thesis.pdf

Quantitative analysis of EEG signals:

Time-frequency methods and Chaos theory

Rodrigo Quian Quiroga

Institute of Physiology - Medical University L

ubeck

and Institute of Signal Processing - Medical University L

ubeck

Aus dem Institut f

ur Physiologie

vertreten in der Technisch-Naturwissenschaftlichen Fakult

at

der Medizinischen Universit

at zu L

ubeck

durch das Institut f

ur Signalverarbeitung und Prozessrechentechnik

Direktoren:

Prof. Dr. med. Wolfgang Jelkmann (Institut f

ur Physiologie)

Prof. Dr.-Ing. Til Aach (Institut f

ur Signalverarbeitung und Prozessrechentechnik)

Quantitative Analyse von EEG-Signalen:

Zeit-Frequenz-Methoden und Chaos-Theorie

Inauguraldissertation

zur

Erlangung der Doktorw

urde

der Medizinischen Universit

at zu L

ubeck

- Aus der Technisch-Naturwissenschaftlichen Fakult

at -

Vorgelegt von


aus Buenos Aires, Argentinien

L

ubeck, 1998

1. Berichterstatter: Prof. Dr.-Ing. Til Aach

2. Berichterstatter: Prof. Dr.-Ing Erol Basar

Tag der m

undlichen Pr

ufung: 04.12.98

Zum Druck genehmigt, L

ubeck, den 18.05.1999

gez. Prof. Dr.-Ing. Erik Maehle.

- Dekan der Technisch-Naturwissenschaftlichen Fakult

at -

ii

Zusammenfassung

Seitdem 1929 die ersten EEGs von Menschen abgeleitet wurden, hat sich das EEG

zu einem der wichtigsten diagnostischen Hilfsmittel in der klinischen Neurophysiologie

entwickelt. Bis jetzt beruht die EEG-Analyse jedoch weitgehend auf der visuellen In-

spektion der EEG-Aufzeichnungen. Dieses Auswertungsverfahren ist sehr subjektiv und

und erschwert statistische Auswertung und Standardisierung. Daher wurden mehrere

Methoden vorgeschlagen, um die im EEG enthaltene Information quantitativ zu erfas-

sen. Unter diesen Verfahren hat sich die Fourier-Transformation als ein sehr n

utzliches

Hilfsmittel erwiesen. Diese kann die Frequenzkomponenten des EEG-Signals charakte-

risieren und hat klinische Bedeutung erlangt. Die Fourier-Transformation hat jedoch

einige Nachteile, die ihre Anwendung einschr

anken. Daher sind andere Methoden erfor-

derlich, um verborgene Information aus dem EEG zu gewinnen.

In dieser Arbeit habe ich Methoden zur Analyse verschiedener Arten von EEG-

Signalen beschrieben, erweitert und vergliechen, und zwar (1) Zeit-Frequenz-Methoden

(Gabor- und Wavelet-Transformation) und (2) Methoden der Chaos-Analyse (Attraktor-

Rekonstruktion, Korrelations-Dimension, Lyapunov-Exponent).

Diese Methoden lieferten hinsichtlich der Quellen und der Dynamik von Grand-Mal-

Anf

allen neue Information, die mit konventionellen Methoden schwierig zu erhalten war.

W

ahrend Grand-Mal-Anf

allen herrschten alpha (815Hz) und theta (47Hz) Rhyth-

men vor, die sp

ater langsamer wurden, was in Beziehung zum Beginn der klonischen

Phase stand. Die Dynamik der Gehirn-Oszillationen in dieser Phase ist von Interesse im

Hinblick auf Prozesse neuronaler Erm

udung, auf ein Ungleichgewicht der Neurotrans-

mitter, auf

Ahnlichkeit mit Tierversuchen und auf Computer-Simulationen. Analysen

mithilfe der Chaos-Theorie zeigten, da Parameter wie die Korrelations-Dimension oder

der Lyapunov-Exponent abnahmen (diese Parameter charakterisieren die Komplexit

at

und die Chaotizit

at des Signals). Diese Ergebnisse zeigten einen

Ubergang zu einem

einfacheren System im Verlauf des epileptischen Anfalls.

Um grundlegende Eigenschaften von Gehirn-Oszillationen zu untersuchen, habe ich

ereigniskorrelierte Potentiale (also Ver

anderungen des EEG aufgrund externer oder in-

terner Reize) analysiert, und zwar mit neueren Methoden der Zeit-Frequenz-Analyse. In

diesem Zusammenhang zeigte die Untersuchung ereigniskorrelierter Alpha-Oszillationen

(also der Alpha-Komponenten ereigniskorrelierter Potentiale) eine topographische Ver-

teilung mit signikanten Latenz-Unterschieden zwischen anterioren und posterioren Elek-

troden. Dies legte nahe, da diese ereigniskorrelierten Alpha-Oszillationen an multiplen

Orten entstehen. Ferner wiesen (a) die Unabh

angigkeit der Alpha-Antworten von der

Bearbeitung einer kognitiven Aufgabe, (b) das deutlichste Auftreten dieser Antworten

iii

an okzipitalen Positionen und (c) die kurze Latenz dieser Antworten auf eine Bezie-

hung zwischen ereigniskorrelierten Alpha-Oszillationen und prim

ar-sensorischer Verar-

beitung hin. Die Untersuchung von Antworten auf bimodale Reize (simultane audi-

torische und visuelle Stimulation) zeigte eine signikante Zunahme der Amplitude im

Vergleich mit unimodalen Reizen. Demzufolge war es m

oglich, eine Beziehung zwischen

Gamma (30 60Hz) Oszillationen und einem Proze anzunehmen, der die Information

tr

agt, da zwei sensorische Wahrnehmungen (im Rahmen der bimodalen Stimulation)

zu ein und demselben Reiz geh

oren.

