Quantitative analysis of EEG signals: Time-frequency methods and ...

Quantitative analysis of EEG signals:
Time-frequency methods and Chaos theory
Rodrigo Quian Quiroga
Institute of Physiology - Medical University Lubeck
and Institute of Signal Processing - Medical University Lubeck

Aus dem Institut fur Physiologie
vertreten in der Technisch-Naturwissenschaftlichen Fakultat
der Medizinischen Universitat zu Lubeck
durch das Institut fur Signalverarbeitung und Prozessrechentechnik
Direktoren:
Prof. Dr. med. Wolfgang Jelkmann (Institut fur Physiologie)
Prof. Dr.-Ing. Til Aach (Institut fur Signalverarbeitung und Prozessrechentechnik)
Quantitative Analyse von EEG-Signalen:
Zeit-Frequenz-Methoden und Chaos-Theorie
Inauguraldissertation
zur
Erlangung der Doktorwurde
der Medizinischen Universitat zu Lubeck
-Ausder Technisch-Naturwissenschaftlichen Fakultat -
Vorgelegt von
Rodrigo Quian Quiroga
aus Buenos Aires, Argentinien
Lubeck, 1998

1. Berichterstatter: Prof. Dr.-Ing. Til Aach
2. Berichterstatter: Prof. Dr.-Ing Erol Basar
Tag der mundlichen Prufung: 04.12.98
Zum Druck genehmigt, Lubeck, den 18.05.1999
gez. Prof. Dr.-Ing. Erik Maehle.
- Dekan der Technisch-Naturwissenschaftlichen Fakultat -
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 Inspektion
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 erfassen.
Unter diesen Verfahren hat sich die Fourier-Transformation als ein sehr nutzliches
Hilfsmittel erwiesen. Diese kann die Frequenzkomponenten des EEG-Signals charakterisieren
und hat klinische Bedeutung erlangt. Die Fourier-Transformation hat jedoch
einige Nachteile, die ihre Anwendung einschranken. Daher sind andere Methoden erforderlich,
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-
Anfallen neue Information, die mit konventionellen Methoden schwierig zu erhalten war.
Wahrend Grand-Mal-Anfallen herrschten alpha (8 ; 15Hz) und theta (4 ; 7Hz)Rhythmen
vor, die spater 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 Ermudung, auf ein Ungleichgewicht der Neurotransmitter,
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 Komplexitat
und die Chaotizitat 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 Veranderungen des EEG aufgrund externer oder interner
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 Verteilung
mit signikanten Latenz-Unterschieden zwischen anterioren und posterioren Elektroden.
Dies legte nahe, da diese ereigniskorrelierten Alpha-Oszillationen an multiplen
Orten entstehen. Ferner wiesen (a) die Unabhangigkeit 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 Beziehung
zwischen ereigniskorrelierten Alpha-Oszillationen und primar-sensorischer Verarbeitung
hin. Die Untersuchung von Antworten auf bimodale Reize (simultane auditorische
und visuelle Stimulation) zeigte eine signikante Zunahme der Amplitude im
Vergleich mit unimodalen Reizen. Demzufolge war es moglich, eine Beziehung zwischen
Gamma (30 ; 60Hz) Oszillationen und einem Proze anzunehmen, der die Information
tragt, da zwei sensorische Wahrnehmungen (im Rahmen der bimodalen Stimulation)
zu ein und demselben Reiz gehoren.
Insbesondere ist diese Arbeit die erste Untersuchung, in der die neue Methode
Wavelet-Entropie fur die Analyse ereigniskorrelierter Potentiale angepat und angewendet
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
Rodrigo Quian Quiroga
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

viii

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) Eventrelated
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 electroencephalography
and a review of previous approaches to its quantitative analysis.
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. Furthermore,
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 analysisoftheEEG.
Chapter x4 describes the theoretical background of the Wavelet Transform. Studies
where the Wavelet Transform is applied to Tonic-Clonic seizures and to eventrelated
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 dierenttype 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 ofLubeck, for giving me the opportunity towork 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. Iwould also liketomention 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.
Iamvery 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 ayoung student of physics
coming up with crazy ideas about Chaos and EEGs. My very special thanks to Dr.
Horacio Garca.
xi

