultaneous 95 % confidence hmits, Tukey method

Figure 4-9. Nonlinearities across recording areas during postictal state for ULSOZ
patients.


procedure, but also will decrease the risk of infection caused by the implanted recording

electrodes. Large sample of patients with ULSOZ and BTSOZ will be required for reliable

estimation of sensitivity and specificity of this method. Correct identification of the focal

area in ULSOZ patients is a challenging task. An obvious question could be whether

any of brain areas where the EEG signals recorded from is close enough to the actual

focal area. If not, it would be very difficult to identify the focal area by an i' 1i, ~i-;' on

these EEG signals. Other issues such as the number of recording areas and number of

recording electrodes in each area could also affect the results of the analysis. In each

of the five ULSOZ patients studied here, EEG signals were recorded from six different

brain areas: left and right hippocampus, subtemporal, and orbitofrontal regions. All five

patients were clinically determined to have focal area in the right hippocampus. During

the interictal state, focal area (right temporal depth) consistently exhibited higher degree

of nonlinearity than in the contralateral temporal depth and subtemporal areas (significant

observations in 4 out of 5 patients). Similar findings were also observed during preictal

and postictal states. These results -ii--.- -1 that it is possible to identify the focal area









interdependencies measures among different brain regions during before and after add-on

AEDs treatment for the patient with ULD.

Given two time series x and y, using method of d. 1iv to obtain delay vectors

(Xn, ..., Xn-(m-l1)) and y, = (x,, ..., Xn-(,r-1)), where n = 1,...N, m is the embedding
dimension and r denotes the time delay [52]. Let r,,j and s,,j, j = 1, k denote the time

indices of the k nearest neighbors of xT and y,/. For each xn, the mean Euclidean distance

to its k neighbors is defined as
k
R() = ,(n r,,j)2 (6-29)
j= 1

and the Y-conditioned mean squared Euclidean distance is defined by replacing the

nearest neighbors by the equal time partners of the closest neighbors of y,
k
RXk)(X|Y) = (, xs,,j)2. (6 30)
j=1

The delay 7 = 5 is estimated by auto mutual information function, the embedding

dimension m = 10 is obtained using Cao's method using 10 sec EEG selected during

interinctal state and a Theiler correction is set to T = 50 [73; 107].

If :, has an average squared radius R(X) = (1/N) N 1 R}N-)(X), then Rek)

Rk)(X) < R(X) if the system are strongly correlated, while R}k)(XIY) R(X) > R(k)(X)
if they are independent. Accordingly, it can be define and interdependence measure

S(k)(X Y) as

S(k)(XY) =- I )(X) (6-31)
N zR (X|Y)
Since Rk)(X Y) > Rk)(X) by construction,

0 < S(k)(Xly) < 1 (6-32)

Low values of Sk(XIY) indicate independence between X and Y, while high values

indicate synchronization.









abnormal features in EEG recordings. Figure 1-7 is taken from Malmivu and Plonse

(1993) [21].

1.6.1 Scalp EEG Recording

Scalp EEG recording is a most common recording method for monitoring the

electrical activity of the brain. See figure 1-8. The recording electrodes are placed on

the scalp of the head and record electrical potential differences between the recording

electrodes. However, recordings acquired from scalp are usually contaminated by multiple

sources of artifacts such as movement artifacts, chewing artifacts, eye movement, vertex

waves and sleep spindles, etc. The international 10-20 electrode placement system is

commonly used for routine scalp EEG recording [24]. Figure 1-8 is taken from Malmivu

and Plonse (1993) [21].

A B Nasion
20% Vertex i 10%


/20% 20%

020%/ P3 20%
I I I IA I
10% 7 1 A\ A C3 C C T 2
ITN T"5p if1-
Nasion 10%

A A 10% T 20%
Preaurical Inion '
g point / Inion \,
V 20%

Inion 10%

Figure 1-8. International 10-20 electrode placement


1.6.2 Subdural EEG Recording

The subdural recordings provide less unwanted information in the signals by placing

the electrodes under the scalp. See figure 1-9. It requires surgical procedure to place

the subdural recording electrodes and the risks of infection increase with the amount









do not elicit larger the amplitude of the action potential. Therefore, the intensity

of a stimulus is encoded in the frequency of action potential rather than in their

amplitude. Third, the action potential "ti i,. !- along the axon without fading out

because the signal is regenerated at each .,li i:ent membrane.

1.6 Recording Electric Brain Activity

The first brain electrical scalp recordings of human was performed by Hans Berger in

1929 [23]. Since then the EEG recordings has been the most common diagnosis tool for

epilepsy. EEG measures the electrical activity of the brain. EEG studies are particularly

important when neurologic disorders are not accompanied by detectable alteration in

brain structure. It is accepted that the neurons in the thalamus pl li an important role

in generating the EEG signals. The synchronicity of the cortical synaptic activity reflects

the degree of synchronous firing of the thalamic neurons that are generating the electrical

activities. However, the purposes of these electrical activities and EEG oscillations are

largely unknown.








Figure 1-6. EEG recording acquired by Hans Berger in 1929


The configurations of EEG recordings pl i, an important role in determining the

normal brain function from abnormal.The most obvious EEG frequencies of an awake,

relaxed adult whose eyes are closed is 8-13 Hz also known as the alpha rhythm. The alpha

rhythm is recorded best over the parietal and occipital lobes and is known to be associated

with decreased levels of attention. When alpha rhythm are presented, subjects commonly

report that they feel relaxed and happy. However, people who normally experience more

alpha rhythm than usual have not been shown to be psychologically different from those

with less. Another important EEG frequencies is the beta rhythm, people are attentive to
























Figure 1-2. Vagus nerve stimulation electrode


Long term follow-up studies showed that prevention of recurrent seizures was

maintained and adverse events decreased significantly over time [15; 16]. Positron emission

tomography and functional MRI studies showed that VNS activates or increases blood

flow in certain areas of the brain such as the thalamus [17; 18]. Cerebrospinal fluid (CSF)

was analyzed in 16 subjects before, 3 months after, and 9 months after VNS treatment

GABA (total and free) increased in low or high stimulation groups, aspartate marginally

decreased and ethanolamine increased in the high stimulation group -ii--:. -ii.:-; an

increased inhibitory effect [19]. Krahl et al., -,ir--- -I. 1 that seizure suppression induced by

VNS may depend on the release of norepinephrine and they observed that acute or chronic

lesions of the "Locus coeruleus" attenuated VNS-induced seizure suppression [20].

1.5 Neuron States and Membrane Potentials

1.5.1 Neuron States

The birth of Electroencephalogram (EEG) has inspired the attempts to extract the

subtle alternations in brain activity. It is known the fluctuations on EEG frequency and

voltage arise from spontaneous interactions between excitatory and inhibitory neurons in

circuit loops. If a neuron is stimulated, the membrane potential will be altered and this

alternation can be classified into two different states Figure 1-3 is taken from Malmivu and

Plonse (1993)[21].









For example two planes of dimension di and d2 embedded in m dimensional space will

intersect if m < di + d2, it is clear that if dl = d2 = d the embedding dimension need at

least 2d + 1 to avoid the intersections in the state space. However, if only s subset of the

degrees of freedom of M is represented in our measurement x(t) = ((t)), it is impossible

to obtain additional information. A technique called method of delay is employ, ,1 to

retreat the information from pervious times with a embedding window r and form a set of

reconstructed delay vector x(t),


x(t) = (x(t),x(t r), x(t 2r),...,x(t (m 1)r)), (2-19)


and the duration of each embedding vector is


r = (m 1) '. (2-20)


A much more general situation for time-1 ,.-.- d variables constitute an adequate

embedding provided the measured variable is smooth and coupled to all the other

variables is proved by Takens, and the number of time lag is at least 2d + 1 [52].

0 : M --+ R2d+l is an open and dense set in the space of pairs of smooth maps (f,h),

where f is the dynamical system measure by function h.

2.5 Lyapunov Exponents

The concept of Lyapunov exponents was first introduced in by A.M. Lyapunov.

Lyapunov developed "Lyapunov Stability "concepts to measure the stability of a

dynamical system. It quantifies the rate of separation of nearby trajectories in the

state space. In this section I describe the method for estimating the Lyapunov exponents.

For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov

exponents. For example, consider two trajectories with nearby initial conditions on an

attracting manifold. Eckmann and Ruelle (1985) pointed out that when the attractor

is chaotic, the trajectories diverge, on average, at an exponential rate characterized by

the largest Lyapunov exponent [53]. For a dynamical system as time evolves the sphere









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represented as

y(w) (C- C- (6-4)

The cross-coherence quantifies the degree of coupling between X and Y at given

frequency w and it is also bounded between -1 and +1.

6.2.2 Partial Directed Coherence

Partial Directed Coherence a frequency domain based Granger-causality technique

which -,i- that an observed time series xj(n) causes another series xi(n), if knowledge

of xj(n)s past significantly improves prediction of xi(n) [90]. However, the reverse case

may or may not be true. To make a quantitative assessment of the amount of linear

interaction and the direction of interaction among multiple time-series, the concept of

Granger-causality can be used to and into the development multi autoregressive model

(\!VAR). The partial directed coherence from j to i at a frequency w is given by:


S(W) ( (65)


where for i = j

A(w) =1 Ya(r)e-j2wr; (6-6)
r=i
and for i / j
P
A-.(a;) Y- aj(r)e -2-w (6-7)
r=l
aij are the multivate auto-regressive (\!AR) coefficient at lag r, obtained by least-square

solution of MAR model
p-1
x = A'x(p r) + c, (6-8)
r=l
here









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in epileptic seizures: evidence for a devil's staircase. Phys Rev E Stat Nonlin Soft
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Improved spatial characterization of the epileptic brain by focusing on nonlinearity.
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[73] L.-Y. Cao. Practical method for determining the minimum embedding dimension of
a scalar time series. P,;-,'.. D, 110(1-2):43-50, 1997.

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for phase-space reconstruction using a geometrical construction. Phys Rev. A,
45:3403-34, 1992.

[75] A.M. Fraser and H.L. Swinney. Independent coordinates for strange attractors from
mutual information. Phys Rev A, 33:1134-1140, 1986.

[76] P. Grassberger and I. Procaccia. ('C! i i:'terization of strange attractors. P,;,-.. 'l/
Review Letters, 50(5):346-349, 1983.

[77] D.S. Broomhead and G.P. King. Extracting qualitative dynamics from experimental
data. Ph,;,-.. D, 20:217-236, 1986.

[78] M.C. Casdagli, L.D. Iasemidis, J.C. Sackellares, S.N. Roper, R.L. Gilmore, and R.S.
Savit. ('! i o i.terizing nonlinearity in invasive EEG recordings from temporal lobe
epilepsy. P,;,-..:',. D, 99:381-399, 1996.

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in interictal eeg recorded with foramen ovale electrodes predicts side of primary
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Predictibility analysis for an automated seizure prediction algorithm. Journal of
Clinical N ;,', -, ... ,,/;/ 23(6):509-520, 2006.





























25


20


15


10


5


0


-5


-10


-15s


-20


0 5 10 15 20
time (s)


Figure 2-7. Y component of Lorenz system


25 30 35









frequency bands across different regions of the brain, leading to certain clinical events

such as evoked potentials [95; 96]. Similarly, it is also believed that phase synchronization

across narrow frequency EEG bands, pre-seizure and at the onset of seizure may provide

useful hints of the spatio-temporal interactions in epileptic brain [33; 97; 35]. Hilbert

transform is used compute the instantaneous parameters ip(t) and Pb(t) of a time-signal.

Consider a real-valued narrow-band signal x(t) concentrated around frequency f,. Define

x(t) as
1
x = x(t) x (6-14)

where x(t) can be regarded as the output of the filter with impulse respond

1
h(t) = < t < oo, (6-15)
'rt

excited by an input signal x(t). This filter is call a Hillbert transformer. Hilbert

transforms are accurate only when the signals have narrow-band spectrum, which is often

unrealistic for most real-world signals. Pre-processing of the signal such as decomposing

it into narrow frequency bands is needed before we apply Hilbert transformation to

compute the instantaneous parameters. Certain conditions need to be checked to define

a meaningful instantaneous frequency on a narrow-band signal. It has been reported

that the distinct differences in the degree of synchronization between recordings from

seizure-free intervals and those before an impending seizure, indicating an altered state of

brain dynamics prior to seizure activity [89].

6.4 Mutual Information

The concept of mutual information dates back to the work of Shannon in 1948 [98].

Generally, mutual information measures the information obtained from observations of

one random event for the other. It is known that mutual information has the capability

to capture both linear and nonlinear relationships between two random variables since

both linear and nonlinear relationships can be described through probabilistic theories.

Here in our model, the mutual information measures how much information of EEG time
























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8.5 LST
RST
S- Seizure Onset






7.5
I(.1
E

7



6.5




0 50 100 Oe ( mi 20 250 300 350
I me (minutes)

Figure 3-2. Average minimum embedding dimension profiles for Patient 1 (seizure 1)


showed stable during the interictal state. In other words, the underlying degree of

freedom is uniformly distributed over the interictal state in the EEG recordings. The

results indicated the lowest minimum embedding dimension were found within the

epileptic zone during interictal state (the RST electrodes in Figs. 2, 3, and 4; the LTD

electrodes in Figs. 5 and 6). The complexity of the EEG recordings from the epileptic

region is lower than that from the brain regions. The values of the minimum embedding

dimension from all brain regions start decrease and converge to a lower value as the

patient proceed from interictal to ictal state. The underlining dynamical changes before

entering ictal period were consistently detected by the algorithm.

3.6 Conclusions

In this chapter, we investigate the degree of complexity for EEG recordings by

estimating the minimum embedding dimension. The algorithm we use for the minimum

embedding estimation is faster and requires less data points to obtain accurate results.









Right Hppocampus


44
20


l10







--
4-




0 ......

0 20 40 60 80 100 120 140 160 180

Figure 4-5. STL, x and T-index profiles during preictal state


4.4 Discussion

In this study, we demonstrated the usefulness of nonlinear dynamics measures and

investigated the degree of nonlinearity for EEG signals in different brain area for epilepsy

patients with and without unilateral seizure onset zone. The degree of nonlinearity was

defined as the distinction of the signal from Gaussian linear processes. The method

combined the estimation of Short-Term Maximum Lyapunov Exponents (STLmax), a

nonlinear discriminating statistic, and surrogate time series techniques. The hypotheses

tested were that (1) there exists difference in signal nonlinearities across recording

brain regions for patients with unilateral seizure onset zone, (2) EEG nonlinearities are

distributed uniformly across recording brain regions for patients with bi-temporal seizure

onset zone, and (3) in patients with unilateral seizure onset zone, the focal area can be

identified by comparisons of EEG nonlinearities among recording brain regions. The









CHAPTER 5
OPTIMIZATION AND DATA MINING TECHNIQUES FOR THE SCREENING OF
EPILEPTIC PATIENTS

5.1 Introduction

Detecting and identifying the important abnormal electroencephalogram (EEG)

complex by visual examination is not only a time consuming task but also requires fully

attentions form the electroencephalographer. In this study, we investigate the possibility

for classifying EEG recordings between seizure free patients and patients still suffering

from seizure attack using the support vector machine (SVM). Two multi-dimensional

SVMs, connectivity SVM (C-SVM) and dynamics SVM (D-SVM), were proposed

to identify the EEG recordings acquired from epileptic patients. The C-SVM uses

connectivity feature that extracted from EEG recording through mutual information

and D-SVM uses three dynamical measures (1. Angular frequency 2. Approximate entropy

3. Short-term largest lyapunov exponent) input for the EEG classification. One hour scalp

EEG recording was acquired from each subject (5 class 1, 5 class 2) in this study. Prior to

C-SVM classification, the independent component analysis (ICA) methods were applied

to remove the noise in the EEG recording for improving the performance od the SVM.

D-SVM achieved 94.7'. accuracy when identifying class 2 subjects compared to 69.!'.

accuracy with C-SVM.

Epilepsy is the most common disorders of nervous systems. Preliminary findings on

the costs of epilepsy show the total cost to the nation for 2.3 million people with epilepsy

was approximately $12.5 billion. The high incidence of epilepsy originates from the fact

that it occurs as a result of a large number of factors, including febrile disturbance,

genetic abnormal mutation, developmental deviation as well as brain insults such as

central nervous system (CNS) infections, hypoxia, ischemia, and tumors.

Neuron or groups of neurons generate electrical signals when interacting or transmitting

information between each other. The EEG recordings capture the local field potential

around electrodes that generate from neuron in the brain. Through visual inspection,









The .-i-viii.ii I ry of above nonlinear interdependencies is the main advantage over

other synchronization measures. This .-i-i'i::. 1 ry property can give directionality of

information transport between different cortical regions. Furthermore the "driver-response

"relationships but can also reflect different properties of brain functions when it is

importance to detect causal relationships. It should be clear that the above nonlinear

interdependencies measures are bivariate measures. Although it quantified the "driver-response

"for given input-the whole input space under study might be driven by other unobserved

sources.

8.4 Statistic Tests and Data Analysis

In this study, the mutual information and nonlinear interdependence measures

were estimated for every 10 seconds (2500 EEG data points) of continuous EEG

recordings. The bootstrap re-sampling approach was adapted for deriving estimates

on the measures[144]. Ten seconds of continuous EEG epoch is randomly sampled from

every channels and this sampling procedure was repeated with replacement for 30 times.

The reference Al and A2 channels (inactive regions) are excluded from the analysis.

Two sample t-test (N 30, a = 0.05) is used to test the statistical differences on mutual

information and nonlinear interdependence during before and after treatment. Low mutual

information and information transport between different brain cortical regions were

observed in our subjects with less severity of ULD. Furthermore, for each patient both

mutual information and information transport between different brain cortical regions

decrease after AEDs treatment. t-test for mutual information are summarized in Table

8-2, the topographical distribution for mutual information is also plot in heatmaps shown

in Figures 8-2.

The significant "driver-response "relationship is reveled by t-test. After t-test the

significant information transport between Fpl and other brain cortical regions is shown in

Fig. 8-1. The edges with an arrow starting from Fpl to other channel denote N(XIY)

is significant larger then N(YIX), therefore Fpl is the driver, and vice versa. The
























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to capture synchronization patterns among different sources (brain areas). The idea of

analyzing EEG recordings from several sources (multiple electrodes) is very crucial since

the knowledge from local information (i.e., single electrode) is very limited. In our future

study, we plan to incorporate the knowledge of general brain communication in the brain

network. For example, [114] demonstrated the evolution of cognitive function through

quantitative analysis of fMRI data.

The proposed framework can provide a global structural patterns in the brain network

and may be used in the simulation study of dynamical systems (like the brain) to predict

oncoming events (like seizures). For example, an ON-OFF pattern of electrode selection

in the maximum clique over one period of time can be modeled as a binary observation in

a discrete state in a Markov model, which can be used to simulate the seizure evolution

in the brain. In addition, the number of electrodes in the maximum clique can be used

to estimate the minimum number of features and explain dynamical models or the

parameters in time series regression. Note that the proposed network model represents an

epileptic brain as a graph, where there exist several efficient algorithms (e.g., maximum

clique, shortest path) for finding special structure of the graph. This idea has enabled

us, computationally and empirically, to study the evolution of the brain as a whole. The

Monte-Carlo Markov C'!i ,i (C' IC) framework may be applicable in our future study

on long term EEG analysis. The MC'\ C framework has been shown very effective in

data mining research [132]. It can be used to estimate the graph or clique parameters

in epileptic processes from EEG recordings. Since long term EEG recordings are very

massive, most simulation techniques are not scalable enough to investigate large-scale

multivariate time series like EEGs. The use of MC'\ C makes it possible to approximate

the brain structure parameters over time. More importantly, the MC'\ C framework

can also be extended to the analysis of multi-channel EEGs by generating new EEG

data points while exploring the data sequences using a Markov chain mechanism. In

addition, we can integrate the MC' IC framework with a B ,i -i ,n approach. This can be










Table 5-2. Results for D-SVM using 5-fold cross validation


D-SVM
Results
D-SVM/C-SVM Dynamical features UNICA
5-fold CV 1 .'. 7'
2 9", 71,',
3 ,
4 '-', 72'

6 92' 5 !',
7 II,' 7, ,
8 95' 7 ,' ,
9 !', 7 '",
10 95', 7.i'.
Average of correctness 94.7'. 69. !'


The SVM has a very long statistical foundation and assure the optimal feasible

solution for a set of training data, given a set of features and the operation of the SVM.

In this study, we attempted to study the separability between abnormal EEG and normal

EEG using different EEG features. We tested the performance on scalp EEG recordings

from normal individuals and abnormal patients. The EEG data was filtered using ICA

algorithm. ICA filters the noise in EEG scalp data, keeps essential structure and makes

better representable EEG data sets. The Euclidean distance based C-SVM was proposed

to evaluate the connectivity among different brain regions. The dynamical features were

generated as input for D-SVM, the classification results of the proposed D-SVM are very

encouraging. The results indicated that D-SVM improves classification accuracy compare

to C-SVM. It gives an average accuracy of 94.7'. The dynamical features provide a subset

in the feature space and improve classification accuracy.









One can obtain the mutual information between X and Y using the following

equation [143]:

I(X;Y)= H(X) + H(Y) H(X,Y), (8-2)

where H(X), H(Y) are the entropies of X, Y and H(X, Y) is the joint entropy of X and

Y. Entropy for X is defined by:

H(X)= p(x) logp(x). (8-3)

The units of the mutual information depends on the choice on the base of logarithm.

The natural logarithm is used in the study therefore the unit of the mutual information is

nat. We first estimate H(X) from X by
N
H(X) = P(X x). (8-4)
i= 1

For X and Y time series we define d = ) xjY,,d = 1, yj\\ as the distances

for xi and yi between every other point in matrix spaces X and Y. One can rank these

distances and find the knn for every xi and yi. In the space spanned by X, Y, similar

distance rank method can be applied for Z = (X, Y) and for every z = (xi, yi) one can

also compute the distances d) = zjl and determine the knn according to some

distance measure. The maximum norm is used in this study:

d max{ -xj, y, -dyj}, d = x, xj. (8-5)

Next let 6-) be the distance between zi and its kth neighbor. In order to estimate the

joint probability density function (p.d.f.), we consider the probability Pk(c) which is the

probability that for each zi the kth nearest neighbor has distance + dc from zi. This

Pk(e) represents the probability for k 1 points have distance less than the kth nearest
neighbor and N k 1 points have distance greater than C) and k 1 points have









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U. Pietrzyk, A. Relic, and I. Podreka. Preictal spect in temporal lobe epilepsy:
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[28] P.D. Adelson, E. N. i,,i.,i M. Scheuer, M. Painter, J. Morgan, and H. Yonas.
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included to extend this study. We test the hypothesis that, in patients with unilateral

seizure onset zone, the degree of nonlinearity are different over the brain regions, and the

focus areas exhibit higher degree of nonlinearity during interictal, preictal, and postictal

periods. Further, for patients with independent bi-temporal seizure onset zones, the

distribution of nonlinearity would be uniform over brain regions.

4.2 Materials and Methods

4.2.1 EEG Description

EEG recordings were obtained from bilaterally placed depth and subdural electrodes

(Roper and Gilmore, 1995), multi-electrode 28-32 common reference channels were used

in this study. Figure 4-1 a inferior transverse views of the brain, illustrating approximate

depth and subdural electrode placement for EEG recordings are depicted. Subdural

electrode strips are placed over the left orbitofrontal (LOF), right orbitofrontal (ROF), left

subtemporal (LST), and right subtemporal (RST) cortex. Depth electrodes are placed in

the left temporal depth (LTD) and right temporal depth (RTD) to record hippocampal

activity.

EEG recordings obtained from eight patients with temporal lobe epilepsy were

included in this study. See Table 4-1. Five patients were clinically determined to have

unilateral seizure onset zone (ULSOZ) and the remaining three patients were determined

to have independent bi-temporal seizure onset zone (BTSOZ). For each patient, three

seizures were included in the EEG recordings. Segments from interictal (at least one

hour before the seizure), preictal (immediately before the seizure onset) and postictal

(immediately after the seizure offset) time intervals corresponding to each seizure were

sampled for testing the hypothesis. Two electrodes from each brain area were included,a

total of 12 electrodes were analyzed for each patient. The EEG recordings were sampled

using amplifiers with input range of 0.6 mV, and a frequency range of 0.5-70Hz. The

recordings were stored digitally on videotapes with a sampling rate of 200 Hz, using an









different scaled spaces (marginal and joint) are not comparable To avoid this problem,

instead of using a fixed k, n(i) + 1 and ny(i) + lare used in obtaining the distances (where

n,(i) and ny(i) are the number of samples contained the bin [x(i) -, x(i) + (] and

[y(i)- ~7, y(i) + i] respectively) in the x-y scatter diagram. The Eq.(8-12) becomes:
SN
H(X) = (N) (n(i) + 1) +logc c, + N log(i). (8-13)
i=1

Finally the Eq.(8-2) is rewritten as:

N
Iknr,(X; Y) = (k) + ^(N) [ (n(i) + 1) + y(n,(i) + 1)]. (8-14)
i=1

8.3.2 Nonlinear Interdependencies

Arnhold et al., (1999) introduced the nonlinear interdependence measures for

characterizing directional relationships (i.e. driver & response) between two time

sequences [106]. Given two time series x and y, using method of delay we obtain the

delay vectorsx = (x,, ..., xn-(m-i),) and y, = (x,, ..., xn-(m-_l)), where n = 1, ...N, m is

the embedding dimension and r denotes the time d. 1v [52]. Let rT,j and s,j, j = 1,..., k

denote the time indices of the k nearest neighbors of x, and y,. For each xn, the mean

Euclidean distance to its k neighbors is defined as
k
R(X) = (x, xr,j)2, (8-15)
j= 1

and the Y-conditioned mean squared Euclidean distance is defined by replacing the

nearest neighbors by the equal time partners of the closest neighbors of y,


R7k)(XlY) = (x xs.,)2. (8 16)
j=1

The delay = 5 is estimated using auto mutual information function, the embedding

dimension m = 10 is obtained using Cao's method and the Theiler correction is set to

T 50 [73; 107].