Insbesondere ist diese Arbeit die erste Untersuchung, in der die neue Methode

Wavelet-Entropie f

ur die Analyse ereigniskorrelierter Potentiale angepat und ange-

wendet wurde. In ereigniskorrelierten Potentialen gehen signikante Abnahmen der

Wavelet-Entropie mit einer kognitiven P300-Antwort einher. Dies zeigte, da diese

P300-Antwort mit einer Ordnung der spontanen EEG-Oszillationen assoziert ist.

iv

Quantitative analysis of EEG signals:

Time-frequency methods and Chaos theory


Institute of Physiology - Medical University Lubeck

and Institute of Signal Processing - Medical University Lubeck

1998

To my family: Mama, Consuelo, Huguito and Elisa

and to my closest friends: Esteban and Samy.

vii

Preface

In this work, I will describe and extend two new approaches that started to be applied

to physiological signals: 1) the time-frequency methods, and 2) the methods based on

Chaos theory. I will discuss their applicability and usefulness mainly in two types of

brain signals: a) EEG recordings from \Grand Mal" epileptic seizures, and b) Event-

related potentials. Moreover, I will compare all these new methods, comparison which

was not performed so far, stressing their advantages over conventional approaches in the

analysis of EEG signals. Furthermore, the results obtained will be closely linked with

physiological interpretations. In particular, this thesis is the rst work where the novel

method \Wavelet entropy" is adjusted and applied to the analysis of evoked responses.

The structure is as follows:

The rst part of the thesis consists in an introduction to basic concepts of elec-

troencephalography and a review of previous approaches to its quantitative analy-

sis. In particular, chapter x1 gives a brief description of the necessary background

of neurophysiology focusing on the concepts needed for understanding the basics

of brain signals, and chapter x2 describes the traditional Fourier analysis and its

main applications to EEGs.

Chapters x3 to x6 are the main part of the thesis, each chapter referring to one of

the new quantitative methods. They all have the same internal structure: 1) they

start with an introduction in which the goal of the method is described, 2) then,

a theoretical background is given, 3) their application to dierent types of EEG

is shown and nally, 4) a physiological interpretation of the results is given and

advantages of the methods are discussed in comparison with other approaches.

More specically, chapter x3 presents the Gabor Transform, a time-frequency

method that solves some of the disadvantages of the Fourier Transform. Fur-

thermore, since in many cases a more detailed information is required, as I will

show with the study of Grand Mal seizures, I will introduce new denitions that

will allow a better quantitative analysis of the EEG.

Chapter x4 describes the theoretical background of the Wavelet Transform. Studies

where the Wavelet Transform is applied to Tonic-Clonic seizures and to event-

related potentials will show the advantages of this new method in the analysis of

EEG signals.

Chapter x5 presents the approach based on the Non-linear Dynamics (Chaos)

theory. I will show its application to dierent type of seizure recordings, correlating

ix

these results with the ones obtained with the methods described in the previous

chapters. I will remark several problems in the implementation of these methods

in the analysis of physiological signals that in many cases lead to pitfalls and

misinterpretations. Furthermore, I will establish some criteria for the analysis of

EEG signals with Chaos methods.

Chapter x6 introduces a new method based on the \information theory", the

Wavelet-entropy, that gives quantitative information about the ordered/disordered

nature of the EEG signals. I will show its application to event-related potentials.

Furthermore, I will show how it avoids several disadvantages of Chaos methods

allowing the study of similar concepts with a completely dierent approach.

Finally, in chapter x7, I will compare the time-frequency and Chaos approaches,

and I will discuss the main physiological results by joining the evidence obtained

with the dierent methods.

Acknowledgments

This work was supported by the Bundesministerium fur Bildung und Forschung

(BMBF), Germany and by the Medical University of Lubeck, Germany. I am very

thankful to Prof. Erol Basar, director of the group of Neurophysiology of the Medical

University of Lubeck, for giving me the opportunity to work under his direction and for

his experienced advice in the development of this work.

I am also very thankful to Prof. Til Aach for his criticisms and corrections to this

thesis, especially in the mathematical formalisms, and to Prof. Rupert Lasser for his

guiding in the mathematical background during the rst stage of my work. I would

like to thanks Dr. Martin Schurmann for two years of invaluable scientic discussions,

non-scientic activities and for his criticisms and comments after a careful reading of

this thesis. I am also very thankful to Dr. Juliana Yordanova and Dr. Vasil Kolev

for very helpful criticisms during the development of this work and for their warm

friendship. During my staying in Lubeck I also appreciated very much the collaboration

with Dr. Atsuko Schutt, Dr. Irina Maltseva, Oliver Sakowitz, Dr. Richard Rascher-

Friesenhausen and Dr. Tamer Demiralp to whom I am also very thankful for software

implementation. I am also very thankful to Prof. Wolfgang Jelkmann, director of the

Institute of Physiology for giving me the opportunity to work at his institute. This

thesis would have not been achieved without the help of the group of neurophysiology

in Lubeck. I would also like to mention the very nice work atmosphere that they created.

