A spatio-temporal approach to epileptic focus localization from array electrocorticography

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A spatio-temporal approach to epileptic focus localization from array electrocorticography
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Thesis (Ph. D.)--University of Florida, 1993.
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Includes bibliographical references (leaves 258-267).
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by Armando Bennett Barreto.
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A SPATIO-TEMPORAL APPROACH TO EPILEPTIC FOCUS
LOCALIZATION FROM ARRAY ELECTROCORTICOGRAPHY















By

ARMANDO BENNETT BARRETO


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

UNIVERSITY OF FLORIDA


1993














ACKNOWLEDGEMENTS


Completion of a graduate study program is undoubtedly a demanding undertaking.

It may also require as much perseverance as intellectual skill. Thus, I will take the liberty

of acknowledging first my debt to those who provided me with guidance and support at a

personal level. I thank my parents, brothers and sister for their example and unconditional

support. I am also indebted to Steve and Jeanne Miller for their friendship through all

these years and, especially, to Ms. MaryLynn Cook whose understanding and enthusiasm

took her contribution to this work far beyond the editorial assistance she so graciously vol-

unteered.

I would like to thank Drs. Jack Smith, Donald Childers and Buna J. Wilder for par-

ticipating on the supervisory committee overseeing this research. Special thanks are due to

Drs. Jose C. Principe and Steven A. Reid for their involvement in this research since its

inception. The many hours spent in the Computational NeuroEngineering Laboratory in

the pursuit of this research were made more enjoyable by the spirit of collaboration and

camaraderie that its members have developed. I thank all of them for their support. In par-

ticular I thank Mr. Karl Gugel for his permanent and indefatigable optimism and Mr. Rus-

sell Walters, who was always willing to share his vast computer expertise with me. I also

want to thank the individuals who were instrumental in the collection and review of the

data involved in this project: Dr. Bruce Richards and Ms. Jerline White at Alachua Gen-

eral Hospital;and Drs. Steven Roper, Robin Gilmore and Richard Schmidt and Mr. David

Juras and Mr. Stephen Eisenshenk at Shands Hospital.














TABLE OF CONTENTS
Page


ACKNOW LEDGEM ENTS ............................................................................................. ii


A B ST R A C T ..................................................................................................................... v


CHAPTERS


I IN TRO D U CTIO N .......................................................................................... 1


1.1 Focal Epilepsy and its Surgical Therapy ........................................... 1
1.2 Intraoperative ECoG .......................................................................... 4
1.3 Target Signal Specification ................................................................ 10
1.4 The Epileptic Focus ........................................................................... 11
1.5 An Example of Focal Interictal Activity (FIA) from Real ECoG ........ 22

II FOCAL INTERICTAL EVENTS IN TIME AND SPACE......................... 26


2.1 Temporal Dimension of Focal Interictal Activity.............................. 26
2.2 The Spatial Dimension of Focal Interictal Activity........................... 43
2.3 A Spatio-Temporal View of Focal Interictal Events.......................... 61

III THE SPATIO-TEMPORAL LAPLACIAN (STL) FOR THE ANALYSIS
OF THE CORTICAL TIME-VARYING POTENTIAL FIELD................... 78


3.1 Three-Dimensional Sample Space from Array ECoG....................... 78
3.2 Average-Reference Field Reconstruction .......................................... 87
3.3 The Search for Coincident Spatial and Temporal Sharpness in
the 3D Sam ple Space .................................................................... 91
3.4 SL: A Measure for Instantaneous Spatial Sharpness......................... 93
3.5 TL: A Measure for Temporal Sharpness .......................................... 109








3.6 Increased Discriminant Power of the Spatio-Temporal Identification
of Focal Interictal Events.............................................................. 122
3.7 A Physical Interpretation for the STL Transformation...................... 138

IV ON-LINE STL IMPLEMENTATION FOR INTRAOPERATIVE
EPILEPTIC FOCUS LOCALIZATION....................................................... 144


4.1 Block Diagram for the System .......................................................... 144
4.2 Information Flow through the System............................................... 148
4.3 Control Flow ...................................................................................... 152

V VALIDATION OF THE STL FOCUS LOCALIZATION APPROACH ....... 187


5.1 Goals of the Validation Procedure ..................................................... 187
5.2 The Validation Protocol ..................................................................... 188
5.3 Data Sets used in the Validation Process ........................................... 192
5.4 Event-by-Event Analysis ................................................................... 199
5.5 Focus Localization Analysis.............................................................. 212

VI CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH ....... 235


6.1 Conceptual Benefits of this Research................................................ 235
6.2 Practical Accomplishments and Limitations..................................... 241
6.3 Proposed Improvements to the STL Intraoperative Focus
Localization System...................................................................... 245
6.4 Extrapolation of the STL Principle to Alternative Recording
Environm ents................................................................................ 251

APPENDIX


LETTER OF INSTRUCTIONS TO THE REVIEW PANEL FOR THE
VALIDATION PROCEDURE .................................................................... 255


REFEREN CES ..................................................................................................... 258


BIOGRAPHICAL SKETCH ................................................................................ 268













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



A SPATIO-TEMPORAL APPROACH TO EPILEPTIC FOCUS
LOCALIZATION FROM ARRAY ELECTROCORTICOGRAPHY



By


ARMANDO BENNETT BARRETO


May 1993


Chairman: Dr. Jose Principe
Major Department: Electrical Engineering


This dissertation reports on the development of an automatic diagnosis tool for the

localization of the epileptic focus in patients undergoing seizure surgery.

A new algorithm, the Spatio-Temporal Laplacian (STL) transformation, is intro-

duced and proposed for the identification and localization of focal interictal events from

electrocorticographic (ECoG) signals. These signals are collected through a planar array

of electrodes placed over the cerebral cortex of the patient during surgery.

The STL transformation is applied to the time-varying potential field reconstructed

from the ECoG signals to selectively emphasize field configurations that are sharp in both

time and space. These configurations are shown to be associated with focal interictal








events, traditionally identified as "spikes" in the ECoG paper chart. In order to demon-

strate this correspondence, a new form of display for the array ECoG data, called the

"Bubble Diagram," was devised. This diagram is a convenient two-dimensional represen-

tation of the time-varying potential field sampled in space and in time.

A Focus Localization System based on the STL transformation has been imple-

mented in a NeXT workstation. The actual STL processing is carried out in the MC 56001

Digital Signal Processor embedded in this computer, allowing on-line operation. The sys-

tem is capable of displaying the field configurations associated with the identified interic-

tal events within two seconds of their occurrence. The distribution of detected interictal

activity is automatically displayed at the end of each recording session, referenced to ana-

tomical landmarks, to facilitate the focus localization task.

The system has been tested in eight surgery cases with encouraging results. The

detailed analysis of the results from three of those, with respect to the opinion of four

human experts, is included in this dissertation.















CHAPTER I
INTRODUCTION



The work presented in this dissertation is aimed at the development of an automatic

tool to help the neurologist and the neurosurgeon in localizing the epileptic "focus" to be

rejected in the surgical procedure applied to eligible epileptic patients.

In this chapter the problem of intraoperative focus localization from the signals ob-

tained through electrocorticography (ECoG), i.e., the measurement of electric potentials on

the exposed cerebral cortex of the patient, is presented in the wider context of focal epi-

lepsy. Additionally, relevant information and hypotheses about the nature and characteris-

tics of the epileptic "focus" are introduced. This information will be used as a basis for the

presentation of the digital signal processing algorithm developed for focus localization in

Chapters III and IV.



1.1 Focal Epilepsy and its Surgical Therapy


Epilepsy is a relatively common neurological disorder, affecting an estimated 1%

of the population [Ki91]. The exact mechanisms causing epileptic symptoms, such as fits

and loss of consciousness, are not fully understood as of yet. However, it has been observed

that the disease is closely related to peculiar disturbances in the electrical activity of the

brain. In particular, abnormally large transients are observed in the electroencephalogram

(EEG) of epileptic patients. When these transients appear in rapid succession within a given








interval, they constitute an electrographic seizure, also referred to as "ictal activity." On the

other hand, when the abnormal transients occur in isolation, they are called "interictal

spikes" due to the shape of the trace that represents them on the paper chart of a polygraph.

Some epileptic patients suffer from generalized seizures, i.e., seizures associated

with abnormal electrical discharges occurring in both cerebral hemispheres simultaneously.

Conversely, a number of epileptic patients have partial seizures, in which the abnormal

electrical activity begins in a defined location of a cerebral hemisphere, called the "epileptic

focus."

Although the therapeutic attempts for epilepsy start with anticonvulsant drugs,

some individuals may not be controlled in this way. Surgical treatment of epilepsy may be

offered to those patients, provided it can be shown that the seizures originate in a specific

area of the brain that can be removed rejecteded") without producing significant neurolog-

ical deficits. Temporal lobectomy is probably the most common surgical procedure in the

management of patients with epilepsy [Ki91].

In considering the possibility of surgical therapy for an epileptic patient suffering

from partial seizures, some of the following tests may be performed:




Review of the history of seizures.

Long-term video EEG recording with surface (scalp) electrodes.

Computerized Tomography (CT) scan, Magnetic Resonance Imaging (MRI) and

Single Photon Emission Computerized Tomography (SPECT) scanning.

Neuropsychological assessment.

Wada Test.








The review of the history of seizures of the patient is aimed at establishing a prelim-

inary localization of the focus in terms of his clinical manifestations. This preliminary lo-

calization is enhanced and complemented with the electrographic findings obtained from

the long-term video EEG monitoring.

In the preoperative evaluation of surgical candidates, electrode montages covering

most of the surface of the scalp are normally used. The basis for these montages is the In-

ternational 10-20 System [Ja58], with, perhaps, some added electrode locations. The lim-

ited number of electrodes used to cover all of the scalp determines a certain electrode

density that, in turn, sets a limit to the accuracy of the proposed focus localization results.

Imaging techniques used in the preoperative evaluation (CAT, MRI, SPECT) allow

the location of underlying structural and metabolic abnormalities that may be closely asso-

ciated with the "epileptic focus" responsible for the symptoms displayed by the patient.

The neuropsychological evaluation may be able to further define the clinical local-

ization of an epileptic focus. Additionally, verbal and nonverbal memory tests are applied

to determine dominance of either hemisphere for these functions.

The Wada test is applied to candidates for temporal lobectomy, to determine if the

temporal lobe that will not be rejected is capable of sustaining memory. It consists of the

temporary inactivation of one or the other cerebral hemisphere by unilateral intracarotid in-

jection of sodium amytal.

All the information gathered from this preoperative evaluation of the surgical can-

didate should result in a decision regarding the feasibility of a cortical resection and, if ap-

propriate, an approximate location of the target area for the resection.








1.2 Intraoperative ECoG


If a patient can indeed be offered surgical therapy for partial epilepsy, the approxi-

mate localization of the focus obtained preoperatively allows the neurosurgeon to deter-

mine the overall area of cortex that should be exposed. The actual location of the areas to

be rejected, however, may be confirmed or redefined according to the improved localization

obtained through electrocorticography (ECoG), performed intraoperatively [G175],

[Ras82],[Sh90],[Oj91].


1.2.1 Advantages of ECoG over Scalp EEG for Focus Localization


The ECoG offers a number of advantages for the focus localization process, with

respect to standard scalp EEG.

An immediate advantage of ECoG is that it is to be applied to a reduced area of cor-

tex already preselected with the information from the scalp EEG and the rest of the preop-

erative tests. This consideration alone permits a significantly higher electrode density,

which enhances localization accuracy in that same proportion. Similarly, the intraoperative

conditions will normally result in the absence of a number of artifacts commonly present

in scalp EEG recordings, such as muscle or eye blink signals.

Additionally, there are other intrinsic advantages in recording from the surface of

the cortex. In particular, the distance from the recording electrodes to the epileptic focus,

i.e., the aggregate of neurons responsible for the epileptiform features in the EEG, is now

smaller. Furthermore, there is only a single type of medium (brain tissue) between the focus

and the recording electrodes. When recording from the scalp, there are at least three distinct

media between the source of the electrographic phenomena and the recording electrodes:








brain tissue, cerebrospinal fluid, skull and scalp. This different situation has a large impact

on the recordings since the skull is approximately 80 times less conductive than brain tis-

sue, causing the signals recorded in the scalp to be attenuated and spatially diffused.

It is in view of these advantages of ECoG over scalp EEG that the intraoperative lo-

calization is used to confirm the overall presence of a focus and to establish its exact loca-

tion, which may then be used to guide the resection. On the other hand, the time available

for the analysis of the ECoG traces is a significant drawback. The whole process of analysis

of the ECoG, from setting up the recording electrodes to a final decision regarding focus

localization, has to be completed in a limited time, normally 30 to 40 minutes.

It is in this type of scenario that the use of an automated tool for the analysis of

ECoG signals obtained from an array of electrodes is proposed to help in the determination

of the cortical areas that should be rejected.



1.2.2 Electrode Systems for Electrocorticography


The electrical signals from the surface of the cerebral cortex are collected by means

of a set of electrodes, placed over the suspected focus location. There are two main types

of ECoG electrode systems. The first system uses "individually movable electrodes"

(IMEs), held in place by an apparatus affixed to the skull. This system allows almost com-

plete placement flexibility, but takes a considerable amount of time to set up. Additionally,

the geometric relationships between the electrodes in this system are not constant and may

not be easy to determine with the necessary accuracy. The second approach to ECoG elec-

trode placement is based on the use of electrode strips, grids, sheets or arrays. The main

advantages of this system are the shortened setup time and the fact that the geometric rela-

tionships between electrodes are fully specified, even beforehand. Although most of these








arrays were initially used for the placement of implanted electrodes, an improved version

designed specifically for its intraoperative use [Re89], has been utilized for this work.

Bipolar recordings (i.e., the measurement of potential differences between elec-

trode sites) have been preferred for the acquisition of data for focus localization, as this

choice eliminates the need for a sufficiently stable reference point.

Figures 1.1 and 1.2 illustrate the two ECoG recording arrays used in the collection

of signals for this work and their dimensions. Three sets of bipolar "derivations" (i.e., dif-

ferential voltage measures) used with the first array are shown in la), lb) and ic). These

predefined sets of bipolar derivations are commonly known as "montages."



1.2.3 Computer-assisted Intraoperative Epileptic Focus Localization.


The previous paragraphs have outlined the scenario in which a computer-based tool

for intraoperative epileptic focus localization has to be considered.