LIST OF TABLES


Table page

2-1 Patient information and EEG description ............ ... .. 59

2-2 Performance for D-SVM ............... .......... .. 59

3-1 Patients and EEG data statistics for complexity analysis . . 62

4-1 Patients and EEG data statistics .................. ....... .. 76

5-1 EEG data description .................. ............. .. 94

5-2 Results for D-SVM using 5-fold cross validation ....... . ...... 103

7-1 Patient information for clustering analysis ................ .... .. 121

8-1 ULD patient information .................. .......... 140

8-2 Topographical distribution for treatment decoupling effect (DE: Decouple Electrode
(DE)) . . . . . . . . .. .. .. 146

8-3 Patient 1 before treatment nonlinear interdependencies . . ... 150

8-4 Patient 1 after treatment nonlinear interdependencies . . .... 151

8-5 Patient 2 before treatment nonlinear interdependencies . . ... 152

8-6 Patient 2 after treatment nonlinear interdependencies . . ... 153

8-7 Patient 3 before treatment nonlinear interdependencies . . ... 154

8-8 Patient 3 after treatment nonlinear interdependencies . . ..... 155

8-9 Patient 4 before treatment nonlinear interdependencies . . ... 156

8-10 Patient 4 after treatment nonlinear interdependencies . . ... 157









LIST OF FIGURES
Figure page

1-1 Vagus nerve stimulation pulse generator .................. ..... 22

1-2 Vagus nerve stimulation electrode .................. ....... .. 23

1-3 Cortical nerve cell and structure of connections ................. .. 24

1-4 Membrane potentials .................. .............. .. 25

1-5 Electrical potentials .................. .............. .. 26

1-6 EEG recording acquired by Hans Berger in 1929 .. . 27

1-7 Basic EEG patterns .................. .............. .. 28

1-8 International 10-20 electrode placement .................. .. 29

1-9 Surbdural electrode placement .................. ......... .. 30

1-10 Depth electrode placement .................. ......... .. .. 31

2-1 R6ssler attractor ............... .............. .. 36

2-2 X component of R6ssler system .............. ....... .. 37

2-3 Y component of R6ssler system .............. ....... .. 38

2-4 Z component of R6essler system .............. ....... .. 39

2-5 Lorenz system . .............. .. .......... ... ..41

2-6 X component of Lorenz system .................. ........ .. 42

2-7 Y component of Lorenz system ............... ....... .. 43

2-8 Z component of Lorenz system ............... ...... .. 44

2-9 Estimation of Lyapunov exponent (Lm,,) .................. .. .. 50

2-10 Temporal evolution of STL.ma ................ ........ .. 51

2-11 Phase/Angular frequency of Lyapunov exponent (Qma) . . .. 52

2-12 Three dimension plot for entropy, angular frequency and Lmax during interictal
state ............... ................ .. 55

2-13 Three dimension plot for entropy, angular frequency and Lmax during preictal
state ............... ................ .. 56

3-1 Electrode placement ............... ............ .. 62









seizures, however, frequently have bilateral hemispheric involvement. Partial onset seizure

can be classified into one of the following three groups:

1. Simple partial onset seizures (consciousness not impaired)

The EEG features for simple partial onset seizures are local contralateral discharge

starting over the corresponding area of cortical representation and the significant

feature for the background EEG is local contralateral discharge. Simple partial onset

seizures may have the following clinical features:

(a) With motor signs

i. Focal motor without march

ii. Focal motor with march (Jacksonian)

iii. Versive

iv. Postural

v. Phonatory(Vocalization or arrest of speech)

(b) With somatosensory or special sensory symptoms (simple hallucinations, e.g.,

tingling, light flashes, buzzing)

i. Somatosensory

ii. Visual

iii. Auditory

iv. Olfactory

v. Gustatory

vi. Vertiginous

(c) With autonomic symptoms or signs (including epigastric sensation, pallor,

sweating, flushing, piloerection and pupillary dilatation)

(d) With p' ii, symptoms (disturbance of higher cerebral function). These

symptoms rarely occur without impairment of consciousness and are much more

commonly experienced as complex focal seizures

i. Dysphasic









TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .................................

LIST O F TABLES . . . . . . . . . .

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

A B ST R A C T . . . . . . . . . .

CHAPTER

1 OVERVIEW OF EPILEPSY ............................

1.1 Introduction . . . . . . . . .
1.2 C auses of Seizure . . . . . . . .
1.3 Classification of Epileptic Seizure (ICES 1981 revision) ...........
1.3.1 Partial Onset Seizures .. .. .. .. ... .. .. .. ... .. .. .
1.3.2 Generalized Seizures .. .. .. .. ... .. .. .. ... .. .. .
1.3.3 Unclassified Seizures . .. .. ... .. .. .. ... .. .. .
1.4 Treatment for Epilepsy .. ........................
1.4.1 Pharmacological Treatment .. ...................
1.4.2 Surgeical Section . . . . . . .
1.4.3 Neurostimulator Implant .. ....................
1.5 Neuron States and Membrane Potentials .. ...............
1.5.1 N euron States . . . . . . . .
1.5.2 M embrane Potentials .. .....................
1.6 Recording Electric Brain Activity .. ...................
1.6.1 Scalp EEG Recording .. .....................
1.6.2 Subdural EEG Recording .. ....................
1.6.3 Depth EEG Recording .. ....................
1.7 Conclusions and Remarks .. ......................

2 EPILEPSY AND NONLINEAR DYNAMICS .. ...............

2.1 Introduction . . . . . . . . .
2.2 Dynamical Systems and State Space .. .................
2.3 Fractal D im ension . . . . . . . .
2.3.1 Correlation Dimension .. ....................
2.3.2 Capacity Dim ension .. ......................
2.3.3 Information Dimension .. ....................
2.4 State Space Reconstruction .. .....................
2.5 Lyapunov Exponents .. .........................
2.6 Phase/Angular Frequency .. ......................
2.7 Approxim ate Entropy .. .........................
2.8 Dynamical Support Vector Machine (D-SVM) .. .............




































To my parents in Taiwan.









Table 4-1. Patients and EEG data statistics
Patient # Gender Age Focus (RH/LH) Length of EEG (hr.)
P1 M 19 RH 6.1
P2 M 45 RH 5.4
P3 M 41 RH 5.8
P4 F 33 RH 5.3
P5 F 38 RH 6.3
P6 M 44 RH/LH 5.5
P7 F 37 RH/LH 4.6
P8 M 39 RH/LH 5.4


analog to digital (A/D) converter with 10 bit quantization. In this study, all the EEG

recordings were viewed by two independent board certified electroencephalographers.



(A) (B)


ROF 2- oo oo4 LOF LTD

RST LST


LOF
LST

RTD LTD


Figure 4-1. 32-channel depth electrode placement


4.2.2 Non-Stationarity

Non-stationarity is an fundamental difficulty for time series analysis. The existing

of non-stationarity in a measured time series will result in no invariant measures.

Stationarity will cause errors for many algorithms when one is trying interpret the results

of an invariant measure. In most cases, one can try to remove the stationarity by using

vi I i, of filters or divided the time series into a number of shorter epochs and assume the

underlying dynamics to be approximately stationary within each divided epochs. In this











( a a'12 % a' )iN


AT (6-9)



\ tNI tN2 NN /
and
xli(p -r)

x2(p r)
(p = x3(p r) (6-10)



XN(P r)
p denotes the depth of the AR model, r denotes the delay and n is the prediction error or

the white noise.

Note that Tr, quantifies the relative strength of the interaction of a given signal source

j with regard to signal i as compared to all of js interactions to other signals. It turns out

that the PDC is normalized between 0 and 1 at all frequencies. If i=j, the Partial Directed

Coherence represents the casual influence from the earlier state to its current state.

The MVAR approaches have been used to determine the propagation of epileptic

intracranial EEG activity in temporal lobe and mesial seizures [2-3, 10, 19-20]. However,

these models strictly require that the measurements be made from all the nodes, or

the directional relationships could be ambiguous. In addition, there remains no clear

evidence of causality relationships among the cortical regions as si,--. -1I "the nature of

synchronization is mostly instantaneous or without any detectable d(. I [91].

The general, nonlinearity are commonly inherent within neuronal recordings, the

above linear measures are typically restricted to measure statistical dependencies up to

the second order [92]. If observations are Gaussian distributed, the 2nd order statistics

are sufficient to capture all the information in the data. However, in practice, EEG data









2.3 Fractal Dimension

The term "fractal" was first introduced by Mandelbrot in 1983. Roughly speaking, a

fractal is a set of points that when looked at smaller scales, resembles the whole set. The

concept of fractal dimension refers to a non-integer or fractional dimension originates from

fractal geometry. Strange attractors often have a structure that is not simple; they are

often not manifolds and actually have a highly fractured character. The dimension that is

most useful takes on values that are typically not integers. These non-integer dimensions

are called fractal dimensions. For any attractor, the dimension can be estimated by

looking at the way in which the number of points within a sphere of radius r scales as

the radius shrinks to zero. The geometric relevance of this observation is that the volume

occupied by a sphere of radius r in the dimension d behaves as

For regular attractors, irrespective to the origin of the sphere, the dimension would be

the dimension of the attractor. But for a chaotic attractor, the dimension varies depending

on the point at which the estimation is performed. If the dimension is invariant under

the dynamics of the process, we will have to average the point densities of the attractor

around it. For the purpose of identifying the dimension in this fashion, we find the number

of points y(k) within a sphere around some phase space location x. This is defined by:

N
n(x,r) er- y(k)- x (2-11)
k-0

where is the Heaviside step function such that (n) = 0 for n < 0, O(n) = 1 for n > 0

This counts all the points on the orbit y(k) within a radius r from the point x and

normalizes this quantity by the total number of points N in the data. Also, we know

that the point density, p(x), on an attractor does not need to be uniform for a strange

attractor. C'!, ... -i the function as n(x; r)q-1 and defining the function C(q; r) of two

variables q and r by the mean of n(x; r)q-1 over the attractor weighted with the natural









distance less than (). Pk(c) is obtained using the multinomial distribution:


Pk() k(N-k 1 d(c) P (1- pi)N-k-1 (86)
( k 1) dc

where pi is the mass of the c-ball. Then the expected value of logpi is


E(logpi) = (k) ((N), (8-7)


where b(-) is the 1:li1,,i,,,a function:


(t)= F(t)-d(t) (8-8)
dt

where F(-) is the gamma function. It holds when b(1) =C where C is the Euler -

Mascheroni constant (C 0.57721). The mass of the c-ball can be approximated (if

considering the p.d.f inside the c-ball is uniform) as


pi(c) a cdCP(X= x), (8-9)

where Cd. is the mass of the unit ball in the dx-dimensional space. From Eq.(8-9) we can

find an estimator for P(X = xi)


log[P(X = xi)] b (k) (N) dE(log(i)) log Cd, (8-10)

finally with Eq(8-10) and Eq(8-4) we obtain the Kozachenko-Leonenko entropy estimator

for X [105]
N
H(X) = (N) (k) + log Cd, + log (i), (8-11)
i= 1
where c(i) is twice the distance from xi to its k-th neighbor in the dx dimensional space.

For the joint entropy we have


N
H(X, Y) = (N) (k) + log(cdcd,) + da Nd log(e(i)) (8-12)
i= 1

The I(X; Y) is now readily to be estimated by Eq.(8-2). The problem with this

estimation is that a fixed number k is used in all estimators but the distance metric in























X9~I
Z -Hs


. z 0 0*z 0 0 *z *z *z *z 0 0 0 0
c-^ w0 ~ t ^co c-o^ w^ ^v ^ ^^io ~ o
^d~d~d~d~d~d~d~d~d~d~d~d0d0d
"-H0^0^0^0^0^0^0^0^0^0^0^0^0^


S-H


-I_


-I-
"$I


oooo2
oOc^
0+1+1


- cd~ cd1 rd*- d c~1 r~*- ~ -
*l~~~~~~~i r~* *i *d~~iidd


S' c =- l* =9 c 0 I
.z *z *z *z *- *z *z
d^dd~d~d~d^^d~d~


--H


oo
o-1
?


oo
u5
0-Hd


0


a ^
0


a ^
0 ?


u0
00 CM
CS
0 -


ao
C
0 -


Ln
0
13
0 -H


o
t?-c






















7
Ow:


0**

0R


11U0


Lyapunov Exponent


Figure 5-2. EEG dynamics feature classification


infeasibilities of the constraints. With this formulation, ones wants to maximize the

margin between two classes by minimizing I| w 1|2. The second term of the objective

function is used to minimize the misclassification errors that are described by the slack

variables ci. Introducing positive Lagrange multipliers ai to the inequality constraints in

D-SVM model, we obtain the following dual formulation:

i 1i j=1 n
min 1 Y iyjaiajXiXj ai (5-15)
n n i=1


Co


r3
0


'I,





C


o* .. -
)^*
2-"Sr0


Erropy


i


'


.


'









where I|| || is some Euclidian distance and is given in this paper by maximum norm.

Define the mean value of all a(i, d) as

N-dr-
E(d) N a(, d). (3-8)
i=

E(d) is depend only on the dimension d and the time delay r The minimum

embedding dimension is founded when El(d) = E(d + 1)/E(d) saturated when d is larger

than some value do if the time series comes from an attractor. The value do + 1 is the

estimated minimum embedding dimension.

3.5 Data Analysis

The first step in the data analysis was to divide the EEG data into non-overlapping

windows of 10.24 seconds in duration for nonstationarity purposes. This procedure was

to ensure that the underlining dynamical properties were approximately stationary. For

each divided window, the first step of estimating the minimum embedding dimension is

to construct the delay coordinates using method of delay proposed by. The time delay

7 was obtained from the first local minimum of the mutual information function. We

used these time delay vectors as inputs to Cao's method for the minimum embedding

dimension estimation. The minimum embedding dimension was calculated over time for

EEG recordings with 29 electrodes at six brain regions (RTD, RST, ROF, LTD, LST,

and LOF) from epilepsy patients. Each brain region contains 46 electrodes; the average

of the minimum embedding dimension da is taken as representation to the underlining

brain dynamics. We shall study the minimum embedding dimension in the following three

different time periods: interictal, ictal and postictal. These three different time period

are defined as follows: 1. interictal state: 1 hour away prior to ictal state 2. preictal

state: 2 minutes data length prior to ictal state 3. postictal state: 1 hour after the ictal

state Figures 3-2,3-3,3-4,3-5 and 3-6 show typical the minimum embedding dimension over

time for six seizures. One can observe the behavior of the average minimum embedding

dimension over time for six brain cortical regions. The minimum embedding dimension










1. Excited state: A neuron state with a less negative intraneural membrane potential

compare to resting state of a neuron. The positive increase in voltage above the

normal resting neuronal potential is called the excitatory p -vi!i itic potential

(EPSP), if this potential rise high enough in the positive direction, it will elicit an

action potential in the neuron.

2. Inhibited state: A neuron state with a more negative intraneural membrane

potential compare to resting state of a neuron. An increase in negative beyond the

normal resting member potential level is call an inhibitory ] .. 1 -vi- i)tic potential

(IPSP).

Afferent nerve fibers




Synaptic knobs





Dendrites -
Cell body







Figure 1-3. Cortical nerve cell and structure of connections


The excitatory neurons excite the target neurons. Excitatory neurons in the

central nervous system are often glutamatergic neurons. Neurons in the peripheral

nervous system, such as spinal motor neurons that synapse onto muscle cells, often use

... I i*lcholine as their excitatory neurotransmitter. However, this is just a general tendency

that may not ahlv- -, be true. It is not the neurotransmitter that decides excitatory or

inhibitory action, but rather it is the 1 ..- I -ii ,tic receptor that is responsible for the









function of c as c -i 0:


Do lm g[N ] (2-15)
e-o log[e]

2.3.3 Information Dimension

The information dimension D1 is a generalization of the capacity that are relative

probability of cubes used to cover the attractor. Let I denotes the information function:
N
I Pi(r) log P(r), (216)
i=1

Ps(r) is the normalized probability of an element i is covered such that C,1 Pi(r) 1.
Information dimension is defined as:


D1 y NPi(r)logPi(r) (27)
i=1 log(r)

D2 < D1 < Do if elements of the fractal is equally likely to be visited in the state space.

2.4 State Space Reconstruction

Most dynamical properties are contained within almost any variable and its time

lags. It is not necessary to reconstruct to entire state space from the measured variable

since the attractor dimension will often evolved in a much smaller dimension. The method

called state-space reconstruction was proposed by Takens (1981) for reconstructing the

state space for a dynamical system. For a series of observations acquired from a dynamical

system, the state space reconstruction transforms the observations into stat space using an

embedding coordinate map 0 : M -- S',


x(t) = ((t)), (2-18)

where m is the embedding dimension. The transform function 0 must be unique (i.e., has

no self intersection). Whitney (1936) proved a theorem which can also be used for finding

the embedding dimension [51]. 0 : M -- R2d+l; 0 embedding is an open and dense set in the space of sm
























4-

p-value = 0.9945









0


1 2 3






Patients

Figure 4-11. Nonlinearities across recording areas during preictal state for BTSOZ patients
1-,







I, 2 3









CHAPTER 4
DISTINGUISHING INDEPENDENT BI-TEMPORAL FROM UNILATERAL ONSET IN
EPILEPTIC PATIENTS BY THE ANALYSIS OF NONLINEAR CHARACTERISTICS
OF EEG SIGNALS

4.1 Introduction

In this present study, we investigate the difference in nonlinear characteristics of

electroencephalographic (EEG) recordings between epilepsy patients with independent

bi-temporal seizure onset zone (BTSOZ) and those with unilateral seizure onset zone

(ULSOZ). Eight adult patients with temporal lobe epilepsy were included in the study, five

patients with ULSOZ and three patients with BTSOZ. The approach was based on the

test of nonlinear characteristics, defined as the distinction from a Gaussian linear process,

in intracranial EEG recordings. Nonlinear characteristics were tested by the statistical

difference of short-term maximum Lyapunov exponent STLax, a discriminating nonlinear

measure, between the original EEG recordings and its surrogates. Distributions of EEG

nonlinearity over different recording brain areas were investigated and were compared

between two groups of patients. The results from the five ULSOZ patients showed that the

nonlinear characteristics of EEG recordings are significantly inconsistent (p < 0.01) over

six different recording brain cortical regions (left and right temporal depth, sub temporal

and orbitofrontal). Further, the EEG recordings acquire from focal regions of the brain

exhibit higher degree of nonlinearities than the homologous contralateral regions and the

nonlinear characteristics of EEG are uniformly distributed over the recording areas in

all three patients with BTSOZ. These results -i-i: -1 that it is possible to efficiently and

quantitatively determine whether an epileptic patient has ULSOZ based on the proposed

nonlinear characteristics analysis. For the ULSOZ patients, it is also possible to identify

the focal area. However, these results will have to be validated in a larger sample of

patients. Success of this study can provide more essential information to patients and

epileptologists and lead to successful epilepsy surgery.









CHAPTER 8
TREATMENT EFFECTS ON ELECTROENCEPHALOGRAM (EEG)

8.1 Introduction

Assessing the severity of myoclonus and evaluating the efficacy of antiepileptic

drugs (AEDs) treatment for patients with Unverricht-Lundborg Disease (ULD) have

traditionally utilized the Unified Myoclonus Rating Scale (UMRS). EEG recordings are

only used as a supplemental tool for the diagnosis of epilepsy disorders. In this study,

mutual information and nonlinear interdependence measures were applied on EEG

recordings to identify the effect of treatment on the coupling strength and directionality

of information transport between different brain cortical regions. Two 1-hour EEG

recording were acquired from four ULD subjects during the time period of before and after

treatment. All subjects in this study are from the same family with similar age (48 3

years) and ULD history (~37.75 years). Our results indicate that the coupling strength

was low between different brain cortical regions in the patients with less severity of ULD.

The effects of the treatment was associated with significant decrease of the coupling

strength. The information transport between different cortex regions were reduced after

treatment. These findings could provide a new insight for developing a novel surrogate

outcome measure for patients with epilepsy when clinical observations could potentially

fail to detect a significant difference.

EEG recording system has been the most used apparatus for the diagnosis of

epilepsy and other neurological disorders. It is known that changes in EEG frequency

and amplitude arise from spontaneous interactions between excitatory and inhibitory

neurons in the brain. Studies into the underlying mechanism of brain function have

-.-.-, -1 .'1 the importance of the EEG coupling strength between different cortical regions.

For example, the synchronization of EEG activity has been shown in relation to memory

process [133; 134] and learning process of the brain [135]. In a pathophysiological study,

different brain synchronization/desynchronization EEG patterns are shown to be induced




















0' i
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10
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10

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10

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Figure 4-2. Degree of nonlinearity during preictal state


and post-ictal (Fig. 4-6) state, respectively, in a ULSOZ patient. Each figure contains

two areas, one from the focal area (top two panels) and another from the homologous

contralateral hippocampus area (bottom two panels). From these figures, it is clear that

the EEG recorded from the focus area exhibits higher distinction from Gaussian linear

processes than those recorded from the homologous contralateral hippocampus area in all

three states. Further, the differences of STLax values in the focus area increased from

interictal to preictal, and reached to the maximum in the postictal state, but the difference

remained the stable in the homologous contralateral area.








If x, has an average squared radius R(X) = (1/N) Rn -'(X), then R,) -
Rk)(X) < R(X) if the system are strongly correlated, while Ref)(X Y) w R(X) > R(k)(X)
if they are independent. Accordingly, it can be define and interdependence measure
S(k)(XIY) as

S(k)(X|Y)= 1 t (X (8-17)
N ,lR (X|Y)
Since Rk)(X Y) > R$ (X) by construction,

0 < s(k)(Xly) < 1 (8-18)

Low values of Sk(XIY) indicate independence between X and Y, while high values
indicate synchronization.
Arnhold et al., (1999) introduced another nonlinear interdependence measure
H(k)(X Y) as

H(k)(XY) -log R ) ,(8-19)
n 1 Rf(XY)'
H(k)(XIY) = 0 if X and Y are completely independent, while it is possible if closest in
Y implies also closest in X for equal time indexes. H(k)(XIY) would be negative if close
pairs in Y would correspond mainly to distant pairs in X. H(k)(XIY) is linear measures
thus is more sensitive to weak dependencies compare to mutual information. Arnhold et
al., (1999) also showed H was more robust against noise and easier to interpret than S.
Since H is not normalized Quiroga et al., (2002) introduced another N(XIY):

1 R(X) Rk)(XIY)
N(k)(XlY) j= N -, (8-20)
1N=l R (X)
which is normalized between 0 and 1. The opposite interdependencies S(YIX), H(YIX),
and N(YIX) are defined in complete analogy and they are in general not equal to
S(XIY), H(XIY), and N(XIY), respectively. Using nonlinear interdependencies on
several chaotic model (Lorenz, Roessler, and Heenon models) Quiroga et al., (2000)
showed the measure H is more robust than S.









(Takens, 1981) also provided that the time 1 .-.- d variables constitute an adequate

embedding provided the measured variables is smooth and couples to all the variables,

and number of time lags is at least 2D + 1 [52]. For the above reasons, we employ .1

a method proposed by Cao (1997) to estimate the minimum embedding dimension of

EEG time series [73]. Like some other exiting methods, Caos method is also under the

concepts of false-nearest-neighbors The false-nearest-neighbors utilized on the fact that

if the reconstruction space has not enough dimensions, the reconstruction will perform

a projection, and hence will not be an embedding of the desired system [74]. As a of

result of giving a to low embedding dimension while processing the embedding procedure,

two points which is far away in the true state space will be mapped into close neighbor

in the reconstruction space. These are then the false neighbors. Caos method does not

require large amount of data points, is not subjective and it is not time-consuming

find the proper minimum embedding dimension. The EEG recordings was divided into

non-overlapping single electrode segments of 10.24 s duration, each of which was estimated

for the minimum embedding dimension. Under the assumption the EEG recordings within

each 10.24 s duration was approximately stationary [56], we evaluated the underlining

dynamical behavior by looking at the minimum embedding dimension over time.