xii

Contents
Zusammenfassung
Preface
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
ix
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
3.4.1 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . 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
4.5.2 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . 51
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
4.6.2 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . 61
Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Deterministic Chaos 68
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
5.5.1 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 Application to intracranially recorded tonic-clonic seizures . . . . . . . . 82
5.6.1 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 82
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 Wavelet-entropy 86
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Application to visual event-related potentials . . . . . . . . . . . . . . . . 89
6.3.1 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . 89
Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . 90
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, Lyapunov
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
(3:5;7:5Hz)rhythms, these rhythms later becoming slower in correlation with the starting
of the clonic phase. The dynamics of the frequency patterns during these seizures
was very interesting in relation to processes of neuronal fatigue, neurotransmitter disbalance,
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 responses
(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 independence
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 unimodal
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 perceptions
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. Clinical
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 activity
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 <
5000) in dierent locations of the head. Measures of the electric potentials can be
recorded between pairs of active electrodes (bipolar recordings) or with respect to a
supposed passive electrode called reference (monopolar recordings). These measures are
mainly performed on the surface of the head (scalp EEG) orby using special electrodes
placed in the brain after a surgical operation (intracranial EEG).
3

Figure 1: 10-20 system of montage of electrodes. Notation: F means frontal, C means
central, P means parietal, T means temporal, O means occipital and A means earlobe
reference. Modied from Reilly, 1993.
Scalp recordings
In scalp, or surface, EEG recordings, the most widely used placement of electrodes is
the so called 10;20 system, consisting in 20 electrodes (or sometimes less) uniformly distributed
along the head, generally referenced to 2 electrodes in the earlobes (Reilly, 1993
see g. 1). Normal scalp EEG recordings are usually taken with the subject relaxed,
in three modalities: 1) with eyes closed 2) with eyes open 3) under hyperventilation,
photostimulation, etc.
One important problem of scalp EEGs are the artifacts (Reilly, 1993). Artifacts are
alterations of the EEG due to head movements, blinking, muscle activity, electrocardiogram,
etc. (see an example in g. 2). Due to the very low amplitude of EEG signals
(usually between 10 ; 200V ), artifacts often contaminate the recordings restricting or
making impossible any analysis or interpretation.
Intracranial recordings
With intracranially implanted electrodes a better resolution can be achieved. Moreover,
possible alterations of the EEG signal in its way from the originating neural struc-
4

Figure 2: Normal 20 channels (10-20 system) eyes open EEG. Vertical lines show one second divisions. The arrow marks the
occurrence of a blinking artifact (best visible in frontal electrodes). Note also the appearance of alpha waves ( 10cycles=sec:),
best dened in parietal and occipital locations
5

tures up to the scalp are overcome and most of the artifacts can be avoided. There are
two maintypes of intracranial electrodes: 1) the subdural electrodes, which are arranged
in grids or strips placed on the brain (Arroyo et al., 1993), and 2) the deep electrodes,
which are arranged in a needle in order to study deep structures (Niedermeyer, 1993b).
The application of intracranial electrodes involves a major surgery and their use is restricted
to very particular cases, as for example the study of epileptic patients who are
candidates for a surgical resection.
1.1.1 Brain oscillations
In his rst publication, Berger (1929) already mentioned the presence of what he called
alpha and beta oscillations. The study of dierent types of rhythmicities of the brain and
their relation with dierent pathologies and functions keep the attention of researchers
since the beginnings of electroencephalography. Brain oscillations were divided in frequency
bands that have been related with dierent brain states, functions or pathologies
(see Niedermeyer, 1993a Steriade, 1993).
Alpha rhythms (7:5 ; 12:5Hz): they appear spontaneously in normal adults
during wakefulness, under relaxation and mental inactivity conditions. They are
best seen with eyes closed and most pronounced in occipital locations (see g. 2
sources and functions of alpha band will be discussed in sec. x4.5).
Beta rhythms (12:5 ; 30Hz): they are best dened in central and frontal locations,
they have less amplitude than alpha waves and they are enhanced upon
expectancy states or tension. They are traditionally subdivided in 1 and 2 oscillations.
Theta rhythms (3:5 ; 7:5Hz): they are enhanced during sleep and they play an
important role in infancy and childhood. In the awake adult, high theta activity
is considered abnormal and it is related with dierent brain disorders.
Delta rhythms (0:5 ; 3:5Hz): they are characteristic of deep sleep stages. Furthermore,
delta oscillations with certain specic morphologies, localizations and
rhythmicities are correlated with dierent pathologies.
Gamma rhythms (30 ; 60Hz): previously not of major interest with regard
to the surface EEG, they gained popularity after the cellular level experiments
of Gray and Singer (1989) and Gray et al. (1989) showing their relation with
linking of stimulus features into a common perceptual information (binding theory).
Gamma oscillations will be further studied in sec. x4.6.
6