From this context, the functional requirements for the system can be outlined as fol-

lows. The system should receive the ECoG signals collected with arrays such as the ones

shown in Figures 1.1 and 1.2 and distinguish from these incoming data any clues that may

point to the location of the epileptic focus. As the recording takes place, the system should

indicate the estimated focus location, such that any necessary adjustment or change in the

recording setup can be performed immediately to obtain the most information from the re-

cording. The estimated localization of the focus should be presented to the user in a graphic

display, preferably in reference to the position of the array with respect to anatomical land-

marks.























a) Ml


b) SMI


c) SM2


Figure 1.1 Montages for the first ECoG array: a) Montage Ml;
b) Montage SM 1; c) Montage SM2





















a) SM21


62 (23) (21 20 (19
chl chll
104, (16 13
h16 ....ch6.....- ... l hl


chl5 chl ch


b) SM22



Lchll .._ch~ll i .. .....................
chl chl

.ch3 -, h& ....... ... h


chl,' chl3


c) SM23


Figure 1.2 Montages for the second ECoG array: a) Montage SM21;
b) Montage SM22; c) Montage SM23.








It is important to note that the time limitations determined by the nature of intraop-

erative ECoG evaluation demands "on-line" processing of the signals, i.e., the results of the

processing need to be generated by the system at the same pace as the data are flowing into

it.

This particular specification was considered central to the problem and was highly

influential in the design of the Signal Processing Algorithm proposed for focus localization,

explained in Chapter III.

The previous definition for the problem of computer-assisted intraoperative focus

localization can be summarized in a generalized block diagram (Figure 1.3). This block di-

agram shows the basic functions that the focus localization system has to perform.


Storage of Raw
Data for future
analysis


Discrimination Data
of segments of Pre-Processing
data relevant to
focus localization l (Potential Field
Reconstruction)


Data Acquisition


Figure 1.3. Generalized block diagram of an intraoperative focus localization system.








1.3 Target Signal Specification


From a medical point of view, the neurosurgeon is interested in locating and resect-

ing the portion of brain tissue where the ictal electroencephalographic activity originates

[G175], since it is this kind of electrical activity that is most commonly associated with clin-

ical manifestations of the patient. On the other hand, ictal activity (seizures) is not always

present in intraoperative ECoG records, even after the intentional activation of the suspect-

ed focus with appropriate drugs [Pr67][Oj91]. Therefore, an intraoperative focus localiza-

tion system can not rely on the presence of ictal activity in the ECoG for its performance.

Although the match between the origins of ictal and interictal activities is still sub-

ject to debate, there is evidence that supports the use of interictal events for focus localiza-

tion. On one hand, there does not seem to be a physiological reason to expect that the

interictal activity should have a significantly different origin than the ictal activity. As a

matter of fact, a number of researchers (e.g., [Pr67],[Pe87]) produced seizures in animal

models by topical and intracortical application of various epileptogenic agents and ob-

served that a number of interictal spikes preceded the seizure, which arose from the same

location.

Additionally, Ojemann [Oj91] points out, "There is the extensive experience from

the Montreal Neurological Institute in achieving seizure control with temporal resections

guided by excision of tissue identified by interictal spikes," and refers the reader to the work

of Rasmussen [Ras82].

So, the use of interictal events as targets for an intraoperative focus localization sys-

tem appears both plausible and practical. Furthermore, within a certain time frame each one

of the fast transients that constitute an electrographic seizure may be found to resemble an








interictal spike. Therefore, a localization system could be designed, in principle, to operate

from both the more commonly available interictal event and the less usual but presumably

more indicative seizure components.

In the work reported here, the interictal events were selected as the targets for the

localization process. This means that the system will try to obtain localization information

from sections of the incoming data that conform with the features of an interictal event

(Chapter II). However, the signal processing algorithm proposed here will also regard what-

ever valuable localization information is conveyed by the fast transients that constitute an

electrographic seizure.



1.4 The Epileptic Focus


Before any attempt at localizing a source of physiologic phenomena, such as the ep-

ileptic focus, can be made, all the relevant anatomical and physiological information avail-

able should be considered. In particular, before a digital signal processing algorithm can be

devised for the detection of interictal events and the localization of their origin, some basic

understanding of the available knowledge about the mechanisms involved in an interictal

event is in order.



1.4.1 Interictal "Spikes" in EEG Traces


The peculiar shape of interictal spikes was noticed even in the early reports on the

electroencephalogram of man written by Hans Berger from 1929 to 1938 [G169]. Transient

signals of abnormally large amplitude were noticed in the EEG traces of individuals with

epilepsy. Even from those initial observations, Berger suspected the peculiar features to be

caused by "periodic discharges of (the) cortical centers. ", giving birth to the primitive








concept of an epileptic focus.

When the fast transients observed in the EEG are produced by a local effect, there

will be patterns in the "spikes" shown simultaneously on different traces of the paper chart.

These are the patterns that are traditionally used to determine the location of the origin of

certain EEG events. Table 1.1, adapted from Duffy et al. [Du89], shows a few examples of

the localizing conclusions that can be derived from patterns found in a chain of bipolar der-

ivations.

The occurrence of simultaneous pen deflections with opposite directions in adjacent

channels is termed a "phase reversal" and is indicative of a local effect. One should take

into account that the voltage difference measured by each bipolar derivation only involves

two samples of the continuous potential field on the surface of the scalp (or the cortex, in

ECoG). From this point of view, one would find that a "phase reversal" such as the one in

Table 1.1 A, really represents a depression ("negativity") in the potential field, under elec-

trode B.

If the head of the subject (or his brain, in the case of ECoG) is idealized as a volume

conductor, the presence of the measured voltage differences has to be associated with the

flow of internal electric currents. Further assuming that the conductive medium does not

present significant inductive or capacitive components, one could propose that the time

course of the events sensed at the surface of the volume conductor will essentially follow

the time course of internal current generation. This leads to a first characterization of the

epileptic focus, built with only the evidence gathered from the surface potential recordings.

As such, the epileptic focus appears to be an aggregate of cells capable of generating large

transient currents inside the brain that have local effects in the potential field measured on

the surface.









Table 1.1
Examples of localizing patterns for bipolar derivations
A B C D
Bipolar Chain & Focus of Maximum Focus of Maximum Focus of Maximum Focus of Maximur
Bipolar Derivations Negativity at B Negativity between Positivity at C Negativity at A
B and C


A->B = 4B OA .- ---- --.-_\-




B B->C = c-" AB




C->D = oD C_


1.4.2 Neurons as Current DiDoles


Citing Bishop, Creutzfeldt and Houchin, and Ball, Gloor concludes that it is now

generally accepted that the principal generators of the EEG are cortical neurons, more par-

ticularly pyramidal neurons" [G185]. Thus, it is important to consider how these individual

cells act as current generators within the conductive medium of the brain and how, under

special circumstances, this current generation may experience a transient magnification.

In particular, it is considered that "the bulk of gross potential recorded from the

scalp results from extracellular current flow associated with summated postsynaptic poten-

tials in synchronously-activated, vertically-oriented pyramidal neurons" [Mar85].

The pyramidal neurons mentioned above are found in the cortical structures of the

brain: the olfactory cortex(paleocortex), the hippocampal formation (archicortex) and the








neocortex or "cerebral cortex." They are oriented perpendicularly with respect to the surface

of the structures, forming layers that run parallel to those surfaces. In the cerebral cortex the

bodies (somata) of the pyramidal neurons are mainly found in the cortical layers III and V,

(about I and 1.5 mm deep from the surface of the cortex). These bodies have projections or

dendritess" above and below ("apical" and "basal" dendrites, respectively). A longer projec-

tion, the axon of the neuron, is capable of transmitting an electrical disturbance (action po-

tential) along its length, with virtually no attenuation, to induce an electrical change in

another neuron. The actual effect on the second neuron takes place through an electrochem-

ical interface, known as synapse. This effect may be one of excitation, inducing the second

neuron to "trigger" an action potential of its own, or one of inhibition, preventing the action

potential from happening. The triggering of an action potential depends on the potential dif-

ference between the inside of the neuron and the surrounding medium ("membrane poten-

tial"). This potential remains at a resting value of approximately -70 mV, but is altered by the

activity of inhibitory and excitatory synapses on the cell (postsynaptic potentials). When an

excitatory synapse becomes active, certain ionic channels will be opened that allow specific

ions, such as +Na, to flow into the cell, making the inside of the cell less negative with respect

to the outside depolarizationn) and bringing the membrane potential closer to the threshold

potential (approximately -55 mV), at which point an action potential will be triggered. The

important point to consider here is that an Excitatory Postsynaptic Potential (EPSP) is

achieved via a net inflow of (positive) current into the cell [She83]. According to a symmet-

ric sequence of events, the activation of an inhibitory synapse makes the inside of the neuron

more negative with respect to its surrounding medium (hyperpolarization) via a net outflow

of (positive) current, at the point of the synapse.

In the case in which there are several excitatory synapses acting on the apical den-

drites of a pyramidal neuron, the net effect can be represented by an equivalent current sink








located in the apical region of the cell. Of course, the conservation of charge requires that

a balancing (positive) current flow out of the neuron in a different region. The simplest as-

sumption is that most of the outflow of current takes place in a particular segment of the

cell, e.g., its basal region. Under this assumption, the pyramidal cell experiencing multiple

excitatory input can be modeled as the conjunction of a current sink (in the apical region)

and a current source (in its basal region), which, in turn, can be viewed as a current dipole.

This current dipole will have its axis oriented perpendicular to the surface of the cortex.



1.4.3 Intracellular and Extracellular ("Field") Recordings


Important knowledge regarding the neuronal behavior in the epileptic focus during

interictal events has been gained through intracellular and extracellular ("field") record-

ings. Most of this kind of research has been carried out in animal models of acute focal ep-

ilepsy. Most commonly, an epileptogenic agent, such as penicillin or strychnine, is applied

to a reduced area of the cerebral cortex [Gum65], [Pr67], [Pe70], [Ay73], [Pe84], [Ra87],

or the hippocampus [Di69a] of animals. A few minutes after the application of the agent,

"spikes" that resemble typical "epileptic (interictal) spikes" from human subjects can be

read in the ECoG at the induced focus.

One of the most relevant findings in this area was the realization by Matsumoto and

Ajmone-Marsane [Mat64] that each interictal event detected in the surface ECoG has an

intracellular associated event. It consists of a sudden 20 to 50 mV depolarization, which

lasts for 50 to 100 milliseconds and is followed by a prolonged hyperpolarization. This

"Paroxysmal Depolarizing Shift" (PDS or DS) has been observed to occur simultaneously

in the neurons of the focus. Then, the two major abnormal properties that characterize the

activity of the focus during an interictal event are: "Each involved neuron exhibits a large








amplitude depolarization (the depolarizationn shift") associated with repetitive (action po-

tential) spike generation, and this excitation arises with virtual synchrony in the majority of

cells in a local population" [Pr86].

A considerable amount of effort has been devoted to the study of this synchronous

"Depolarization Shift." In particular, this effect has been proposed to be of synaptic origin.

It has been suggested [Di69b], [Ay73] that the PDS is in fact a magnified Excitatory Post

Synaptic Potential (EPSP). Further support for this view has been offered more recently

[Jo81], on the basis of new experiments. Also, these researchers confirmed that the PDS in-

volves inward currents that are larger than the ones associated with normal EPSPs. When

this information about PDS in individual cells is considered in conjunction with the syn-

chrony of the PDSs of the neurons in the focus, the explanation for the large transient cur-

rents causing the observed "spikes" becomes clear.

While the events recorded right at the induced epileptic focus seem to be dominated

by excitatory interactions, it was found [Pr67] that most of the cells in the areas around the

induced focus showed prominent inhibition during the surface epileptiform discharges

("surround inhibition"). The inhibition was present in the form of large Inhibitory Post Syn-

aptic Potentials (IPSPs) observed in intracellular recordings from neurons in the periphery

of the focus, 10 to 40 milliseconds after the onset of the surface "spike." Prince and Wilder

also noted that such inhibitory fields "were not only large laterally but also in vertical ex-

tent." The observation of inhibitory effects surrounding the "central focus" have found fur-

ther support in multiple experiments with acute induced foci. Dichter and Spencer [Di69a]

mapped the topographic distribution of excited and inhibited neurons in and around the

penicillin-induced focus in the hippocampus and corroborated the presence of the "inhibi-

tion ring." Schwartzkroin et al. [Schw77] noted that, for acute cobalt-induced foci in the

cortex of cats, the deeper cells beneath the focus were, in fact, inhibited, while the more








superficial layers showed excitatory patterns of activity. Reid and Sypert [Rei80] analyzed

the three-dimensional distribution of neuronal activity (inhibition/excitation) at the time of

interictal events in acute foci induced in the cerebral cortex of cats using ferric chloride

(FeC13) as the epileptogenic agent. They confirmed both the presence of "surround inhibi-

tion" and the inhibition of the neurons in the deeper layers of the cortex, below the "central

focus." Elger and Speckman ([E183],[EI87]) confirmed both types of inhibition, forming an

"envelope" around the "central focus" in penicillin-induced cortical foci. They used the

term "vertical inhibition" to describe the inhibition of cells located in the deeper layers of

the cortex, beneath the "central focus."

In several of these experiments, the investigators found that the "inhibitory enve-

lope" acted as a restraining barrier that prevented the spatial and temporal enhancement of

the interictal events. It was also observed that the evolution of the interictal events into a

seizure would normally involve the "breaking" of the inhibitory restraint. Prince and

Wilder, for example, noted, "During ictal episodes, the behavior of neurons in the area of

the surround inhibition changed; these cells became recruited into the epileptiform dis-

charge, showing typical DSs."

These observations are significant for the development of a focus localization sys-

tem in a number of ways. First, they portray a more complete image of the epileptic focus

at the time of an interictal event. In particular, the presence of significant IPSPs around the

PDSs present in the "central focus" will determine a characteristic pattern of surface cur-

rents, flowing from the surrounding area into the "central focus." This will produce larger

surface potential differences (steeper potential gradients) than those expected from the DSs

in the "central focus" alone. Second, the self-restraining feature of interictal events pre-

cludes the use of detection methods based on remote sensing that rely on the uniform, or

otherwise predictable, propagation of an initial perturbation through a given medium (e.g.








"beamforming"). Lastly, these observations show that, in at least these experiments, ictal

activity can have its origin in the same location from where the preceding interictal events

arose, thus justifying the use of interictal events as targets for a focus localization system.



1.4.4 Relationship between the Current Generation at the Focus and the Changes of the
Surface Potential Fields.


Once a more complete image of the current generation phenomena associated with

focal interictal events has been formed, some initial steps toward a basic quantification of

their effects on the surface potential field may be taken.

From the multiple accounts obtained from intracellular studies, we can picture the

neurons at the "central focus," during the "apex" of the interictal event, as a layer of current

dipoles oriented perpendicular to the surface of the cortex, with their current sinks closer to

the surface than their current sources. Simultaneously, the inhibited neurons in the sur-

rounds of the focus will act as dipoles of the opposite polarity. Figure 1.4 a) portrays a sim-

plified version of this situation. Taking into account the resulting flow of superficial

currents caused by this distribution of dipoles and the fact that, in a conductor, (positive)

current flows from a point of higher potential to a point of lower potential, the configuration

of the surface potential field at the time of maximum PDS intensity may be expected to look

as indicated in Figure 1.4 b).