The remaining of this chapter is organized as follows. In Sections. 2 and 3, we

describe the data information and explain the algorithm for estimating the minimum

embedding dimension estimation. The results from two patients with a total number of six

temporal lobe epilepsy (TLE) are given in Section 4. In Section 5, we discuss the results of

our findings with respect the use of this algorithm and the function of nonlinear dynamical

measurements in the area of seizure control.

3.2 Patient and EEG Data Information

Electrocardiogram (EEG) recordings from bilaterally placed depth and subdural

electrodes (Roper and Glimore, 1995) in patients with medically refractory partial seizures

of mesial temporal origin were analyzed in this study. Electrode placement. A Inferior
























E



E



LTD
RTD
6 LOF
ROF
LST
RST
Sezure Onset

40 60 80 100 120 140 160 180 200 22 240
Time (minutes)


Figure 3-4. Average minimum embedding dimension profiles for Patient 1 (seizure 3)



large-scale multi-quadratic 0-1 programming problems. Our results also are compatible

with the findings about the nature of transitions to ictal state in invasive EEG recordings

from patients with seizures of mesial temporal origin. The development of multi-quadratic

0-1 programming modeling is in progress.



























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2.9 Statistical Distance

In this section, we introduced Tj index, a statistical measure to estimate the

difference of EEG recordings in the dynamical measures. The Tj index at time t between

the dynamical profiles of EEG recording at i and j is defined as:


S--ijt
T(t)= D | x ,
N1


(2-42)


where I||Dl ~ denotes the absolute value of the average of all paired differences


Lt) t( w(t)),


SDAj I- (L'


(2-43)









can be described through probabilistic theories. Here in our model, the C\ ll measures

how much information of EEG time series acquired from electrode x is presented by

electrode y and vice versa. Let X be the set of data points where its possible realizations

are xl,, x2, 3,..., x with probabilities P(xi), P(x2), P(x3),.... The Shannon entropy H(X)

of X is defined as

H(X) p Inpi. (7 1)
i=1
Shannon entropy measures the uncertainty content of X. It is ahv-,- positive and

measured in bits, if the logarithm is taken with base 2. Now let us consider another set

of data points Y, where all possible realizations of Y are yi, y2, Y3,... yn with probabilities

P(y ), P(y2), P(3),.... The degree of synchronicity and connectivity between X and Y
can be measured by the joint entropy of X and Y, defined as


H(X, Y) 7 (7-2)
i,j
where pij which is the joint probability of X = Xi and Y = Yj. The cross information

between X and Y, CMI(X, Y), is then given by

CMI(X,Y) = H(Y)- H(XY) H(X)- H(YX) (7-3)

S H(X) + H(Y) H(X,Y) (7-4)

px(x)py( y)

The cross mutual information is nonnegative. If these two random variables X, Y are

independent, fxy(x,y) = fx(x)fy(y), then CMI(X,Y) = 0, which implies that there is no

correlation between X and Y. The probabilities are estimated using the histogram based

box counting method. The random variables representing the observed number of pairs

of point measurements in histogram cell (i,j), row i and column j,are respectively kij, ki.

and k.j. Here, we assume the probability of a pair of point measurements outside the area

covered by histogram is negligible, therefore ,j Pj = 1 [75; 129].









by a higher degree of synchronization among EEG electrodes within the same region,

during the pre-seizure state. We postulate that the active connection may be driven by

seizure evolution, regulating abnormal communications in the epileptogenic brain areas

or vulnerable areas in the brain network. To test these hypotheses, we herein propose

network-theoretical methods through a multivariate statistical analysis of EEGs to study

the seizure development by investigating the topological structure of the brain connectivity

network. Epileptic seizures involve the synchronization of large populations of neurons

[116]. Measuring the connectivity and synchronicity among different brain regions through

EEG recordings has been well documented [99; 117; 69]. The structures and the behaviors

of the brain connectivity have been shown to contain rich information related to the

functionality of the brain [118; 67; 68]. More recently, the mathematical principles derived

from information theory and nonlinear dynamical systems have allowed us to investigate

the synchronization phenomena in highly non-stationary EEG recordings. For example, a

number of synchronization measures were used for analyzing the epileptic EEG recordings

to reach the goals of localizing the epileptogenic zones and predicting the impending

epileptic seizures [99; 106; 119; 38; 120]. These studies also -i-.: -1 that epilepsy is a

dynamical brain disorder in which the interactions among neuron or groups of neurons

in the brain alter abruptly. Moreover, the characteristic changes in the EEG recordings

have been shown to have clear associations with the synchronization phenomena among

epileptogenic and other brain regions. When the conductivities between two or among

multiple brain regions are simultaneously considered, the univariate analysis alone will

not be able to carry out such a task. Therefore it is appropriate to utilize multivariate

analysis. Multivariate analysis has been widely used in the field of neuroscience to study

the relationships among sources obtained simultaneously. In this study, the cross mutual

information (C' \I) approach is applied to measure the connectivity among brain regions

[75]. The C' \ I approach is a bivariate measure and has been shown to have ability for









ACKNOWLEDGMENTS

I would like to thank Dr. Panagote M. Pardalos, Dr. Basim M. Uthman, Dr. J. Chris

Sackellares, Dr. Deng-Shan Shiau, and Dr. W.A. C('!i i 1wongse for their guidance and

support. During the past five years, their energy and enthusiasm have motivated me to

become a better researcher. I would also like to thank Dr. Van Oostrom, Dr. Paul Carney

and Dr. Steven Roper for serving on my supervisory committee and for their insightful

comments and '-I'- -i in

I have been very lucky to have the chance to work with many excellent people in

the Brain Dynamics Laboratory (B.D.L) and Center for Applied Optimization (C.A.O).

I would like to thank Linda Dance, Wichai Suharitdamrong, Dr. Sandeep Nair and Dr.

Michael Bewernitz from B.D.L. and Dr. Altannar Chinchuluun, Dr. Onur Seref, Ashwin

Arulselvan, Nikita Boyko, Dr. Alla Kammerdiner, Oleg Shylo, Dr. Vitally Yatsenko and

Petros Xanthopoulos in C.A.O for their valuable friendship and kindness assistance.

I thank the staff in the Department of Neurology at the Gainesville VA Medical

Center including David Juras and Scott Bearden. I thank Joy Mitchell from North Florida

Foundation for Research and Education, Inc. for providing financial support during

later part of my Ph.D. study. I also like to acknowledge the financial support provided

by National Institutes of Health (NIH), Optima Neuroscience Inc. and North Florida

Foundation for Research and Education, Inc. (NFFRE). There are many friends I have

made at the University of Florida who gave me their help and support but were not

mention here, I would like to take this opportunity to thank all of them for being with me

throughout this wonderful journey.

Lastly, I wish to thank my lovely family in my home country Taiwan my father

Ming-Chih Liu, my mother Chmni-Wei Lin, my elder sister Tsai-Lin, Liu and my younger

brother C('i i"--Hao, Liu for their selfless sacrifice and constant love.




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IwouldliketothankDr.PanagoteM.Pardalos,Dr.BasimM.Uthman,Dr.J.ChrisSackellares,Dr.Deng-ShanShiau,andDr.W.A.Chaovalitwongsefortheirguidanceandsupport.Duringthepastveyears,theirenergyandenthusiasmhavemotivatedmetobecomeabetterresearcher.IwouldalsoliketothankDr.VanOostrom,Dr.PaulCarneyandDr.StevenRoperforservingonmysupervisorycommitteeandfortheirinsightfulcommentsandsuggestions.IhavebeenveryluckytohavethechancetoworkwithmanyexcellentpeopleintheBrainDynamicsLaboratory(B.D.L)andCenterforAppliedOptimization(C.A.O).IwouldliketothankLindaDance,WichaiSuharitdamrong,Dr.SandeepNairandDr.MichaelBewernitzfromB.D.L.andDr.AltannarChinchuluun,Dr.OnurSeref,AshwinArulselvan,NikitaBoyko,Dr.AllaKammerdiner,OlegShylo,Dr.VitaliyYatsenkoandPetrosXanthopoulosinC.A.Ofortheirvaluablefriendshipandkindnessassistance.IthankthestaintheDepartmentofNeurologyattheGainesvilleVAMedicalCenterincludingDavidJurasandScottBearden.IthankJoyMitchellfromNorthFloridaFoundationforResearchandEducation,Inc.forprovidingnancialsupportduringlaterpartofmyPh.D.study.IalsoliketoacknowledgethenancialsupportprovidedbyNationalInstitutesofHealth(NIH),OptimaNeuroscienceInc.andNorthFloridaFoundationforResearchandEducation,Inc.(NFFRE).TherearemanyfriendsIhavemadeattheUniversityofFloridawhogavemetheirhelpandsupportbutwerenotmentionhere,Iwouldliketotakethisopportunitytothankallofthemforbeingwithmethroughoutthiswonderfuljourney.Lastly,IwishtothankmylovelyfamilyinmyhomecountryTaiwanmyfatherMing-ChihLiu,mymotherChing-WeiLin,myeldersisterTsai-Lin,LiuandmyyoungerbrotherChang-Hao,Liufortheirselesssacriceandconstantlove. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1OVERVIEWOFEPILEPSY ............................ 14 1.1Introduction ................................... 14 1.2CausesofSeizure ................................ 15 1.3ClassicationofEpilepticSeizure(ICES1981revision) ........... 16 1.3.1PartialOnsetSeizures .......................... 16 1.3.2GeneralizedSeizures .......................... 19 1.3.3UnclassiedSeizures .......................... 20 1.4TreatmentforEpilepsy ............................. 20 1.4.1PharmacologicalTreatment ....................... 20 1.4.2SurgeicalSection ............................ 21 1.4.3NeurostimulatorImplant ........................ 21 1.5NeuronStatesandMembranePotentials ................... 23 1.5.1NeuronStates .............................. 23 1.5.2MembranePotentials .......................... 25 1.6RecordingElectricBrainActivity ....................... 27 1.6.1ScalpEEGRecording .......................... 29 1.6.2SubduralEEGRecording ........................ 29 1.6.3DepthEEGRecording ......................... 30 1.7ConclusionsandRemarks ........................... 31 2EPILEPSYANDNONLINEARDYNAMICS ................... 33 2.1Introduction ................................... 33 2.2DynamicalSystemsandStateSpace ..................... 34 2.3FractalDimension ................................ 45 2.3.1CorrelationDimension ......................... 46 2.3.2CapacityDimension ........................... 46 2.3.3InformationDimension ......................... 47 2.4StateSpaceReconstruction .......................... 47 2.5LyapunovExponents .............................. 48 2.6Phase/AngularFrequency ........................... 50 2.7ApproximateEntropy .............................. 51 2.8DynamicalSupportVectorMachine(D-SVM) ................ 53 5

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............................... 56 2.10Cross-Validation ................................ 57 2.11PerformanceEvaluationofD-SVM ...................... 57 2.12PatientInformationandEEGDescription .................. 58 2.13Results ...................................... 58 2.14Conclusions ................................... 59 3QUANTITATIVECOMPLEXITYANALYSISINMULTI-CHANNELINTRACRANIALEEGRECORDINGSFROMEPILEPSYBRAIN ................. 60 3.1Introduction ................................... 60 3.2PatientandEEGDataInformation ...................... 61 3.3ProperTimeDelay ............................... 63 3.4TheMinimumEmbeddingDimension ..................... 65 3.5DataAnalysis .................................. 66 3.6Conclusions ................................... 67 4DISTINGUISHINGINDEPENDENTBI-TEMPORALFROMUNILATERALONSETINEPILEPTICPATIENTSBYTHEANALYSISOFNONLINEARCHARACTERISTICSOFEEGSIGNALS ..................... 72 4.1Introduction ................................... 72 4.2MaterialsandMethods ............................. 75 4.2.1EEGDescription ............................ 75 4.2.2Non-Stationarity ............................. 76 4.2.3SurrogateDataTechnique ....................... 77 4.2.4EstimationofMaximumLyapunovExponent ............. 78 4.2.5Pairedt-Test ............................... 79 4.3Results ...................................... 80 4.4Discussion .................................... 84 5OPTIMIZATIONANDDATAMININGTECHNIQUESFORTHESCREENINGOFEPILEPTICPATIENTS ............................. 91 5.1Introduction ................................... 91 5.2EEGDataInformation ............................. 93 5.3IndependentComponentAnalysis ....................... 94 5.4DynamicalFeaturesExtraction ........................ 95 5.4.1EstimationofMaximumLyapunovExponent ............. 95 5.4.2Phase/AngularFrequency ....................... 96 5.4.3ApproximateEntropy .......................... 97 5.5DynamicalSupportVectorMachine ...................... 98 5.6ConnectivitySupportVectorMachine ..................... 100 5.7TrainingandTesting:CrossValidation .................... 102 5.8ResultsandDiscussions ............................ 102 6

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............. 104 6.1Introduction ................................... 104 6.2SecondOrderSynchronizationMeasures ................... 105 6.2.1CrossCorrelationFunction ....................... 105 6.2.2PartialDirectedCoherence ....................... 106 6.3PhaseSynchronization ............................. 108 6.4MutualInformation ............................... 109 6.5NonlinearInterdependencies .......................... 112 6.6Discussions ................................... 114 7CLUSTERINGELECTROENCEPHALOGRAM(EEG)SIGNALSTOSTUDYMESIALTEMPORALLOBEEPILEPSY(MTLE) ................ 117 7.1Introduction ................................... 117 7.2EpilepsyasaDynamicalBrainDisorder ................... 120 7.3DataInformation ................................ 121 7.3.1MultivariateAnalysisonEEGSignals ................. 121 7.3.2BrainSynchronization ......................... 122 7.4Graph-TheoreticModelingforBrainConnectivity .............. 124 7.4.1Cross{MutualInformation(CMI) ................... 124 7.4.2SpectralPartitioning .......................... 128 7.4.3MaximumCliqueAlgorithm ...................... 130 7.4.4ImplicationsoftheResults ....................... 132 7.5DiscussionandFutureWork .......................... 134 8TREATMENTEFFECTSONELECTROENCEPHALOGRAM(EEG) ..... 137 8.1Introduction ................................... 137 8.2DataInformation ................................ 139 8.3SynchronizationMeasures ........................... 140 8.3.1MutualInformation ........................... 140 8.3.2NonlinearInterdependencies ...................... 143 8.4StatisticTestsandDataAnalysis ....................... 145 8.5ConclusionandDiscussion ........................... 146 REFERENCES ....................................... 158 BIOGRAPHICALSKETCH ................................ 169 7

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Table page 2-1PatientinformationandEEGdescription ...................... 59 2-2PerformanceforD-SVM ............................... 59 3-1PatientsandEEGdatastatisticsforcomplexityanalysis ............. 62 4-1PatientsandEEGdatastatistics .......................... 76 5-1EEGdatadescription ................................ 94 5-2ResultsforD-SVMusing5-foldcrossvalidation .................. 103 7-1Patientinformationforclusteringanalysis ..................... 121 8-1ULDpatientinformation ............................... 140 8-2Topographicaldistributionfortreatmentdecouplingeect(DE:DecoupleElectrode(DE)) 146 8-3Patient1beforetreatmentnonlinearinterdependencies .............. 150 8-4Patient1aftertreatmentnonlinearinterdependencies ............... 151 8-5Patient2beforetreatmentnonlinearinterdependencies .............. 152 8-6Patient2aftertreatmentnonlinearinterdependencies ............... 153 8-7Patient3beforetreatmentnonlinearinterdependencies .............. 154 8-8Patient3aftertreatmentnonlinearinterdependencies ............... 155 8-9Patient4beforetreatmentnonlinearinterdependencies .............. 156 8-10Patient4aftertreatmentnonlinearinterdependencies ............... 157 8

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Figure page 1-1Vagusnervestimulationpulsegenerator ...................... 22 1-2Vagusnervestimulationelectrode .......................... 23 1-3Corticalnervecellandstructureofconnections .................. 24 1-4Membranepotentials ................................. 25 1-5Electricalpotentials ................................. 26 1-6EEGrecordingacquiredbyHansBergerin1929 .................. 27 1-7BasicEEGpatterns ................................. 28 1-8International10-20electrodeplacement ....................... 29 1-9Surbduralelectrodeplacement ............................ 30 1-10Depthelectrodeplacement .............................. 31 2-1Rosslerattractor ................................... 36 2-2XcomponentofRosslersystem ........................... 37 2-3YcomponentofRosslersystem ........................... 38 2-4ZcomponentofRoesslersystem ........................... 39 2-5Lorenzsystem ..................................... 41 2-6XcomponentofLorenzsystem ........................... 42 2-7YcomponentofLorenzsystem ........................... 43 2-8ZcomponentofLorenzsystem ........................... 44 2-9EstimationofLyapunovexponent(Lmax) ...................... 50 2-10TemporalevolutionofSTLmax 51 2-11Phase/AngularfrequencyofLyapunovexponent(max) ............. 52 2-12Threedimensionplotforentropy,angularfrequencyandLmaxduringinterictalstate .......................................... 55 2-13Threedimensionplotforentropy,angularfrequencyandLmaxduringpreictalstate .......................................... 56 3-1Electrodeplacement ................................. 62 9

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.... 67 3-3AverageminimumembeddingdimensionprolesforPatient1(seizure2) .... 68 3-4AverageminimumembeddingdimensionprolesforPatient1(seizure3) .... 69 3-5AverageminimumembeddingdimensionprolesforPatient2(seizure4) .... 70 3-6AverageminimumembeddingdimensionprolesforPatient2(seizure5,6) ... 71 4-132-channeldepthelectrodeplacement ........................ 76 4-2Degreeofnonlinearityduringpreictalstate ..................... 81 4-3DegreeofNonlinearityduringpostictalstate .................... 82 4-4STLmaxandT-indexprolesduringinterictalstate ................ 83 4-5STLmaxandT-indexprolesduringpreictalstate ................. 84 4-6STLmaxandT-indexprolesduringpostictalstate ................ 85 4-7NonlinearitiesacrossrecordingareasduringinterictalstateforULSOZpatients 86 4-8NonlinearitiesacrossrecordingareasduringperictalstateforULSOZpatients 86 4-9NonlinearitiesacrossrecordingareasduringpostictalstateforULSOZpatients. 87 4-10NonlinearitiesacrossrecordingareasduringinterictalstateforBTSOZpatients 88 4-11NonlinearitiesacrossrecordingareasduringpreictalstateforBTSOZpatients 89 4-12NonlinearitiesacrossrecordingareasduringpostictalstateforBTSOZpatients 90 5-1Scalpelectrodeplacement .............................. 93 5-2EEGdynamicsfeatureclassication ......................... 99 5-3Supportvectormachines ............................... 101 5-4Connectivitysupportvectormachine ........................ 102 7-1EEGepochsforRTD2,RTD4andRTD6(10seconds) .............. 126 7-2ScatterplotforEEGepoch(10seconds)ofRTD2vs.RTD4andRTD4vs.RTD6 ......................................... 126 7-3Cross-mutualinformationforRTD4vs.RTD6andRTD2vs.RTD4 ...... 127 7-4Completeconnectivitygraph(a);afterremovingthearcswithinsignicantconnectivity(b) ........................................... 128 7-5Spectralpartitioning ................................. 129 10

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.............................. 131 7-7ElectrodeselectionusingthemaximumcliquealgorithmforCase1 ....... 132 7-8ElectrodeselectionusingthemaximumcliquealgorithmforCase2 ....... 133 7-9ElectrodeselectionusingthemaximumcliquealgorithmforCase3 ....... 133 8-1NonlinearinterdependencesforelectrodeFP1 ................... 147 8-2Pairwisemutualinformationbetweenforallelectrodes-beforev.s.aftertreatment 149 11

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Thebasicmechanismsofepileptogenesisremainunclearandinvestigatorsagreethatnosinglemechanismunderliestheepileptiformactivity.Dierentformsofepilepsyareprobablyinitiatedbydierentmechanisms.Thequanticationforpreictaldynamicchangesamongdierentbraincorticalregionshavebeenshowntoyieldimportantinformationinunderstandingthespatio-temporalepileptogenicphenomenainbothhumansandanimalmodels. Intherstpartofthisstudy,methodsdevelopedfromnonlineardynamicsareusedfordetectingthepreictaltransitions.Dynamicalchangesofthebrain,fromcomplextolesscomplexspatio-temporalstates,duringpreictaltransitionsweredetectedinintracranialelectroencephalogram(EEG)recordingsacquiredfrompatientswithintractablemesialtemporallobeepilepsy(MTLE).Thedetectionperformancewasfurtherenhancedbythedynamicssupportvectormachine(D-SVM)andamaximumcliqueclusteringframework.ThesemethodsweredevelopedfromoptimizationtheoryanddataminingtechniquesbyutilizingdynamicfeaturesofEEG.Thequantitativecomplexityanalysisinmulti-channelintracranialEEGrecordingsisalsopresented.Thendingssuggestthatitispossibletodistinguishepilepsypatientswithindependentbi-temporalseizureonsetzones(BTSOZ)fromthosewithunilateralseizureonsetzone(ULSOZ).Furthermore,fortheULSOZpatients,itisalsopossibletoidentifythelocationoftheseizureonsetzoneinthebrain.Improvingclinician'scertaintyinidentifyingthe 12

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RecentadvancesinnonlineardynamicsperformedonEEGrecordingshaveshowntheabilitytocharacterizechangesinsynchronizationstructureandnonlinearinterdependenceamongdierentbraincorticalregions.Althoughthesechangesincorticalnetworksarerapidandoftensubtle,theymayconveynewandvaluableinformationthatarerelatedtothestateofthebrainandtheeectoftherapeuticinterventions.Traditionally,clinicalobservationsevaluatingthenumberofseizuresduringagivenperiodoftimehavebeengoldstandardforestimatingtheecacyofmedicaltreatmentinepilepsy.EEGrecordingsareonlyusedasasupplementaltoolinclinicalevaluations.Inthelaterpartofthisstudy,aconnectivitysupportvectormachine(C-SVM)isdevelopedfordierentiatingpatientswithepilepsythatareseizurefreefromthosethatarenot.Tothatend,aquantitativeoutcomemeasureusingEEGrecordingsacquiredbeforeandafteranti-epilepticdrugtreatmentisintroduced.OurresultsindicatethatconnectivityandsynchronizationbetweendierentcorticalregionsathigherorderEEGpropertieschangewithdrugtherapy.Thesechangescouldprovideanewinsightfordevelopinganovelsurrogateoutcomemeasureforpatientswithepilepsywhenclinicalobservationscouldpotentiallyfailtodetectasignicantdierence. 13

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1 ].Epilepsyoccursinallages,thehighestincidencesoccurininfantsandinelderly[ 2 ; 3 ].Thecauseofepilepsyresultedfromalargenumberoffactors,includingheadinjury,stroke,braintumor,centralnervoussysteminfections,developmentalanomaliesandhypoxia{ischemia. Thehallmarkofepilepsyisrecurrentseizures.Seizuresaremediatedbyabruptdevelopmentofrhythmicringoflargegroupsofneuronsinthecerebralcortex.Theserhythmicdischargesmaybeginlocallyincertainregionofonecerebralhemisphere(partialseizures)orbeginsimultaneouslyinbothcerebralhemispheres(generalizedseizure).Partialonsetseizuresmayremainconnedwithinaparticularregionofthebrainandcausenochangeinconsciousnessandrelativelymildcognitive,sensory,motororautonomicsymptoms(simplepartialseizures)ormayspreadtocauseimpairedconsciousnessduringtheseizure(complexpartialseizure)alongwithavarietyofmotorsymptoms,suchassuddenandbrieflocalizedbodyjerkstogeneralizedtonic-clonicactivity(secondarygeneralizedseizures).Primarilygeneralizedseizuresinvolvebothhemispheresofthebrain,causealteredinlossofconsciousnessduringtheoccurrence 14

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1. 2. 3. 4. (a) Alteredlevelsofsodium,calcium,ormagnesium(electrolyteimbalance) (b) Kidneyfailurewithincreasedureaintheblood(uremia)orchangesthatoccurwithkidneydialysis (c) Lowbloodsugar(hypoglycemia)orelevatedbloodsugar(hyperglycemia) (d) Loweredoxygenlevelinthebrain(hypoxia) (e) Severeliverdisease(hepaticfailure)andelevationofassociatedtoxins 15

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6. 7. 8. 16

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1. (a) Withmotorsigns i. Focalmotorwithoutmarch ii. Focalmotorwithmarch(Jacksonian) iii. Versive iv. Postural v. Phonatory(Vocalizationorarrestofspeech) (b) Withsomatosensoryorspecialsensorysymptoms(simplehallucinations,e.g.,tingling,lightashes,buzzing) i. Somatosensory ii. Visual iii. Auditory iv. Olfactory v. Gustatory vi. Vertiginous (c) Withautonomicsymptomsorsigns(includingepigastricsensation,pallor,sweating,ushing,piloerectionandpupillarydilatation) (d) Withpsychicsymptoms(disturbanceofhighercerebralfunction).Thesesymptomsrarelyoccurwithoutimpairmentofconsciousnessandaremuchmorecommonlyexperiencedascomplexfocalseizures i. Dysphasic 17

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Dysmnesia iii. Cognitive(e.g.,dreamystates,distortionsoftimesense) iv. Aective(fear,anger,etc.) v. Illusions(e.g.,macropsia) vi. Structuredhallucinations(e.g.,music,scenes) 2. (a) Withimpairmentofconsciousnessatonset i. Withimpairmentofconsciousnessonly ii. Withautomatisms (b) Simplepartialonsetfollowedbyimpairmentconsciousness i. Withsimplefocalfeaturesfollowedbyimpairedconsciousness ii. Withautomatisms 3. (a) Simplepartialseizureevolvingtogeneralizedseizures (b) Complexpartialseizureevolvingtogeneralizedseizures (c) Simplepartialseizureevolvingtocomplexfocalseizuresevolvingtogeneralizedseizures 18

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1. (a) (b) 2. 3.