1.1.2 EEG in Epilepsy
One important application of the EEG is to the study of epilepsy. The appearance of
abrupt high amplitude peaks (spikes), abnormal rhythmicities, \slowing" of the recording
and several other paroxysms are a general landmark of it, helping to identify, classify
and localize the seizures.
Epilepsy is a disorder of the normal brain function that aects about 1% of the
population, characterized by an excessive and uncontrolled activity of either a part or
the whole central nervous system. Historically, the EEG has been the most useful tool for
its evaluation, now complemented with the advances in imaging techniques, especially
the ones achieved by the Magnetic Resonance Imaging (MRI).
Given that ictal recordings (i.e. recordings during a seizure) were rarely obtained,
EEG analysis of epileptic patients usually relied on interictal ndings. In those interictal
EEGs, seizures are usually activated with photostimulation, hyperventilation and other
methods. However, one disadvantage of these stimulation techniques is that provoked
seizures do not necessarily have the same behavior as the spontaneous ones. The introduction
of long term Video-EEG recordings has been an important milestone providing
not only the possibility to analyze ictal events, but also contributing with valuable information
in those candidates evaluated for epilepsy surgery (see Quian Quiroga et al.,
1997a Porter and Sato, 1993 Kaplan and Leser, 1990 Gotman et al., 1985 Meierkord et
al., 1991). In this setting and following strict protocols, seizures are elicited by gradually
reducing antiepileptic drugs.
Classication
The classication of epileptic seizures is a very dicult task and leaded to several
controversies. I will present a simplied classication (for further details see Niedermeyer,
1993c). Tonic-Clonic seizures will be described with more detail since these are
the ones to be studied in the following chapters. Seizures can be classied in:
1. Partial seizures: they have a focal origin they can also evolve to a generalized
seizure (secondarily generalized).
Simple partial seizures: consciousness is not impaired. Depending on the
zone of the brain involved, they are characterized by focal motor movements,
sensory symptoms (simple hallucinations), autonomic symptoms (epigastric
sensation, sweating, pupillary dilatation, etc.) or psychic symptoms.
Complex partial seizures: involve a loose of consciousness. They are characterized
by dierent psychical sensations and motor automatisms. Since there
are many types of these seizures, the EEG is very variable but it can be gen-
7

erally characterized by low frequency discharges (3 ; 6 Hz) localized in some
region of the brain. They can begin as simple partial seizures.
2. Generalized seizures: the whole brain is involved.
Absence seizures (Petit Mal epilepsy): Consist of a sudden lapse of consciousness
with impairment of mental functions. They last about 5-20 seconds and
the EEG is characterized by generalized 3Hz spike-wave discharges (a spike
followed by aslow half period oscillation).
Tonic-clonic seizures (Grand Mal epilepsy): described below in more detail.
Other type of generalized seizures are the myoclonic, clonic and atonic ones.
Tonic-clonic seizures
A tonic-clonic (Grand Mal) seizure normally lasts about 40 ; 90 sec and it is characterized
by violent muscle contractions. An initial tonic phase with massive spasms
(i.e. extreme muscular tension without movement) is supplanted some seconds later by
the clonic phase with violent exor movements and characteristic rhythmic contractions
of the entire body until the ending of the seizure (Niedermeyer, 1993c). During these
seizures consciousness is impaired. Visual inspection of scalp recordings of these events
is often troublesome because the signal is obscured by muscle artifacts (see g. 3). In
most cases the analysis is conned to the interpretation of electrical abnormalities that
either precede or follow the tonic-clonic activity, thus neglecting the ictal phase (see
sec. x3.4). Therefore, since these seizures are very dicult to study with traditional
approaches, there will be one of the main type of signals to be studied with the methods
described in the following chapters.
1.2 Event related potentials (ERP)
EEG activity is present in a spontaneous way or can be generated as a response to an
external stimulation (e.g. tone or light ash) or internal stimulation (e.g. omission of
an expected stimulus). The alteration of the ongoing EEG due to these stimuli is called
event related potential (ERP), in the case of external stimulation also called evoked
potential (EP).
Owing to the low amplitude of the ERPs in comparison with the ongoing EEG, it is
a common practice to average several responses in order to visualize the evoked activity.
The reason for averaging is that ERPs have a denite latent period determined from
the stimulation time, and that they have a similar pattern of response which is more or
less predictable under similar conditions (Basar, 1980).
8