My special thanks to Dipl.-Ing. Martin Gehrmann, Dipl.-Ing. Ferdinand Greitschus,

Gabriele Huck, Betina Stier and Gabriela Fletschinger. I am especially thankful to

Beate Nurnberg for her constant support and personal help.

x

I would certainly like to remember all my colleagues/friends from Argentina. A

very special thanks to Dr. Osvaldo Rosso and Dr. Susana Blanco, from the Chaos

and Biology group at the University of Buenos Aires, for giving me the rst push in my

steps as a physicist and also for their friendship and constant scientic and non-scientic

support. I also appreciated further collaboration with Alejandra Figliola of the same

group.

I am very thankful to Dr. Adrian Rabinowicz, director of the Epilepsy department of

the Institute of Neurological Investigations (FLENI) for teaching me what I know about

epilepsy. I can not forget all the support and constant good mood of the people of

the Neurophysiology department at FLENI foundation, with whom I had the pleasure

to work with during my research stage in Argentina. Many thanks to Isabel, Jorge,

Claudia, Sonia, Cecilia, Sandra, Alexandra, Mary, Monica, Dr. Ribero, Dr. Estelles,

Dr. Nogues, Dr. Camarotta, Dr. Navarro Correa and I hope I am not forgetting

somebody.

Finally I would like to thank the one who introduce me in this fascinating world

of Neurophysiology, the one who encouraged and supported a young student of physics

coming up with crazy ideas about Chaos and EEGs. My very special thanks to Dr.

Horacio Garca.

xi

Contents

Zusammenfassung iii

Preface ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Summary 1

1 Outline of Neurophysiology: Brain signals 3

1.1 Electroencephalogram (EEG) . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Brain oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.2 EEG in Epilepsy . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Event related potentials (ERP) . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Relation between EEG and ERP . . . . . . . . . . . . . . . . . . . . . . 11

2 Fourier Transform 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Fourier Transform in EEG analysis . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Frequency bands . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Topographical mapping . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Frequency analysis of evoked responses . . . . . . . . . . . . . . . 18

2.3.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Gabor Transform (Short Time Fourier Transform) 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Application to intracranially recorded tonic-clonic seizures . . . . . . . . 25

3.3.1 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Application to scalp recorded tonic-clonic seizures . . . . . . . . . . . . . 30


Statistical analysis: plateau criteria . . . . . . . . . . . . . . . . 31

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

xiii

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Wavelet Transform 36

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . 38

4.2.2 Dyadic Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . 38

4.2.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.4 B-Splines wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.5 Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Short review of wavelets applied to the study of EEG signals . . . . . . . 44

4.4 Application to scalp recorded tonic-clonic seizures . . . . . . . . . . . . . 46

4.4.1 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Application to alpha responses of visual event-related potentials . . . . . 50

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . 52

Comparison between wavelets and conventional digital ltering . 52

4.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Application to gamma responses of bisensory event-related potentials . . 61

4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61



4.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Deterministic Chaos 68

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


5.2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.2 Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.3 Calculation of the Correlation Dimension . . . . . . . . . . . . . . 70

5.2.4 Problems arising when calculating the Correlation Dimension . . 71

5.2.5 Lyapunov Exponents and Kolmogorov Entropy . . . . . . . . . . 72

xiv

5.2.6 Calculating Lyapunov Exponents . . . . . . . . . . . . . . . . . . 73

5.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Short review of Chaos analysis of EEG signals . . . . . . . . . . . . . . . 75

5.4.1 Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Application to scalp recorded EEGs . . . . . . . . . . . . . . . . . . . . . 79


5.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6 Application to intracranially recorded tonic-clonic seizures . . . . . . . . 82


5.6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 82

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Wavelet-entropy 86

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


6.3 Application to visual event-related potentials . . . . . . . . . . . . . . . . 89



6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 General Discussion 101

7.1 Physiological considerations . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.1.1 Dynamics of Grand Mal seizures . . . . . . . . . . . . . . . . . . . 101

7.1.2 Event-related responses . . . . . . . . . . . . . . . . . . . . . . . . 102

7.1.3 Are EEG signals chaos or noise? . . . . . . . . . . . . . . . . . . . 103

7.2 Comparison of the methods . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2.1 Fourier Transform vs. Gabor Transform . . . . . . . . . . . . . . 104

7.2.2 Gabor Transform vs. Wavelet Transform . . . . . . . . . . . . . . 105

7.2.3 Wavelet Transform vs. conventional digital ltering . . . . . . . . 107

7.2.4 Chaos analysis vs. time-frequency methods (Gabor, Wavelets) . . 108

7.2.5 Wavelet-entropy vs. frequency analysis . . . . . . . . . . . . . . . 108

7.2.6 Wavelet-Entropy vs. Chaos analysis . . . . . . . . . . . . . . . . . 109

Conclusion 110

xv

A Time-frequency resolution and the Uncertainty Principle 111

A.1 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A.2 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.3 Time-frequency resolution of the Fourier, Gabor and Wavelet Transform 113

References 117

Biographical sketch 129

xvi

Summary

Since the rsts recordings in humans performed in 1929, the EEG has become one of

the most important diagnostic tools in clinical neurophysiology, but up to now, EEG

analysis still relies mostly on its visual inspection. Due to the fact that visual inspection

is very subjective and hardly allows any statistical analysis or standardization, several

methods were proposed in order to quantify the information of the EEG. Among these,

the Fourier Transform emerged as a very powerful tool capable of characterizing the

frequency components of EEG signals, even reaching diagnostical importance. However,

Fourier Transform has some disadvantages that limit its applicability and therefore,

other methods for extracting \hidden" information from the EEG are necessary.