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4. 5. 6. 1.4.1PharmacologicalTreatment 20

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4 ].TheevidencefrombothexperimentalandclinicalstudieswhichsuggeststhatlossofecacyofAEDsmaydevelopduringtheirlong-termuseinaminorityofpatients[ 5 ].Themostcommonmechanismisanincreaseintherateofmetabolismofthedrug.Forinstance,many"rstgeneration"AEDs,includingphenobarbital(PB),phenytoin(PHT),andcarbamazepine(CBZ),stimulatetheproductionofhigherlevelsofhepaticmicrosomalenzymes,causingmorerapidremovalandbreakdownoftheseAEDsfromthecirculation[ 6 ]. 7 ].However,studieshavealsoshownthatonlyindividualwithunilateralonsetzoneproducessignicantreductionsinseizurefrequencywithcurrentsurgicalroutines[ 8 ]. 9 ]basedonpreviousobservationsthatVNSblockedinterictalEEGspiking[ 10 ]ordesynchronizedEEGinthethalamusorcortex[ 11 ; 12 ]incats.ZabaraextendedthisideatodemonstrateanticonvulsantactivityofVNSindogs[ 9 ].Lateron,McLachlandemonstratedsuppressionofinterictalspikesandseizuresby 21

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13 ].Furthermore,chronicintermittentVNShasbeenshowntopreventrecurrenceofepilepticseizuresinmonkeys[ 14 ].Theseobservationsleadtoclinicaltrialstoinvestigatethefeasibility,safety,andecacyofVNSinhumanpatients.Thersthumanimplantwasperformedin1988andtherstrandomizedactivecontrolstudywasperformedin1992.EncouragingresultsfromthersttwopilotstudiesleadtorandomizeddoubleblindclinicaltrialsthatresultedinFDAapprovalofVNSasadjunctivetherapyforintractableepilepsyinJuly,1997.ImplantationoftheNCPsystem(VNSpulsegeneratorandelectricalleads)isnormallyperformedundergeneralanesthesiaasanoutpatientbasis.Theprocedureusuallytakestwohoursandrequirestwoskinincisions;oneintheleftupperchestforhousingthepulsegeneratorinapocketundertheskinandtheotherisinleftneckareaabovethecollarbonetogainaccesstothevagusnerveandplacetwosemi-circularelectrodesandaneutraltetheraroundtheleftvagusnerve(Figure 1-1 ). Figure1-1. Vagusnervestimulationpulsegenerator 22

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Vagusnervestimulationelectrode Longtermfollow-upstudiesshowedthatpreventionofrecurrentseizureswasmaintainedandadverseeventsdecreasedsignicantlyovertime[ 15 ; 16 ].PositronemissiontomographyandfunctionalMRIstudiesshowedthatVNSactivatesorincreasesbloodowincertainareasofthebrainsuchasthethalamus[ 17 ; 18 ].Cerebrospinaluid(CSF)wasanalyzedin16subjectsbefore,3monthsafter,and9monthsafterVNStreatmentGABA(totalandfree)increasedinloworhighstimulationgroups,aspartatemarginallydecreasedandethanolamineincreasedinthehighstimulationgroupsuggestinganincreasedinhibitoryeect[ 19 ].Krahletal.,suggestedthatseizuresuppressioninducedbyVNSmaydependonthereleaseofnorepinephrineandtheyobservedthatacuteorchroniclesionsofthe"Locuscoeruleus"attenuatedVNS-inducedseizuresuppression[ 20 ]. 1.5.1NeuronStates 1-3 istakenfromMalmivuandPlonse(1993)[ 21 ]. 23

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2. Figure1-3. Corticalnervecellandstructureofconnections Theexcitatoryneuronsexcitethetargetneurons.Excitatoryneuronsinthecentralnervoussystemareoftenglutamatergicneurons.Neuronsintheperipheralnervoussystem,suchasspinalmotorneuronsthatsynapseontomusclecells,oftenuseacetylcholineastheirexcitatoryneurotransmitter.However,thisisjustageneraltendencythatmaynotalwaysbetrue.Itisnottheneurotransmitterthatdecidesexcitatoryorinhibitoryaction,butratheritisthepostsynapticreceptorthatisresponsibleforthe 24

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Membranepotentials actionoftheneurotransmitter.Inhibitoryneuronsinhibittheirtargetneurons.Inhibitoryneuronsareofteninterneurons.Theoutputofsomebrainstructures(neostriatum,globuspallidus,cerebellum)areinhibitory.TheprimaryinhibitoryneurotransmittersareGABAandglycine.Modulatoryneuronsevokemorecomplexeectstermedneuronmodulation.Theseneuronsuseneurotransmitterssuchasdopamine,acetylcholine,serotoninandothers[ 22 ].Theeectofsummingsimultaneouspostsynapticpotentialsbyactivatingmultipleterminalsonwidelyspacedareasofthemembraneiscalledspatialsummationandsuccessivepresynapticdischargesfromasinglepresynapticterminal,thistypeofsummationiscalltemporalsummation. 1-4 andgure 1-5 istakenfromMalmivuandPlonse(1993)[ 21 ].Thesourcesofelectricalpotentialcanbecategorizedintothefollowingfourdierentcategorizes: 25

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Electricalpotentials 1. 2. 3. 4. 26

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23 ].SincethentheEEGrecordingshasbeenthemostcommondiagnosistoolforepilepsy.EEGmeasurestheelectricalactivityofthebrain.EEGstudiesareparticularlyimportantwhenneurologicdisordersarenotaccompaniedbydetectablealterationinbrainstructure.ItisacceptedthattheneuronsinthethalamusplayanimportantroleingeneratingtheEEGsignals.Thesynchronicityofthecorticalsynapticactivityreectsthedegreeofsynchronousringofthethalamicneuronsthataregeneratingtheelectricalactivities.However,thepurposesoftheseelectricalactivitiesandEEGoscillationsarelargelyunknown. Figure1-6. EEGrecordingacquiredbyHansBergerin1929 ThecongurationsofEEGrecordingsplayanimportantroleindeterminingthenormalbrainfunctionfromabnormal.ThemostobviousEEGfrequenciesofanawake,relaxedadultwhoseeyesareclosedis8-13Hzalsoknownasthealpharhythm.Thealpharhythmisrecordedbestovertheparietalandoccipitallobesandisknowntobeassociatedwithdecreasedlevelsofattention.Whenalpharhythmarepresented,subjectscommonlyreportthattheyfeelrelaxedandhappy.However,peoplewhonormallyexperiencemorealpharhythmthanusualhavenotbeenshowntobepsychologicallydierentfromthosewithless.AnotherimportantEEGfrequenciesisthebetarhythm,peopleareattentiveto 27

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BasicEEGpatterns anexternalstimulusorarethinkinghardaboutsomething,thealpharhythmisreplacedbylower-amplitude,high-frequency(>13Hz).Thisisbetarhythmoscillations.ThistransformationisalsoknownasEEGarousalandisassociatedwiththeactofpayingattentiontostimuluseveninadarkroomwithnovisualinputs.Atransientisaneventwhichclearlystandsoutagainstthebackground.Asharpwaveisatransientwith70ms{200msinduration.Aspikewaveisatransitionwithlessthan70msinduration.Aspikethatfollowbyaslowwaveiscalledaspike-and-wavecomplex,whichcanbeseeninpatientswithtypicalabsenceseizure.Inthecaseshavingtwoormorespikeoccurinsequenceformingmultiplespikecomplexcallpolyspikecomplex,iftheyarefollowedbyaslowwave,theyarecalledpolyspike-and-wavecomplex.Spikeandsharpwavesare 28

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1-7 istakenfromMalmivuandPlonse(1993)[ 21 ]. 1-8 .Therecordingelectrodesareplacedonthescalpoftheheadandrecordelectricalpotentialdierencesbetweentherecordingelectrodes.However,recordingsacquiredfromscalpareusuallycontaminatedbymultiplesourcesofartifactssuchasmovementartifacts,chewingartifacts,eyemovement,vertexwavesandsleepspindles,etc.Theinternational10-20electrodeplacementsystemiscommonlyusedforroutinescalpEEGrecording[ 24 ].Figure 1-8 istakenfromMalmivuandPlonse(1993)[ 21 ]. Figure1-8. International10-20electrodeplacement 1-9 .Itrequiressurgicalproceduretoplacethesubduralrecordingelectrodesandtherisksofinfectionincreasewiththeamount 29

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Figure1-9. Surbduralelectrodeplacement 1-10 .Thedepthrecording 30

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Figure1-10. Depthelectrodeplacement 25 ].Likemanyphenomenaoccurinnature,thereexistcertainbuildupperiodpriortosomemajorevents. 31

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26 { 31 ].Fromabovendings,itisreasonabletohypothesizethataseizureisstartingfromsmallabnormallydischargingfromneuronsthatrecruitandentraintheneighboringneuronsintoalargerorfullseizure.Thishypothesisisparticularlyclearforthefocalonsetseizures.Theserecruitment,entrainmentandtransitionphenomenatakeplaceinabrainstate,thepreictalstate.Recently,forpatientswithMTLEseveralauthorshaveshownitispossibletodetectthepreictaltransitionsusingquantitativeEEGanalysis[ 32 { 39 ]. Inchapter2,weinvestigatedtheexistenceofpreictalstates.WequantiedanddetectedthechangesinEEGdynamicsthatareassociatedwithpreictalstateinEEGrecordings.Threedierentdynamicalmeasureswereused:1.LargestLyapunovexponent,ameasureisknownformeasuringthechaoticityofthesteadystateofadynamicalsystem,2.PhaseinformationofthelargestLyapunovspectrum,basedontheoryfromtopologyandinformationtheoryand3.Approximateentropy,amethodformeasuringtheregularityorpredictabilityoftimeseries.Thedynamicssupportvectormachine(D-SVM)wassubsequentlyintroducedforimprovingtheperformanceofthepreictaldetection.Inchapter3,weanalyzedthecomplexityofEEGrecordingsusingmethodsdevelopedfromnonlineardynamicsandshowedtheEEGcomplexitychangespriortotheseizureonsets.Inthechapter4,weinvestigatedthedierencesinnonlinearcharacteristicsbetweenpatientswithindependentbi-temporalseizureonsetzone(BTSOZ)andpatientswithunilateralseizureonsetzone(ULSOZ).Inchapter5,weintroducedthesupportvectormachinestoclassifyEEGrecordingsacquiredfromseizurefreeandnoneseizurefreepatients.Inchapter6,wediscusseddierentmethodsforspatiotemporalEEGtimeseriesanalysis.Inchapter7,weproposedamaximumcliqueframeworktostudythebraincorticalnetworks.Inthechapter8,westudiedthemedicaltreatmenteectsonthestructureofbraincorticalnetworks. 32

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Thedevelopmentofpreictaltransitionscanbeconsideredasasuddenincreaseofsynchronousneuronalringinthecerebralcortexthatmaybeginlocallyinaportionofonecerebralhemisphereorbeginsimultaneouslyinbothcerebralhemisphere.Byobservingtheoccurrenceofepilepticseizures,itisreasonabletobelievethattherearemultiplestatesexistinaepilepticbrainandthesequencesofthestatesarenotdeterministic.ThepreictaltransitionsaredetectableEEGdynamicalchangesbyapplyingmethodsdevelopedfromnonlineardynamics.Severalgroupshavereportedthatseizuresarenotsuddentransitionsinandoutoftheabnormalictalstate;instead,seizuresfollowacertaindynamicaltransitionthatdevelopsovertime[ 32 { 39 ]see[ 40 ; 41 ]forreview.InanstudyofPijnetal.in1991,authorswereabletodemonstratedecreaseinthevalueofcorrelationdimensionatseizureonsetintheratmodel.Inearly1990s,Iasemidisetal.,rstestimatedthelargestLyapunovexponentandreportedseizurewasinitiateddetectabletransitionperiodbyanalyzingspatiotemporaldynamicsoftheEEGrecordings;thistransitionprocessischaracterizedby:(1)progressiveconvergenceofdynamicalmeasuresamongspecicanatomicalareas\dynamicalentrainment"and(2)followingtheovershotbrainresettingmechanismduringpostictalstate.Martinerieetal.,(1998) 33

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42 ].Mormannetal.,(2003)detectedthepreictalstatebasedondecreasein\synchronization"measures[ 43 ]. Thebasictextofnonlineardynamicsandnonlineardynamicalmodelsarepresentedinthefollowingsections.Nonlineardynamicalmeasuresnamely(thelargestLyapunovexponent(Lmax),Phase/Angularfrequency(),Approximateentropy(ApEn))wereusedfordetectingthepreictaltransitionsinintracranialEEGrecordingsacquiredfrompatientswithintractableMTLE.Sincetheunderlyingdynamicsofpreictaltransitionsischangingfromcasetocase,thisdemandssophisticateanalyticaltoolswhichhavetheabilityforidentifyingthechangesofbraindynamicswhenpreictaltransitionsoccur.Thepreictaldetectionperformancewasfurtherimprovedbyproposeddynamicssupportvectormachine(D-SVM),aclassicationmethoddevelopedfromoptimizationtheoryanddataminingtechniques.Thedetectionperformancesweresummarizedinthelaterpartofthischapter. 34

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1. 2. whereTdenotestheperiodofthiscycle. 3. 4. 2-1 2-5 forRosslerandLorenzattractor). Recallanattractorisasetofstatevariables;geometricallyanattractorcanbeapoint,acurve,amanifold,orevenacomplicatedsetwithafractalstructureknownasthe\strangeattractor".Describingtheseattractorshasbeenoneoftheachievementsofchaos 35

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Rosslerattractor theory.Inthefollowingsections,severalmethodsforqualifyingthedynamicalattractorsaredescribed;resultsonapplicationforrealworldEEGrecordingarealsoincluded. 44 ] _x=(y+z); _y=x+ay; _z=b+xzcz: (2{6) (x;y;z)2<3 Initialconditions:[00.01-0.01]. 36

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XcomponentofRosslersystem 37

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YcomponentofRosslersystem 38

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ZcomponentofRoesslersystem 39

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45 ].ThestrangeattractorinthiscaseisafractalofHausdordimensionbetween2and3.Grassberger(1983)hasestimatedtheHausdordimensiontobe2.060.01andthecorrelationdimensiontobe2.050.01[ 46 ]. (2{7) _y=x(z)y _z=xyz (2{10) (x;y;z)2<3 Initialconditions:[00.01-0.01]; Length:40seconds; Samplerate:50Hz. 40

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Lorenzsystem 41

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XcomponentofLorenzsystem 42

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YcomponentofLorenzsystem 43

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ZcomponentofLorenzsystem 44

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Forregularattractors,irrespectivetotheoriginofthesphere,thedimensionwouldbethedimensionoftheattractor.Butforachaoticattractor,thedimensionvariesdependingonthepointatwhichtheestimationisperformed.Ifthedimensionisinvariantunderthedynamicsoftheprocess,wewillhavetoaveragethepointdensitiesoftheattractoraroundit.Forthepurposeofidentifyingthedimensioninthisfashion,wendthenumberofpointsy(k)withinaspherearoundsomephasespacelocationx.Thisisdenedby: whereistheHeavisidestepfunctionsuchthat(n)=0forn<0,(n)=1forn0 Thiscountsallthepointsontheorbity(k)withinaradiusrfromthepointxandnormalizesthisquantitybythetotalnumberofpointsNinthedata.Also,weknowthatthepointdensity,(x),onanattractordoesnotneedtobeuniformforastrangeattractor.Choosingthefunctionasn(x;r)q1anddeningthefunctionC(q;r)oftwovariablesqandrbythemeanofn(x;r)q1overtheattractorweightedwiththenatural 45

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ThisC(q;r)iswellknowncorrelationintegralandthefractaldimensionofthesystemisestimated: forsmallarthatthefunctionlog[C(q,r)]behavelinearlywithlog[r]fortruedimension. CorrelationDimensionhasbeenshowntohavetheabilityincapturingthepreictaltransitioninmanystudies.Forexample,A.BabloyantzandA.Destexhe,(1986)showntheexistenceofchaoticattractorinphasespacefromEEGacquiredfromanapatientwithabsenceseizure[ 47 ].Pijnetal.,(1997)showedintemporallobeepilepsypatientsthatepilepticseizureactivityoften,butnotalways,emergesasalow-dimensionaloscillation[ 48 ].Itwasalsofoundthatcorrelationdimensiondecreasesindeepsleepstages,thusreectingasynchronizationoftheEEG.AdecreaseincorrelationdimensionhasbeenrelatedtotheabnormalsynchronizationbehaviorsonEEGrecordingsinepilepsyandotherpathologiessuchasAlzheimers,dementia,Parkinson[ 48 { 50 ]. 46

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log[]: Informationdimensionisdenedas: log(r);(2{17)D2D1D0ifelementsofthefractalisequallylikelytobevisitedinthestatespace. wheremistheembeddingdimension.Thetransformfunction#mustbeunique(i.e.,hasnoselfintersection).Whitney(1936)provedatheoremwhichcanalsobeusedforndingtheembeddingdimension[ 51 ].#:M!<2d+1;#embeddingisanopenanddensesetinthespaceofsmoothmaps. 47

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andthedurationofeachembeddingvectoris Amuchmoregeneralsituationfortime-laggedvariablesconstituteanadequateembeddingprovidedthemeasuredvariableissmoothandcoupledtoalltheothervariablesisprovedbyTakens,andthenumberoftimelagisatleast2d+1[ 52 ].#:M!<2d+1isanopenanddensesetinthespaceofpairsofsmoothmaps(f,h),wherefisthedynamicalsystemmeasurebyfunctionh. 53 ].Foradynamicalsystemastimeevolvesthesphere 48

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54 ].Forandimensionalsystem,astimeevolves,theorderLyapunovexponentiscorrespondingtothemostexpandedtothemostcontractedprincipalaxes. Iasemidisetal.rstusedthemaximumLyapunovexponenttoshowtheEEGrecordingsexhibitabrupttransientdropsinchaoticitybeforeseizureonset[ 55 { 61 ].ThemaximumLyapunovexponentisdenedby: jXi;j(0)j;(2{22) whereti=t0+(j1)t,withandi2[1;Na]andt2[0;t],tismaximumevolutiontimeforXi;j. Xi;j(0)=X(ti)X(tj);(2{24) isperturbationoftheducialorbitatti,and Xi;j(t)=X(ti+t)X(tj+t);(2{25) istheevolutionofXi;j(0)aftert. TheX(ti)isvectoroftheducialtrajectoryt(X(t0)),wheret=t0+(i1)t,X(t0)=x(t0;:::;X(t0+(p1))T,andX(tj)isaproperlychosenvectoradjacenttoX(ti)inthestatespace.Naisnecessarynumberofiterationsforthesearchthroughreconstructedstatespace(withembeddingdimensionpanddelay),fromNdatapoints 49

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EstimationofLyapunovexponent(Lmax) anddurationT.IfDtisthesamplingperiod,then Iftheevolutiontimetisgiveninsecond,thentheunitofLisbit=sec.TheselectionofpisbasedfromTakens'embeddingtheoremandwasestimatedfromepochesduringictalEEGrecordings.Takens'embeddingtheoremisdened: usingtheabovedenedfx,evenifoneonlyobservesonevariablex(t)fort!1,onecanconstructanembeddingofthesystemintoap=2m+1dimensionalstatespace. ThedimensionoftheictalEEGattractorisfoundbetween2to3inthestatespace.ThereforeaccordingtoTakens'theembeddingdimensionwouldbeatleastp=23+1=7.Theselectionofischosenassmallaspossibletocapturethehighestfrequencycomponentinthedata. 50

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TemporalevolutionofSTLmax =1 whereNisthetotalnumberofphasedierencesestimatedfromtheevolutionofX(ti)toX(ti+(t))inthestatespace,and i=jarccos(X(ti)X(ti+t) 62 ].Itcandierentiatebetweenregularandirregulardataininstanceswheremomentstatistics(e.g.meanandvariance)approachesfailtoshowasignicantdierence.Applicationsincludeheartrateanalysis 51

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Phase/AngularfrequencyofLyapunovexponent(max) inthehumanneonateandinepilepticactivityinelectrocardiograms(Diambra,1999)[ 63 ].Mathematically,aspartofageneraltheoreticalframework,ApEnhasbeenshowntobetherateofapproximatingaMarkovchainprocess[ 62 ].Mostimportantly,comparedApEnwithKolmogrov-Sinai(K-S)Entropy(Kolmogrov,1958),ApEnisgenerallyniteandhasbeenshowntoclassifythecomplexityofasystemviafewerdatapointsviatheoreticalanalysisofbothstochasticanddeterministicchaoticprocessesandclinicalapplications[ 62 ; 64 { 66 ].HereIgivebriefdescriptionaboutApEncalculationforatimeseriesmeasuredequallyintimewithlengthn.SupposeS=s1;s2;:::;snisgivenandusethemethodofdelayweobtainthedelayvectorx1;x2;:::;xnm+1inRm: 52

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m(rf)=nm+1Xi=1lnCmi(rf) Finallytheapproximateentropyisgivenby: Theparameterrfcorrespondstoanapriorixeddistancebetweenneighboringtrajectoryandrfischosenaccordingtothestandarddeviationestimatedfromdata.Hence,rfcanbeviewedasalteringlevelandtheparametermistheembeddingdimensiondeterminingthedimensionofthephasespace.Heuristically,ApEnquantiesthelikelihoodthatsubsequencesinSofpatternsthatarecloseandwillremaincloseonthenextincrement.ThelowerApEnvalueindicatesthatthegiventimeseriesismoreregularandcorrelated,andlargerApEnvaluemeansthatitismorecomplexandindependent. D-SVMperformsclassicationbyconstructinganN-dimensionalhyperplanethatseparatesthedataintotwodierentclasses.ThemaximalmarginclassierruleisusedtoconstructtheD-SVM.TheobjectiveofmaximalmarginD-SVMistominimizethebond 53

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Ahyperplane(w;b)iscalledacanonicalhyperplanesuchthat 2kwk2+C subjectto whereCisaparametertobechosenbytheuser,wisthevectorperpendiculartotheseparatinghyperplane,bistheosetandarereferringtotheslackvariablesforpossibleinfeasibilityoftheconstraints.Withthisformulation,oneswantstomaximizethemarginbetweentwoclassesbyminimizingkwk2.Thesecondtermoftheobjectivefunctionisusedtominimizethemisclassicationerrorsthataredescribedbytheslackvariablesi.IntroducingpositiveLagrangemultipliersitotheinequalityconstraintsinDSVMmodel,weobtainthefollowingdualformulation: min1 2i=1Xnj=1XnyiyjijxixjnXi=1i(2{39) s.t. 0iC;i=1;:::n:(2{41) Thesolutionoftheprimalproblemisgivenbyw=Piiyixi,wherewisthevectorthatisperpendiculartotheseparatinghyperplane.Thefreecoecientbcanbefound 54

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2-12 and 2-13 showthe3Dplotforforentropy,angularfrequency,andLmaxduringinterictal(100datapointsdynamicalfeatures2hourspriortoseizureonset)andpreictalstate(100datapointsdynamicalfeaturessampled2minspriortoseizureonset). Figure2-12. Threedimensionplotforentropy,angularfrequencyandLmaxduringinterictalstate 55

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Threedimensionplotforentropy,angularfrequencyandLmaxduringpreictalstate wherek 56

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TN+1;t T];(2{44) whereNdenotesnumberofLmaxinthemovingwindowand^ijtdenotesthestandarddeviationofthesampleDijwithinw(t).Asymptotically,Tij(t)followsthet-distributionwithN1degreeoffreedom.WeusedN=30(i.e.averagesof30paireddierencesofvaluesfromdynamicalprolespermovingwindow). AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyapreictalEEGepochasapreictalEEGsample. AclassicationresultisconsideredtobetruenegativeiftheD-SVMclassifyainterictalEEGepochasainterictalEEGsample. 57

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AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyainterictalEEGepochasapreictalEEGsample. AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyapreictalEEGepochasainterictalEEGsample. TheperformanceoftheD-SVMisevaluateusingsensitivityandspecicity: Sensitivity=TP 58