Figure 3: Scalp EEG of 18 channels during a Tonic-Clonic seizure. Note how the artifacts obscure the recording completely.
Vertical lines show one second divisions.
9

Modalities and paradigms
There are mainly three modalities of stimulation (for more details see Regan, 1989):
1. Auditory: stimuli are single tones of a determined frequency, or clicks with a
broadband frequency distribution.
2. Visual: stimuli are produced by a single light or by the reversal of a pattern as
for example a checkerboard.
3. Somatosensory: stimuli are elicited by electrical stimulation of peripheral nerves.
The rst two modalities can be combined in what is called bimodal stimulation.
Bimodal stimulation will be studied in sec. x4.6.
Sequence of stimuli are arranged in paradigms in order to study the responses to
dierent tasks. The most widely used paradigms are:
1. No-task evoked potentials: subjects are relaxed and instructed to perceive the
stimuli without performing any task.
2. Oddball paradigm: two dierent stimuli are presented in a pseudorandom order.
A frequent NON-TARGET stimulus is presented in 75% of the trials, and a deviant
stimuli called TARGET in the remaining 25% of the trials. Subjects are instructed
to pay attention to the appearance of the TARGET stimuli.
Transient responses: description of P100 and P300 peaks.
Responses can be classied in: 1) transient (occurring after the delivery of a shortlasting
stimulus, with interstimulus intervals greater than the responses), 2) driven (generated
with rapidly repeating stimuli), or 3) sustained (occurring throughout the duration
of a continuous stimulus). In the following we will only study transient responses.
As an example, gure 4 shows the averaged transient responses to pattern visual ERPs
generated with an oddball paradigm. One second pre- and one second post-stimulation
EEG recorded from the left occipital electrode are plotted. A positive peak at about
100ms after stimulation (P100) can be observed upon NON-TARGET and TARGET
stimulation. Since this peak does not depend on the task, it has a relatively short
latency (100ms) and it is best dened in the primary sensory visual cortex (occipital
locations), it is mostly related with primary sensory processing (Regan, 1989). Another
peak at about 450ms after stimulation (P300) appears only upon TARGET stimulation.
Since this response is task dependent and has long latency, it is traditionally related to
cognitive processes as signal matching, decision making, attention, memory updating,
etc. (Polich and Kok, 1995 Regan, 1989 Basar-Eroglu et al., 1992).
10

Figure 4: Averaged responses upon NON TARGET and TARGET stimuli of a left
occipital electrode in pattern visual evoked potentials
1.3 Relation between EEG and ERP
In contrast with the conventional view of ERPs as signals being added to the noisy
EEG, Basar (1980) pointed out that ERPs might arise from the ongoing EEG by means
of a resonance process. Resonance is awell known phenomenon of physics: if a driving
is applied to a system, for example a pendulum, the system will show very large and
increasing outputs if the driving frequency is similar to the natural frequency of the
system (for a physical description see Feynman et al., 1964). Basar (1980) introduced
the following concept for understanding the relation between EEG and ERP: In the
EEG several oscillations are occurring at the same time in a non-synchronized way, and
when a stimulus is applied, some of these frequencies can be enhanced by a resonance
phenomenon. Furthermore, it was assumed that these dierent enhanced oscillations
are related to transmitting information through the brain, having dierent \meanings"
and \functions" (see sec. x2.3.3). According to a classication of Galambos (1992)
event related oscillations can be evoked (originated by and time locked to the stimulus),
induced (originated by, but not time locked to the stimulus) or emitted (originated by
an internal process rather than by the stimulus).
11