In this work, I described, extended and compared methods of analysis of dierent

types of EEG signals, namely time-frequency methods (Gabor Transform and Wavelet

Transform) and Chaos methods (attractor reconstruction, Correlation dimension, Lya-

punov exponents, etc.).

Time-frequency methods provided new information about sources and dynamics of

Grand Mal (Tonic-clonic) seizures, something very dicult to obtain with conventional

methods. Grand Mal seizures were rst dominated by alpha (7:5 12:5Hz) and theta

(3:57:5Hz) rhythms, these rhythms later becoming slower in correlation with the start-

ing of the clonic phase. The dynamics of the frequency patterns during these seizures

was very interesting in relation to processes of neuronal fatigue, neurotransmitter dis-

balance, similarity with animal experiments and computer simulations. The analysis

with Chaos theory showed a decrease in parameters as the Correlation Dimension or

the maximum Lyapunov exponent, parameters that characterize the complexity and

\chaoticity" of the signal. These results showed a transition to a more simple system

during epileptic seizures.

In order to study basic features of brain oscillations, I analyzed event-related re-

sponses (i.e. alterations of the ongoing EEG due to an external or internal stimuli) with

recent methods of time-frequency analysis. In this context, the study of event-related

alpha oscillations (i.e. event-related responses in the alpha range) showed that these

responses were distributed along the scalp with signicant dierences in their delays

between anterior and posterior electrodes. This result implied that several sources were

involved in the origin of the event-related alpha oscillations. Furthermore, their indepen-

dence on the performance of a cognitive task, the best denition in occipital locations

and the short latency of the responses pointed towards a relation between event-related

alpha oscillations and primary sensory processing.

The study of the responses upon bimodal stimulation (simultaneous visual and audi-

1

tory stimulation) showed a signicant increase of amplitude in comparison with the uni-

modal ones. Then, it was possible to conjecture a relation between gamma (30 60Hz)

oscillations and a process responsible of carrying the information that two sensory per-

ceptions of bimodal stimulation correspond in fact to the same stimulus.

In particular, this thesis is the rst work where the novel method \Wavelet entropy",

a measure of the distribution of the signal in the frequency domain, was adjusted and

applied to the analysis of event-related responses. In event-related potentials, signicant

decreases in the wavelet entropy correlated with the P300 cognitive response showed that

this response was associated with an ordering of the spontaneous EEG oscillations.

2

1 Outline of Neurophysiology: Brain signals

This chapter presents some basic topics of neurophysiology necessary for understanding

the experiments and results to be described in the following chapters. In this context,

the concepts exposed and the detail of their treatment are not expected to provide

a complete background on neurophysiology. On the contrary, this chapter is focused

on describing the electroencephalogram and event-related potentials (ERPs), especially

applied to the study of epilepsy and brain oscillations. Despite the wide application

of these issues, some fundamental points are still controversial and due to the complex

behavior of these signals, they are dicult to resolve with traditional approaches, thus

being ideal candidates to be studied with new quantitative methods.

1.1 Electroencephalogram (EEG)

The EEG was originally developed as a method for investigating mental processes. Clin-

ical applications soon became visible, most notably in epilepsy, and it was only with the

introduction of ERP recordings that EEG correlates of sensory and cognitive processes

nally became popular. The rst recordings of brain electrical activity were reported by

Caton in 1875 in exposed brains of rabbits and monkeys, but it was not until 1929 that

Hans Berger (Berger, 1929) reported the rst measurement of brain electrical activity in

humans. EEG visual patterns were correlated with functions, dysfunctions and diseases

of the central nervous system, then emerging as one of the most important diagnostical

tools of neurophysiology.

The electroencephalogram (EEG) can be roughly dened as the mean electrical ac-

tivity of the brain in dierent sites of the head. More specically, it is the sum of the

extracellular current ows of a large group of neurons. Since the generation of the EEG

from the action potentials of the neurons is beyond the scope of this thesis, for further

details I suggest the comprehensive works of Steriade et al. (1990), Lopes da Silva

(1991), Steriade (1993), Speckermann and Elger (1993), Pedley and Traub (1990) and

Basar (1980).

EEG recordings are achieved by placing electrodes of high conductivity (impedance

>

>

>

>

:

sin[ (k + 1=2) ]; if d = 1;

2

1=2

cos[ !

mh

(k + 1=2) ]; if d is even;

2

1=2

sin[ !

mh

(k + 1=2) ] if d is odd;

cos[ 2(k + 1=2) ] if d = 2

m

;

(38)

with 1 d 2

m

, 0 k < 2

m

and h = [[d=2]], where [[ ]] denotes the integer part. It

can be demonstrated that M

(m)

is a 2

m

2

m

dimensional orthogonal matrix (Serrano,

1996).

Then, we can dene the new set of elemental functions in order to expand r

(m;l)

j

(n)

as a 2

m

dimensional vector obtained from

(m;l)

j;d

( n ) =

l+2

m

1

X

k=l

M

(m)

dk

j;k

( n ) (39)

for 1 d 2

m

.