The effect of a single current dipole, immersed in a homogeneous conductive me-

dium (conductivity a) on the potential of a measurement point is given by [Nu81]:



S= ( ) (1- 1)
47Cy R1 R2
























AflC


11


Figure 1.4


Current and voltage effects of a focal interictal event.
a) Hypothetical current dipole distribution in and around
the focus; b) Surface potential configuration in and around
the focus.


In equation (1-1) RI and R2 are the distances from the source and the sink to the

observation point, respectively, and I is the intensity of both the source and the sink.

Equation (1-1), however, assumes that both the current dipole and the measurement

point are within the volume conductor. In the case of EEG or ECoG measurements, the

measurement points are located in the interface between the medium of conductivity o,

where the current dipole is immersed, and an external medium of much smaller conductiv-

ity, i.e., the air. This fact causes a distortion of the current flow in the interface. A method

has been designed (the "method of images" [Nu81]) to account for such distortion under








these circumstances. According to this compensation, the potential values caused by a po-

tential dipole at the interface between both media will be twice the one indicated by equa-

tion (1-1). Therefore, the relationship changes to:



2DtaR R2) (1-2)


As mentioned earlier, the DSPs in the neurons of the central focus not only are ex-

traordinarily large, but they also are produced in many cells at the same time. Therefore,

the study of the effects of interictal events on the surface potentials should account for these

multiple sources acting simultaneously. Currently, there are two approaches used to model

this consideration. The first and simplest approach lumps the multiple single-cell dipoles

into a single "large" dipole with a unique direction and spatial location. The alternative ap-

proach uses the more physiologically accurate assumption of a layer of individual neurons

acting in synchrony: the "dipole layer."

The study of the potential fields caused by a single dipole focus will generally be

based on an expression related to equation (1-2). On the other hand, the determination of

the surface potential fields caused by a dipole layer at a given test point requires the con-

sideration of the individual contributions produced by the elements of the layer, weighted

according to their particular geometric relationships to the test point (superposition). This

brings about the concept of "Spatial Convolution" ([Ka81], [Fr80]), which relates the di-

pole layer configuration at the focus to the surface potential field observed at a given instant.

A somewhat abbreviated way to relate the dipole layer configuration to the surface potential

field that it causes has been proposed by Gloor [G185]. This approach uses the "Solid Angle

theorem, introduced by Woodbury [Woo60], which characterizes the dipole layer by the

equivalent potential e present across its two surfaces (positive and negative surfaces of the








dipole layer) and states, "The potential P generated by a dipole layer in a volume conductor

measured at any point within this conductor is proportional to the solid angle subtended by

the dipole layer at the point of measurement:


P =- 0 (1-3)
411

where Q is the solid angle subtended by the dipole layer at the measuring point."

This approach may be very useful in estimating the resulting surface potential field pro-

duced by a given dipole layer, by simple inspection. However, this is only a different way

to look at the same concept of spatial convolution, as Woodbury resorts to the superposition

of effects from elementary dipoles in his derivation of this theorem [Woo60].

A standard measure for the strength of a current dipole is its dipole moment p, de-

fined as the product of the intensity I of its source or sink (assumed the same) and the dis-

tance d between source and sink (the orientation of this vector is from sink to source, along

the axis of the dipole):



Ipl = Id (1-4)


The kinds of surface potential fields resulting from the consideration of a single di-

pole focus and a dipole layer focus may be different. As an example, Figure 1.5 shows the

results of simulated surface potential fields for a) a single dipole with a dipole moment of

89 4-n units and b) a dipole layer with 89 elementary dipoles of moment 47t each, evenly

distributed in the circular area indicated (1 distance unit between every two elementary di-

poles). For ease of visualization, the dipoles were simulated with their current sources on

top, which produces a convex surface potential pattern, opposite to the concave pattern nor-

mally associated with most epileptic "spikes." Examination of this figure reveals that the








dipole layer model is capable of producing smoother deflections of the field, for a given fo-

cus depth, than the single dipole model. Another point to consider is that the dipole layer

model is more versatile in that it can be made to equate the single dipole model (making the

"spread" of the elementary dipoles zero) and in that it can associate asymmetrical surface

potential fields to asymmetric dipole layers.

It is important to note that whatever the modeling approach used, the activation of

a local current generation will result in a "deflection" convexityy or concavity) in the sur-

face potential field. In other words, a local effect will introduce "sharpness" in the potential

landscape. The magnitude of this "spatial sharpness" observed in the potential field is, in

general terms, directly proportional to the intensity of the generator and inversely propor-

tional to its depth and spread.



1.5 An Example of Focal Interictal Activity (FIA) from Real ECoG


The discussion of the internal mechanisms involved in the generation of an electro-

graphic "spike" in the preceding sections can now be used to confront the traditional form

of display for the ECoG (on the polygraph) with an alternative form, based on the concept

of a time-varying surface potential field.

As an example, we can consider Figure 1.6. In part a) of this figure the traces cor-

responding to one second of data from the four bipolar derivations connected to electrode

20 in Array # 1 (Figure 1.1) are shown. Two instances of combined phase reversals that in-

dicate a local negativity under electrode 20, according to montage Ml in Figure 1.1, have

been boxed. Figure 1.6 b), on the other hand, shows the reconstruction of the instantaneous

surface potential field at the time of sample number 217, in the form of a contour level map.

In these maps, points with the same potential are connected to form an isopotential line.





23

(The details of how this field reconstruction is obtained from the ECoG derivations will be

explained in Chapter III.) The circular patterns found in the contour level map of Figure 1.6

b) confirm that, effectively, the event that the electroencephalographer (EEGer) normally

acknowledges as a "local spike" involved a local deflection (negativity) of large magnitude

in the surface potential field sensed by the array.

It is important to note that the field representation of the interictal event is, by its

very nature, a form of display in which the localization of the origin of interictal activity

should be much easier.











A


Measuring
.,. plane -


S dipole layer 3

.............
dipole


c)

Figure 1.5 Simulation of surface potential fields due to a) a single vertical
dipole and b) a circular dipole layer with the same total dipole
moment; c) Conditions for the simulations.




















































Figure 1.6 Focal Interictal Event represented by a) Four bipolar derivations
connected to the same electrode and b) a contour level map.















CHAPTER II
FOCAL INTERICTAL EVENTS IN TIME AND SPACE


Using the background information provided in Chapter I, this chapter discusses the

temporal and spatial features that have been observed in association with the surface poten-

tial field perturbations caused by focal interictal events. After the available information that

characterizes focal interictal events in time and space is reviewed, some of the previously

proposed detection and localization methods, based on those features, are considered. In

each case, an effort is made to highlight the most reliable and indicative features of the sur-

face potential fields associated with interictal events. Towards the end of the chapter, the

basic premise of the research work presented here, namely that focal interictal events

should be considered as spatio-temporal events, is introduced, and the possibility of build-

ing a spatio-temporal model for them is considered.



2.1 Temporal Dimension of Focal Interictal Activity


Ever since the first links were established between epilepsy and electroencephalo-

graphic recordings, the sudden character of epileptiform discharges, as displayed in EEG

traces, has been their most distinctive identifier. In his seventh report on the electroenceph-

alogram of man (1933), Berger published his finding of "especially striking" features in the

EEG of a female patient showing clonic jerks of the right arm, hand and fingers. He de-

scribed these special features as "sudden drops in potential" recorded around the left pre-

central convolution of the patient and was able to demonstrate that such electrographic

features were time-locked with the twitching of her fingers [G169].

26








Even today, one of the most significant features used by the experts in evaluating

the EEG records of a patient suspected of suffering epilepsy is the presence of abrupt tran-

sitions in the potential recorded from the scalp under different conditions such as normal

relaxation, hyperventilation and photic stimulation.



2.1.1. Temporal Characterization of an Interictal "spike"


The most widely accepted "definition" of an interictal "spike" is the one offered by

Chatrian et al. [Ch74]:



"A transient, clearly distinguished from background activity, with pointed peak at

conventional paper speeds and a duration from 20 to under 70 msec, i.e., 1/50 to 1/14 sec,

approximately. Main component is generally negative relative to other areas. Amplitude is

variable."



This definition is complemented with the definition of a "sharp wave" offered by the

same authors:



"A transient, clearly distinguished from background activity, with pointed peak at

conventional paper speeds and duration of 70-200 msec, i.e., over 1/14-1/5 sec, approxi-

mately. Main component is generally negative to other areas. Amplitude is variable."

Thus, essentially, spikes and sharp waves could be grouped together as transients

lasting from 20 to 200 milliseconds, with the morphological characteristics mentioned

above.

The main problem that has been pointed out about these definitions is their lack of

"functional" specifics that could be used to encode them in a detection system [Got87],








[Kt87], [Pa90]. In particular, Panych and Wada [Pa90] raise the question as to how it is that

the spike or sharp wave is "clearly distinguished" from the background. They also associate

the ambiguity in the definition with the common significant disagreement in the visual as-

sessment of EEG recordings among experts, or even by the same EEGer in repeated ap-

praisals of an EEG record [Gos74].

A number of attempts have been made to find quantitative and specific features that

characterize EEG traces associated with interictal events. One of the earliest of such studies

is the one carried out by Kooi [Ko66], by observation of interictal "spikes" and "sharp

waves" from the paper chart of the EEG machine. He considered that the morphological

features of interest were the ones indicated in the triangular approximation of a generic in-

terictal spike reproduced in Figure 2.1.

He analyzed 18 fast transients, paying special attention to the following parameters:



1) Amplitude, from a to b.

2) Amplitude, from b to d.

3) Duration, from a to c.

4) Duration, from a to d.

5) Rise time, from a to b.

6) Descent time, from b to d.

7) Peak angle (theta) at 50 jAV = 7mm and 1000 ms = 3 mm.

8) Segmental "velocity" (slope), from a to b.

9) Segmental "velocity" (slope), from b to d.

10) Overall "velocity" (total voltage excursion/ time), from a to d.


























Figure 2.1 Triangular approximation of a generic interictal "spike"


Out of the 18 sharp waves studied, Kooi selected six as definitely epileptiform. He

found that all of the selected transients reached "segmental velocities" (rising or falling

slopes) of more than 2 lpV/msec. He suggested that the "segmental velocity" could be the

more indicative feature of transients of epileptic origin, since "it provides a measure where-

by degrees of wave "sharpness" or "spikeness" can be quantified and compared". On the

other hand, he also emphasized the value of the "peak angle" as a distinctive measure of

epileptiform transients, since it "interlocks duration and amplitude". However, he acknowl-

edged that one major shortcoming of this feature is its direct dependence on channel am-

plification (gV/mm) and paper speed (ms/mm), the quantification of "sharpness" through

a mathematical concept such as the second time derivative of the signal not being discussed

in his paper.

It is interesting to note that, in spite of his expressed preference for the maximum

"segmental velocity" as the parameter of choice for classification, Kooi selected 8 of the 9








waveforms with the smallest "peak angle" as epileptiform. (The ninth was left out due to

its duration of 85 ms, in adherence to the "old" definition of a "spike" [Br6l], which set its

maximum duration at 80 ms).

Kooi also indicates that, contrary to the common view that the rising phase of a

sharp wave is steeper than its decent [Hi63][Do61], "instances may also be observed in

which the reverse is true, for example, slow rise with rapid descent".

The capability of digitizing EEG "spikes" allowed Ktonas and Smith [Kto74] to

perform a closer quantitative analysis of epileptiform transients. They reported on the anal-

ysis of 16 "abnormal spikes" gathered from 3 epileptic patients. This research was prima-

rily aimed at finding quantifiable descriptors of the sharpness and duration of epileptic

spikes, which were thought to be their most distinctive characteristics. The descriptors were

defined in terms of the evolution of the first time derivative (slope) of the wave form and

points P1 and P2 (same as points a and b in Figure 2.1), the closest minima to the left and

to the right of the spike "peak" (maximum), respectively. The six descriptors studied in that

work were:



S = The maximum spike slope before the transient reaches its peak.

S2 = The maximum spike slope after the transient reaches its peak.

S3 = The time it takes for the spike to reach its peak after it attained maximum slope.

S4 = The time it takes for the spike to reach maximum slope after it reaches its peak.

S5 = The sum of S3 and S4.

S6 = Time difference between PI and P2.

S6' = Time difference between P1 and the point of maximum spike slope after the

peak of the spike.








These researchers pointed out that S5, i.e the time that it takes for the spike to switch

from maximum slope in a certain direction to maximum slope in the opposite direction, was

"inversely proportional to the spike sharpness". Table 2.1 summarizes the results observed,

indicating the mean values and standard deviations, obtained from Table 1 in their original

paper.



Table 2-1: Parameter values found by Ktonas and Smith [Kto74]

Standard
Parameter Mean Value Standard
Deviation

Sl( V/ms) 4.8 2.87
S2(pV/ms) 5.56 4.15
S3(ms) 9.31 2.93
S4(ms) 12.81 3.03
S5(ms) 22.13 4.89
S6(ms) 53.63 8.22
S6'(ms) 34.93 6.44


In general terms, these observations matched and further validated those by Kooi,

enhancing the accuracy of the measurements to the I millisecond level (the sampling inter-

val used) and eliminating the presence of measurement errors due to pen curvature and dy-

namics.

Two particular points that were reinforced with these new measurements were: a)

Epileptiform transients will normally reach maximum slopes above the 2 gV/msec, as es-

tablished by Kooi, and b) the "falling" slope of the spike (S2) may, in fact, be steeper than

the "rising" slope (Sl) of some spikes. Ktonas and Smith further explained that it may be

the longer value of S4, with respect to S3, that gives the EEGer the impression that sharp

waves have a "fast increase" and "slow decrease".








From Table 2.1, one can notice that the slopes exhibited by the transients have sig-

nificant variability from one case to the other, therefore making it difficult to use these val-

ues to characterize the temporal expression of interictal events. On the other hand, the

variability of the parameters representing sharpness (S5) and duration (S6 and S6') of the

spike seem to be more consistent from one transient to the other.

Guedes de Oliveira et al. [Gu83] investigated waveform parameters similar to those

studied by Ktonas and Smith, but used a comparative approach intended to highlight the

features that could best differentiate between epileptiform and sharp but non-epileptiform

transients. In order to do this, they subjected a set of transients or "candidate epileptiform

events", picked-out due to their large value of second time derivative, from 10 epileptic pa-

tients, to classification by 8 EEGers. Two subsets of transients were distinguished. Those

transients selected by at least 7 of the 8 scorers conformed the "epileptiform" set. Transients

not selected by any of the raters were grouped in the "non-epileptiform" set.

The quantification of temporal characteristics was carried out on both sets and in-

volved measurements on the signals themselves as well as on their first and second time de-

rivatives. The six parameters defined by Ktonas and Smith [Kto74] (Si through S6) were

included and complemented with the following:



PP1 = Amplitude difference between the peak (maximum) of the spike and the pre-

ceding minimum.

PP2 = Amplitude difference between the peak (maximum) of the spike and the suc-

ceeding minimum.