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PatientinformationandEEGdescription Patientno.GenderAgeFocusregion(s)NumberofseizureLengthofrecording 1F45R.H128days2F42R.H/L.H1812days3M30R.H65days4M39R.H/L/H43days5M52R.H95days6F65R.H76days Table2-2. PerformanceforD-SVM Patientno.SensitivitySpecicity 191.3%91.4%293.2%95.2%390.9%91.7%494.1%96.4%584.7%92.0%689.3%94.5% resultsalsoconrmthatthepreictalandinterictalbrainstatearedierentiableusingapproachesdevelopedfromnonlineardynamics. 59

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67 { 71 ].Itisknownthatadynamicalsystemwithddegreeoffreedommayevolveonamanifoldwithalowerdimension,sothatonlyportionsofthetotalnumberofdegreeoffreedomareactuallyactive.Forasimplesystemwithlimitcycles,itisobviousthattime-delayembeddingproduceanequivalentreconstructionofthetruestate.AccordingtoembeddingtheoremfromWhitney(1936),anarbitraryD-dimensioncurvedspacecanbemappedintoaCartesian(rectangular)spaceof2d+1dimensionswithouthavinganyselfintersections,hencesatisfyingtheuniquenessconditionforanembedding[ 51 ].Saueretal.(1991)generalizedWhitneysandTakens'theoremtofracturalattractorswithdimensionDfandshowedtheembeddingspaceonlyneedtohaveadimensiongreaterthan2Df[ 72 ].Althoughitispossibleforafractaltobeembeddedinanotherfractal,weonlyconsidertheintegerembedding.Takensdelayembeddingtheorem 60

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52 ].Fortheabovereasons,weemployedamethodproposedbyCao(1997)toestimatetheminimumembeddingdimensionofEEGtimeseries[ 73 ].Likesomeotherexitingmethods,Caosmethodisalsoundertheconceptsoffalse-nearest-neighborsThefalse-nearest-neighborsutilizedonthefactthatifthereconstructionspacehasnotenoughdimensions,thereconstructionwillperformaprojection,andhencewillnotbeanembeddingofthedesiredsystem[ 74 ].Asaofresultofgivingatolowembeddingdimensionwhileprocessingtheembeddingprocedure,twopointswhichisfarawayinthetruestatespacewillbemappedintocloseneighborinthereconstructionspace.Thesearethenthefalseneighbors.Caosmethoddoesnotrequirelargeamountofdatapoints,isnotsubjectiveanditisnottime-consumingndtheproperminimumembeddingdimension.TheEEGrecordingswasdividedintonon-overlappingsingleelectrodesegmentsof10.24sduration,eachofwhichwasestimatedfortheminimumembeddingdimension.UndertheassumptiontheEEGrecordingswithineach10.24sdurationwasapproximatelystationary[ 56 ],weevaluatedtheunderliningdynamicalbehaviorbylookingattheminimumembeddingdimensionovertime. Theremainingofthischapterisorganizedasfollows.InSections.2and3,wedescribethedatainformationandexplainthealgorithmforestimatingtheminimumembeddingdimensionestimation.Theresultsfromtwopatientswithatotalnumberofsixtemporallobeepilepsy(TLE)aregiveninSection4.InSection5,wediscusstheresultsofourndingswithrespecttheuseofthisalgorithmandthefunctionofnonlineardynamicalmeasurementsintheareaofseizurecontrol. 61

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PatientsandEEGdatastatisticsforcomplexityanalysis Patient#GenderAgeFocus(RH/LH)LengthofEEG(hr.)Numberofseizure P1M19RH20h37m05s3P2M33LH09h43m57s3 transverseandBlateralviewsofthebrain,illustratingthedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodesripsareplacedovertheleftorbitfrontal(LOF),rightorbitofrontal(ROF),leftsubtemporal(LST),andrightsubtemporal(RST)cortexseeFigure 3-1 .TheEEGrecordingdataforepilepsypatientswereobtainedaspartofpre-surgicalclinicalevaluation.TheyhadbeenobtainedusingaNicoletBMSI4000and5000recordingsystem,usings0.1Hzhigh-passanda70hzlowpasslter.Eachrecordingcontainsatotalnumberof28to32intracranialelectrodes(8subduraland6hippocampaldepthelectrodesforeachcerebralhemisphere).Priortostorage,thesignalsweresampledat200Hzusingananalogtodigitalconverterwith10bitsquantization.Therecordingswerestoreddigitallyontohighdelityvideotype.Twoepilepsysubjects(seeTable 3-1 )wereincludedinthisstudy. Figure3-1. Electrodeplacement 62

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75 ].Theconceptofmutualinformationisgivenasbelow Mutualinformationisoriginatedfrominformationtheoryandithasbeenusedformeasuringinterdependencebetweentwoseriesofvariables.LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.samplingrate)isxed.Themutualinformationbetweenobservationsxiandyiisdenedas: wherePx;y(xi;yj)isthejointprobabilitydensityofxandyevaluatedat(xi;yj)andPx(xi),Py(yj)arethemarginalprobabilitydensitiesofxandyevaluatedatxiandyjrespectively.Theunitofmutualinformationisinbit,whenbased2logarithmistaken. Ifxandyarecompletelyindependent,thejointprobabilitydensityPx;y(xi;yj)equalstotheproductofitstwomarginalprobabilitiesandthemutualinformationbetween 63

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FraserandSwinney(1986)showedonecouldobtaintheaveragemutualinformationbetweentwotimeseriesSandQwithlengthns1;s2;:::;snandq1;q2;:::qnusingtheentropiesH(S),H(Q)andH(S;Q).Supposewehaveobservedtherstseriesofinterestwithasetofnoutcomess1;s2;:::;sn;andeachoutcomeisassociatedwithprobabilitiesPs(s1);Ps(s2);:::Ps(sn)andsameasQ. Theaverageamountofuncertaintythatameasurementofsreducestheuncertaintyofqisgiven Inotherwords,\Byknowingameasurementofs,howmanybitsonaveragecanbepredictedaboutq?" SupposeavariablevisinvestigatedbybeingsampledwithsamplingintervalTs.LetsuchprocessbethecontextofsystemSandsystemQ,letsbethemeasurementofvattimet,andletqbethemeasurementattimet+Ts.UsingthesemeasurementtodenesystemsSandQ,mutualinformationI(Q;S)canbecalculated.Thus,mutualinformationbecomesafunctionofTs.Forthisproblem,mutualinformationwillbethenumberofbitsofv(t+Ts)thatcanbepredicated,onaverage,whenv(t)isknown.OnewantstopickTsshouldbechosensothatv(t+Ts)isasunpredictableaspossible.Maximumunpredictabilityoccursatminimumofpredictability;thatis,attheminimuminthemutualinformation.Becauseoftheexponentialdivergenceofchaotictrajectories, 64

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52 ].Thereareseveralclassicalalgorithmsusedtoobtaintheminimumembeddingdimension[ 76 ; 74 ; 77 ].Theclassicalapproachesusuallyrequirehugecomputationpowerandvastamongofdata.Anotherlimitationofthesealgorithmsisthattheyusuallysubjectivetodierenttypesofdata.WeevaluatedtheminimumembeddingdimensionoftheattractorsfromtheEEGbyusingCaosmethod.Thenotionsherefollowed\Practicalmethodfordeterminingtheminimumembeddingdimensionofascalartimeseries".Supposethatwehaveatimeseries(x1;x2;x3;:::;xN).Applyingthemethodofdelayweobtainthetimedelayvectorasfollows: wheredistheembeddingdimensionandisthetime-delayandyi(d)meanstheithreconstructedvectorwithembeddingdimensiond.Similartotheideaofthefalsenearestneighbormethod,dening kyi(d)yn(i;d)(d)k;i=1;2;:::;Nd(3{7) 65

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3-2 3-3 3-4 3-5 and 3-6 showtypicaltheminimumembeddingdimensionovertimeforsixseizures.Onecanobservethebehavioroftheaverageminimumembeddingdimensionovertimeforsixbraincorticalregions.Theminimumembeddingdimension 66

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AverageminimumembeddingdimensionprolesforPatient1(seizure1) showedstableduringtheinterictalstate.Inotherwords,theunderlyingdegreeoffreedomisuniformlydistributedovertheinterictalstateintheEEGrecordings.Theresultsindicatedthelowestminimumembeddingdimensionwerefoundwithintheepilepticzoneduringinterictalstate(theRSTelectrodesinFigs.2,3,and4;theLTDelectrodesinFigs.5and6).ThecomplexityoftheEEGrecordingsfromtheepilepticregionislowerthanthatfromthebrainregions.Thevaluesoftheminimumembeddingdimensionfromallbrainregionsstartdecreaseandconvergetoalowervalueasthepatientproceedfrominterictaltoictalstate.Theunderliningdynamicalchangesbeforeenteringictalperiodwereconsistentlydetectedbythealgorithm. 67

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AverageminimumembeddingdimensionprolesforPatient1(seizure2) Itiscomputationallyecientandcertainlylesstimeconsumingcomparedtosomeclassicalproceduresforestimatingembeddingdimensionestimation.TheresultsofthisstudyconrmthatitispossibletopredictanseizurebasedonnonlineardynamicsofmultichannelintracranialEEGrecordings.Formajorityofseizuresthespatiotemporaldynamicalfeaturesofthepreictaltransitionaresimilartothatoftheprecedingseizure.Thissimilaritymakesitpossibletoapplyoptimizationtechniquestoidentifyelectrodesitesthatwillparticipateinthenextpreictaltransition,basedontheirbehaviorduringthepreviouspreictaltransition.Atpresenttheelectrodeselectionproblemsweresolvedecientlyandthesolutionswereoptimallyattained.However,futuretechnologymayallowphysicianstoimplantthousandsofelectrodesitesinthebrain.Thisprocedurewillhelpustoobtainmoreinformationandallowtohaveabetterunderstandingabouttheepilepticbrain.Therefore,inordertosolveproblemswithlargernumberofrecordingelectrodes,thereisaneedtodevelopcomputationallyfastapproachesforsolving 68

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AverageminimumembeddingdimensionprolesforPatient1(seizure3) large-scalemulti-quadratic0{1programmingproblems.OurresultsalsoarecompatiblewiththendingsaboutthenatureoftransitionstoictalstateininvasiveEEGrecordingsfrompatientswithseizuresofmesialtemporalorigin.Thedevelopmentofmulti-quadratic0{1programmingmodelingisinprogress. 69

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AverageminimumembeddingdimensionprolesforPatient2(seizure4) 70

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AverageminimumembeddingdimensionprolesforPatient2(seizure5,6) 71

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72

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1 ].Epilepsymaybetreatedwithdrugs,surgery,aspecialdiet,oranimplanteddeviceprogrammedtostimulatethevagusnerve(VNStherapy).ForthepatientwhodoesnotreacttoAnti{epilepticdrugs(AEDs)orothertypesofthetremens,epilepsysurgeryisoftenconsideredbecauseitoersthepotentialforcureofseizuresandsuccessfulpsychosocialrehabilitation.Theusefulnessofresectivesurgeryforthetreatmentofcarefullyselectedpatientswithmedicallyintractable,localization-relatedepilepsyisclear.Seizure-freeratefollowingtemporallobectomyareconsistently65%to70%inadults.Epilepsysurgeryliesoncarefullyevaluationofthecandidatesforsurgery,surgicalinterventionmaybecarriedoutwithahighprobabilityofsuccessiftheareaofseizureonsetisconsistentlyandrepeatedlyfromthesameportionofthebrain.Thefocuslocalizationprocedurebecomesoneofthemostimportanttasksinthepre-surgicalexamination.Someseizuresareresultedfromcorticaldamage.Neuroimagingcanhelpinidentifyingandlocalizingthedamageregionsinthebrainandtherefore,thefocus.Currently,brainmagneticresonanceimaging(MRI)providesthebeststructuralimagingstudy.However,themostcommontoolinepilepsydiagnosisiselectroencephalogram(EEG).TheEEGrecordingsreectinteractionsbetweenneuronsinthebrain.Forroutineper-surgicalevaluation,thepatienthastostayintheepilepsymonitoringunit(EMU)for4to5daysandisexpectedtoobtain4seizuresintherecording.However,insomecasesthehospitalstaymaybelongerinordertohaveenoughseizurestoidentifythefocusarea.Therefore,aecientlymethodfordeterminingthesuitabilityofapatientbasedontheanalysisoftheshorterEEGrecordingswouldnotonlyimprovetheoutcomeoftheseizurecontrolbutalsoreducethepatients'nancialburdenforthepre-surgicalevaluationprocedures.Ingeneral,EEGcapturesthespatiotemporalinformationoftheunderlyingneuronsactivitiesnearbytherecordingelectrodes.ItisacceptedthatEEGrecordingcontainsnon-linearmechanismsatmicroscopiclevel.ManystudieshaveshownthepresencesofnonlinearityintheEEGrecordingsfrombothhumans 73

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78 ].Andrzejaketal.(2006)showedbyfocusingonnonlinearityandacombinationofnonlinearmeasureswithsurrogatesappearsasthekeytoasuccessfulcharacterizationofspatiotemporaldistributionofepilepticprocess[ 71 ].B.Weberetal.(1998)alsoshownevidencesfortheusefulnessofnonlineartimeseriesanalysisforthecharacterizationofthespatio-temporaldynamicsoftheprimaryepileptogenicareainpatientswithTLE[ 79 ].UsingcorrelationdimensionK.Lehnertzetal.(1995)reportedthevarianceoftheEEGdimensionduringinterictalallowedtheprimaryepileptogenicareatobecharacterizedinexactagreementwiththeresultsofthepresurgicalwork-up[ 49 ].ManyresearchersalsohaveshownthattheintracranialEEGrecordingsexhibitcertaincharacteristicsthataresimilartochaoticsystems.Forexample,Sackellares,Iasemidisetal.rstusedthemaximumLyapunovexponent,ameasureofchaoticity,toshowtheEEGrecordingsexhibitabrupttransientdropsinchaoticitybeforeseizureonset[ 32 ; 60 ; 80 ].Byfollowingtheconceptofspatiotemporaldynamicalentrainment(i.e.,similardegreeofchaoticitybetweentwoEEGsignals),thisgroupfurtherreportedthat,duringtheinterictalstate,thenumberofrecordingsitesentrainedtotheepileptogenicmesialtemporalfocuswassignicantlylessthanthatofthehomologouscontralateralelectrodesites[ 57 ].TheseresultsuggestedthatitispossibletoidentifytheepileptogenicfocusbyexaminingthedynamicalcharacteristicsoftheinterictalEEGsignals.Therstpartofthisstudy,aphaserandomizationsurrogatedatatechniquewasusedtogeneratesurrogateEEGsignals.ByrandomizingthephaseoftheFourieramplitudes,allinformationwhichisnotcontainedinthepowerspectrumislost.ThesurrogateEEGwillhavethesamelinearpropertiesthusthesamepowerspectrumandequalcoecientsofalinearautoregressive(AR)model.IftheoriginaldistributionofLmaxissignicantlydierentfromitssurrogate,itwillbetheevidenceforthenonlinearity.Inthesecondpartofthisstudy,eightadultpatientswithtemporallobeepilepsywere 74

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4.2.1EEGDescription 4-1 ainferiortransverseviewsofthebrain,illustratingapproximatedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodestripsareplacedovertheleftorbitofrontal(LOF),rightorbitofrontal(ROF),leftsubtemporal(LST),andrightsubtemporal(RST)cortex.Depthelectrodesareplacedinthelefttemporaldepth(LTD)andrighttemporaldepth(RTD)torecordhippocampalactivity. EEGrecordingsobtainedfromeightpatientswithtemporallobeepilepsywereincludedinthisstudy.SeeTable 4-1 .Fivepatientswereclinicallydeterminedtohaveunilateralseizureonsetzone(ULSOZ)andtheremainingthreepatientsweredeterminedtohaveindependentbi-temporalseizureonsetzone(BTSOZ).Foreachpatient,threeseizureswereincludedintheEEGrecordings.Segmentsfrominterictal(atleastonehourbeforetheseizure),preictal(immediatelybeforetheseizureonset)andpostictal(immediatelyaftertheseizureoset)timeintervalscorrespondingtoeachseizureweresampledfortestingthehypothesis.Twoelectrodesfromeachbrainareawereincluded,atotalof12electrodeswereanalyzedforeachpatient.TheEEGrecordingsweresampledusingamplierswithinputrangeof0.6mV,andafrequencyrangeof0.5{70Hz.Therecordingswerestoreddigitallyonvideotapeswithasamplingrateof200Hz,usingan 75

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PatientsandEEGdatastatistics Patient#GenderAgeFocus(RH/LH)LengthofEEG(hr.) P1M19RH6.1P2M45RH5.4P3M41RH5.8P4F33RH5.3P5F38RH6.3P6M44RH/LH5.5P7F37RH/LH4.6P8M39RH/LH5.4 analogtodigital(A/D)converterwith10bitquantization.Inthisstudy,alltheEEGrecordingswereviewedbytwoindependentboardcertiedelectroencephalographers. Figure4-1. 32-channeldepthelectrodeplacement 76

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55 ]. 81 ].Therearethreedierentproceduresforsurrogatedata. 1. Surrogatesarerealizationsofindependentidenticallydistributed(iid)randomvariableswiththesamemean,variance,andprobabilitydensityfunctionastheoriginaldata.Theiidsurrogatesweregeneratedbyrandomlypermutingintemporalorderthesamplesoftheoriginalseries.Thisshuingprocesswilldestroythetemporalinformationandthusgeneratedsurrogatesaremainlyrandomobservationdrawn(withoutreplacement)fromthesameprobabilitydistributionasoriginaldata. 2. Fouriertransform(FT)surrogatesareconstrainedrealizationsoflinearstochasticprocesseswiththesamepowerspectraastheoriginaldata.FTsurrogateserieswereconstructedbycomputingtheFToftheoriginalseries,bysubstitutingthephaseoftheFouriercoecientswithrandomnumbersintherangewhilekeepingunchangedtheirmodulus,andbyapplyingtheinverseFTtoreturntothetimedomain.Torendercompletelyuncoupledthesurrogatepairs,twoindependentwhitenoiseswhereusedtorandomizetheFourierphases. 3. Autoregressive(AR)surrogatesaretypicalrealizationsoflinearstochasticprocesseswiththesamepowerspectraastheoriginalseries.BygeneratingaGaussiantimeserieswiththesamelengthasthedata,andreorderedittohavethesamerankdistribution.TaketheFouriertransformofthisandrandomizethephases(FT).Finally,thesurrogateisobtainedbyreorderingtheoriginaldatatohavethesame 77

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Inthispresentedstudy,weemployedthesecondalgorithmtogeneratethesurrogatedata.Byshuingthephasesbutkeepingtheamplitudeofthecomplexconjugatepairsatthesametimethesurrogateswillhavethesamepowerspectrum(autocorrelation)asthedata,butwillhavenononlineardeterminism.ForeachEEGepoch,tensurrogateswillbeproducedtoinsuredFourierphasesarecompletelyrandomized.Thissurrogatealgorithmhasbeenappliedandcombiningwithcorrelationintegral,ameasuresensitivetoawidevarietyofnon-linearities,wasusedfordetectionfornonlinearitybyCasdigalietal.(1996)[ 78 ]. 55 ; 54 ].Inthissection,wewillonlygiveashortdescriptionandbasicnotationofourmathematicalmodelsusedtoestimateSTLmax.First,letusdenethefollowingnotation. 78

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LetLbeanestimateoftheshorttermmaximumLyapunovexponent,denedastheaverageoflocalLyapunovexponentsinthestatespace.Lcanbecalculatedbythefollowingequation: jXi;j(0)j:(4{1) 79

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wherek overamovingwindoww(t)denedas TN+1;t T](4{4) whereNdenotesnumberofLmaxinthemovingwindowand^ijtdenotesthestandarddeviationofthesampleDijwithinw(t).Asymptotically,Tij(t)followsthet-distributionwithN1degreeoffreedom.WeusedN=30(i.e.averagesof30paireddierencesofLmaxvaluespermovingwindow).SinceeachLmaxvaluewasderivedfroma10.24secondofEEGepoch,thelengthofourmovingwindowisapproximatelyabout5minutes.AcriticalvalueT 4-2 4-3 showtheLmaxprolefrombothoriginalanditssurrogatesrespectively,foranalysispurposes,herewedeneinterictalstateisthetimedurationatleast60minspriortoseizureonset,preictalisthetimeduration10minspriortoseizureonset,nallypostictalis5minsafterseizureonset.Theresultsclearshowedthenonlinearitywasdetectedinthreedierentstates,thisfurtherclariedLmaxwascapabletocaptureinformationwhichisnotcontainedinlinearARmodel.ThedierencesbetweenoriginalLmaxanditsurrogateswasfoundlargestintheEEGrecordedfromepilepticfocusregions. TheSTLmaxvalues(thediscriminatingstatistic)estimatedfromoriginalEEGanditssurrogates,andT-indexprolesduringinterictal(Fig. 4-4 ),pre-ictal(Fig. 4-5 ) 80

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Degreeofnonlinearityduringpreictalstate andpost-ictal(Fig. 4-6 )state,respectively,inaULSOZpatient.Eachgurecontainstwoareas,onefromthefocalarea(toptwopanels)andanotherfromthehomologouscontralateralhippocampusarea(bottomtwopanels).Fromthesegures,itisclearthattheEEGrecordedfromthefocusareaexhibitshigherdistinctionfromGaussianlinearprocessesthanthoserecordedfromthehomologouscontralateralhippocampusareainallthreestates.Further,thedierencesofSTLmaxvaluesinthefocusareaincreasedfrominterictaltopreictal,andreachedtothemaximuminthepostictalstate,butthedierenceremainedthestableinthehomologouscontralateralarea. 81

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DegreeofNonlinearityduringpostictalstate Figures 4-7 (A), 4-8 (A)and 4-9 (A)showthedegreeofnonlinearity(quantiedmeanT-indexvalues)inEEGforULSOZpatients,duringinterictal,preictalandpostictalstates,respectively.Figures 4-7 (B), 4-8 (B)and 4-9 (B)showMultiplecomparisonsofnonlinearitiesineachpairofrecordingareas(A=LTD,B=RTD,C=LST,D=RST,E=LOF,F=ROF). Theresultsdemonstratedthatthenonlinearitieswereinconsistentacrossrecordingareasforallvepatients.TheresultsfromANOVAshowedthatthereexistsignicantlyrecordingareaeectsonthedegreeofnonlinearityinallthreestates(p-values=0.0019,0.0012,0.0015forinterictal,preictalandpostictal,respectively).Further,multiplecomparisons(showninFigures 4-7 b, 4-8 band 4-9 b)revealedthatsignicantlydierences 82

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(p-value<0.05)inthedegreeofnonlinearityexistsbetweenthefocusarea(RightHippocampus,RTD)anditshomologouscontralateralbrainarea(LeftHippocampus,LTD)inthreestates,withhigherdegreeofnonlinearityinfocalarea.ThedegreeofnonlinearityacrossrecordedareasforthreeBTSOZpatientswasshowninFigures 4-10 4-11 and 4-12 forinterictal,preictalandpostictalstates,respectively.Itisobservedthatthedegreeofnonlinearitywasuniformlydistributedoverrecordedareas.TheresultsfromANOVArevealedthattherecordingareaeectsonthedegreeofnonlinearityinstateswerenotsignicant(p-values=0.9955,0.9945,0.9975forinterictal,preictalandpostictal,respectively). 83

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84

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resultsofthisstudysuggestthatthedistributionofEEGsignalnonlinearitiesacrossrecordingbrainareasinULSOZpatientsisdierentfromBTSOZpatients.IneachofthevetestpatientswithULSOZ,theEEGnonlinearitiesweresignicantlyinconsistentamongrecordingareas.Ontheotherhand,theEEGnonlinearitieswereuniformlydistributedacrossbrainareasineachofthethreetestpatientswithBTSOZ.Theseresultswereconsistentduringtheinterictal,preictalandpostictalperiods.Thusitmaybepossibletoecientlyandquantitatively,withashortdurationofEEGrecording,determinewhetheranepilepticpatienthasunilateralfocalareathathe/shecouldbeacandidateforepilepsysurgerytreatment.Iftheseresultscanbevalidatedinalargesamplepatient,thedurationofEEGmonitoringprocedureforaBTSOZepilepticpatientcouldbegreatlyshortened.ThiswillnotonlyreducethecostoftheEEGmonitoring 85

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NonlinearitiesacrossrecordingareasduringinterictalstateforULSOZpatients Figure4-8. NonlinearitiesacrossrecordingareasduringperictalstateforULSOZpatients 86