Clearly, these functions constitute a new local orthonormal basis covering the interval

under analysis 2

j

l n 2

j

(l + 2

m

). Therefore we can give a second description of

the local signal as

43

r(m;l)

j

( n ) =

2

m

X

d=1

D

(m;l)

j

( d )

(m;l)

j;d

( n ) : (40)

and the corresponding coecients are easily computed as

D

(m;l)

j

( d ) =

l+2

m

1

X

k=l

M

(m)

dk

C

j

( k ) ; (41)

where 1 d 2

m

.

The trigonometric wavelet packets

(m;l)

j;d

(n) have zero mean, oscillate on the interval

2

j

l n 2

j

(l + 2

m

) and decay with exponential ratio. Moreover, their wave-forms

resemble modulated sines or cosines. In fact, it can be demonstrated that each Fourier

transform

^

(m;l)

j;d

(!) is centered at the fundamental frequency !

mh

, when d = 2h or

d = 2h+ 1. Moreover,

^

(m;l)

j;d

(!) = 0 on the other fundamental frequencies.

In other words, the coecients fD

(m;l)

j

(d)g can be considered as the discrete Fourier

spectrum for the local signal r

(m;l)

j

(n). Summing up, we can resume in the double set

of coecients fC

j

(k); D

(m;l)

j

(d)g the time-scale-frequency information of the local signal

r

(m;l)

j

(n).

Finally, to analyze the complete function r

j

(n), that is, the details at level j, we

choose some partition in local components r

(m

i

;l

i

)

j

(n), according the structure of the

signal,

r

j

( n ) =

X

m

i

r

(m

i

;l

i

)

j

( n ) ; (42)

where the sequence of index l

i

veries l

i+1

= l

i

+ 2

m

i

. Then, we implement the above

refereed time-scale-frequency technique for each local signal.

4.3 Short review of wavelets applied to the study of EEG sig-

nals

Several works applied the Wavelet Tranform to the study of EEGs and ERPs (see a

review in Unser and Aldroubi, 1996; or in Samar et al., 1995). One rst line of appli-

cations is for pattern recognition in the EEG. This is achieved by correlating dierent

transients of the EEG with wavelet coecients of dierent scales. Schi et al. (1994a)

used a multiresolution decomposition implemented with B-Splines mother functions for

extracting features of EEG seizure recordings. They showed a better performance of

wavelets in comparison with Gabor Transform, and a similar resolution of the multires-

olution decomposition compared with the continuous Wavelet Transform but with a

44

marked decrease in computational time. Other works also used this approach for auto-

matic detection of spike complexes characteristic of epilepsy, thus helping in the analysis

of EEG recordings from epileptic patients (Schi et al., 1994b; Senhadji et al., 1995;

Clark et al., 1995).

Demiralp et al. (1999) used coecients in the delta frequency band for detecting

P300 waves in single trials of an auditory oddball paradigm. Furthermore, they used

this approach for making a selective averaging of the single trials, thus obtaining a better

denition of the P300. Basar et al. (1999) reported the utility of Wavelet Transform for

classifying dierent type of single sweep responses to cross-modality stimulation.

A digital ltering of ERPs based on the Wavelet Transform was proposed by Bertrand

et al. (1994). They used the method as a noise reduction technique, reporting better

results than the ones obtained with Fourier based methods, especially when applied to

non-stationary signals. The main goal of this type of ltering is to extract the event-

related responses from the single sweeps by eliminating the contribution of the ongoing

EEG. This procedure would avoid the necessity of averaging the single sweeps. In this

context, Bartnik et al. (1992) characterized the event-related responses from the wavelet

coecients, then using selected coecients for isolating the event-related responses from

the background EEG in the single trials. A similar approach has being later proposed

by Zhang and Zheng (1997).

Akay et al. (1994) used the Wavelet Transform for characterizing electrocortical

activity of fetal lambs, reporting much better results than the ones obtained with the

Gabor Transform. Thakor et al. (1993) analyzed somatosensory EPs of anesthetized

cats with cerebral hypoxia by using the multiresolution decomposition. They report

that selected coecients are sensitive to neurological changes, but having comparable

results than the ones obtained with Fourier based methods. Ademoglu et al. (1997) used

wavelet analysis for discriminating between normal and demented subjects by studying

the N70-P100-N130 complex response to pattern reversal visual evoked potentials (N70

and N130 are negative peaks of the ERP with a latency of 70 and 130ms respectively).

Kolev et al. (1997) used the multiresolution decomposition for studying the presence

of dierent functional components in the P300 latency range in an auditory oddball

paradigm. Basar et al. (1999) used the wavelet decomposition for studying the alpha

responses to cross-modality stimulation, reporting similar results than the ones obtained

with digital ltering.

45

Figure 15: Scalp EEG seizure recording.

4.4 Application to scalp recorded tonic-clonic seizures

4.4.1 Material and Methods

An EEG time series corresponding to a tonic-clonic seizure of an epileptic patient was

analyzed. Scalp electrodes were applied following the 10-20 international system. The

signal was digitized at 409:6 Hz through a 12 bit A/D converter and ltered with an

antialiasing eight pole lowpass Bessel lter with a cuto frequency of 50 Hz. Then, it

was digitally ltered with a 1 50 Hz bandwidth Butterworth lter and stored, after

decimation, at 102:4 Hz in a PC hard drive. The recording included one minute of the

EEG before the seizure onset and two minutes which included the ictal and post-ictal

phases. All 3 minutes were analyzed at the right central (C4) electrode, choosing this

electrode after visual inspection of the EEG as the one with the least amount of artifacts.