C = Maximum absolute value of the second time derivative reached by the spike.








In order to relate these direct measurements to the background of the original signal

and its derivatives, the standard deviations for the signal, the first and second time deriva-

tives were calculated and represented by: 0, Gd, Odd, respectively. With these values, the

following indirect measures were found:



PPlm = PPl/a
PP2m = PP2/a
Ml = Sl/ad
M2 = S2/ad
CM = C/Odd


The mean values and standard deviations of these parameters in each of the two

populations of transients, epileptiform and non-epileptiform, were calculated. These statis-

tics were then used to rank the discriminating power of each of these parameters in sepa-

rating epileptiform from non-epileptiform transients. The authors computed the

"Mahalanobis Distance"



MahalanobisDistance = mep- mnepl (2-1)
02 +n
ep nep


A long Mahalanobis distance in between the two populations, for a given measure,

indicates that the difference of mean values (mep, mnep)of that particular measure is not

likely to be explained exclusively in terms of the combined variance (ep2 + anep2). This

may, therefore, point to a significant underlying difference for that parameter in-between

the two populations. Table 2.2 has been adapted from the results published by Guedes de

Oliveira et al.










Table 2-2: Parameter Ranking according to Guedes de Oliveira et al. [Gu83]

m in a in
mm mnG in
Paramet Ranking M- .. Non- Non-
Ranking epilepti epileptic .
er Distance pti epilepti epilepti
formform form

ICMI 1 0.8495 10.52 6.27 4.62 1.29
IM21 2 0.7279 8.05 4.89 3.77 1.12
IM11 3 0.6832 6.97 3.97 3.59 0.98
IPP2ml 4 0.6303 7.31 3.96 3.85 1.82
S6 5 0.3328 103.20 31.64 76.98 32.63
S4 6 0.1998 21.17 10.29 15.42 7.72
S5 7 0.1806 36.99 13.01 29.65 11.40
IPPlml 8 0.1771 5.04 3.21 3.54 1.55

From this table, it can be observed that the maxima attained for the first (IMll and

IM21) and second (ICMI) time derivatives, with respect to the background displayed a sig-

nificantly greater discriminating power in this study. Based on these observations, a system

was developed to compute the first 3 parameters of the table and compare against the fol-

lowing thresholds:




IM11 > 2.9
IM21 > 2.7
ICMI > 5.3



to accept a transient as epileptiform. It was found that the performance (matching with clas-

sification by EEGers) on a test set of transients did not improve with the inclusion of the

fourth parameter but was enhanced when requiring the wave duration, S6, to be more than

40 milliseconds.








In general terms, the results reported by Guedes de Oliveira et al. seem to indicate

that the most distinctive features of epileptiform transients are their fast slopes and the sud-

den character of the change of direction that these fast slopes exhibit. It is also apparent that

epileptiform waves are set apart from other transients by their sustained time course, which

normally spans over more than 40 milliseconds. Once more the mean values and standard

deviations obtained for the slope parameters Ml and M2 in epileptiform transients do not

seem to support the hypothesis of a consistent asymmetry in these slopes that could be ex-

ploited for detection purposes.



Ktonas [Kt87] proposed 4 parameters of the morphology of spikes as the most char-

acteristic ones. These parameters are:



1) Amplitude from peak (maximum) to succeeding minimum.



2) Duration, measured between the inflection points to each side of the peak. This

corresponds to S5 among the parameters originally used by Ktonas and Smith [Kto74].



3) Maximum slope in the interval from peak (maximum) to the succeeding mini-

mum of the spike.



4) A measure of the sharpness of the spike given as:


(SL1 + SL2)
SHRP2 = (SL +L2) (2-2)
DUR2


where SL1 and SL2 are the maximum slopes attained by the spike before and after its peak,

and DUR2 is the duration, measured as explained in 2).








Summarizing, it seems that the three most important concepts in distinguishing ep-

ileptiform spikes recorded from the scalp from other transients are their sharpness, their

maximum slopes and their duration. At this point it is worth pointing out that, if the even-

tual asymmetries of spikes, (which have not been clearly established), are ignored in favor

of a generic triangular shape, then the concept of "sharpness" at the peak of the spike is di-

rectly proportional to its slopes and inversely proportional to its duration.

While the morphology of spikes recorded from the scalp has been repeatedly ana-

lyzed, only a few similar studies have been carried out for the quantification of spikes re-

corded from subdural electrodes. Blume and Lemieux [B187] performed a comparative

study, measuring the main morphological features of 200 spikes recorded with subdural

electrodes and with scalp electrodes, simultaneously.

In their study they considered each spike as being composed of 2 half waves, the

first one, FHW, from the preceding minimum (point a, in Figure 2.1) to the peak (point b,

in Figure 2.1) and the second one, SHW, from the peak to the succeeding minimum (point

d, in Figure 2.1). They had the following three definitions for the slope of each half wave:



Slopel = half wave amplitude / half wave duration

Slope2 = amplitude difference between middle point of the half wave and peak of

spike divided by 0.5 of half wave duration.

Slope3 = maximum instantaneous slope found in the half wave.


Table 2.3 summarizes the results they published.









Table 2-3: Average parameter values found by Blume and Lemieux [B187]

Parameter Subdural Scalp
Amplitude FHW (p.V) 608 82
Amplitude SHW (pV) 759 115
Duration FHW (p.sec) 25 31
Duration SHW (.sec) 29 35
Slope 1 FHW (pV / msec) 27.5 2.5
Slope 1 SHW (pV / msec) 26.9 2.6
Slope 2 FHW (gV / msec) 27.3 2.3
Slope 2 SHW (pV / msec) 30.0 3.0
Slope 3 FHW (pV / msec) 43.1 4.4
Slope 3 SHW (pV / msec) 41.7 4.9


From these measurements the authors concluded that the average amplitudes of

scalp-recorded electrodes were only 13% and 15% of their subdural counterparts, for the

first and second half waves, respectively. The first and second half waves of scalp-recorded

spikes were found to last 24% and 21% longer than when recorded by subdural electrodes,

respectively. This meant that the slopes observed in spikes recorded subdurally were almost

an order of magnitude larger than those perceived, for the same pool of spikes, in scalp re-

cordings. These authors also point out that the spikes recorded subdurally appeared to have

a steeper first half wave, while the scalp-recorded spikes seem to be slightly closer to sym-

metry.

Although Blume and Lemieux did not list explicit differences of sharpness in spikes

recorded from subdural electrodes with respect to those obtained from the scalp electrodes,

one may conclude, given the averages shown in Table 2.3 for the duration and amplitude of








the half waves, that the spikes recorded subdurally are significantly sharper than those

found in scalp recordings.

The morphological differences noted were explained by the authors in terms of a

model of spikes that considers them as propagating locally. They reasoned that the same

"propagating" spike would be "seen" by an electrode on the scalp for longer time than by

a subdural electrode. Their 1987 publication, however, does not include any example of

scalp-subdural matched spikes that could further substantiate this idea in terms of align-

ment between the two waveforms.



2.1.2 Temporal methods for the detection of interictal events


A number of automatic or semiautomatic systems have been devised that measure

one or several of the features described above in the EEG signals, attempting to identify the

occurrence of a "spike" on a single channel by the increase of the measured parameters over

certain thresholds. Very complete reviews of these methods have been presented by Ktonas

[Kt87], Gotman[Got87] and more recently by Panych and Wada [Pa90].

Perhaps one of the earliest attempts made to develop an automatic system for spike

detection on a single trace of EEG signals was that of Walter et al. [Wal73]. They used an-

alog circuitry to obtain the second time derivative signal from two EEG channels and com-

pare them to thresholds. The two signals that could be processed at the same time were

treated independently. The results of the electronic processing were sent back to the poly-

graph for display and to obtain a permanent record. Although this system was aimed at the

"semiautomatic quantification of sharpness of EEG phenomena" in general, the authors il-

lustrate in their report how the measure of sharpness could be used to detect the occurrence

of spike discharges in an epileptic patient. Even in this early system the high sensitivity of








a method based on measurements of sharpness to muscle potentials (EMG) is pointed out

as a possible source of problems.

Smith [Sm74] reported on the implementation of a system that detected spikes on

the bases of the changes of the first derivative of the EEG through time. This system would

require the fulfillment of three sequential conditions on the magnitude and timing of the

first time derivative of the EEG, before a detection was indicated:



a) The first derivative of the wave must remain above a certain magnitude threshold

Ml for at least TO milliseconds, then

b) the first derivative of the wave must change sign and reach the same magnitude

level Ml within Ts milliseconds, and, finally,

c) the first derivative of the signal must keep the same sign and remain above the

Ml level for, at least TO milliseconds.



Step b) in this sequence is closely related to parameter S5 as defined by Ktonas and

Smith[Sm74] and represents an alternative implementation of sharpness measurement that

prevents the undesired amplification of high frequency noise inherent to the standard eval-

uation sharpness through the second time derivative of the signal. Such magnification of

high frequency components would make the system very susceptible to indicate the pres-

ence of a spike where there is only a high frequency noise wave (a "false positive").

Even with this modified method for the quantification of sharpness, the author ac-

knowledged that false detections could result from the presence of EMG waves, which may

fulfill all three requirements in the system. In that regard, Smith pointed out the necessity

of taking into account other elements of reinforcing or opposing evidence. For example, in-








terchannel relationships such as phase reversals in adjacent bipolar derivations could verify

a detection, while the detection of artifacts like EMG activity, by an independent sub-

system, could deny the detection.

Some years later, Vera and Blume [Ve78] reported on the implementation of an on-

line spike detection system based on virtually the same algorithm as the one by Smith

[Sm74], but using an adjustable value for the slope thresholds (MI).

Several other methods have been proposed that attempt to improve the detections

by involving the interplay of several spike characteristics and adjusting the thresholds ac-

cording to the local background EEG activity, as reported by Gotman[Got87].

Carrie ([Ca72a],[Ca72b]) measured the amplitude or the sharpness (second time de-

rivative) of each wave and compared them to the average of duration or sharpness of the

128 preceding waves. Harner and Ostergen [Har76] compared the amplitude of each wave

to the average amplitude of waves of similar duration found in the background. Gevins et

al. [Ge76] used curvature at the apex of the waves as discriminant criterion, and obtained a

threshold level from the average curvature of the waves in the first 4 seconds of the record-

ing. Goldberg et al. [Go73] used a two-pass process. The first pass determined the duration

and amplitude of background waves. In the second pass that information was used to auto-

matically adjust the threshold for the amplitude of the high-pass filtered (7 Hz) EEG. Frost

[Fro79] proposed a multi-stage method where the second time derivative was compared

with background levels, several thresholds for wave duration were then used and, finally,

thresholds were used for the absolute amplitude of the spikes.

The system implemented by Gotman and Gloor [Got76] based its detections on sev-

eral morphological measures from the graphically pre-processed EEG signals. The graphic

pre-processing involved the transformation of the signal into a series of "linear segments"








obtained by connecting successive extrema with straight lines. This pre-processing con-

cluded by grouping the segments in "half waves" that neglected small riding waves. The

actual detection process was carried out in several stages. First, the amplitude of each half-

wave was measured relative to the average amplitude of the half-waves in the preceding 5

seconds. The duration of each half wave was also measured and, if two adjacent half-waves

fell within a set of thresholds for relative amplitude and duration, then the sharpness of the

apex for these two half-waves, relative to background, was also measured, and a criterion

involving relative amplitude and sharpness was used to determine if the feature under anal-

ysis was to produce a detection.

In addition to these so-called "anthropomimetic" methods, in which the evaluation

of direct morphological parameters presumably performed by the EEGer is emulated, there

are at least two other types of approaches for interictal spike detection that have been at-

tempted: the Template Matching and Parametric methods [Got87].

In the template matching methods a prototype time series or "template" is continu-

ously compared with a "sliding window" of the incoming EEG. The likeness of the tem-

plate and the sliding window is normally quantified through the cross-correlation function

of the two time series. If this index exceeds a certain pre-defined threshold, a detection is

indicated. Matched filtering is a special type of template matching where the template is

computed based on background statistics and a priori knowledge of the spectral character-

istics of the transients [Pa90]. The main problem with this approach was pointed out by

Barlow and Dubinsky [Ba76]: "Template matching of a given waveform detects rather well

a specific waveform (e.g. a spike and slow-wave complex), but by the same token, depar-

tures from the template that may occur in the same EEG are not detected with as high a level

of correlation coefficient between EEG and template, and hence may not be identified".

This means that the selection of waveform and span (duration) of the template are extreme-








ly critical and hard to optimize. Additionally, the selection of an appropriate threshold for

the measured correlation between EEG and template is also critical to the performance and

difficult to define. Attempts to circumvent these problems have been made by using an av-

erage of 20 spikes to design the matched filter [Sa71], letting the operator of the system se-

lect a "representative" recorded spike at the beginning of each processing session [Po79],

and using matched filtering in combination with parametric methods (please see below)

[Pf78].

Parametric methods for the detection of interictal spikes do not monitor indices de-

rived directly from the morphology of the EEG signal in question. Instead they pay atten-

tion to the evolution of parameters derived from the EEG as a time series. Perhaps the best

known of these methods is the "inverse filtering" procedure proposed by Lopes da Silva et

al. [Lo76] for the detection of "non-stationarities" in the EEG. This approach is based on

the idea proposed by Zetterberg [Ze69] that the behavior of background EEG can be de-

scribed as "the output of an autoregressive filter model having a stationary input noise with

normal distribution" ([Lo76]). If this assumption were true, the values of the coefficients in

such an autoregressive filter, embodying the mechanisms of the generation of the EEG,

could be estimated using system identification procedures. Lopes da Silva et al. used a least

squares estimating procedure to obtain 15 coefficients for the autoregressive filter. From

these coefficients the "inverse filter" was calculated and fed with the EEG signal. The out-

put of the inverse filter was expected to display a normal distribution for background EEG

activity with eventual departures from normality caused by events not pertaining to the

background EEG. Since these events could be seen as disruptions of the normality assump-

tion adopted for the background EEG, the authors labeled them "non-stationarities" and

were indicated in the system using a chi-square test on the smoothed output of the inverse

filter. The second big assumption intrinsic to this method of spike detection was that most








of the non-linearities in the EEG could be associated with the occurrence of interictal

events. This latter assumption is perhaps the one that has been more heavily contended by

the critics of the method, since the application of the system has ordinarily resulted in a sig-

nificant number of false detections. To reduce these false positives some forms of post-pro-

cessing have been tried. Birkemeier et al. [Bi78] subjected the output of the inverse filter to

double differentiation and noticed improvement in the overall performance. Pfurtscheller

and Fischer [Pf78] improved the selectivity of the inverse filter by combining it with a

matched filter using multiple templates.