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NonlinearitiesacrossrecordingareasduringpostictalstateforULSOZpatients. procedure,butalsowilldecreasetheriskofinfectioncausedbytheimplantedrecordingelectrodes.LargesampleofpatientswithULSOZandBTSOZwillberequiredforreliableestimationofsensitivityandspecicityofthismethod.CorrectidenticationofthefocalareainULSOZpatientsisachallengingtask.AnobviousquestioncouldbewhetheranyofbrainareaswheretheEEGsignalsrecordedfromiscloseenoughtotheactualfocalarea.Ifnot,itwouldbeverydiculttoidentifythefocalareabyananalysisontheseEEGsignals.Otherissuessuchasthenumberofrecordingareasandnumberofrecordingelectrodesineachareacouldalsoaecttheresultsoftheanalysis.IneachoftheveULSOZpatientsstudiedhere,EEGsignalswererecordedfromsixdierentbrainareas:leftandrighthippocampus,subtemporal,andorbitofrontalregions.Allvepatientswereclinicallydeterminedtohavefocalareaintherighthippocampus.Duringtheinterictalstate,focalarea(righttemporaldepth)consistentlyexhibitedhigherdegreeofnonlinearitythaninthecontralateraltemporaldepthandsubtemporalareas(signicantobservationsin4outof5patients).Similarndingswerealsoobservedduringpreictalandpostictalstates.Theseresultssuggestthatitispossibletoidentifythefocalarea 87

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NonlinearitiesacrossrecordingareasduringinterictalstateforBTSOZpatients inpatientswithULSOZ.Furtherstudiesonalargersampleofpatientstovalidatetheseresultsarewarranted.Successofthisstudywillprovidemoremuch-neededinformationtoguideelectroencephalographerandcliniciantoimprovethelikelihoodofsuccessfulsurgery. 88

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NonlinearitiesacrossrecordingareasduringpreictalstateforBTSOZpatients 89

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NonlinearitiesacrossrecordingareasduringpostictalstateforBTSOZpatients 90

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Epilepsyisthemostcommondisordersofnervoussystems.Preliminaryndingsonthecostsofepilepsyshowthetotalcosttothenationfor2.3millionpeoplewithepilepsywasapproximately$12.5billion.Thehighincidenceofepilepsyoriginatesfromthefactthatitoccursasaresultofalargenumberoffactors,including,febriledisturbance,geneticabnormalmutation,developmentaldeviationaswellasbraininsultssuchascentralnervoussystem(CNS)infections,hypoxia,ischemia,andtumors. Neuronorgroupsofneuronsgenerateelectricalsignalswheninteractingortransmittinginformationbetweeneachother.TheEEGrecordingscapturethelocaleldpotentialaroundelectrodesthatgeneratefromneuroninthebrain.Throughvisualinspection, 91

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SVMhasbeensuccessfullyimplementedforbiomedicalresearchonanalyzingverylargedatasets.MoreoverSVMhasbeenrecentlyappliedfortheuseofepilepticseizurepredictionandithasbeenshowntoachieved76%sensitivityand78%specicityforEEGrecordingsfrom3patients[ 82 ].NurettinAcirandCuneytGuzelisintroducedatwo-stageprocedureSVMfortheautomaticepilepticspikesdetectioninamulti-channelEEGrecordings[ 83 ].BrunoGonzalez-Velldnetetal.,reporteditispossibletodetecttheepilepticseizuresusingthreefeaturesoftheelectroencephalogram(EEG),namely,energy,decay(damping)ofthedominantfrequency,andcyclostationarityofthesignals[ 84 ].Alongwiththisdirections,theabnormalEEGidenticationproblemcanbemodeledasbinaryclassicationproblem{\normalorabnormal".EmbeddedwithneuronnetworkandconnectivityconceptswerstproposedanddescribedanapplicationofconnectivitysupportvectormachineC-SVM,C-SVMisbasedonnetworkmodelingconceptsandconnectivitymeasurestocomparetheEEGsignalsrecordedfromdierentbrainregions.AdetailowchartoftheproposedC-SVMframeworkisgiveninFigure??.WealsousesthreedynamicalfeaturesofEEG1.Angularfrequency2.Approximateentropy3.Short-termlargestlyapunovexponenttoconductthedynamicalSVMinthesecondpartofthisstudy. 92

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Scalpelectrodeplacement 5-1 showsthelocationoftheelectrodesonthescalp. Table 8-1 showstheEEGdescriptionfrom10subjects,EEGsignalswererecordedatsamplingrate250Hz.Forconsistency,weanalyzeandinvestigateEEGtimeseriesusingbipolarelectrodesonlyfrom18standardchannelsforeverypatient.EEGrecordingsfromeachsubjectswereinspectedbycerticatedelectroencephalographers.Werandomlyanduniformlysampletwo30-secondEEGepochsfromeachsubject.SinceEEGrecordingsweredigitizedatthesamplingrateof250Hz,thelengthofeachEEGepoch7,500points. 93

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EEGdatadescription EEGdata Patient Duration(minutes) Length(points) A1 28.71 430,650 A2 29.87 448,025 A3 20.89 313,375 A4 30.19 452,875 A5 29.94 449,150 N1 31.50 472,464 N2 32.90 493,464 N3 28.06 420,964 N4 21.90 328,464 N5 33.33 499,464 Total 287.29 4,309,395 85 ].TheICAalgorithmsconsiderthehigher-orderstatisticsoftheseparatedatamapsrecordedatdierenttimepoints,withnoregardforthetimeorderinwhichthemapsoccur.[ 86 { 88 ].Assumingalinearstatisticalmodel,wehave wherexandyarerandomvectorswithzeromeanandnitecovariance;Aisarectangularmatrixwithatmostasmanycolumnsasrows. TheElementsofvectorxaretheindependentcomponents,x1;:::;xn,whicharenlinearmixturesobserved.Elementsofvectoryaretheindependentcomponents,y1;:::;ym.Withoutlossofgenerality,zeromeanassumptioncanalwaysbemade.Iftheobservablevariablesxidonothavezeromean,itcanalwaysbecenteredbysubtractingthesamplemeanthatproducesthezero-meanmodel.Ahaselementsaijfori=1;:::;nandj=1;:::;m.Valuesofnandmmaybedierent. 94

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Inthisstudy,WeusedtheGaussianKerneltoimprovetheperformanceofC-SVM.TheGaussianisdenedasK(xi;xj)=exp(jjxixjjj2 5.4.1EstimationofMaximumLyapunovExponent 55 ].Inthissection,wewillonlygiveashortdescriptionandbasicnotationofourmathematicalmodelsusedtoestimateSTLmax.First,letusdenethefollowingnotation. 95

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LetLbeanestimateoftheshorttermmaximumLyapunovexponent,denedastheaverageoflocalLyapunovexponentsinthestatespace.Lcanbecalculatedbythefollowingequation: jXi;j(0)j:(5{3) =1 whereNisthetotalnumberofphasedierencesestimatedfromtheevolutionofX(ti)toX(ti+(t))inthestatespace,and i=jarccos(X(ti)X(ti+t) 96

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62 ].Itcandierentiatebetweenregularandirregulardataininstanceswheremomentstatistics(e.g.meanandvariance)approachesfailtoshowasignicantdierence.Applicationsincludeheartrateanalysisinthehumanneonateandinepilepticactivityinelectrocardiograms(Diambra,1999)[ 63 ].Mathematically,aspartofageneraltheoreticalframework,ApEnhasbeenshowntobetherateofapproximatingaMarkovchainprocess[ 62 ].Mostimportantly,comparedApEnwithKolmogrov-Sinai(K-S)Entropy(Kolmogrov,1958),ApEnisgenerallyniteandhasbeenshowntoclassifythecomplexityofasystemviafewerdatapointsviatheoreticalanalysisofbothstochasticanddeterministicchaoticprocessesandclinicalapplications[ 62 ; 64 { 66 ].HereIgivebriefdescriptionofApEncalculationforatimeseriesmeasuredequallyintimewithlengthn,S=s1;s2;:::;snisgivenbyrstformasequenceofvectorx1;x2;:::;xnm+1inRmusing: wheremisgivenasanintegerandrfisapositiverealnumber.ThevalueoflisthelengthofcomparedsubsequencesinS,andrfspeciesatolerancelevel. m(rf)=nm+1Xi=1lnCmi(rf) Finallytheapproximateentropyisgivenby: 97

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Ahyperplane(w;b)iscalledacanonicalhyperplanesuchthat 2kwk2+C subjectto 98

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EEGdynamicsfeatureclassication infeasibilitiesoftheconstraints.Withthisformulation,oneswantstomaximizethemarginbetweentwoclassesbyminimizingkwk2.Thesecondtermoftheobjectivefunctionisusedtominimizethemisclassicationerrorsthataredescribedbytheslackvariablesi.IntroducingpositiveLagrangemultipliersitotheinequalityconstraintsinD-SVMmodel,weobtainthefollowingdualformulation: min1 2i=1Xnj=1XnyiyjijxixjnXi=1i(5{15) 99

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0iC;i=1;:::n(5{17) Thesolutionoftheprimalproblemisgivenbyw=Piiyixi,wherewisthevectorthatisperpendiculartotheseparatinghyperplane.Thefreecoecientbcanbefoundfromi(yi(wxi+b)1)=0,foranyisuchthatiisnotzero.D-SVMmapagivenEEGdatasetofbinarylabeledtrainingdataintoahighdimensionalfeaturespaceandseparatethetwoclassesofdatalinearlywithamaximummarginhyperplaneinthedynamicalfeaturespace.Inthecaseofnonlinearseparability,eachdatapointxintheinputspaceismappedintoadierentspaceusingsomenonlinearmappingfunction'.Anonlinearkernelfunction,k(x;x),canbeusedtosubstitutethedotproduct<'(x);'(x)>.ThiskernelfunctionallowstheD-SVMtooperateecientlyinanonlinearhigh-dimensionalfeaturespacewithoutbeingadverselyaectedbydimensionalityofthatspace. LetGbeanundirectedgraphwithverticesV1;:::;Vn,whereVirepresentselectrodei.Thereisanedge(link)withtheweightwijforeverypairofnodesViandVjcorrespondingtotheconnectivityofthebraindynamicsbetweenthesetwoelectrodes.Theconnectivityorsynchronizationcanbeviewedasbeenactivatedbyinteractionsbetweenneuronsinthelocalcircuitryunderlyingtherecordingelectrodes.Figure 5-4 representsahypotheticalbraingraphinwhicheachconnectedpathdenotestheunderlyingconnectivity.Withthisgraphmodel,theattributesofC-SVMinputsarethepair-wisedrelationbetweentwotimeseriesprolesratherthantimestampsofatimeseriesprole.Inthiscontext,the 100

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Supportvectormachines inputofC-SVMisthedegreeofconnectivitybetweendierentbrainregions.Givenntimeseriesdatapoints,eachwithmtimestamps,theproposedframeworkwilldecreasethenumberattributesby2(n1)=mtimes.Letlbethetotalnumberofdatapoints,thedimensionalitycanbereducedfromA2
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Connectivitysupportvectormachine 5-2 .TheproposedC-SVMwithoutGaussiankernelproducedaverageaccuracyof69.4%andD-SVMusingdynamicalfeaturesobtainedfromEEGrecordingsproduced94.7%inclassication. 102

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ResultsforD-SVMusing5-foldcrossvalidation D-SVM Results D-SVM/C-SVM DynamicalfeaturesUNICA 5-foldCV1 46%72% 2 92%76% 3 94%68% 4 98%72% 5 96%66% 6 92%54% 7 96%70% 8 95%76% 9 94%70% 10 95%70% Average%ofcorrectness 94.7%69.4% TheSVMhasaverylongstatisticalfoundationandassuretheoptimalfeasiblesolutionforasetoftrainingdata,givenasetoffeaturesandtheoperationoftheSVM.Inthisstudy,weattemptedtostudytheseparabilitybetweenabnormalEEGandnormalEEGusingdierentEEGfeatures.WetestedtheperformanceonscalpEEGrecordingsfromnormalindividualsandabnormalpatients.TheEEGdatawaslteredusingICAalgorithm.ICAltersthenoiseinEEGscalpdata,keepsessentialstructureandmakesbetterrepresentableEEGdatasets.TheEuclideandistancebasedC-SVMwasproposedtoevaluatetheconnectivityamongdierentbrainregions.ThedynamicalfeaturesweregeneratedasinputforD-SVM,theclassicationresultsoftheproposedD-SVMareveryencouraging.TheresultsindicatedthatD-SVMimprovesclassicationaccuracycomparetoC-SVM.Itgivesanaverageaccuracyof94:7%.Thedynamicalfeaturesprovideasubsetinthefeaturespaceandimproveclassicationaccuracy. 103

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89 ].Synchronizationcanbequantiedinbothspaceandtimedomain.Foramulti-variatesystem,understandingtheinteractionsamongitsvariousvariables,whosebehaviorcanberepresentedalongtimeastime-sequences,presentsmanychallenges.Oneofthekeyaspectsofhighlysynchronizedsystemswithspatialextentistheirabilitytointeractbothacrossspaceandtime,whichcomplicatestheanalysisgreatly.Inbiologicalsystemssuchasthecentralnervoussystem,thisdicultyiscompoundedbythefactthatthecomponentsofinteresthavenonlinearcomplicateddynamicsthatcandictateoverallchangesinthesystembehavior.Theexactgureofhowtoquantifytheinformationexchangesinasystemremainsambiguous.Studiesonmulti-variatetimeseriesanalysishaveresultedindevelopmentofawiderangeofsignal-processingtoolsforquanticationofsynchronizationinsystems.However,thegeneralconsensusonhowtoquantifythisphenomenonislargelyuncertain.Intheliterature,synchronizationbetweenvariablescanbecategorizedasidenticalsynchronization,phasesynchronizationandgeneralizedsynchronization.Inthefollowingchapters,Iundertakeanin-depthanalysisofpreictalandinterictalsynchronizationbehavior,focusingonEEGrecordingsfrompatientswithtemporallobeandgeneralizedseizures. 104

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6.2.1CrossCorrelationFunction where=N1;N;:::;0;:::;N1.ThiscrosscorrelationissymmetricRxy()=Rxy().Itcanalsobeshownthat wherex; yaretheestimatedmeanofx(n)andy(n). Cross-correlationcoecientbetweenx(n)andy(n)isdenedascrosscovariancenormalizedbytheproductofthesquarerootofthevariancesoftwoobservedseries,asfollows: wherexyarethestandarddeivationofx(n)andy(n).Cross-correlationcoecientboundedbetween-1and+1. Theasthefrequencydomain,denex(!)=F(x(n))andY(!)=F(y(n))astheFouriertransformequivalentsofx(n)andy(n).Ifthecross-spectrumCxy(!)andauto-spectrumsCxx(!)andCyy(!)aredenedbynormalizedcross-coherenceandcanbe 105

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(Cxx(!)Cyy(!))1 2:(6{4) Thecross-coherencequantiesthedegreeofcouplingbetweenXandYatgivenfrequency!anditisalsoboundedbetween-1and+1. 90 ].However,thereversecasemayormaynotbetrue.Tomakeaquantitativeassessmentoftheamountoflinearinteractionandthedirectionofinteractionamongmultipletime-series,theconceptofGranger-causalitycanbeusedtoandintothedevelopmentmultiautoregressivemodel(MVAR).Thepartialdirectedcoherencefromjtoiatafrequency!isgivenby: wherefori=j Aij(!)=1pXr=1aij(r)ej2!r;(6{6) andfori6=j here 106

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and pdenotesthedepthoftheARmodel,rdenotesthedelayandnisthepredictionerrororthewhitenoise. Notethat!quantiestherelativestrengthoftheinteractionofagivensignalsourcejwithregardtosignaliascomparedtoallofjsinteractionstoothersignals.ItturnsoutthatthePDCisnormalizedbetween0and1atallfrequencies.Ifi=j,thePartialDirectedCoherencerepresentsthecasualinuencefromtheearlierstatetoitscurrentstate. TheMVARapproacheshavebeenusedtodeterminethepropagationofepilepticintracranialEEGactivityintemporallobeandmesialseizures[2-3,10,19-20].However,thesemodelsstrictlyrequirethatthemeasurementsbemadefromallthenodes,orthedirectionalrelationshipscouldbeambiguous.Inaddition,thereremainsnoclearevidenceofcausalityrelationshipsamongthecorticalregionsassuggested\thenatureofsynchronizationismostlyinstantaneousorwithoutanydetectabledelay"[ 91 ]. Thegeneral,nonlinearityarecommonlyinherentwithinneuronalrecordings,theabovelinearmeasuresaretypicallyrestrictedtomeasurestatisticaldependenciesuptothesecondorder[ 92 ].IfobservationsareGaussiandistributed,the2ndorderstatisticsaresucienttocapturealltheinformationinthedata.However,inpractice,EEGdata 107

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78 ]. 93 ].Conceptually,iftherhythmsofonesignalareinharmonywiththatoftheother,thetwosignalsareknowntobephaselocked.Phasesynchronizationcanthereforebedenedasthedegreetowhichtwosignalsarephaselocked.ForthisHuygens'classicalcase,phasesynchronizationisusuallydenedaslockingofthephasesoftwooscillators. wherenandmareintegers,'a(t)and'b(t)denotethephasesoftheoscillators,andn;misdenedastheirrelativephase. Rosenblumetal,(1996)generalizedtheabovephaselockingformulabytheweakerconditionofphaseentrainment[ 94 ]: byevenweakerconditionoffrequencylocking: dtmgdbt dt=0;(6{13)edenotesaveragingovertime,and'n;mtherelativefrequencyofthesystem. Mostofrealworldsignalshavebroadspectra.Forexample,EEGsignalrecordingsareusuallyintherangeof0.1to1000Hzeventhoughtheyareusuallybandpasslteredbetween0.1and70Hzsinceamajorportionoftheenergyiscontainedinthatspectrum.TheEEGcanbeclassiedroughlyintove(5)dierentfrequencybands,namelythedelta(0-4Hz),theta(4-8Hz),alpha(8-12Hz),Beta(12-16Hz)andtheGamma(16-80Hz)frequencybands.FreemandemonstratedevidenceofphaselockingbetweenEEG 108

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95 ; 96 ].Similarly,itisalsobelievedthatphasesynchronizationacrossnarrowfrequencyEEGbands,pre-seizureandattheonsetofseizuremayprovideusefulhintsofthespatio-temporalinteractionsinepilepticbrain[ 33 ; 97 ; 35 ].Hilberttransformisusedcomputetheinstantaneousparameters'a(t)and'b(t)ofatime-signal.Considerareal-valuednarrow-bandsignalx(t)concentratedaroundfrequencyfc.Denex(t)as where excitedbyaninputsignalx(t).ThislteriscallaHillberttransformer.Hilberttransformsareaccurateonlywhenthesignalshavenarrow-bandspectrum,whichisoftenunrealisticformostreal-worldsignals.Pre-processingofthesignalsuchasdecomposingitintonarrowfrequencybandsisneededbeforeweapplyHilberttransformationtocomputetheinstantaneousparameters.Certainconditionsneedtobecheckedtodeneameaningfulinstantaneousfrequencyonanarrow-bandsignal.Ithasbeenreportedthatthedistinctdierencesinthedegreeofsynchronizationbetweenrecordingsfromseizure-freeintervalsandthosebeforeanimpendingseizure,indicatinganalteredstateofbraindynamicspriortoseizureactivity[ 89 ]. 98 ].Generally,mutualinformationmeasurestheinformationobtainedfromobservationsofonerandomeventfortheother.Itisknownthatmutualinformationhasthecapabilitytocapturebothlinearandnonlinearrelationshipsbetweentworandomvariablessincebothlinearandnonlinearrelationshipscanbedescribedthroughprobabilistictheories.Hereinourmodel,themutualinformationmeasureshowmuchinformationofEEGtime 109

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99 { 104 ]. Kraskovetal.,2004introducedtwoclassesofimprovedestimatorsformutualinformationM(X;Y)fromsamplesofrandompointsdistributedaccordingtosomejointprobabilitydensity(x;y).Incontrasttoconventionalestimatorsbasedonhistogramapproach,theyarebasedonentropyestimatesfromknearestneighbourdistances.LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.,samplingperiod)isxed.Thenthemutualinformationisgivenby: wherepx(i)=Ridx,py(i)=Ridyand \Ri"denotestheintegraloverbini.Ifnx(i)andny(j)arethenumberofdatapointsintheithbinofXandjthbinofY;n(i;j)isthenumberofdatapointsintheintersectionbin(i;j).Theprobabilitiesareestimatedaspx(i)nx(i)=N,px(j)nx(j)=Nandp(i;j)nx(j)=N.Ratherthenbinapproachthemutualinformationcanbeestimatedfromk-nearestneighborstatistics. WerstestimatebH(X)fromXby 110

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Nextlet(i) 2bethedistancebetweenzianditskthneighbor.Inordertoestimatethejointprobabilitydensityfunction(p:d:f:),weconsidertheprobabilityPk()whichistheprobabilitythatforeachzithekthnearestneighborhasdistance(i) 2dfromzi.Thisprobabilitymeansthatk1pointshavedistancelessthanthekthnearestneighborandNk1pointshavedistancegreaterthan(i) 2andk1pointshavedistancelessthan(i) 2.Pk()isobtainedusingthemultinomialdistribution: wherepiisthemassofthe-ball.Thentheexpectedvalueoflogpiwillbe: where()isthedigammafunction: where()isthegammafunction.Itholdsthat(1)=CwhereCistheEuler-Mascheroniconstant(C0:57721).Themassofthe-ballcanbeapproximated(ifweconsidertheprobabilitydensityfunctioninsidetheballisthesame)by: 111

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6{23 )wecanndanestimatorforP(X=xi): log[P(X=xi)](k)(N)dE(log(i))logcdx;(6{24) nallywithEq( 6{24 )andEq( 6{25 )weobtaintheKozachenko-LeonenkoentropyestimatorforX[ 105 ]: ^H(X)=(N)(k)+logcdx+dx where(i)istwicethedistancefromxitoitsk-thneighborinthedxdimensionalspace.Forthejointentropywehave: ^H(X;Y)=(N)(k)+log(cdxcdy)+dx+dy TheI(X;Y)isnowcanbeobtainedbyEq.( 6{16 ).Theproblemwiththismethodisthataxedkisusedinallestimatorsbutthedistancemetricindierentscaledspaces(marginalandjoint)arenotcomparable.Toavoidsuchproblem,insteadofusingaxedk,nx(i)+1andny(i)+1areusedinobtainingthedistances(wherenx(i)andny(i)arethenumberofsamplescontainedthebin[x(i)(i) 2;x(i)+(i) 2]and[y(i)(i) 2;y(i)+(i) 2]respectively)inthex{yscatterdiagram.TheEq.( 6{26 )becomes: ^H(X)=(N)(nx(i)+1)+logcdx+dx FinallytheEq.( 6{16 )isrewrittenas: 106 ].Inthissection,weinvestigatedirectionalrelationshipsusingthenonlinear 112

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Giventwotimeseriesxandy,usingmethodofdelaytoobtaindelayvectorsxn=(xn;:::;xn(m1))andyn=(xn;:::;xn(m1)),wheren=1;:::N,mistheembeddingdimensionanddenotesthetimedelay[ 52 ].Letrn;jandsn;j,j=1;:::;kdenotethetimeindicesoftheknearestneighborsofxnandyn.Foreachxn,themeanEuclideandistancetoitskneighborsisdenedas andtheY-conditionedmeansquaredEuclideandistanceisdenedbyreplacingthenearestneighborsbytheequaltimepartnersoftheclosestneighborsofyn Thedelay=5isestimatedbyautomutualinformationfunction,theembeddingdimensionm=10isobtainedusingCao'smethodusing10secEEGselectedduringinterinctalstateandaTheilercorrectionissettoT=50[ 73 ; 107 ]. IfxnhasanaveragesquaredradiusR(X)=(1=N)PNn=1R(N1)n(X),thenR(k)nR(k)n(X)R(k)(X)iftheyareindependent.Accordingly,itcanbedeneandinterdependencemeasureS(k)(XjY)as SinceR(k)n(XjY)R(k)n(X)byconstruction, 0
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whichisnormalizedbetween0and1.TheoppositeinterdependenciesS(YjX),H(YjX),andN(YjX)aredenedincompleteanalogyandtheyareingeneralnotequaltoS(XjY),H(XjY),andN(XjY),respectively.Usingnonlinearinterdependenciesonseveralchaoticmodel(Lorenz,Roessler,andHeenonmodels)Quirogaetal.,(2000)showedthemeasureHismorerobustthanS. Theasymmetryofabovenonlinearinterdependenciesisthemainadvantageoverothernonlinearmeasuressuchasthemutualinformationorthephasesynchronization.Thisasymmetrycangiveinformationabout\driver-response"relationshipsbutcanalsoreectdierentpropertiesofdynamicalsystemswhenitisimportancetodetectcausalrelationships.Itshouldbeclearthattheabovenonlinearinterdependenciesmeasuresarebivariatemeasures.Althoughitquantiedthe\driver-response"forgiveninput-thewholeinputspaceunderstudymightbedrivenbyotherunobservedsystem(s). 114