Wavelet Transform was applied by using a cubic Spline function as mother wavelet.

The multiresolution decomposition method (Mallat, 1989) was used for separating the

signal in 7 frequency bands: B

1

= 25:8 51:2Hz;B

2

= 12:8 25:2Hz;B

3

= 6:4

12:8Hz;B

4

= 3:2 6:4Hz;B

5

= 1:6 3:2Hz;B

6

= 0:8 1:6Hz;B

7

= 0:4 0:8Hz).

4.4.2 Results

Figure 15 shows 90sec of the Tonic-Clonic seizure studied. The whole recording was

already shown in g. 11. In this case seizure starts at second 10 and ends at second 85.

Due to the fact that the aim of this work was to analyze middle and low frequencies

46

brain activity during an epileptic seizure, we eliminated B

1

and B

2

bands, both con-

taining high frequency artifacts that obscure the EEG (see sec. x3.4). The relative band

intensity ratio (RIR) (dened as in sec. x3.2 but in this case from the wavelet scales)

had a similar behavior as the one showed with Gabor Transform in gure 12.

Frequency bands B

3

and B

4

were chosen for performing an analysis with Wavelets

Packets, these bands being important in the development of the tonic-clonic seizures as

showed in Chapter x3.4 (see also Quian Quiroga et al., 1997b).

B

3

band coecients were segmented with sliding windows of l = 32 samples corre-

sponding to time intervals of t = 2:5 sec. Discrete sets of frequencies between 6:4 and

12:8 Hz with intervals of 0:4 Hz were obtained as showed in g. 4.4.2 (squared values

shown).

From second 50, we can see an increase of the activity in nearly all the packets. Due

to the good time-frequency resolution of the Wavelet Packets it is possible to follow the

evolution of the frequency peaks. For example, the peak marked with an arrow in the

wavelet packet corresponding to 8:4Hz at about second 50, is also visible in the packets

corresponding to 8:0, 7:6 and 7:2Hz, appearing with higher amplitude and some delay.

Then, this peak originated in the 8:4Hz packet, or probably in higher frequencies but

with lower amplitude, is evolving with time to lower frequencies.

Figure 17 shows the Wavelet Packets corresponding to the B4 band (squared values

shown). They were generated by using l = 16 samples corresponding to time intervals

of t = 2:5 sec and discrete sets of frequencies between 3:2 and 6:4 Hz with intervals

of 0:4 Hz. Note that the j = 4 level has half dispersion in frequencies compared with

the j = 3 level and for this reason we used a window of 16 samples in order to obtain

the same denition.

In the B4 band there is a very clear peak, marked with an arrow, after second 60

in the frequencies around 3 4Hz, this increase being correlated with the starting of

the clonic phase of the seizure. This peak is also visible, but appearing earlier in time,

in the higher frequency packets (also marked with an arrow). Although this behavior is

predictable with a visual inspection of the EEG, it is very interesting to note that this

peak is in fact the same peak described in the gure 4.4.2 but appearing more delayed.

Analyzing both gures together, we can observe very clearly how this high amplitude,

low frequency peak (3 4Hz) at about 65sec was in fact rst observed in the higher

frequencies (about 9Hz), then evolving with time to lower frequencies until reaching

a very high amplitude when the clonic phase starts, this evolution being very dicult

to identify from a visual inspection of the EEG or by using traditional methods as the

spectrograms (see discussion of sec. x3.5).

47

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

7.2 Hz 7.6 Hz

8.0 Hz 8.4 Hz 8.8 Hz 9.2 Hz

9.6 Hz 10 Hz 10.4 Hz 10.8 Hz

11.2 Hz 11.6 Hz 12.0 Hz 12.4 Hz

Figure 16: Wavelet packets for the scale level B3.

48

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

0 20 40 60 80

0

1

2

3

3.2 Hz 3.6 Hz 4.0 Hz 4.4 Hz

4.8 Hz 5.2 Hz 5.6 Hz 6.0 Hz

6.4 Hz

Figure 17: Wavelet packets for the scale level B4.

49

4.4.3 Discussion

The high time-frequency resolution achieved with Wavelet Packets allowed a very de-

tailed study of the time evolution of the frequency peaks during the seizure. In fact, it

was possible to establish that the high amplitude peaks of about 34Hz, characteristic

of the clonic activity, were generated in higher frequencies. This result is in agreement

with the dynamic described with the Gabor Transform. In this case, with the RIR (see

sec. x3.4) I showed how during the seizure the relevant brain activity was dominated by

alpha and theta bands, until the ending of the seizure, when delta activity dominated

again.

Then, it is reasonable to conjecture that the violent clonic contractions characteristic

of the clonic phase of the Grand Mal seizures are the response to brain oscillations

generated in higher frequencies, but owing to the fact that the muscles can not contract

so fast, the muscle activity is limited to a tonic contraction until the oscillations become

slower and the muscles are capable to oscillate in resonance with them.

It is interesting to note that the frequency behavior described is in agreement with

simulations of thalamic slices performed by Wang and coworkers (Wang et al., 1995;

Golomb et al., 1996), who observed a slowing from 10Hz to 4Hz in their simulations

after the suppression of GABA

A

inhibitors. This result is also in agreement with in vitro

experimental results of Bal et al. (1995a,b). These experiments suggest that it would

be very interesting to investigate if the \slowing" of the frequencies observed during

the seizures can be related with a variation in the concentration of neurotransmitters,

especially of the GABA inhibitors, possible due to a neuronal fatigue provoked by the

abnormal ring rate of the neurons during the seizure.