Overall, it seems as if the most peculiar characteristic of the temporal pattern exhib-

ited by interictal events is the "explosive" nature of these events. This could be the reason

why the first and second time derivatives of the EEG derivations (slopes and sharpness) are

involved, in one way or the other, in the majority of spike detection methods. On the other

hand, the need for additional evidence that could be considered to promote or veto the iden-

tification of a given transient as epileptiform seems to be another constant in the develop-

ment of recent detection approaches.




2.2 The Spatial Dimension of Focal Interictal Activity


The quantitative spatial characteristics of interictal events have been far less re-

searched than their temporal aspects. This is especially true when stated in reference to hu-

man subjects. It is somewhat surprising that this should be so, since the understanding of

the EEG signals as being the reflection of field events can be found even in the early reports

of the EEG literature. Of particular interest is the paper by Shaw and Roth [Sha55] in which

the authors propose quantitative models for the surface potential fields caused by radial and

tangential dipoles immersed at variable depths in a conductive sphere and suggest their use








for the analysis of EEG phenomena. Interestingly, even in this early report the suggestion

is made that the depth of the dipole generator could be determined from the quantification

of the surface potential, foreshadowing the development of the "inverse solution" methods

of generator localization. The authors, however, are also quick to point out that, even with

complete knowledge of the surface potential distribution and the medium, the "inverse so-

lution" is not unique.

It may be that the more stringent instrumental requirements involved in the spatial

study of EEG activity delayed the appearance of this kind of analysis. Such extended in-

strumentation requirements include the need for a relatively large number of electrodes and

amplifiers, the accurate determination of geometric relationships between electrode sites

and special display devices.

In fact, most of the research carried out to determine the spatial location of EEG

generators has been developed in the field of evoked response potentials (ERPs). In this

area, the signals of interest are time-locked to stimuli that occur at times determined by the

experimenter. This feature allows the use of averaging to increase the signal to noise ratio

to the levels required for several of the localization methods.

The transient and asynchronous (i.e. not time-locked to a predictable signal) nature

of interictal events makes the use of such averaging techniques difficult and calls for forms

of spatial analysis that can be applied to single events.



2.2.1 The spatial nature of a "focus"


The concept of a "focus" has an immediately implicit spatial connotation, indicat-

ing the center of activity of a phenomenon or the place of origin of a disturbance. Thus, the

idea of "focus" is necessarily one of a spatial nature, having a location and a spatial extent

attached to it.








Furthermore, it has been postulated ([G175], [G185]) that the generation of EEG

phenomena recorded as waves of short duration, such as spikes and sharp waves, must in-

volve high levels of synchrony in the activity of a local population of cells, i.e. the interictal

"focus", since otherwise the effects would not be noticed at the scalp surface.

Particularly when the problem being approached is the one of finding the sections

of brain tissue responsible for abnormal electrical activity shown in the EEG traces, the

consideration of the spatial aspect associated with the problem is essential. Gloor [G175]

summarizes the importance of this spatial view of focal EEG phenomena: "The localization

of scalp-derived potentials must be viewed as a problem of analyzing the distribution and

configuration of electrical fields on the scalp from which information on the localization of

the neuronal generator involved in the genesis of these potentials can be inferred. The in-

terpreter must make the mental effort to translate the set of potential-versus-time plots pro-

vided by a number of EEG channels into meaningful potential maps. He must therefore be

able to visualize the three-dimensional configuration of potential fields of significant elec-

trographic events on the scalp if he is to avoid drawing misleading localizing conclusions".



2.2.2 Local changes in the potential field of the cortex due to focal activity


As explained in Chapter I, the changes in the surface potential field sensed with the

ECoG electrodes are really the result of the combined effects of current sources established

by groups of neurons acting in synchrony. If we considered a single current dipole, the

cause (current generation) and the effect (distribution of potential values) would be related

by equation (1-1), [or equation (1-2)] in the more practical case of measures taken from the

external surface of the conductive medium. However, the group of synchronously active

neurons may be considered to be distributed in a dipole layer. This dipole layer has a spatial

extent of its own, which, in fact, represents the "focus", at least with regard to its electrical








manifestation. The study of the effects caused by a dipole layer should consider it as formed

by a number of elementary dipoles, each one contributing according to equation (1-2), once

its particular location with respect to the surface point under analysis is taken into account.

This superposition of contributions weighted by the relative position of the elementary di-

poles is termed "Spatial Convolution" ([Ka81], [Fr80]). The point to be noted here is that

the spatial extent of the observed surface potential configuration is not the extent of the di-

pole layer causing it, but is closely related to it.

If one considers a focal radial (vertical) generator, be it a dipole or a dipole layer of

reduced spatial extent (in comparison with the extent of the surface area observed), one

would find that its effect on the surface potential is the introduction of a concavity or a con-

vexity. This effect has already been illustrated in the results of the simulations displayed in

Figure 1.5. However, a better insight into the nature of these deflections of the potential

field can be gained through the analysis of Figure 2.2, adapted from [G185]. In this figure a

qualitative picture of the potential field (O(x,y,z)) generated around a current dipole is indi-

cated by means of isopotential lines. In fact, the figure represents a section of the (three di-

mensional) field generated around the dipole, according to a plane that contains the axis of

the dipole. Thus, the isopotential lines drawn here are really the intersections of isopotential

surfaces with the plane of analysis. The picture is complemented by the inclusion of "cur-

rent lines" [Nu81] representing the current density vector field (J(x,y,z)) associated with the

dipole. The relative closeness of neighboring current lines is used to indicate the magnitude

of J, and their direction, at any given point is the direction of J at that point. This situation

is depicted under the assumption that the dipole is immersed in a homogeneous medium of

conductivity a, extending infinitely in all directions. Under this conditions, the current den-

sity field is related to the Electric field (vector) through the point form of Ohm's Law








J = GE (2-3)

which implies that, for these conditions, the "current lines" have the same configuration as

the "lines of electric field" [Nu81] or "lines of force" [Ha77], representative of the electric

field, E.


The electric field is, in turn, related to the (scalar) potential field 4, as:


E = -VV ) (2-4)

From these two equations we have:


J = -V) (2-5)


This equation is important in understanding Figure 2.2 and in forecasting the sur-

face potential configurations that can be expected from local current generators. In essence,

it indicates that current will flow from a point of higher potential to a point of lower poten-

tial (opposite direction of the gradient).This is the reason why the lines representing the cur-

rent density vector are perpendicular to the isopotential lines (and, in fact, perpendicular to

the isopotential surfaces). Equation (2-5) also implies that, for a given conductivity, more

current will flow between two points if the potential difference between them (potential gra-

dient) is larger. Conversely, the patterns of current density considered in a given measure-

ment surface determine the potential field configurations in that same surface.

In particular, one could consider the potential distribution on a measuring surface

like the one labeled S in Figure 2.2 (a rectangular surface, perpendicular to the plane of

the drawing). By means of the current lines alone, one could tell that steeper gradients will

be found at the center of S where the current lines intersecting S are more concentrated.

Also, since current flows from the outside of Sl to its center, we can expect a local "nega-








tivity" (concavity) at the center of the potential field in Si. These observations are corrob-

orated by the potential profile for S drawn at the top of Figure 2.2 on the bases of the

intersections of the isopotential lines with that measuring surface. The potential profile for

S2, a measuring plane located farther away from the dipole, is also shown. Comparison of

both profiles reinforces the general notion that the potential configurations caused by local

generators will be sharper when the generator is closer to the measuring surface.

In the case of electrocorticographic measurements, however, the current lines can-

not flow from one side to the other of the measuring plane (array plane). In fact, the mea-

surements are being obtained from the interface between the brain, with a nonzero

conductivity a and the air, which has a comparatively negligible conductivity. This situa-

tion has been approached analytically, assuming that the conductive medium extends infi-

nitely below the measuring plane, using the "method of images" [Nu81]. The result is that

the actual potential values under the revised conditions will be twice as large as the ones for

a single homogeneous medium extending to both sides of the measuring plane. The as-

sumption regarding the extent of the conductive medium is still a reasonable approximation

if the generators are located close to the interface. In the case of ECoG that means that the

generators would be close to the periphery of the head. In the case of deeper generators, the

magnification of surface potentials, with respect to the infinite homogeneous medium may

be larger. For example, if the generator was placed in the center of a conductive sphere sur-

rounded by air, this magnification factor would be three [Nu81].

The relationship between a local generator and the characteristic potential land-

scape that it creates at the surface was studied by Hjorth [Hj76] and compared to the chang-

es produced in an elastic membrane by a force applied to it, in a perpendicular direction.

By means of his analogy, he described the situation as follows: "The force, representing a

source component, creates the changes in the level and the slope of the membrane, and also,








as secondary effects, changes that are not directly subjected to the primary force". This de-

scription, which matches the observations noted above for Figure 2.2, served as the basis

for his development of the "Source derivations" for the study of local effects in the surface

potential field (Section 2.2.3.1).

If the dipole moment caused during an ictal event is assumed to appear distributed

over a certain area, instead of concentrated in a single (large) dipole, as in the more plausi-

ble model of a dipole layer, the resulting field configuration is likely to be less sharp than

that for a lumped representation.An example for this statement was presented in Figure 1.5.

It is important to note, however, that in either case the deflection of the field (concavity or

convexity) is still present.

One aspect in which a diagram like the one shown in Figure 2.2 can be more useful

than the elastic membrane analogy is in that it is easily adapted to the study of tangential or

horizontal dipoles. One needs only consider a measuring surface running parallel to the axis

of the dipole (or perpendicular to the plane of the dipole layer). Such a surface is labeled

S3 in Figure 2.2, and one can use the isopotential lines to determine the potential profile

corresponding to a "tangential dipole". There are some important aspects to notice in doing

this. First, that there is indeed a characteristic field pattern associated with a tangential di-

pole. In fact, this is the typical observation in EEG records from patients with Benign Ro-

landic Epilepsy of Childhood (BREC) [Gr84]. Second, even these patterns include "sharp"

points (maxima and minima) in the field, although the "sharpness" is comparatively less in

any of these points than in the single "sharp" point produced by the radial (vertical) config-

uration, for the same depth. And, third, given the reduced sharpness of the field it is harder

for a human observer, looking at only the independent EEG traces, to perform a mental re-

construction of this peculiar field configuration than it is for the case of the radial dipole

(layer). This last remark offers an explanation to the fact that traditional EEG recordings








seem to be less useful in detecting tangential dipoles than the magnetoencephalogram

(MEG), which in fact performs better for tangential dipoles than for radial dipoles ([An92],

[Ka81]). This also opens the question of the extent to which the sensitivity of EEG mea-

surements for tangential dipoles can be enhanced through methods of analysis that monitor

the potential landscape as a whole and not as isolated channels.

Much less quantitative information is available regarding the spatial dimension of

ictal events than regarding their temporal dimension. One of the main questions to be an-

swered in this respect is the "minimum extent" of the potential field deflections observed

in association with interictal events (the equivalent to the minimum duration of a "spike" in

time). This information is relevant in determining the minimum required values for inter-

electrode distance and total area sampled.

It may be expected that the deflections of the surface potential fields due to sponta-

neous foci in humans must be broader than those observed from acute induced foci. In the

induced foci the application of the epileptogenic agent is restricted to a very limited cortical

zone and the effects (interictal events) are recorded only a few minutes later. For this reason,

it may be speculated that the neurons directly affected by the agent (induced focus) are con-

centrated in a very small volume of brain tissue. Under conditions like these, Petsche et alt.

[Pe84] recorded interictal spikes at the cortical surface that spanned over more than 36 mm2

(a 6 mm x 6 mm square). Spontaneous foci in humans, however can reasonably be expected

to have considerable variation in size, depending on, for example, the stage of advance of

the pathology.



2.2.3 Spatial localization of focal activity.



Two major families of methods have been proposed for the localization of "focal"

generators. Both techniques rely on the relationships between the current generators and the

surface potential configurations established through the conductive medium a.






51












S. .. -200) V



































+300jV
-400pV..



















00 V
-surfaces S, S2 and S3.S2



+400pV



+300gV


+2OOV


Figure 2.2 Current lines around a dipole immersed in a volume conductor
and potential profiles caused by them in three arbitrary measuring
surfaces Sl, S2 and S3.








The first method of spatial analysis for the EEG or ECoG considered here is the

"source derivation" technique, developed by Hjorth ([Hj75],[Hj76],[Hj79]). This approach

is specifically aimed at the detection of sharp features introduced in the surface potential

landscape by a local generator. As such, it is relevant to the method for localization of in-

terictal foci developed here and will be explained in some detail.



2.2.3.1 Hiorth's "Source Derivation"


In order to explain the foundation and meaning of the source derivation, a simplified

scenario may be considered for the case in which a radial (vertical) generator is active. One

can assume, initially, that the radial generator is close enough to the surface, such that a

large percentage of the total current driven by the generator flows in the superficial layers

of the conductive medium. Under that assumption, one can proceed to analyze the effects

of the superficial current on the two-dimensional field monitored by an array of surface

electrodes. In this partial view of the situation, the generator appears as an "isolated" source

or sink of current, defining surface current lines that diverge from it or converge to it, re-

spectively. According to equation (2-5), these current patterns will produce a deflection of

the surface potential field. For example, if the most superficial pole of the generator (dipole)

was a current sink, the convergent pattern of currents will define a minimum in the surface

potential field, at the surface location that is closest to that sink. This minimum can, in turn,

be associated with a spatially "sharp" point in the field.

The main premise in this approach to localization is that a superficial generator can

be characterized by the surface current patterns and the resulting sharpness in the surface

potential field.

The surface current configuration can be thought of as a two-dimensional vector

space, J(x,y). From this point of view the degree of "convergence" or "divergence" of the








current pattern at each point of the field can be quantified by the divergence operator, ap-

plied to that current density field. The result is a scalar filed, which will be designated here

as L(x,y), and is the foundation for the "source derivation":


L = VJ (2-6)

Furthermore, the value of this characteristic of the field can be obtained directly

from the knowledge of the surface potential filed 4(x,y), by substituting (9) in (10):


L = V (-aV 0) = -oV (V 0) (2-7)

The application of the gradient operator to a scalar field, followed by the application

of the divergence operator to the resulting vector field has been synthesized as a single op-

erator, the "Laplacian" of the original scalar field. This composite operator is indicated with

the "nabla squared" notation (V2 ). Thus, we have that the new field, L(x,y) can be ob-

tained directly from the potential field ((x,y) as:


L = V24 (2-8)

It is important to keep in mind the immediate association between the existence of

a point of convergence for the surface currents and the minimum in the surface potential

filed. This relationship becomes critical when considering a numerical approximation of

the Laplacian operator for a potential field that is only sampled at discrete points.

For a continuous scalar field referred to a rectangular coordinate system, such as

O(x,y), the Laplacian is calculated as the sum of both second partial derivatives in orthog-

onal directions:


V2 (x, y) = 2- + 2 (2-9)
X2 Ty








In the case of a field that has only been sampled at discrete locations, such as the

values obtained from the ECoG array, the above expression has to be approximated. For an

initial assumption, the field may be represented by its value at locations defined by a grid

were the spacing in both orthogonal directions is the same. For further simplification of the

explanation, let us assume that the spacing between sample points (electrode locations) is

unity. Such a situation is depicted in Figure 2.3, where a local current source has been con-

sidered, only because the resulting convexity of the potential field is easier to visualize.