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108 ].Incognitivetaskstudies,In1980,Freemanfound\moreregular"spatiotemporalactivitiesinEEGforabriefperiodoftimewhentheanimalinhaledafamiliarodoruntiltheanimalexhaled[ 109 ; 96 ].Someearlierstudiesalsoindicatedthesignicantroleofsynchronizationforphysiologicalsystemsinhumans;thedetectablealtersinsynchronizationphenomenahavebeenassociatedtoanumberofchronic,acutediseasesorthenormalityofbrain.[ 110 ; 111 ].Intheeldofepilepsyresearch,severalauthorshavesuggesteddirectrelationshipbetweenaltersinsynchronizationphenomenaandonsetoftheepilepticseizuresusingEEGrecordings.Forexample,Iasemidisetal.,(1996)reportedfromintracranialEEGthattheentrainmentinthelargestLyapunovexponentsfromcriticalcorticalregionsisanecessaryconditionforonsetofseizuresforpatientswithtemporallobeepilepsy[ 112 ; 61 ];LeVanQuyenetal.,showedepilepticseizurecanbeanticipatedbynonlinearanalysisofdynamicalsimilaritybetweenrecordings[ 35 ].Mormannetal.,showedthepreictalstatecanbedetectedbasedonadecreaseinsynchronizationonintracranialEEGrecordings[ 89 ; 43 ].ThehighlycomplexbehaviorontheEEGrecordingsisconsideredtonormalityofbrainstate,whiletransitionsintoalowercomplexitybrainstateareregardedasapathological 115

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113 ].Innextchapters,wewillusetheconceptsandmathematicalframeworksintroducedinthischaptertoclustertheEEGbetweennormalandabnormalepochesandtodetecttheeectinEEGresultedfromAEDstreatments. 116

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Neuralactivityismanifestedbyelectricalsignalsknownasgradedandactionpotentials.Berger'sdemonstrationin1929hasshownthatitispossibletorecordtheelectricalactivityfromthehumanbrain,particularlytheneuronslocatednearthesurfaceofthebrain[ 23 ].Whileweoftenthinkofelectricalactivityinneuronsintermsofactionpotentials,theactionpotentialsdonotusuallycontributedirectlytotheelectroencephalogram(EEG)recordings.Infact,forscalpEEGrecordings,theEEGpatternsaremainlythegradedpotentialsaccumulatedfromhundredsofthousandsofneurons.TheEEGpatternsvarygreatlyinbothamplitudeandfrequency.TheamplitudeoftheEEGreectsthedegreeofsynchronousringoftheneuronslocatedaroundtherecordingelectrodes.Ingeneral,thehighEEGamplitudeindicatesthatneuronsare 117

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Althoughthebrainmayhaveoriginallyemergedasanorganwithfunctionallydedicatedregions,recentevidencesuggeststhatthebrainevolvedbypreserving,extending,andre-combiningexistingnetworkcomponents,ratherthanbygeneratingcomplexstructuresdenovo[ 114 ; 115 ].Thisissignicantbecauseitsuggests:(1)thebrainnetworkisarrangedsuchthatthefunctionalneuralcomplexessupportingdierentcognitivefunctionssharemanylow-levelneuralcomponents,and(2)thespecicconnectiontopologyofthebrainnetworkmayplayasignicantroleinseizuredevelopment.Thislineofthinkingisalsosupportedby[ 70 ],whichdemonstratesthatspecicconnectedstructuresareeithersignicantlyabundantorrareincorticalnetworks.Ifseizuresevolveinthisfashion,thenweshouldbeabletomakesomespecicempiricalhypothesesregardingtheevolutionofseizures,thatmightbeborneoutbyinvestigatingthesynchronizationbetweentheactivityindierentbrainareas,asrevealedbyquantitativeanalysisofEEGrecordings.Thegoalofthisstudyistotestthefollowingtwohypotheses.First,weshouldexpectthebrainactivityintheorbitofrontalareasarehighlycorrelatedwhiletheactivityinthetemporallobeandsubtemporallobeareasarehighlycorrelatedwiththeirownside(leftonlyorrightonly)duringthepre-seizureperiod.Thehighcorrelationcanbeviewedasarecruitmentoperationinitiatedbyanepileptogenicareathrougharegularcommunicationchannelinthebrain.Notethattheconnectionofthesebrainareashasbeenalong-standingprincipleinnormalbrainfunctionsandwebelievethatthesameprincipleshouldholdinthecaseofepilepsyaswell.Second,weshouldexpectsomebrainregionstobeconsistentlyactive,manifested 118

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116 ].MeasuringtheconnectivityandsynchronicityamongdierentbrainregionsthroughEEGrecordingshasbeenwelldocumented[ 99 ; 117 ; 69 ].Thestructuresandthebehaviorsofthebrainconnectivityhavebeenshowntocontainrichinformationrelatedtothefunctionalityofthebrain[ 118 ; 67 ; 68 ].Morerecently,themathematicalprinciplesderivedfrominformationtheoryandnonlineardynamicalsystemshaveallowedustoinvestigatethesynchronizationphenomenainhighlynon-stationaryEEGrecordings.Forexample,anumberofsynchronizationmeasureswereusedforanalyzingtheepilepticEEGrecordingstoreachthegoalsoflocalizingtheepileptogeniczonesandpredictingtheimpendingepilepticseizures[ 99 ; 106 ; 119 ; 38 ; 120 ].Thesestudiesalsosuggestthatepilepsyisadynamicalbraindisorderinwhichtheinteractionsamongneuronorgroupsofneuronsinthebrainalterabruptly.Moreover,thecharacteristicchangesintheEEGrecordingshavebeenshowntohaveclearassociationswiththesynchronizationphenomenaamongepileptogenicandotherbrainregions.Whentheconductivitiesbetweentwooramongmultiplebrainregionsaresimultaneouslyconsidered,theunivariateanalysisalonewillnotbeabletocarryoutsuchatask.Thereforeitisappropriatetoutilizemultivariateanalysis.Multivariateanalysishasbeenwidelyusedintheeldofneurosciencetostudytherelationshipsamongsourcesobtainedsimultaneously.Inthisstudy,thecrossmutualinformation(CMI)approachisappliedtomeasuretheconnectivityamongbrainregions[ 75 ].TheCMIapproachisabivariatemeasureandhasbeenshowntohaveabilityfor 119

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101 ; 97 ].ThebrainconnectivitygraphisthenconstructedwhereverticesinthegraphrepresenttheEEGelectrodes. EverydistinctpairofverticesisconnectedbyanarcwiththelengthequaltotheconnectivityquantiedbyCMI.Afterconstructingabrainconnectivitygraph,whichisacompletegraph,wethenremovearcsofconnectivitybelowaspeciedthresholdvaluetopreserveonlystrongcouplingsofelectrodepairs.Finally,weemployamaximumcliquealgorithmtondamaximumcliqueinwhichthebrainregionsarestronglyconnected.Themaximumcliquesizecanbe,inturn,usedtorepresenttheamountoflargestconnectedregionsinthebrain.Themaximumcliquealgorithmreducesthecomputationaleortforsearchingintheconstructedbrainconnectivitygraph.Theproposedgraph-theoreticapproachoersaneasyprotocolforinspectingthestructuresofthebrainconnectivityovertimeandpossiblyidentifyingthebrainregionswhereseizuresareinitiated. 120

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Table7-1. Patientinformationforclusteringanalysis PatientGenderAgeNumberofSeizureDurationofEEGNumberofelectrodesonsetzonerecordings(days)seizures 1Male2926R.Hippocampus6.07192Male3730L./R.Hippocampus9.8811 121

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Therefore,weusemultivariatemeasuresforquantifyingthesynchronizationfromonly2electrodesatatime.Specically,asimplesignalprocessingusedtocalculatethesynchronizationbetweenelectrodepairsisemployedinthisstudy.Thenweapplyadataminingtechniquebasedonnetwork-theoreticalmethodstothemultivariateanalysisofEEGdata. 121 ].Althoughtheinformationfromcross-correlationfunctionandfrequencycoherencehasbeenshowntobeidentical[ 122 ],thesimilaritybetweentwoEEGsignalsindierentfrequencybandssuchasdelta,theta,beta,alphaandgamma,isstillcommonlyusedtoinvestigateEEG 122

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123 ; 121 ].Forexample,[ 124 ]usedfrequencycoherencemeasurestoinvestigatetheinteractionsbetweenmediallimbicstructuresandtheneocortexduringictalperiods(seizureonsets).Inanotherstudyby[ 125 ],thecoherencepatternofcorticalareasfromepilepticbrainwasinvestigatedtoidentifyacorticalepilepticsystemduringinterictal(normal)andictal(seizure)periods. Althoughlinearmeasuresareveryusefulandcommonlyused,theyareinsensitivetononlinearcouplingbetweensignals,andnon-linearitiesarequitecommoninneuralcontexts.TobeabletoinvestigatemoreoftheinterdependencebetweenEEGelectrodes,nonlinearmeasuresshouldbeapplied.NonlinearmeasureshavebeenwidelyusedtodeterminetheinterdependenceamongEEGsignalsfromdierentbrainareas.Forexample,[ 106 ]and[ 35 ]studiedthesimilaritybetweenEEGsignalsusingnonlineardynamicalsystemapproaches.Theyappliedatime-delayembeddingtechniquetoreconstructatrajectoryofEEGinphasespaceandusedtheideaofgeneralizedsynchronizationproposedby[ 126 ]tocalculatetheinterdependenceandcausalrelationshipsofEEGsignals. Weproposeanapproachtoinvestigateandquantifythesynchronizationofthebrainnetwork,specicallytailoredtostudythepropagationofepileptogenicprocesses.[ 127 ]investigatedthispropagation,wheretheaverageamountofmutualinformationduringtheictalperiod(seizureonset)wasusedtoidentifythefocalsiteandstudythespreadofepilepticseizureactivity.Subsequently,[ 128 ]appliedtheinformation-theoreticapproachtomeasuresynchronizationandidentifycausalrelationshipsbetweenareasinthebraintolocalizeanepileptogenicregion.Here,weapplyaninformation-theoreticapproach,calledcross-mutualinformation,whichcancapturebothlinearandnonlineardependencebetweenEEGsignals,toquantifythesynchronizationbetweennodesinthebrainnetwork.Inordertogloballymodelthebrainnetwork,werepresentthebrainsynchronizationnetworkasagraph. 123

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70 ].Inanearlierstudy,[ 115 ]demonstratedthatthebrainevolvedahighlyecientnetworkarchitecturewhosestructuralconnectivity(ormotif)iscapableofgeneratingalargerepertoireoffunctionalstates.Inanotherrecentstudy,thebrainnetworkgraphwasinvestigatedtoverifythatthere-useofexistingneuralcomponentsplayedasignicantroleintheevolutionarydevelopmentofcognition[ 114 ].Applyingnetwork/graph-theoreticmethodstoEEGsignals,wecanmodelthebrainconnectivity/synchronizationnetworkasacompletegraphG(V;E),whereVisasetofverticesandEisasetofedges.Vertices(alsocallednodes)arerepresentedbyEEGelectrodes(alsoreferredaschannels).Edges(alsocalledarcs)arerepresentedbythesynchronization/similaritybetween2EEGelectrodeswhosedegreescorrespondtotheedgeweights.Inshort,abrainconnectivitynetworkcanthenbeconstructedasagraphwhoseverticesareEEGelectrodesandtheweightededgesarethecouplingstrengthofelectrodepairs.Everypairofverticesisconnectedbyaweightededge.Inthisstudy,wefocusonthestructuralchangesinthebrainconnectivitynetworkthatmayberelatedtotheseizureevolution.Thestructuralchangescouldberepresentedbyconnectivityfractions/partitionsthroughaggregationandsegregationofthebrainnetwork.Ininthisstudy,weproposetwonetwork-theoreticapproaches,spectralpartitioningandmaximumclique,toidentifyindependent/segregatedandclusteredbrainareas. 98 ].Generally,CMImeasurestheinformationobtainedfromobservationsofonerandomeventfortheother.ItisknownthatCMIhasthecapabilitytocapturebothlinearandnonlinearrelationshipsbetweentworandomvariablessincebothlinearandnonlinearrelationships 124

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ShannonentropymeasurestheuncertaintycontentofX.Itisalwayspositiveandmeasuredinbits,ifthelogarithmistakenwithbase2.NowletusconsideranothersetofdatapointsY,whereallpossiblerealizationsofYarey1;y2;y3;:::ynwithprobabilitiesP(y1);P(y2);P(y3);::::.ThedegreeofsynchronicityandconnectivitybetweenXandYcanbemeasuredbythejointentropyofXandY,denedas wherepXYijwhichisthejointprobabilityofX=XiandY=Yj:ThecrossinformationbetweenXandY,CMI(X;Y),isthengivenby (7{3) =H(X)+H(Y)H(X;Y) (7{4) =ZZpXY(x;y)log2pXY(x;y) Thecrossmutualinformationisnonnegative.IfthesetworandomvariablesX;Yareindependent,fXY(x;y)=fX(x)fY(y),thenCMI(X;Y)=0,whichimpliesthatthereisnocorrelationbetweenXandY.Theprobabilitiesareestimatedusingthehistogrambasedboxcountingmethod.Therandomvariablesrepresentingtheobservednumberofpairsofpointmeasurementsinhistogramcell(i;j),rowiandcolumnj,arerespectivelykij;ki:andk:j.Here,weassumetheprobabilityofapairofpointmeasurementsoutsidetheareacoveredbyhistogramisnegligible,thereforePi;jPij=1[ 75 ; 129 ]. 125

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EEGepochsforRTD2,RTD4andRTD6(10seconds) ScatterplotforEEGepoch(10seconds)ofRTD2vs.RTD4andRTD4vs.RTD6 Figure 7-3 showstheCMIvaluesmeasuredfromrightmesialtemporaldepth(R(T)D)regions.Figure 7-2 displaysthescatterplotsfortheEEGrecordedfromthesame(R(T)D)brainregion.Fromthescatterplot,itisclearthatEEGrecordingsbetweenR(T)D2andR(T)D4haveweaklinearcorrelationwhichhavealsoyieldedlowerCMIvaluesinFigure 7-3 .ThestrongerlinearrelationshipisdiscoveredbetweenR(T)D4andR(T)D6andthislinearcorrelationpatternhasresultedinhigherCMIvalues.PriortomeasuringtheCMI,werstdividedEEGrecordingsintosmallernon-overlappingEEGepochs.Thesegmentationprocedureiswidelyutilizedtosubduethenon-stationarynatureoftheEEGrecordings.ThechangesofEEGpatterntendtoappearverybriey, 126

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Cross-mutualinformationforRTD4vs.RTD6andRTD2vs.RTD4 examplesincludesharpwavetransients,spikes,spike-wavecomplexes,andspindles.WorkingonshorterEEGepochswillinsurethestationarityfortheunderlyingprocessesandthusanychangeintheconnectivitycanbedetected.Therefore,aproperlengthofEEGepochshastobedeterminedformeasuringtheconnectivityamongEEGrecordings.WechosethelengthoftheEEGepochsequalto10.24seconds(2048points),whichhasalsobeenutilizedinmanyperviousEEGresearchstudies[ 58 ; 130 ].ThebrainconnectivitymeasuredusingCMIformthecompletegraph,inwhicheachnodehasanarctoeveryotheradjacentvertex.Intheprocedureforremovingtheinsignicantarcs(weakconnectionbetweenbrainregions),werstestimatedanappropriatethresholdvaluebyutilizingthestatisticaltests.Wedeterminedthisthresholdbyobservingthestaticalsignicanceoverthecompleteconnectivitygraph,thisthresholdvaluewassettobeavaluewherethesmallnoiseiseliminated,butyettherealsignalisnotdeleted[ 131 ]. 127

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Completeconnectivitygraph(a);afterremovingthearcswithinsignicantconnectivity(b) 7-5 (a)and 7-5 (b). Notethatthismatrixissymmetricbecausethemutualinformationmeasurehasnocouplingdirection.Ineachrowandcolumnofthisbitmap,thecolorrepresentsthesynchronizationlevel.Notethatwewillignorethediagonalofthematrixbecausewecanalwaysndaveryhighlevelofself-synchronization. Afterapplyingthenormalizedcut,wecalculatedaneigenvectorcorrespondingtothesecondsmallesteigenvalue.Subsequently,weseparatedelectrodesintotwogroupswiththeminimumcutorseparationwithminimumcostbyapplyingthethresholdvalueat0.UsingtheeigenvectorinFigure 7-5 c,electrodeswereseparatedintotwoclusters.Itiseasytoobserveaclearseparationofthesetwoclustersthroughthevalueofeigenvector,inwhichasharptransientfromR(S)T4toL(O)F1isusedasaseparatingpoint.TherstgroupofsynchronizedelectrodesisfromL(S)T,L(T)D,R(S)TandR(T)Dareas.ThesecondgroupisfromL(O)FandR(O)Fareas.Afterthe 128

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Spectralpartitioning rstiteration,itisclearthatthesynchronizationintheLD-LT-RD-RTclusterisnotuniformthroughoutallelectrodesinthecluster.Therefore,weconsequentlyperformedanotheriterationofspectralpartitioningontheLD-LT-RD-RTclustertondhighlysynchronizedgroupsofelectrodeswithinthecluster.Thisprocedurecanbeviewedasahierarchicalclustering.Afterrearrangingtheelectrodesbasedonthesynchronizationlevel,wefoundtwosub-clustersofelectrodesinthebitmapshowninFigure 7-5 b.AsshowninFigure 7-5 d,thevalueofeigenvectorindicatesthattherearetwosub-clusterswithintheLD-LT-RD-RTclusterbyapplyingthethresholdof0.TheLD-LT-RD-RTclusterwasseparatedintotwosub-clusters:LD-LTandRD-RT.Thisobservationsuggeststhatthereexistsahighlysynchronizedpatterninthesamesideoftemporallobeaswellasintheentireorbitofrontalarea.Thisndingcanbeconsideredasaproofofconceptthattheseizureevolutionalsofollowsaregularcommunicationpatterninthebrainnetwork. 129

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Weanalyzed3epochsof3-hourEEGrecordings,2hoursbeforeand1hourafteraseizure,fromPatient2whohadtheepileptogenicareasonbothrightandleftmesialtemporallobes.Figures 7-7 and 7-8 demonstratetheelectrodeselectionofthemaximumcliquegroupduringtwohoursbeforeandonehouraftertheseizureonset.Duringtheperiodbeforetheseizureonset,bothguresmanifestedapatternwherealltheLDelectrodeswereconsistentlyselectedtobeinthemaximumclique.Duringtheseizureonset,thesizeofthemaximumcliqueincreasesdrastically.Thisisveryintuitivebecause,intemporallobeepilepsy,allofthebrainareasarehighlysynchronized.WevisuallyinspectedtherawEEGrecordingsbeforeandduringtheseizureonsetsandfoundasimilarsemiologicalpatternoftheseizureonset-electrodesfromtheL(T)Dareas 130

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begin sortallnodesbasedonvertexordering LIST=orderednodes cbc=0currentbestcliquesize depth=0currentdepthlevel enter-next-depth(LIST,depth) end begin 1m=thenumberofnodesintheLIST 2depth=depth+1 3foranodeinpositioniintheLIST 4ifdepth+(m-i)cbcthen 6else 8ifnoadjacentnodethen 10else 12end 15ifdepth=1 16deletenodeifromLIST 17end Maximumcliquealgorithm initiatedahighlyorganizedrhythmicpatternsandthepatternsstartedtopropagatethroughoutallthebrainareas.Weinitiallyspeculatedthattheepileptogenicareascouldbetheonesthatarehighlysynchronizedlongbeforeaseizureonset.Inthepreviouscase,weobservedthattheL(T)Delectrodesaretheonethatstartedtheseizureevolution.However,inafurtherinvestigationofEEGrecordingsfromthesamepatient,wefoundsomecontrastresults.InFigure 7-9 ,theelectrodeselectionpatternofthemaximumcliquedemonstratesaveryhighlysynchronizedgroupofelectrodesinbothleftandrightorbitofrontalareasduringthe2-hourperiodprecedingtheseizure.AftervisualinspectionontherawEEGrecordings,thisseizurewasinitiatedbytheR(T)Darea.Generally,itis 131

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ElectrodeselectionusingthemaximumcliquealgorithmforCase1 132

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ElectrodeselectionusingthemaximumcliquealgorithmforCase2 ElectrodeselectionusingthemaximumcliquealgorithmforCase3 processesslowlydevelopthemselvesthrougharegularcommunicationchannelinthebrainnetwork,ratherthanabruptlydisrupt,collapse,orchangethewaybrainscommunicate.Fromthisobservation,wepostulatethatthisphenomenonmaybeareectionofneuronalrecruitmentinseizureevolution.Thisobservationconrmsourrsthypothesis.In 133

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134

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114 ]demonstratedtheevolutionofcognitivefunctionthroughquantitativeanalysisoffMRIdata. Theproposedframeworkcanprovideaglobalstructuralpatternsinthebrainnetworkandmaybeusedinthesimulationstudyofdynamicalsystems(likethebrain)topredictoncomingevents(likeseizures).Forexample,anON-OFFpatternofelectrodeselectioninthemaximumcliqueoveroneperiodoftimecanbemodeledasabinaryobservationinadiscretestateinaMarkovmodel,whichcanbeusedtosimulatetheseizureevolutioninthebrain.Inaddition,thenumberofelectrodesinthemaximumcliquecanbeusedtoestimatetheminimumnumberoffeaturesandexplaindynamicalmodelsortheparametersintimeseriesregression.Notethattheproposednetworkmodelrepresentsanepilepticbrainasagraph,wherethereexistseveralecientalgorithms(e.g.,maximumclique,shortestpath)forndingspecialstructureofthegraph.Thisideahasenabledus,computationallyandempirically,tostudytheevolutionofthebrainasawhole.TheMonte-CarloMarkovChain(MCMC)frameworkmaybeapplicableinourfuturestudyonlongtermEEGanalysis.TheMCMCframeworkhasbeenshownveryeectiveindataminingresearch[ 132 ].ItcanbeusedtoestimatethegraphorcliqueparametersinepilepticprocessesfromEEGrecordings.SincelongtermEEGrecordingsareverymassive,mostsimulationtechniquesarenotscalableenoughtoinvestigatelarge-scalemultivariatetimeserieslikeEEGs.TheuseofMCMCmakesitpossibletoapproximatethebrainstructureparametersovertime.Moreimportantly,theMCMCframeworkcanalsobeextendedtotheanalysisofmulti-channelEEGsbygeneratingnewEEGdatapointswhileexploringthedatasequencesusingaMarkovchainmechanism.Inaddition,wecanintegratetheMCMCframeworkwithaBaysianapproach.Thiscanbe 135

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136

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EEGrecordingsystemhasbeenthemostusedapparatusforthediagnosisofepilepsyandotherneurologicaldisorders.ItisknownthatchangesinEEGfrequencyandamplitudearisefromspontaneousinteractionsbetweenexcitatoryandinhibitoryneuronsinthebrain.StudiesintotheunderlyingmechanismofbrainfunctionhavesuggestedtheimportanceoftheEEGcouplingstrengthbetweendierentcorticalregions.Forexample,thesynchronizationofEEGactivityhasbeenshowninrelationtomemoryprocess[ 133 ; 134 ]andlearningprocessofthebrain[ 135 ].Inapathophysiologicalstudy,dierentbrainsynchronization/desynchronizationEEGpatternsareshowntobeinduced 137

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136 ]. Inepilepsyresearch,severalauthorshavesuggesteddirectrelationshipbetweenchangeinsynchronizationphenomenaandonsetofepilepticseizuresusingEEGrecordings.Forexample,Iasemidisetal.,reportedfromintracranialEEGthatthenonlineardynamicalentrainmentfromcriticalcorticalregionsisanecessaryconditionforonsetofseizuresforpatientswithtemporallobeepilepsy[ 112 ; 37 ; 60 ];LeVanQuyenetal.,showedepilepticseizurecanbeanticipatedbynonlinearanalysisofdynamicalsimilaritybetweenrecordings[ 35 ];Mormannetal.,showedthepreictalstatecanbedetectedbasedonadecreaseinsynchronizationonintracranialEEGrecordings[ 89 ; 43 ].ThehighlycomplexbehavioronEEGrecordingsisconsideredasnormalityofbrainstate,whiletransitionsintoalowercomplexitybrainstateareregardedasapathologicalnormalitylosses.Synchronizationpatternswerealsofoundtodiersomewhatdependingonepilepticsyndromes,withprimarygeneralizedabsenceseizuresdisplayingmorelong-rangesynchronyinfrequencybands(3-55Hz)thangeneralizedtonicmotorseizuresofsecondary(symptomatic)generalizedepilepsyorfrontallobeepilepsy[ 113 ]. Inthisstudy,mutualinformationandnonlinearinterdependencemeasureswereappliedontheEEGrecordingstoidentifytheeectoftreatmentonthecouplingstrengthanddirectionalityofinformationtransportbetweendierentbraincorticalregions[ 137 ; 100 { 102 ; 104 ].TheEEGrecordingswereobtainedfrompatientswithULD. ULDisonetypeofProgressiveMyoclonicEpilepsy(PME);arareepilepsydisorderwithcomplexinheritance.ULDwasrstdescribedbyUnverrichtin1891andLundborgin1903[ 138 ; 139 ].AEDsismainstayforthetreatmentofULDwithoverallunsatisfactoryecacy.Duetotheprogressionoftheseverityofmyoclonus,theecacyofAEDstreatmentisdiculttobedeterminedclinicallyespeciallyinthelaterstagesofthedisease.TheEEGrecordingsforULDsubjectsusuallydemonstrateabnormalslowbackground,generalizedhigh-amplitude3{5Hzspikewavesorpolyspikeandwave 138