4.5 Application to alpha responses of visual event-related po-

tentials

4.5.1 Introduction

EEG alpha rhythms can be dened as oscillations between 8 and 13 Hz, with an am-

plitude usually below 50V and localized over posterior regions of the head. Alpha

rhythms appear spontaneously during wakefulness, best seen with eyes closed, under

relaxation and mental inactivity conditions (Niedermeyer, 1993a).

Although alpha oscillations have been widely studied, their sources and functional

correlates are still under discussion. Sources of alpha rhythms have been investigated

leading to controversies whether they are thalamic, cortical or whether other structures

are involved in their generation (Adrian, 1941; Andersen and Andersson, 1968; Lopes da

50

Silva et al.,1973a,1973b; Lopes da Silva and Storn van Leewen, 1977; Basar et al., 1997).

Moreover, many studies were performed in order to understand their functional mean-

ings. These studies showed that alpha rhythms could be correlated even to sensory or

cognitive processes depending on the task performed and generators involved, therefore

not having an unique and specic function (for a review, see Basar et al., 1997).

4.5.2 Material and Methods

In 10 voluntary healthy subjects (no neurological decits, no medication known to aect

the EEG) two types of experiments were performed:

1. No-task visual evoked potential (VEP): subjects were watching a checkerboard

pattern (sidelength of the checks: 50'), the stimulus being a checker reversal (N =

100 stimuli).

2. Oddball paradigm: subjects were watching the same pattern as above. Two dif-

ferent stimuli were presented in a pseudorandom order. NON-TARGET stimuli

(75%) were pattern reversal, and TARGET stimuli (25%) consisted in a pattern

reversal with horizontal and vertical displacement of one-half of the square side

length. Subjects were instructed to pay attention to the appearance of the target

stimuli (N = 200 stimuli).

The inter-stimulus interval varied pseudo-randomly between 2.5 and 3.5 s. After

each pattern reversal, the reverted pattern was shown for one second, then the pattern

was re-reverted. Recordings were made following the international 10 20 system in

seven dierent electrodes (F3, F4, Cz, P3, P4, O1, O2) referenced to linked earlobes.

Data were amplied with a time constant of 1:5sec: and a low-pass lter at 70Hz. With

each stimulus, a single sweep of EEG data was recorded, i.e.: for each single sweep, 1sec:

pre- and post-stimulus EEG were digitized with a sampling rate of 250Hz and stored

in a hard disk.

After visual inspection of the data, 30 sweeps free of artifacts were randomly selected

for each type of stimuli (VEP, NON-TARGET and TARGET) for future analysis. A

Wavelet Transform was applied to each single sweep using a quadratic B-Spline function

as mother wavelet. The multiresolution decomposition method (Mallat, 1989) was used

for separating the signal in frequency bands, dened in agreement with the traditional

frequency bands used in physiological EEG analysis. After a ve octave wavelet de-

composition, components corresponding to the alpha band (8 16Hz) were analyzed.

For each subject the alpha components of the 30 single sweeps were averaged. Finally,

results for each subject were averaged to obtain a \grand average". The temporal reso-

51

lution of the scale corresponding to the alpha band was of 64 ms

5

. However it should be

remarked that the \real" resolution will depend on the characteristics of the signal and

the mother function (i.e. how the mother function matches the signal; see section 4.2.4).

In this respect, the optimal resolution of B-Splines was shown with numerical computa-

tions (Unser et al., 1992). It is also interesting to note that non-redundancy is important

for increasing the computational speed.

Statistical analysis

The wavelet coecients processed were the ones obtained after averaging the re-

sponses of the 30 single trials for each electrode and subject. Then, wavelet coecients

were rectied and the maximum coecients and their time delay with respect to the

stimulus occurrence were computed in the rst 500 ms post-stimulation. The analysis

was limited to the rst 500 ms owing to the fact that no subject showed meaningful

alpha responses after this time. Comparison between modalities and electrodes were

done by using a multiple factor ANOVA test.

Comparison between wavelets and conventional digital ltering

Figure 18 gives some examples of single-trial evoked potentials, showing the compar-

ison of results obtained with Wavelet Transform and with digital ltering. In addition,

the gure shows the relation between the wavelet coecients (used for all statistical

analyses) and the waveforms reconstructed from the wavelet coecients for a specic

level (which will be shown in the gures). I would like to remark that the sweeps se-

lected do not necessary show a clear event-related response, but they are suitable for

showing the better resolution achieved with the multiresolution decomposition based on

the Wavelet Transform in comparison with conventional digital ltering. The digital

lter used was an ideal lter (i.e. a digital lter based on band pass ltering in the

Fourier domain as used in several earlier papers, Basar, 1980) with the lter limits set

in agreement with the limits obtained with the multiresolution decomposition for the

alpha band.