In order to approximate equation (2-9) for this discretely sampled field, the contin-

uous second order derivatives will be substituted with second order differences. When the

unitary grid aperture is considered, we get:




V2 2 -= (d2 +d;) (2-10)
dx2 dy


In particular, if we were to calculate the approximated Laplacian for the field loca-

tion labeled as "B" in Figure 2.3, the second (central) differences in x and y would be com-

puted as:


d2 B--C A-B) (2-11)


d2 = (B OE) (D- B) (2-12)

and the approximate Laplacian is


V2IB (B ) (A B) ) + ( (B ) (D B)) (2-13)


which can be rearranged as:







V2IB (B A) + (B O + (B- D) + B E) (2-14)

This kind of approximation could be implemented for any of the field positions lo-

cated inside the array, but not for field locations on the boundaries of the area being moni-

tored. The form of equation (2-14), however, suggests a generalization of this
approximation that makes use of information from all the field samples and, therefore, is

applicable to any of the field positions.

Hjorth [Hj79] proposed that the four terms in equation (2-14), represent the poten-

tial gradient directed to B, from each of the discrete field locations surrounding it. Then a

new approximation for the Laplacian at B can be found as the average potential gradient
from all the other field locations to B.

Thus, the computation of this transformation (Hjorth's "Source derivation") for
electrode i, from a total of m electrodes, is accomplished as:


v2 1 rij (2-15)


where rij is the distance between electrodes i and j. This can also be written as:

V2 1i y (I ) = Ni (2-16)
jei ii j i.i. j i ri

Notice that if the potential field is a 2-D impulse under electrode i (1 under electrode

i and 0 everywhere else), this formula yields a value Ni as the approximation to the Lapla-

cian. To make this "(spatial) impulse response" the same (1.0) for all electrodes, the right

side of equation (2-16) must be normalized, dividing by Ni in each case:


v2 0 -" b = j (2-17)
NtV"y ~








The m equations of the form (2-17) that have to be solved to complete the field

transformation are readily expressed as a single matrix equation:



KL I [-b- m x (2-18)
Lnmx 1 m-bi]Jxm mxl

where the by are as defined implicitly in equation (2-17), except for the bii which are all -

1. The Oj in the column vector of the right side of (2-18) are the measured potential values,

"row-scanned" in a specific order and the Lj are the transformed field values, in the same

arrangement as the 4j.

Hjorth implemented this scheme for scalp EEG, using analog amplifiers and adders

to determine each of the elements of the output vector in the above equation from the orig-

inal referential derivation signals [Hj75][Hj76]. The resulting analog signals were plotted

using a standard polygraph. In these reports, he showed how the tracings for the source der-

ivation were better fitted to differentiate local activity than both the referential and bipolar

derivations. Although not constrained to epileptic foci, his 1976 report already included

two cases of source derivation localization in epileptic subjects. In that stage, the ultimate

localization of the focal generators still required the observer to associate a particular

source derivation trace in the polygraph with the physical location of the corresponding

electrode.


As the availability and performance of computer display systems increased, the re-

sults of source derivation analysis have been presented as topographic maps, which will di-

rectly indicate the location of the focal generator as a maximum in the map [Ro89],[In90].

This technique, however, does not provide any explicit results as to the depth of the focal

generator in the brain.





57









B



C












Plane of the
.. ..... .: ... .a r r a y


















Figure 2.3 Hypothetical vertical current generator and the corresponding
deflection of the surface potential field.








On the other hand, it is worth emphasizing two of the most important virtues of this

method of analysis. The source derivation is a reference invariant operator, since all the

terms in equation (2-15) are differences of potential values referred to any given reference.

If the reference potential used were to be increased by a fixed amount, that change would

not alter any of the terms in equation (2-15) and, therefore, would not change the source

derivation value for the electrodes. Additionally, the method of analysis is robust in that it

does not bear a heavy dependency on physical parameters that may vary from subject to

subject or even from instant to instant. The only parameter that is intrinsically involved in

the foundation of the method is a, the conductivity of the medium. Even the restrictions im-

posed by the method on this parameter are few: it is required that a be a fixed real number

throughout the medium. Furthermore, the conductivity value need not be accurately esti-

mated, since it acts only as a (uniform) scaling factor for the resulting L(x,y) field.


2.2.3.2 Inverse solution methods


As mentioned above, the idea of finding the three dimensional location of a dipole

responsible for the surface potential field observed at a given time, has been pondered for

many years now. Shaw and Roth [Sha55] obtained quantitative descriptions for a number

of dipole locations and orientations within a conductive sphere (intended to model the brain

of the subject), based on the equations of Wilson and Bayley [Wi50]. Their report is one of

the earliest applications of the "forward problem", namely the determination of the surface

potential field given the position and strength of a dipole or dipole distribution, to EEG. On

the other hand, the determination of the location and strength of one or several dipoles that

could result in a potential landscape like the one actually measured is termed the "inverse

problem". Essentially, the attempts at the solution of the "inverse problem" involve an iter-

ative procedure to minimize the differences between the potential landscape generated by








a proposed set of dipoles and the one actually observed, at a given time. In each iteration,

the method finds the potential field that the estimate of the dipole field configuration for that

iteration would produce, (solving the "forward problem"), a certain measure of error (e.g.

the mean of the squares of the point-to-point differences) between that resulting field and

the measured field is determined and an adjustment to the locations and strengths of the di-

pole configuration estimate is performed, based on the particular heuristics of the imple-

mentation in question. The procedure continues until a minimum error is reached, at which

time the strengths and locations of the current estimate are considered for the final solution.

Methods based on the above general description are collectively referred to as "Dipole Lo-

calization Methods" (DLM).

Evidently, the goal of this kind of generator localization methods are far more am-

bitious than those of the "source derivation" methods. On the other hand, the eventual ful-

fillment of those goals is conditioned to the accuracy and availability of a number of

information items that must be known a priori. Nunez [Nu90], lists some of the aspects on

which the accuracy of these methods rests:



"1) The accuracy of the head model

2) The number of (scalp) recording sites. There must be at least seven electrodes (in-

cluding the reference), but typically many more are used ([Fe87]).

3) The signal to noise ratio of the recording.

4)The location of the dipole. For example, superficial dipoles are probably more

easily located than deep dipoles.

5) The orientation of the dipole. For example, dipoles located near brain ventricles

or skull holes may cause quite different errors in imperfect head models, depending on local

current source directions.








6) The computer algorithm, including built-in constraints.

7) The initial guess as to the location and orientation of the dipole. (This may or may

not be important to the final solution obtained.)"



Additionally, there is the intrinsic limitation of the method regarding the possible

existence of a number of dipole configurations resulting in the same instantaneous potential

field and the complications arising from iterative minimization, such as the presence of lo-

cal minima in the cost function.

Advocates of these methods, such as Fender[Fe87], propose that "if reasonable as-

sumptions can used to restrict the possible sources in some way, then the position changes

and it is possible to determine the sources uniquely from surface potential methods". Some

of those assumptions may include the number of dipoles to be fitted to the surface poten-

tials. This, however is one more decision that would have to be made a priori, on a mostly

heuristic basis.

The vast majority of applications for Dipole Localization Methods can be found in

the area of ERP research. There have been, however, some attempts to use them for the off-

line localization of epileptic foci.

Schneider [Sch72] applied the basic DLM method to scalp recordings from a 13-

year-old epileptic patient, attempting the minimization of the difference between the mea-

sured value of the bipolar derivations and the values found using the equations of Wilson

and Bayley [Wi50] (a single dipole in a single conductive sphere). He reported a high de-

gree of consistency of the localization and direction of the dipole corresponding to several

spikes. However, he could not contrast the location found against any anatomical or phys-

iological evidence for the dipole, located about halfway between the surface and the center

of the sphere. Actually, in a series of 15 patients suffering "petit mal" epilepsy, he frequent-








ly found the dipole located "very near the center of the brain... But it is physically impos-

sible to admit the existence of a located source, as the intracerebral recordings have never

detected high deep intensities". Some methodological inaccuracies might have been in-

volved in the results, since he took the potential difference between the points limiting the

first half of the spike in each bipolar derivation as the value to be considered for that deri-

vation, irrespective of the times when these extreme points occurred, therefore portraying

a potential field that never existed.

More recently, Wong and Weinberg [Wo88] applied the DLM method to referential

signal obtained using the International 10/20 system of electrodes from 13 children diag-

nosed with BREC. The field configuration determined by the spikes in many BREC patients

is stereotypic and resembles the one that could be produced by a tangential dipole, which

may have instigated the use of the DLM for localization. The processing was carried out

off-line, including averaging and "manual" selection and alignment of 20 artifact-free

spikes from each subject. They found that, although the projections of the localizations onto

the scalp were concentrated in a tight cluster, the results "could not be interpreted easily

because of the ambiguity of radius (or depth) estimation". A mention is made of the known

inaccuracies in their DLM model, which included "poor geometric matching of the head

(children vs. adult head size and shape), mismatched skull/scalp attenuation, and most im-

portantly the assumption of only one independent source".



2.3 A Spatio-Temporal View of Focal Interictal Events


The previous sections have summarized important facts about the temporal and spa-

tial aspects of interictal events. Mention has also been made of detection or localization ap-

proaches that capitalize on those peculiar characteristics of interictal events in the domains








of time and space, independently. In this section, a comprehensive spatio-temporal view of

focal interictal events is advanced that will serve as the basis for the focus localization al-

gorithm proposed in Chapter III, the Spatio-Temporal Laplacian.



2.3.1 Spatio-Temporal nature of focal interictal events


At a macroscopic scale, like the one involved in scalp EEG recordings, it may seem

as if the detection of an interictal spike at the different electrode positions were an all-or

nothing phenomenon, i.e. either the spike was sensed by the electrode or it was not. A closer

look at these events, obtained for example with the "micro-EEG" by Petsche et al. [Pe84],

using a grid of electrodes separated by 2 mm in each direction, and with enough temporal

resolution (sampling rate), can reveal that the surface phenomenon follows a peculiar evo-

lution in time and in space. This is further reassured by intracellular recordings, such as the

ones obtained by Dichter and Spencer [Di69a] from the "central focus" and its surround-

ings in an (acute) animal model of focal epilepsy. They, in fact, observed that the Paroxys-

mal Depolarizing Shift (PDS) appeared at different times and with different strengths in

cells located at different relative positions with respect to the "central focus". These con-

siderations suggest the existence of a fundamental relationship between the temporal and

spatial manifestations of a focal interictal event.

Those interlocked spatial and temporal aspects of interictal events can also be found

in the surface potential sensed with the array shown in Figures 1.1 and 1.2. In fact the re-

construction of the surface potential fields from the measured voltage references, detailed

in Chapter III, and their display as a rapid succession of contour level maps suggested that

both temporal and spatial patterns exist in the evolution of the field.

As an example of how residual information associated with the occurrence of a








spike may be left behind by processing approaches that are exclusively temporal or exclu-

sively spatial, we can consider Figures 2.4 and 2.5. Both of these figures represent a series

of 5 contour level maps, covering an interval around the "peak" of the spike shown in Fig-

ure 1.6.

The values used to plot Figure 2.4 were obtained by reconstructing the field for each

of the instants shown and approximating the second time derivative of the reconstructed

electrode potentials, n, as (more will be said regarding this approximation in Chapter III):



2 5
+ 2 + # (2-19)
t j=


The results obtained are then re-organized in the form of a matrix, according to the

electrode positions in the array. One should note that this transformation only involved po-

tential samples from the same site, i.e. is a merely temporal transformation. In examining

the sequence of resulting contour levels, one finds that even these results show a particular

spatial structure, especially around the time of occurrence of the "peak" of the spike in elec-

trode 20. It is important to note that the "temporal processing" carried out on the signals

may have facilitated the recognition of the "apex" of the spike, but it is not regarding this

extra source of information.

Conversely, the contour level maps in Figure 2.5 were obtained by reconstructing

the potential field for each sampling time, applying the "source derivation" [equation (2-

18)] to each instantaneous potential field and displaying the results in two dimensions. In

this case, one verifies that the suspected projection of the focus on the cortical surface has

been clearly indicated by the concentric contours in sample 218, but, at the same time, one

finds that such pattern, indicative of a field concavity, displays a peculiar evolution in time.








As a matter of fact, this evolution happens to be closely related to the one displayed by the

spikes in the traces from bipolar derivations connected to electrode 20 (Figure 1.6). Once

more, this suggests that spatial processing performed individually on instantaneous poten-

tial field reconstructions may not be exhausting the available clues for the detection and lo-

calization of an active interictal focus.



2.3.2 Spatio Temporal electric models of an epileptic focus during an interictal event.


It is usually considered that a detection (identification) system "must be designed

around some form of mathematical model of the signal and non-signal activity" [Sm74]. In

previous sections, "models" for the temporal and spatial aspects of surface potentials asso-

ciated with an interictal event have been shown. For the temporal aspects, a triangular or

pseudo-triangular transient has been proposed, and the similarity of this waveform or its pa-

rameters (slope, sharpness, etc.) to those of an incoming wave may qualify the latter as a

detection. In the spatial analyses presented before, each instantaneous potential field con-

figuration is compared against the model of "sharp" potential landscape that a test dipole,

at a given location, would produce. Otherwise, a more general spatial model of a local gen-

erator can be described as a point of convergence or divergence for the current density in

the plane of measurement.

In this section, three progressively simpler models for the electrical activity of an

epileptic focus, during an interictal event will be proposed. It will also be emphasized that,

while a very detailed model could result in a more accurate representation of the electrical

activity at the focus, instrumental and computational limitations point to the consideration

of only the essential characteristics found in all the models for the design of an identifica-

tion method.








2.3.2.1 Minimal Assumptions



It has been indicated above that most types of spatial analysis implicitly invoke a

number of assumptions regarding the geometry and electrical properties of the brain. The

spatio-temporal models of electrical activity at the focus presented below make use of the

following assumptions:



The brain tissue will be regarded as a passive, conductive medium, except for the

interictal focus itself.

The conductivity of the medium is an (unknown) real constant throughout the me-

dium, i.e. the medium is purely resistive and homogeneous.