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140 ; 141 ].Furthermore,AEDsassociatedgeneralizedslowingofEEGbackgroundrhythmshasbeenreportedwithhighlyvariablefrompatienttopatient[ 142 ].However,itwasdiculttodeterminehowstrongthecorrelationbetweenEEGslowinganddiseaseprogressionwassincetheintensicationofdrugtreatmentduringthelaterstagesofillnessmighthavecontributedtotheEEGslowing[ 140 ]. TheclinicalobservationshavebeenthemostcommonmethodforevaluatingtheinuenceandeectivenessofAEDsinterventionsinpatientswithepilepsyandotherneurologicaldisorders.Morespecically,ecacyoftreatmentisusuallymeasuredbycomparingtheseizurefrequencyduringtreatmenttoanitebaselineperiod.EEGrecordingsaremainlyusedassupplementaldiagnostictoolsinmedicaltreatmentevaluations.Otherthancountingthenumberofseizuresasameasurefortreatmenteectthereiscurrentlynoreliabletoolforevaluatingtreatmenteectsinpatientswithseizuredisorders.AquantitativesurrogateoutcomemeasureusingEEGrecordingsforpatientswithepilepsyisdesired. Thisrestofthischapterisorganizedasfollows.ThebackgroundofthepatientsandtheparametersoftheEEGrecordingsaregiveninsectionII.InsectionIII.Themethodsforidentifythecouplingstrengthanddirectionalityofinformationtransportbetweendierentbraincorticalregionsaredescribed.Thequantitativeanalysis,statisticaltestsandresultsarepresentedinSectionIV.TheconclusionanddiscussionaregiveninSectionV. 139

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Table8-1. ULDpatientinformation 1Female47998482Male451080653Male501250664Male51116854 8.3.1MutualInformation 103 ].LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.,samplingperiod)isxed.Thenthemutualinformationisgivenby; 140

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143 ]: whereH(X);H(Y)aretheentropiesofX;YandH(X;Y)isthejointentropyofXandY.EntropyforXisdenedby: Theunitsofthemutualinformationdependsonthechoiceonthebaseoflogarithm.Thenaturallogarithmisusedinthestudythereforetheunitofthemutualinformationisnat.WerstestimatebH(X)fromXby ForXandYtimeserieswedened(x)ij=kxixjk;d(y)ij=kyiyjkasthedistancesforxiandyibetweeneveryotherpointinmatrixspacesXandY.Onecanrankthesedistancesandndtheknnforeveryxiandyi.InthespacespannedbyX;Y,similardistancerankmethodcanbeappliedforZ=(X;Y)andforeveryzi=(xi;yi)onecanalsocomputethedistancesd(z)ij=kzizjkanddeterminetheknnaccordingtosomedistancemeasure.Themaximumnormisusedinthisstudy: Nextlet(i) 2bethedistancebetweenzianditskthneighbor.Inordertoestimatethejointprobabilitydensityfunction(p:d:f:),weconsidertheprobabilityPk()whichistheprobabilitythatforeachzithekthnearestneighborhasdistance(i) 2dfromzi.ThisPk()representstheprobabilityfork1pointshavedistancelessthanthekthnearestneighborandNk1pointshavedistancegreaterthan(i) 2andk1pointshave 141

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2.Pk()isobtainedusingthemultinomialdistribution: wherepiisthemassofthe-ball.Thentheexpectedvalueoflogpiis where()isthedigammafunction: where()isthegammafunction.Itholdswhen(1)=CwhereCistheEuler-Mascheroniconstant(C0:57721).Themassofthe-ballcanbeapproximated(ifconsideringthep:d:finsidethe-ballisuniform)as wherecdxisthemassoftheunitballinthedx-dimensionalspace.FromEq.( 8{9 )wecanndanestimatorforP(X=xi) log[P(X=xi)](k)(N)dE(log(i))logcdx;(8{10) nallywithEq( 8{10 )andEq( 8{4 )weobtaintheKozachenko-LeonenkoentropyestimatorforX[ 105 ] ^H(X)=(N)(k)+logcdx+dx where(i)istwicethedistancefromxitoitsk-thneighborinthedxdimensionalspace.Forthejointentropywehave ^H(X;Y)=(N)(k)+log(cdxcdy)+dx+dy TheI(X;Y)isnowreadilytobeestimatedbyEq.( 8{2 ).Theproblemwiththisestimationisthataxednumberkisusedinallestimatorsbutthedistancemetricin 142

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2;x(i)+(i) 2]and[y(i)(i) 2;y(i)+(i) 2]respectively)inthex{yscatterdiagram.TheEq.( 8{12 )becomes: ^H(X)=(N)(nx(i)+1)+logcdx+dx FinallytheEq.( 8{2 )isrewrittenas: 106 ].Giventwotimeseriesxandy,usingmethodofdelayweobtainthedelayvectorsxn=(xn;:::;xn(m1))andyn=(xn;:::;xn(m1)),wheren=1;:::N,mistheembeddingdimensionanddenotesthetimedelay[ 52 ].Letrn;jandsn;j,j=1;:::;kdenotethetimeindicesoftheknearestneighborsofxnandyn.Foreachxn,themeanEuclideandistancetoitskneighborsisdenedas andtheY-conditionedmeansquaredEuclideandistanceisdenedbyreplacingthenearestneighborsbytheequaltimepartnersoftheclosestneighborsofyn Thedelay=5isestimatedusingautomutualinformationfunction,theembeddingdimensionm=10isobtainedusingCao'smethodandtheTheilercorrectionissettoT=50[ 73 ; 107 ]. 143

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SinceR(k)n(XjY)R(k)n(X)byconstruction, 0
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144 ].TensecondsofcontinuousEEGepochisrandomlysampledfromeverychannelsandthissamplingprocedurewasrepeatedwithreplacementfor30times.ThereferenceA1andA2channels(inactiveregions)areexcludedfromtheanalysis.Twosamplet-test(N=30,=0:05)isusedtotestthestatisticaldierencesonmutualinformationandnonlinearinterdependenceduringbeforeandaftertreatment.LowmutualinformationandinformationtransportbetweendierentbraincorticalregionswereobservedinoursubjectswithlessseverityofULD.Furthermore,foreachpatientbothmutualinformationandinformationtransportbetweendierentbraincorticalregionsdecreaseafterAEDstreatment.t-testformutualinformationaresummarizedinTable 8-2 ,thetopographicaldistributionformutualinformationisalsoplotinheatmapsshowninFigures 8-2 Thesignicant\driver-response"relationshipisreveledbyt-test.Aftert-testthesignicantinformationtransportbetweenFp1andotherbraincorticalregionsisshowninFig. 8-1 .TheedgeswithanarrowstartingfromFp1tootherchanneldenoteN(XjY)issignicantlargerthenN(YjX),thereforeFp1isthedriver,andviceversa.The 145

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Table8-2. Topographicaldistributionfortreatmentdecouplingeect(DE:DecoupleElectrode(DE)) (1)Fp1F3,C4,P4,F7Fp2,F3,F8,T5F3,F7F3,P3,Fz,T5T4,T5,O1F8,T4,Fz(2)Fp2C3,C4,F8,T4Fp1,F4,T6,O2F8C3,C4,T5,P4T5,PzCz(3)F3Fp1,C4,P4,O2Fp1,C3,P3,PzFp1Fp1,F7(4)F4C4,P4O2Fp2,P4,O2,FzCzC4,Fz(5)C3Fp2,C4,P3,O1F3,P3,O1P3,O1Fp2,P3,O1(6)C4Fp1,Fp2,F3,F4,C3P4,T6,O2P4,O2Fp1,Fp2,F3A1N/AN/AN/AN/AA2N/AN/AN/AN/A(7)P3C3,O2F3,C3,T3,PzO1Fp1,C3,Cz(8)P4Fp1,F3,F4F4,C4C4Fp2,(9)O1Fp1,C3C3,T5,T3,F7P3,PzC3(10)O2F3,F4,P3Fp2,F4,C4C4Pz(11)F7Fp1,Fz,PzC3,T5,O1Fp1F3(12)F8Fp2Fp1,C4,P4Fp2Fp2,(13)T3NONEP3,O1NONENONE(14)T4Fp1,Fp2,CzFp2,O1NONEFp2,(15)T5Fp1,Fp2Fp2,C4NONEFp2,Cz(16)T6NONEC4NONENONE(17)FzF7Fp1,F4CzFp1,Fp2,F3(18)PzFp2,F7,CzF3,P3,CzO2O2(19)CzT4,PzPzF4,FzFp2,T5,P3 146

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NonlinearinterdependencesforelectrodeFP1 beforeandaftertreatment.AlthoughtheresultsindicatethatthemutualinformationandnonlinearinterdependenciesmeasurescouldbeusefulindeterminingthetreatmenteectsforpatientswithULD.Toprovetheusefulnessoftheproposedstudy,alargerpatientpopulationisneeded.Theapproachesinthisstudyareabivariatemeasures,sinceamultivariatemeasureisnoteasytomodelandhasnotbeenresolved.Thedecouplingbetweenfrontalandoccipitalcorticalregionsmaybecausedbydecreaseddrivingforcedeepinsidethebrain.Inotherwords,theeectofthetreatmentmayreducethecouplestrengthbetweenthalamusandcortexinULDsubjects.Nevertheless,thelimitationsmustbementioned,ithasbeenreportedthatitisnecessarytotakeintoaccountthe 147

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145 { 147 ]. 148

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Pairwisemutualinformationbetweenforallelectrodes-beforev.s.aftertreatment 149

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Patient1beforetreatmentnonlinearinterdependencies Fp1-F30.370.441.401.380.690.700.880.100.080.390.320.100.090.05 Fp1-F40.200.340.770.770.410.440.590.070.270.630.360.200.180.25 Fp1-C30.230.310.880.750.490.480.710.060.040.400.240.140.110.09 Fp1-C40.130.220.460.400.280.270.420.090.130.400.280.230.180.28 Fp1-P30.170.260.820.560.400.350.580.040.060.480.280.170.110.12 Fp1-P40.150.250.620.410.320.280.500.080.060.600.250.190.140.15 Fp1-O10.140.360.400.210.170.100.150.040.990.510.200.200.100.26 Fp1-O20.130.230.390.210.220.140.180.030.020.280.140.150.090.21 Fp1-F70.390.371.461.420.680.680.840.110.110.400.480.170.170.16 Fp1-F80.200.240.650.600.370.360.400.060.060.330.230.180.140.33 Fp1-T30.260.311.191.080.590.570.760.080.050.520.470.170.170.20 Fp1-T40.150.240.590.460.300.260.330.040.130.510.320.230.170.29 Fp1-T50.180.280.760.540.380.340.560.050.120.490.330.190.170.22 Fp1-T60.120.210.320.160.180.100.180.020.020.240.110.140.090.17 Fp1-Fz0.270.671.131.180.570.640.800.101.510.650.340.170.110.09 Fp1-Cz0.190.360.740.650.420.420.630.050.380.380.230.170.140.11 Fp1-Pz0.160.270.630.410.350.290.490.040.060.490.220.190.140.17

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Patient1aftertreatmentnonlinearinterdependencies Fp1-F30.390.481.561.370.720.700.870.110.070.490.250.100.080.05 Fp1-F40.240.320.870.830.460.500.710.080.070.580.340.200.150.14 Fp1-C30.220.300.880.810.500.500.720.070.040.290.180.120.090.08 Fp1-C40.190.250.620.660.360.420.630.060.040.320.220.180.120.16 Fp1-P30.160.270.540.490.330.330.580.050.030.250.140.130.100.09 Fp1-P40.140.240.470.420.270.290.530.040.040.320.170.170.100.14 Fp1-O10.130.230.340.270.180.180.440.030.030.300.150.150.100.12 Fp1-O20.120.190.230.210.100.110.310.030.040.210.130.130.090.16 Fp1-F70.200.170.600.620.280.290.360.260.230.770.840.360.370.44 Fp1-F80.210.200.870.430.490.5340.640.050.090.630.220.220.250.47 Fp1-T30.260.310.930.950.510.540.720.120.080.440.380.240.190.23 Fp1-T40.150.210.480.380.250.220.320.040.060.480.300.220.180.35 Fp1-T50.150.250.520.460.310.300.580.040.030.250.150.130.100.08 Fp1-T60.140.230.330.260.180.180.380.040.040.320.180.180.120.21 Fp1-Fz0.270.381.031.060.550.600.790.100.060.510.320.160.110.09 Fp1-Cz0.190.280.690.680.400.440.650.070.040.370.180.170.090.10 Fp1-Pz0.160.260.500.460.300.310.570.060.030.270.140.130.090.08

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Patient2beforetreatmentnonlinearinterdependencies Fp1-F30.530.581.901.750.790.780.860.140.130.350.350.070.080.09 Fp1-F40.400.511.581.400.710.680.790.090.080.400.450.110.130.14 Fp1-C30.340.471.371.400.540.630.740.090.070.430.460.230.120.11 Fp1-C40.250.411.031.040.430.490.640.080.130.440.560.210.190.19 Fp1-P30.280.391.161.380.460.580.690.070.050.460.540.260.160.13 Fp1-P40.231.000.951.090.430.490.620.063.440.420.550.200.180.20 Fp1-O10.220.340.871.150.330.470.570.070.070.430.570.290.200.19 Fp1-O20.220.340.921.040.420.460.540.050.060.370.490.170.170.22 Fp1-F70.430.471.681.760.710.740.850.150.150.440.390.130.110.11 Fp1-F80.300.401.301.240.610.600.680.060.100.390.410.140.150.29 Fp1-T30.370.411.451.890.590.700.750.110.120.500.540.250.150.13 Fp1-T40.250.370.961.050.410.470.580.070.080.430.580.230.200.25 Fp1-T50.250.311.091.410.440.550.630.060.050.420.620.230.210.17 Fp1-T60.210.360.951.150.390.490.550.060.140.470.570.270.190.24 Fp1-Fz0.4310.631.781.550.760.720.860.1155.360.280.360.070.100.07 Fp1-Cz0.320.441.391.330.610.630.750.070.040.370.470.150.150.13 Fp1-Pz0.240.350.921.040.370.490.610.060.060.400.510.240.170.17

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Patient2aftertreatmentnonlinearinterdependencies Fp1-F30.330.461.261.300.660.690.830.050.030.210.160.060.050.03 Fp1-F40.210.310.780.720.440.440.620.050.050.230.250.110.130.11 Fp1-C30.160.280.550.490.350.340.550.030.020.120.120.070.070.06 Fp1-C40.140.240.390.390.220.250.340.030.010.080.170.060.110.10 Fp1-P30.130.240.290.230.170.160.140.030.030.090.080.070.060.13 Fp1-P40.130.230.290.270.160.180.120.030.020.110.120.080.080.11 Fp1-O10.120.220.240.260.130.16-0.030.030.010.120.110.090.080.20 Fp1-O20.130.230.270.300.140.20-0.080.030.030.100.120.080.080.14 Fp1-F70.290.331.121.090.540.530.660.180.200.650.630.310.300.37 Fp1-F80.220.230.780.510.450.300.510.050.060.280.240.120.150.25 Fp1-T30.200.280.750.720.430.430.640.030.040.130.140.070.080.07 Fp1-T40.140.190.420.290.240.150.190.030.040.120.150.080.140.25 Fp1-T50.140.240.330.180.190.110.180.020.030.110.080.060.060.14 Fp1-T60.130.120.280.280.160.180.030.020.020.100.080.080.060.09 Fp1-Fz0.310.431.211.180.620.640.790.070.050.300.240.090.070.04 Fp1-Cz0.150.230.400.430.240.290.450.040.040.090.150.060.080.07 Fp1-Pz0.130.230.260.260.140.180.120.020.020.070.100.060.070.11

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Patient3beforetreatmentnonlinearinterdependencies Fp1-F30.370.492.001.940.750.750.840.110.090.941.030.160.160.13 Fp1-F40.320.891.551.560.640.670.680.0932.560.840.860.180.160.21 Fp1-C30.241.031.521.430.600.630.700.083.590.730.710.200.190.23 Fp1-C40.150.260.780.810.360.400.40.110.170.720.700.280.290.34 Fp1-P30.165.370.880.820.400.400.480.0424.10.670.620.240.220.25 Fp1-P40.150.710.880.790.370.380.370.062.640.700.690.220.200.28 Fp1-O10.159.500.780.690.360.350.330.0528.220.620.550.220.210.32 Fp1-O20.040.050.660.800.220.210.330.040.050.660.800.220.210.33 Fp1-F70.391.682.082.040.750.720.830.136.701.031.210.180.240.21 Fp1-F80.210.281.191.310.490.5340.640.090.090.921.020.280.250.26 Fp1-T30.220.401.271.330.520.530.640.100.511.001.170.250.280.23 Fp1-T40.198.141.171.250.510.540.610.0830.120.760.880.220.220.29 Fp1-T50.1714.481.231.250.500.480.550.0743.450.961.110.240.250.26 Fp1-T60.0614.220.740.810.240.200.280.164.320.900.990.380.440.38 Fp1-Fz0.372.661.811.730.730.720.790.126.680.740.840.120.18 Fp1-Cz0.1727.751.101.050.450.490.530.0688.790.680.730.190.190.18 Fp1-Pz0.643.620.950.920.400.410.442.7218.610.710.790.240.240.27

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Patient3aftertreatmentnonlinearinterdependencies Fp1-F30.370.492.001.940.750.750.840.110.090.941.030.160.160.13 Fp1-F40.320.891.551.560.640.670.680.0932.560.840.860.180.160.21 Fp1-C30.241.031.521.430.600.630.700.083.590.730.710.200.190.23 Fp1-C40.150.260.780.810.360.400.40.110.170.720.700.280.290.34 Fp1-P30.165.370.880.820.400.400.480.0424.10.670.620.240.220.25 Fp1-P40.150.710.880.790.370.380.370.062.640.700.690.220.200.28 Fp1-O10.159.500.780.690.360.350.330.0528.220.620.550.220.210.32 Fp1-O20.040.050.660.800.220.210.330.040.050.660.800.220.210.33 Fp1-F70.391.682.082.040.750.720.830.136.701.031.210.180.240.21 Fp1-F80.210.281.191.310.490.5340.640.090.090.921.020.280.250.26 Fp1-T30.220.401.271.330.520.530.640.100.511.001.170.250.280.23 Fp1-T40.198.141.171.250.510.540.610.0830.120.760.880.220.220.29 Fp1-T50.1714.481.231.250.500.480.550.0743.450.961.110.240.250.26 Fp1-T60.0614.220.740.810.240.200.280.164.320.900.990.380.440.38 Fp1-Fz0.372.661.811.730.730.720.790.126.680.740.840.120.18 Fp1-Cz0.1727.751.101.050.450.490.530.0688.790.680.730.190.190.18 Fp1-Pz0.643.620.950.920.400.410.442.7218.610.710.790.240.240.27

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Patient4beforetreatmentnonlinearinterdependencies Fp1-F30.450.521.631.280.730.680.780.070.050.280.230.080.060.07 Fp1-F40.250.350.950.790.490.470.590.080.060.310.220.140.110.09 Fp1-C30.210.280.660.530.360.340.470.040.020.280.280.160.130.13 Fp1-C40.140.210.410.350.230.220.280.100.120.310.290.180.170.21 Fp1-P30.170.240.470.360.240.230.210.060.030.220.180.140.110.15 Fp1-P40.160.470.430.290.210.190.180.051.290.210.170.130.100.18 Fp1-O10.170.230.450.270.210.170.030.050.030.260.200.160.120.14 Fp1-O20.160.230.460.300.220.180.060.050.040.240.200.140.130.20 Fp1-F70.450.501.761.510.770.720.870.100.110.280.260.060.050.05 Fp1-F80.360.391.270.910.650.540.740.050.070.380.290.110.130.11 Fp1-T30.210.800.810.610.440.380.550.051.980.250.260.120.130.10 Fp1-T40.180.280.510.330.270.210.350.070.060.310.250.190.160.18 Fp1-T50.170.250.450.240.200.160.150.050.030.260.190.180.120.14 Fp1-T60.160.240.370.230.170.140.100.060.040.260.150.180.100.14 Fp1-Fz0.410.481.471.190.700.650.770.060.040.220.220.060.060.06 Fp1-Cz0.180.630.610.490.280.300.400.062.100.280.250.190.130.13 Fp1-Pz0.170.240.480.290.230.200.160.060.020.230.160.160.100.15

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Patient4aftertreatmentnonlinearinterdependencies Fp1-F30.320.461.241.250.640.670.820.040.030.270.150.080.050.03 Fp1-F40.210.310.760.680.430.420.600.060.050.240.260.110.130.11 Fp1-C30.180.290.580.520.350.360.550.030.020.120.120.070.060.07 Fp1-C40.150.240.380.400.220.260.320.030.020.080.150.060.100.10 Fp1-P30.140.250.330.220.200.160.150.030.030.090.080.070.050.15 Fp1-P40.120.250.270.270.150.170.090.020.020.100.090.070.060.10 Fp1-O10.140.220.250.220.150.14-0.170.030.010.140.110.100.080.18 Fp1-O20.130.240.260.280.140.17-0.090.030.030.080.140.070.090.15 Fp1-F70.380.431.461.440.700.690.860.100.120.210.190.070.070.05 Fp1-F80.230.240.910.590.500.320.450.050.080.250.280.100.180.31 Fp1-T30.200.290.720.690.410.410.630.040.030.150.140.110.080.08 Fp1-T40.150.200.440.280.260.150.190.020.040.170.150.100.130.27 Fp1-T50.130.250.300.170.180.110.190.020.030.070.060.050.050.13 Fp1-T60.130.230.280.250.160.150.010.020.020.100.090.080.060.10 Fp1-Fz0.300.421.191.170.610.630.780.050.030.240.200.080.060.04 Fp1-Cz0.150.250.410.420.240.280.430.040.040.100.140.080.070.10 Fp1-Pz0.130.250.280.250.150.170.100.020.030.080.090.060.060.10

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D.V.Moretti,C.Miniussi,G.B.Frisoni,C.Geroldi,O.Zanetti,G.Binetti,andP.M.Rossini.Hippocampalatrophyandeegmarkersinsubjectswithmildcognitiveimpairment.ClinicalNeurophysiology,118(12):2716{2729,2007. [137] R.QuianQuiroga,A.Kraskov,T.Kreuz,andP.Grassberger.Performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals.Phys.Rev.E,65:041903,2002. [138] H.Unverricht.DieMyoclonie.FranzDeutick,Leipzig,1891. [139] H.B.Lundborg.DieprogressiveMyoclonus-Epilepsie(UnverrichtsMyoclonie).AlmqvistandWiksell,Uppsala,1903. [140] N.K.Chew,P.Mir,M.J.Edwards,C.Cordivari,D.M.,S.A.Schneider,H.-T.Kim,N.P.Quinn,andK.P.Bhatia.Thenaturalhistoryofunverricht-lundborgdisease:Areportofeightgeneticallyprovencases.MovementDisorders,Vol.23,No.1:107{113,2007. [141] E.Ferlazzoa,A.Magauddaa,P.Strianob,N.Vi-Hongc,S.Serraa,andP.Gentonc.Long-termevolutionofeeginunverricht-lundborgdisease.EpilepsyResearch,73:219{227,2007. [142] M.C.Salinsky,B.S.Oken,andL.Morehead.Intraindividualanalysisofantiepilepticdrugeectsoneegbackgroundrhythms.ElectroencephalographyandClinicalNeurophysiology,90(3):186{193,1994. [143] T.M.CoverandJ.A.Thomas.ElementsofInformationTheory.Wiley,NewYork,1991. [144] M.R.Chernick.BootstrapMethods:APractitioner'sGuide.Wiley-Interscience,1999. [145] M.SteriadeandF.Amzica.Dynamiccouplingamongneocorticalneuronsduringevokedandspontaneousspike-waveseizureactivity.JournalofNeurophysiology,72:2051{2069,1994. [146] E.SitnikovaandG.vanLuijtelaar.Corticalandthalamiccoherenceduringspikewaveseizuresinwag/rijrats.EpilepsyResearch,71:159{180,2006. [147] E.Sitnikova,T.Dikanev,D.Smirnov,B.Bezruchko,andG.vanLuijtelaar.Grangercausality:Cortico-thalamicinterdependenciesduringabsenceseizuresinwag/rijrats.JournalofNeuroscienceMethods,170(2):245{254,2008. 168

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Chang-ChiaLiuearnedhisB.S.degreeinspring2000fromtheDepartmentofIndustrialEngineering,Da-YehUniversityinTaiwan.HewenttotheUnitedStatesinspring2002andjoinedtheUniversityofFloridainfall2002.HereceiveddualM.S.degreesfromDepartmentsofIndustrialandSystemsEngineeringandJ.CraytonPruittFamilyBiomedicalEngineeringinspring2004andfall2007,respectively.Hisresearchinterestsincludeglobaloptimization,timeseriesanalysis,chaostheory,andnonlineardynamicswithapplicationsinbiomedicine. 169