As a general remark it can be stated that with the wavelet coecients a better

resolution and localization of the features of the signal is achieved. In between the

vertical dashed lines in sweep #1 three oscillations in the alpha range are shown, having

the last oscillation a larger amplitude as observed in the original sweep. This is well

resolved with the wavelet coecients and also in the reconstructed form. However, this

ne structure of this train of alpha oscillations is not resolved by digital ltering; i.e.

reading a maximum from this curve is imprecise. In sweep #2, in between the dashed

5

From sec. 4.2.2 the time steps b

j;k

are 2

j

data points, and since the alpha band corresponds to

j = 4 and the sampling rate was of 250Hz we have t = 2

4

=250Hz = 64ms

52

Originalsweep

Digitalfiltering

Waveletcoeff.

Waveletreconstr.

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-150

0

150

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-150

0

150

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-150

0

150-1.0 -0.5 0.0 0.5 1.0

-150

0

150-1.0 -0.5 0.0 0.5 1.0

-150

0

150

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

-1.0 -0.5 0.0 0.5 1.0

-50

0

50

sweep #1 sweep #5sweep #2 sweep #4sweep #3

Figure 18: Examples of the better performance of the Wavelet Transform in comparison with an \ideal" digital ltering for ve

single sweeps.

53

vertical lines, is showed a transient with a frequency clearly lower than the range of alpha

band. The digital ltering does not resolve this transient and it spuriously \interpolates"

alpha oscillations in continuity with the ones that precede or follow the transient. On

the other hand, the wavelet coecients show a decrease in this time segment, this

phenomenon being also visible in the reconstructed form. Something similar occurs in

sweep #3 with the transient marked with an arrow. In fact in this last case, the transient

is due to the cognitive P300 wave typically obtained upon target stimuli. With wavelets

it is visible that, as in the original signal, there is no important contribution of alpha

oscillations in this time range, the digital ltering having not enough resolution for

resolving this. The better time-frequency resolution of wavelets (in this case a better

frequency localization for a certain time range) can be also seen in sweep #4. In the

original signal, in between the vertical dashed lines there is a marked oscillation of

about 4 6Hz, corresponding to the theta band. The digital ltered signal shows an

alpha oscillation not present in the original signal. On the other hand, due to its better

resolution the wavelet coecients and the reconstructed signal show a clear decrease

for this time range. Finally, sweep #5 shows a ringing eect (i.e. spurious oscillations

appearing before the stimulation time point due to time resolution limitations). The

oscillation before the stimulation time, marked with an arrow, appears in the digital

ltered signal with more amplitude than in the original signal, this eect being overcome

with wavelets.

4.5.3 Results

The grand average wideband ltered (0:5 100Hz) event-related potentials are shown

in gure 19. The P100 response is clearly visible upon all stimuli types at about 100ms,

best dened in occipital locations, where it reaches amplitudes about 8V . In the case

of target stimulation, a marked positive peak appears between 400 500ms, according

to the expected cognitive (P300) response (see sec. x x1.2).

Figure 20 shows the wavelet components in the alpha band (for brevity, \alpha

responses") for the subject JA (for a better visualization of the responses, the signal

reconstructed from the alpha band wavelet coecients is shown). One second pre- and

one second post-stimulation EEG are plotted. Alpha components corresponding to the

pre-stimulus EEG have about 5V and upon all stimulus types, amplitude enhancements

are clearly marked in posterior locations reaching values up to 20V . Furthermore, in

posterior electrodes responses upon TARGET stimulation are prolonged compared with

the other two stimulus types.

Results for the grand average of the 10 subjects (g. 21) are similar to the ones

outlined for the rst subject. Amplitude increases were distributed over the whole scalp

54

VEP non-target target

-15

0

15

-15

0

15

-15

0

15

F3 F4

Cz

P3 P4

O1 O2

F3 F4

Cz

P3 P4

O1 O2

F3 F4

Cz

P3 P4

O1 O2

V

-1sec 0 1sec

V

-1sec 0 1sec

V

-1sec 0 1sec

Figure 19: Grand average of the wide-band frequency responses.

55

Electrode F3 F4 Cz P3 P4 O1 O2

Mean 6.27 6.72 9.80 8.96 8.85 12.65 11.63

SEM 0.56 0.64 0.99 0.85 0.85 1.36 1.22

F3 XXX - < 0.05 - - < 0.01 < 0.01

F4 XXX < 0.05 - - < 0.01 < 0.01

Cz XXX - - < 0.05 -

P3 XXX - < 0.01 -

P4 XXX < 0.01 < 0.05

O1 XXX -

O2 XXX

Table 2: Multiple factor ANOVA comparison of the maximum alpha band wavelet

coecients for the factor electrode. Note: SEM means standard error of the mean, -

means no signicance

for the three stimulation types, best dened in the occipital electrodes. The multiple

factor ANOVA test showed no signicant dierences between stimuli type. Electrode

dierences, instead, were signicant, conrming the predominant localization of the

enhancements in the occipital locations, with a lower response in the anterior electrodes

(see table 2).

It is also interesting to note that in some of the subjects, responses in posterior elec-

trodes upon TARGET stimulation are prolonged in comparison with the NON-TARGET

and VEP ones. This coherent alpha activity extended up to a second post-stimulation.

With the other stimulus types, enhancements have an abrupt decay after 200 300ms

post-stimulus. However, we should remark that this result was not consistent for the

whole group.

The delay of the maximum response in occipital electrodes was about 180ms after

stimulation (see table 3). In parietal electrodes the maximum appeared about 30ms

later, and in central and frontal electrodes between 50 100ms after the occipital one.

After applying a multiple factor ANOVA test, we veried statistically that frontal and

central responses were signicantly delayed in comparison to the occipital ones (p

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