While most of these assumptions are fundamentally false, they may be justified in

the context of interictal event localization. Disregarding the impact of capacitive or induc-

tive components in the volume conductor used to represent the brain is common practice

[Nu81]. Although it is well known that inside the brain there will be regions with substanc-

es of varied conductivities (cerebro spinal fluid, blood, white matter, cortex), the complica-

tion of mapping these for each individual case justifies the use of a uniform, "average"

conductivity. On the other hand, this same complication makes it difficult to find a reliable

estimate for that "average" conductivity. Once the medium has been accepted as homoge-

neous, the assumption of a passive behavior for the surroundings of the focus can be viewed

from at least two perspectives. First, in a homogeneous conductor, a linear summation of

all the sources that are active at a given time can be expected. This observation is important

not only for the determination of the surface potential field caused by the current sources

modeling the focus, but also to postulate the near-cancellation of other sources that are not

acting in synchrony.









sample 210





sample 214


sample 218




sample 222


sample 226

Time

Figure 2.4 Results of the temporal processing of the potential field
displayed as a sequence of maps of the transformed field.
(The data corresponds to the event represented in Figure 1.6)


T
8 msec


IF













sample 210



8 msec



sample 214







sample 218







sample 222







sample 226


Time

Figure 2.5 Results of (static) spatial processing of the potential field
displayed as a sequence of maps of the transformed field.
(The data corresponds to the event represented in Figure 1.6)








Second, it has been shown [Mat64], [Pr67], [Ay73],[Jo81], that the cellular event

involved in an interictal event, i.e. the Paroxysmal Depolarizing Shift, has a larger magni-

tude than the "normal" postsynaptic potentials. Thus the collective effect of a large number

of cells undergoing these comparatively large transients can be expected to be predominant

over the activity in cells that are not involved in the interictal event. The obvious violation

of passivity by the cells in the surroundings of the "central focus" (i.e. the surround and ver-

tical inhibitions) will be assigned to the model itself, instead of accounting for it as a prop-

erty of the medium.



2.3.2.2 A general model for the epileptic focus during an interictal event


If one considers the region of the brain under study as a three-dimensional dipole

moment field that changes through time, a fairly general model for the epileptic focus dur-

ing an interictal event is obtained. Under the assumption of homogeneity, this model can be

represented as three orthogonal fields:



x =Px (x, y,z,t) i (2-20)
y = py (x, y, z, t) j (2-21)
pz = z (x, y, z, t) k (2-22)



This model implies that every volume differential at a given time may have associ-

ated with it a dipole moment with arbitrary strength and orientation, allowing for maximum

freedom in the representation of the activity of a focus during the event. If this model is

made discrete with a fine enough granularityy", each individual discrete dipole may actu-

ally represent a group of neurons behaving almost identically. Figure 2.6 a) shows this mod-

el for a particular dipole configuration and a qualitative estimate of the resulting surface

potential pattern.








In spite of the completeness and versatility of this model, instrumental and compu-

tational limitations make it impractical for its use in focus localization. For example, It has

been defined that, under noise-free conditions, six surface potential measurements are nec-

essary to determine the strength, orientation and position of a single dipole [Fe87]. In real,

noisy situations, this number may have to be doubled. Therefore an array like the one used

for this work, sampling the potential field in only 12 sites, could hardly be used to determine

the characteristics of a few dipoles.



2.3.2.3 A single-laver model for the epileptic focus during an interictal event


A great simplification in the model is achieved if the dipole moment field is restrict-

ed to exist in only one layer, parallel to the surface of the volume conductor, with the further

limitation that the dipoles can only be perpendicular to both surfaces:




Pz = Pz (x, y, zo, t) k (2-23)

This simplification is plausible when considering, for example, a region of the cor-

tex located in the "crown" (top portion) of a gyrus. In fact, Freeman [Fr80], used this model

to find the patterns of neuronal activity (dipole moment field) in the olfactory bulb of rab-

bits, in response to different odors. His application of this method in that experiment is par-

ticularly suitable, since the olfactory bulb is not convoluted, and anatomical evidence was

available a priori to determine the value of z0.

With the restrictions specified by this model, the problem of determining the values

of a discrete dipole moment field from a discrete surface potential field one instant a time

- can be approached through the method of "Spatial Deconvolution" ([Ka81], [Fr80]). This

method is capable of resolving Pz for as many dipoles as surface potential samples are avail-








able, at a particular instant in time. Figure 2.6 b) portrays a possible instance of this model

for the "peak" of an interictal event and a qualitative representation of the resulting sur-

face potential field.



2.3.2.4 Individual dipole models


One step further in the simplification process results in the representation of the

focus as a unique dipole. The use of this approximation on a instant-by-instant bases has

already been described in section 2.2.3.2 ("Inverse Solution Methods"), along with the

different sources of error that may deteriorate the results. The physiological plausibility

of this model has been repeatedly questioned [Sch72], [Won88]. A pictorial representa-

tion of this model and its instantaneous effect on the surface potential field is shown in

Figure 2.6 c).

Recently, an extension of the Dipole localization Methods has been proposed

([Sc85],[Sc86]), which includes the temporal dimension in its conception. This method is

now known as the Spatio-Temporal Source Modeling (STSM), and is based on the

premise that physiological "sources", or groups of neurons acting in synchrony, are not

activated at single instants of time, but follow a certain activation pattern. Thus, a se-

quence of surface potential configurations should be explained by the same set of dipoles,

which keep their location and orientation constant through the interval under analysis,

varying only in their strengths. This results in an augmented minimization problem, op-

erated on a "spatio-temporal data matrix" i.e. the surface potential samples collected at

different locations, at successive sampling instants. The output of this minimization prob-

lem will now be a set of parameters describing the constant position and orientation of

each dipole and strength vs. time activation function for each of them.








This newer approach to dipole modeling has also been applied mainly to the off-line

localization of generators responsible for evoked potentials, where pre-processing steps

such as manual signal alignment and averaging are possible. Although the improvement in

performance gained by involving the temporal dimension of the events is significant

[Ac88], this has also increased the number and impact of potential sources of error. Achim

et al. [Ac91] have studied some of these added problems appearing in STSM. They point

out that the minimization problem involved in STSM often presents local minima that yield

sub-optimal solutions, this being more critical for larger numbers of dipoles to be localized.

Those solutions may introduce significant errors in the location of the dipoles. Furthermore,

they indicate that with the existence of local minima, the susceptibility of the method to the

initial "guess" for the parameter that will be iteratively improved is also increased. They

also warn about the presence of spatio-temporally structured noise in the EEG signals that

may distort the localization process. As a result of their studies Achim et al. attempted to

set some guidelines that could be followed to minimize the impact of these sources of error.

In general, their conclusion seemed to be that the impact of the initial estimations on the

final results is hardly unavoidable, since, they say "... in many instances our procedure

stopped at suboptimal local minima that it could not escape. In those cases, the further use

of a variety of initial approximations lead to solutions that fitted the data more closely, iden-

tifying the former solutions as suboptimal". Of course, this second level of iteration, involv-

ing the direct intervention of the observer is very computationally expensive and definitely

suited for off-line implementation only.

Ebersole[Eb91] applied the STSM principle to scalp-recorded interictal events. In

his study, up to two dipoles were fitted to averaged surface fields from 8 to 32 individual

spikes, from patients with complex partial epilepsy. He reports having found "stable" (con-

sistent) dipole locations for a certain type of potential field configurations, although he ac-








knowledge that alternative solutions with better fits were discarded because "the resultant

equivalent dipole pairs are anatomically unrealistic". In fact, the chosen solutions were

probably too deep. They were anatomically interpreted only in the context of an admitted

"depth inaccuracy" on the part of the localization method. Incorporating pertinent depth

modifications, he suggested that the dipoles could represent activity in the infero-mesial

and lateral temporal cortex.



2.3.3 Effects of the interictal events in the potentials at the cortical surface.


After having analyzed the three levels of modeling for an active interictal focus, it

is important to attempt a generalization of the overall impact that any of these models of

dipole distribution (Figure 2.6) will have on the surface potential field measured by the

ECoG array.

It is proposed here that the activation of an interictal focus can be characterized by

the introduction of a larger degree of sharpness at some point in the potential field. Further-

more, it is proposed that the spatial sharpness of the field at that point evolves according to

the "sharp" time course inherent in the genesis of an interictal event.

While the temporal "sharpness" mentioned above seems to be relatively invariant

for some given recording conditions (scalp vs. cortical recordings), the degree of spatial

sharpness introduced by the active interictal focus will vary according to the actual extent

of the aggregate of neurons working in synchrony and the depth of the focus (deeper foci

will produce comparatively less sharp surface fields).

In agreement with these observations, a simplified model for an active interictal fo-

cus can be proposed for its use as a framework in the preceding chapters of this dissertation.

The model is shown in Figure 2.7.





73











0 *" "













O .. .. .... .. .."""
0 ';.O:-:, 0 .0-0"













C) .
b) .... .o::: ... -.. ..















Figure 2.6 Levels of representation for the distribution of dipole moment in
an active epileptic focus, a) Volumetric distribution of dipole
elements; b) Vertical dipole elements distributed in a single layer;
c) Individual dipole model with arbitrary orientation.










































I(t)





b) t tO t2

Figure 2.7 Simple electrical model proposed for the activation of an interictal
focus, a) Physical location of the model elements; b) Temporal
pattern of activation of the equivalent current source I(t).








The two major elements of the model are:


a) A time-varying current source (in the electrical engineering sense, i.e. a source-

sink combination), meant to represent the collective dipole moment field established by the

synchronous PDS of the cells in the focus and their balancing current sources. No assump-

tions are made regarding the actual size or shape of this source, for as long as it is "local"

(i.e. much smaller than the array). On the other hand, the current source is expected to have

an equivalent direction, in which the flow of current will be stronger, e.g. Figure 2.7 illus-

trates a vertical or radial current source. The generic time course assigned to this current

source has been proposed on the bases of:



The time course of the PDS, as measured in intracellular recordings ([Mat64],

[Di69a], [Ay73])



The observed shape of the main phase of interictal spikes, as discussed in section

2.1.1.



b) A volume conductor characterized by a constant, real, unknown conductivity a.

The need and practical justification for this assumption has already been discussed in sec-

tion 2.3.2.1.



Some further assumptions will be invoked for the particular intended application of

this model:

1) A plane boundary for the conductive volume is assumed in the area covered by

the array. The distortion in the potential field derived from this assumption was studied by

Nunez[Nu81]. He found that if the "local" effect of a (radial) current source immersed in a








spherical volume conductor were to be analyzed disregarding the curvature of the surface,

a corrective term (1 d2/2) should be used. Here d = 1 r1la, r1 being the distance from the

center of the conductive sphere to the center of the source in question and a representing

the radius of the sphere. He concluded that "If the current source depth is much less than

the radius of the sphere the use of plane geometry is quite accurate for estimating surface

potentials, provided one is interested only in potentials at surface locations close to the di-

pole". The preoperative evaluation of the subject and the size of the array make it very like-

ly that these conditions will be met in the intended application.



2) A large portion of the current sinking into the source I(t) flows beneath the plane

of the array. This assumption is indicated in Figure 2.7 by the "Minimal dispersion" label

and it determines the extent of the potential field distortion at the surface that has been

caused by I(t). The assumption holds better for sources close to the surface, since the re-

striction for current flow defined by the interface with the external medium (a = 0) forces

most of the current to be superficial.



2.3.4 Time-varying potential field representation of focal interictal events sensed at the cor-
tical surface


In terms of the potential field as sampled by an array of surface electrodes, the mod-

el translates into a field configuration similar to the one shown in Figure 4, with an overall

magnitude that varies according to I(t). Thus, it is proposed that an acceptable representa-

tion for the surface interictal phenomenon is the following two-dimensional time-varying

potential field:


O(x,y,t)





77

or, accounting for the temporal and spatial discretization necessary for computer

processing of the data:



<(xi,yj,n)


The remainder of this dissertation will deal with these representations of the inter-

ictal events.















CHAPTER III
THE SPATIO-TEMPORAL LAPLACIAN (STL) FOR THE ANALYSIS OF THE COR-
TICAL TIME-VARYING POTENTIAL FIELD


In this chapter the spatio-temporal view of the focal interictal events proposed in

chapter II is further explored and its consequences for the potentials measured through the

ECoG array (Figures 1.1 and 1.2) are examined. A novel pictorial representation of the ar-

ray ECoG data through time (The "Bubble Diagram") is proposed for the display of the 3-

dimensional ((xi,yj,n) sample space associated with the time-varying surface potential field

monitored with the array. The "Bubble Diagrams" are used to develop an algorithm for the

detection of the surface manifestation of focal interictal activity, on the basis of the spatial

and temporal sharpness that it introduces in 0(xi,yj,n). This algorithm, the "Spatio-Tempo-

ral Laplacian", is presented as a mapping of O(xi,yj,n) into STL(xi,yj,n), where the spatio-

temporal samples associated with the time and place of occurrence of focal interictal events

are expected to be selectively emphasized.



3. 1 Three-dimensional Sample Space from Array ECoG


Chapter II presented the focal interictal events as spatio-temporal phenomena and

highlighted the need to account for both their spatial and temporal characteristics in a de-

tection or localization scheme.

The problem remains of representing the measured time-varying surface potential

field in a way that is both indicative to a human observer and suitable for quantitative ma-








nipulations. As the field is defined in a plane and varies through time, the most natural rep-

resentation would be an animated surface, changing according to the subsequent values

acquired by the potential at the sampled (xi,yj) locations. This representation, however, is

not easy to implement in a static, two-dimensional medium, such as a piece of paper. The

evolution of the field through time can alternatively be represented as a series of static sur-

face portraits, printed in sequence, to be "integrated" in the mind of the observer. In this

same category, a series of "contour level" or "topographic maps" is capable of representing

the time-varying potential fields leaving the "time integration" task to the observer.

Figure 3.1 shows all the 16 bipolar derivation signals obtained from the array illus-

trated in Figure 1.1 a) (montage M 1), in a 1-second interval. The occurrence of a focal spike

(same one as in Figure 1.6), has been indicated with an arrow. Figure 3.2 shows a sequence

of surface representations of the potential fields in the interval delimited by the vertical

lines in Figure 3.1. Figure 3.3 shows a similar sequence of contour level maps. (The partic-

ular way in which the potential values for each electrode position were reconstructed from

the differential measurements is the subject of the next section.)

In comparing these three forms of representation for the same event, one finds that

Figure 3.1 makes the temporal evolution of the events evident but requires that the observer

perform the spatial integration of the data, as discussed before (Section 2.2.1). This is an

obvious shortcoming when attempting the localization of focal interictal events.

On the other hand, Figures 3.2 and 3.3 allow a much easier localization of the sur-

face point closest to the focus. However, the time evolution of the event has to be interpreted

by the observer (time has no direct correspondence with distance in the sequence).




















r\j~


Figure 3.1 One-second segment of a 16-channel ECoG file recorded using
montage Ml. The two black lines indicate an interval containing
a focal interictal event. The arrow indicates the apex of the "spike".













* *


K,


Figure 3.2 Sequence of potential field portraits representing the interval
marked in Figure 3.1 by two black vertical lines.













-L*



*


. .


Figure 3.3 Sequence of contour level maps representing the interval
marked in Figure 3.1 by two black vertical lines. Black
contours represent potentials below the instantaneous
average reference.








An alternative 2-dimensional representation is proposed for the purpose of objec-

tively evaluating the spatio-temporal characteristics of the time-varying potential field. In

this representation, termed the "Bubble Diagram", each electrode potential value is repre-

sented by a sphere, with its radius determined by the magnitude (absolute value) of the elec-

trode potential and its shading (dark or light gray) determined by the polarity (sign) of the

electrode potential. All 12 electrode potentials measured simultaneously can be represented

using this convention and the corresponding spheres arranged according to the electrode

positions in the array. In this way, the status of the surface potential field at that particular

sampling instant will be indicated. Although this "slice" representation is not as general as

the corresponding contour level map for that same sampling instant, such representation

can be drawn in perspective, from a number of angles. This, in turn, gives it the added ad-

vantage of being "stackable" in the sense that a number of subsequent "slices" can be su-

perimposed with a certain amount of horizontal and vertical shift to indicate the time

relationship between them. Then, the use of perspective drawing allows the two-dimension-

al representation of a field that depends on 3 variables: x, y and n (time).

Figure 3.4 portrays the "Bubble Diagram" representation of a hypothetical focal in-

terictal event. Here it was assumed that the potential field would evolve from a fairly flat

distribution (top of the figure), with all its values close to zero, into a "spatially sharp" dis-

tribution, in the middle "slice" of the figure (all the potential values are assumed to be of

the same polarity). Then the field would experience a return to a flat configuration (bottom

of the figure), as predicted by the time course of the I(t) activation in the model for the focus

during the interictal event (Figure 2.7).

This hypothetical "Bubble Diagram" shows how the sampling of the surface poten-

tial field at certain fixed points in space and specific instants in time results in a 3-dimen-

sional "sample space", also described by O(xi,yj,n). In addition, Figure 3.4 shows how the








Bubble Diagram makes explicit the characteristic spatial and temporal sharpness that have

been proposed as inherent features of focal interictal events. These have been emphasized

in the drawing with auxiliary lines.

Although this type of representation can only be applied to situations where the

number of spatial samples obtained at any given time (active electrode locations) is rela-

tively small, it may be a very good alternative to handle the problem of representing array

ECoG or even scalp EEG data.

A somewhat simplified "Bubble Diagram" representation of the same event shown

in Figures 3.2 and 3.3 is displayed in Figure 3.5. Here, the time representation evolves from

the top-left to the bottom-right, covering the same time interval indicated by the vertical

lines of Figure 3.1. The data represented by consecutive slices were collected at intervals

of 2 ms, as indicated.

In this representation the circles in dark gray indicate "negative" polarities, while

the circles in light gray indicate "positive" polarities. (These polarities are really defined

with respect to an average reference, as explained in the next section.)

Although not as clear-cut as the ideal case represented in Figure 3.4, both the spatial

and temporal sharpness in the potential for electrode 20 are distinguishable in this single

representation. The isolated "slice" for the temporal sample in the middle of the selected

interval (inset at the top-right) further emphasizes the spatial sharpness that is reached by

the potential of electrode 20 at the "apex of the spike". The temporal sharpness for the same

electrode potential, on the other hand, can be detected directly form the increasing-decreas-

ing "envelope" formed by the successive potential samples obtained from that electrode

site.












FOCAL EPILEPTIC ACTIVITY in a 3-D SAMPLE SPACE
0 RADIUS is proportional to absolute value of instantaneous electrode voltage


n-2




n-1




n




n+1


n+2


Figure 3.4


00


"Bubble Diagram" representation for a hypothetical
focal interictal event.


time























kLd~ L


00
0 0
~ ~ o __C





9.,
I ," ,.-:T 1 ,,s



,o


o

o





o 0 : ,
Ii U

0 ~
I 4
0
o|










ofo
**
**
0

,_ *
*rI
Sas




I
U
I


a"'
_____ .. ;.i


o o 0


0*0








One point worth mentioning here is that the three-dimensional sample space repre-

sented by the Bubble Diagrams contains all the physical evidence collected by the surface

ECoG array regarding the occurrence of a particular interictal event. This means that the

values of the sample space in an interval around a given time of analysis constitute the most

complete input for any spatio-temporal detection or localization algorithm.

In particular, the following sections will detail the definition and justification of a

transformation designed to objectively isolate sets of spatio-temporal samples displaying

coincident spatial and temporal sharpness. These instances will be considered possible oc-

currences of focal interictal events.



3.2 Average-Reference Field Reconstruction


The display of any of the field representations mentioned above (surface represen-

tation, contour level maps or each slice of the Bubble Diagram), requires the reconstruction

of the surface potential as a field from the differential measurements obtained at every sam-

pling instant through the bipolar derivations. This stage of data analysis is not without its

complications when EEG or ECoG signals are being considered.

The electric potential is in itself a differential concept: "To find the electric potential

difference between two points A and B in an electric field, we move a test charge q0 from

A to B, always keeping it in equilibrium, and we measure the work (WAB) that must be done

by the agent moving the charge. The electric potential difference (Vg VA) is defined from


WAB
VB- VA = WAB (3-1)
qo

[Ha77]. Physicists, however, have found a way to assign an electric potential value

to each point of, for example, a surface: "Usually point A is chosen to be at a large (strictly,








an infinite) distance from all charges, and the electric potential at this infinite distance is

arbitrarily taken as zero. This allows us to define the electrical potential at a point. Putting

VA = 0 in equation (3-1) and dropping the subscripts leads to


V = w (3-2)
qo

where W is the work that an external agent must do to move the test charge qo from

infinity to the point in question" [Ha80]. The reason why an infinitely distant point was cho-

sen as a reference for potential is that its electric condition will not be affected by charge

transfers taking place in any finite distance from the observation point. In practice, however,

it is not possible to use an infinitely distant reference.

Instead, researchers have tried to define an inactive "reference" for the measure-

ment of brain potentials, using one or several electrode leads to define it. Since such "ref-

erence electrode" cannot be truly fixed in its potential, the attempt is made to locate a point

of minimal electric activity. The EEG literature displays a number of alternatives that have

been tried in the search for a "quiet" or inactive reference point for the measurements. None

of these, however, has been completely satisfactory [G175], [Ka81], [Nu88], [Le87], thus

the popularity of bipolar (differential) measurements.

The differential measurements obtained through the bipolar derivations can be used

to reconstruct the surface potential field, arbitrarily assigning any of the electrodes as a "ref-

erence electrode", i.e. assuming that the potential at that electrode location remained un-

changed and making it a relative zero. In particular, if one chose electrode 1 as the

pseudoreference, the resulting reconstructed electrode potentials would be obtained by

"cascading" appropriate differential measurements, in such a way the intermediate elec-

trode potentials in the chain would be cancelled, yielding the potential difference between

the electrode in question and electrode 1:









Oil = (Ot>- =) = (j--;) + (i--h) + "'+ (a-il) (3-3)



This type of field reconstruction presents at least two immediate shortcomings.

First, it implicitly cancels electrode 1 as a sensing element, since the temporal evolution of

the potential in that electrode site is overridden by the assumption made. The second prob-

lem with the approach is that, if indeed there were activity local to the position of electrode

1, it would be mistakenly reconstructed as activity (of the opposite polarity) spread to most

of the area of the array, except for the area around electrode 1. It can be seen that for certain

types of analysis both the real field configuration and the one reconstructed with an arbi-

trary reference are equivalent. These forms of (instantaneous) analysis are thus referred to

as "reference invariant" or "reference-free" methods [Leh80],[Le87]. The source derivation

is one such "reference invariant" form of analysis, since it involves exclusively the calcula-

tion of potential differences that will be the same regardless of the level at which the refer-

ence is selected.



An alternative referencing scheme that circumvents both of the problems presented

above is the so called "average reference". In this scheme electrode potential values are first

calculated with respect to an arbitrary point, e.g. electrode 1, as indicated by equation (3-

3). In this case, however, the reconstruction procedure continues by calculating the (instan-

taneous) average of the electrode potentials with respect to the pseudoreference



1 12
avgl = (i (3-4)
i= 1








and then using this average as the reference. This is accomplished by subtracting the aver-

age value from each of the electrode potentials with respect to electrode 1 found through

equation (3-3)



i,, = Oi oavgl (3-5)

After these last steps, all the reconstructed electrode potentials, even the one for

electrode 1, may acquire a non-zero value, and display a temporal evolution. The temporal

evolution of the electrode potentials reconstructed in this way will not really be the evolu-

tion of the surface potential at the electrode sites themselves. Instead, the evolution of these

potentials with respect to the average potential value across the area covered by the array

will be obtained. Although this is also a limitation of the field reconstruction process, it was

estimated that, for the particular purposes of this study, average-reference reconstruction

may still be the most suitable approach. The reason for favoring this type of reconstruction

is that the target events, focal interictal events, are expected to result in localized increases

or decreases of potential that will make some electrode potentials reach values that are sig-

nificantly different from the ones for surrounding electrodes and, hopefully, from the ones

of most of the electrodes in the array. If this assumption holds, then the target events involve

electrode potentials that differ significantly from the average over the array at the time of

occurrence of the event.

Accordingly, all the electrode potential reconstructions presented in this work will

be those obtained through average-reference reconstruction, unless noted otherwise.

The Bubble Diagram representation uses exclusively the reconstructed potential

values at the electrode sites. On the other hand, the surface portraits (e.g. Figure 3.2) and

the contour level maps (e.g. Figure 3.3) require the generation of interpolated values to ob-








tain a relatively smooth representation. In this work, interpolation was performed using the

method by Walter et al. [Wa84]. In this method, the interpolated value is obtained as a

weighted sum of the (reconstructed) potentials of the 4 nearest measurement points:



4interp = l + W2+ W3+ W44 (3-6)

The weights are given by:

N
wi = (3-7)
(di)


where di is the distance between the location of the point to be interpolated and the ith (i =

1,2,3,4) measured point, and N, is calculated from all 4 of these distances:


N = 1 1 1 (3-8)

(da) 2 (d2 2 (d3) 2 (d4 2


3. 3 The Search for Coincident Spatial and Temporal Sharpness in the 3D Sample Space.


Once the effect of the coincident spatial and temporal sharpness on the three-dimen-

sional sample space has been noted, the definition of an analytic transformation to filter (se-

lectively emphasize) those occurrences is desirable. In the particular application at hand the

transformation capable of such spatio-temporal filtering should be suitable for numerical

implementation in a Digital Signal Processor, in a way that allows for its "on-line" appli-

cation to the intraoperative focus localization problem.

By virtue of the amount of knowledge already available regarding the temporal as-

pects of interictal phenomena and the spatial characteristics of local generators, it was








deemed convenient to develop the transformation sought as a composite of transformations

assessing each type of "sharpness" independently. These subtransformations used to eval-

uate the degree of spatial and temporal "sharpness" of each spatio-temporal sample will be

called SL ("Spatial Laplacian") and TL ("Temporal Laplacian"), respectively. Then, they

can be described in terms of their domains and ranges:



SL (xi, yj, n) = SL ( (xi, yj, n) ) (3-9)


TL (xi, yj, n) = TL (4 (xi, yj, n) ) (3-10)


where SL(xi,yj,n) and TL(xi,yj,n) are the three-dimensional ranges of 0(xj,yj,n) under the

SL and TL transformations, respectively.

In determining the value of SL for a given spatio-temporal sample, its relationship

to the rest of the samples recorded at that time (i.e. in the same "slice" of the Bubble Dia-

gram) will be considered. On the other hand, in gauging the temporal sharpness exhibited

by that spatio-temporal sample, through TL, a number of the potential values recorded in

the same electrode before and after the time of analysis will be involved. Thus, it can be

seen that, when the composite transformation including SL and TL is integrated, a larger

and more diverse set of spatio-temporal samples will be used in assessing the potentiality

of a given spatio-temporal sample as representative of a focal interictal event.

If both SL and TL are successful in evaluating the degrees of spatial and temporal

sharpness associated with each spatio-temporal sample, then a simple method to determine

the degree to which both clues of focal interictal activity support each other is the point-to-

point product of the images of the two subtransformations. The resulting measure of coin-

cident spatial and temporal sharpness in the three-dimensional sample space, the Spatio-

Temporal Laplacian (STL), is then defined as:









STL (xi, yj, n) = SL (xi, yj, n) x TL (xi, yi, n) (3-11)


Thus, for a spatio-temporal sample to have a comparatively large STL image, it

must have both a large SL image (i.e. it must display significant spatial sharpness) and a

large TL image (i.e. it must exhibit significant temporal sharpness). In cases in which only

one of the two types of sharpness is present the lack of support from the other component

will make the product comparatively smaller. This would be the case of slowly-varying lo-

cal potentials or transient changes that appear simultaneously (to the temporal sampling

resolution) in an area including several of the electrodes. This mechanism of cross-confir-

mation should result in an intrinsic reduction of false positives since a more comprehensive

set of spatio-temporal conditions needs to be met before a large value of STL can be as-

signed to a spatio-temporal sample. The definition of the STL transformation in terms of

the SL and TL subtransformations, applied to (xi,yj,n) is illustrated in Figure 3.6.

It should be noted that the STL transformation represents a new approach to the lo-

calization of the origin of interictal events. Traditionally, in the visual inspection process

carried out by the human expert, and in most automatic detection and localization schemes,

the time of occurrence of the event is determined first and only then interchannel relation-

ships are considered to confirm the event and to specify its spatial origin. The STL trans-

formation attempts to answer the questions of the place and time of occurrence of the event

simultaneously, involving spatial and temporal relationships in the calculation of the STL

image for each spatio-temporal sample.



3.4 SL: A Measure for Instantaneous Spatial Sharpness.


Mathematically, the spatial sharpness of a field at any given point can be measured

through the Laplacian of the field, at that point. In essence, this operator "provides a mea-








sure of the difference between the (spatial) average of the field in the immediate neighbor-

hood of the point and the precise value of the field at that point" [Da87]. Thus, it can be

expected that the deflection of the potential introduced during a focal interictal event, caus-

ing a reduced sector of the field to acquire potential values significantly different from those

of the surroundings, should be captured in a large value of the Laplacian for those points.

Section 2.2.3.1 has already dealt with the definition of the Laplacian for a field ex-

pressed in the rectangular coordinates (x,y) [equation (2-9)] and how this operator can be

implemented numerically through the "source derivation" transformation proposed by

Hjorth. This transformation has been applied in a variety of situations ([Hj75],[Hj76],

[Ro89], [Wo88], [Nunez, 1990], [In90]), where its value in determining the position of lo-

cal generators has been confirmed.

Some of the features of the "source derivation" approach to the measurement of spa-

tial sharpness in the surface potential field that make it a likely candidate for use as the SL

subtransformation are:


a) The feasibility of the assumptions on which the source derivation is based, within

the context of data collection from the ECoG arrays.



b) The physiological support behind this method, added to its capability of detecting

the "geometric" sharpness of the two-dimensional potential field at each instant in time.


c) Its reference-invariant character. Since the source derivation deals exclusively

with reconstructed potential differences, it should not be affected by the relative placement

of the reference level, on an instant-by-instant basis.