I will summarize here some of the experimental evidence supporting this view (for
more details see Basar, 1980, 1998, 1999 Basar et al., 1999):
1. In three year old children, neither spontaneous, nor evoked 10Hz activity is obtained
(Kolev et al, 1994, Basar-Eroglu et al., 1994). Then, according to the
resonance theory, in this case alpha rhythm does not belong to the natural brain
frequencies. In other words, alpha oscillations can not be evoked because there is
no spontaneous alpha activity.
2. High amplitude pre-stimulus 10Hz activity reduces the evoked potential amplitudes
(Basar, 1980 Basar and Stampfer, 1985 Basar et al., 1992), thus showing
an inverse relation between the pre-stimulus EEG and the evoked responses. An
inuence of pre-stimulus theta EEG was also reported on visual evoked potentials
(Basar et al., 1998). Then, since the background EEG inuences the evoked
responses, it can not be merely considered as additive noise.
3. Converging results support the view that evoked or induced oscillations can be
correlated, at least partially, with dierent functions, as it will be described in
sec. x2.3.3, thus showing that the frequency analysis of the evoked responses is not
arbitrary.
In conclusion, following the resonance theory, the ERP can be seen as evoked synchronization,
frequency stabilization, frequency selective enhancement and/or phase reordering
of the ongoing EEG, and it should not be interpreted as an additive component
to a noisy background EEG.
12

2 Fourier Transform
2.1 Introduction
Time signals can be represented in many dierent ways depending on the interest in
visualizing certain given characteristics. Among these, the frequency representation is
the most powerful and standard one. The main advantage over the time representation
is that it allows a clear visualization of the periodicities of the signal, in many cases
helping to understand underlying physical phenomena. Frequency analysis, developed
by Jean Baptiste Fourier (1768-1830), reached innumerable applications in mathematics,
physics and natural sciences. Furthermore, the Fourier Transform is computationally
very attractive since it can be calculated by using an extremely ecient algorithm called
the Fast Fourier Transform (FFT Cooley and Tukey, 1965).
This chapter starts with a brief mathematical background of Fourier Transform and
then, applications of Fourier analysis to EEG signals will be reviewed. Four main
applications will be described: analysis of frequency bands, topographical mapping,
analysis of evoked responses and coherence.
2.2 Theoretical background
Under mild conditions, the Fourier Transform describes a signal x(t) (I will consider
only real signals) as a linear superposition of sines and cosines characterized by their
frequency f.
where
x(t) =
X(f) =
Z +1
;1
Z +1
;1
X(f) e i2ft df (1)
x(t) e ;i2ft dt (2)
are complex valued coecients that give the relative contribution of each frequency f.
Equation 2 is the continuous Fourier Transform of the signal x(t). It can be seen as an
inner product of the signal x(t) with the complex sinusoidal mother functions e ;i2ft ,
i.e.
X(f) = (3)
Its inverse transform is given by eq. 1 and since the mother functions e ;i2ft are
orthogonal, the Fourier Transform is nonredundant and unique.
Let us consider in the following that the signal consists of N discrete values, sampled
every time .
13

x(n) =fx 0 x 1 :::x N;1 g = fx j g (4)
where x j is the measurement taken at a time t j = t 0 + j.
The discrete Fourier Transform of this signal is dened as:
and its inverse as:
X(k) =
N;1
X
n=0
x(n) e ;i2kn=N k =0:::N ; 1 (5)
x(n) = 1 N
N;1
X
k=0
X(k) e i2kn=N (6)
where, since we are considering real signals, the following relation holds: X(k) =X (N;
k), and the discrete frequencies are dened as:
f k =
k
(7)
N
Note that the discrete Fourier Transform gives N=2 independent complex coecients,
thus giving a total of N values as in the original signal and therefore being nonredundant.
Clearly, thefrequency resolution will be
f = 1
N
(8)
Aliasing
The frequency f N = 1
2 , corresponding to k = N 2
in eq. 7, is called the Nyquist
frequency and it is the highest frequency that can be detected with a sampling period
. If the signal x(n) has frequencies above the Nyquist frequency, these ones will be
processed just as though they are in the range f k < f N , thus giving a spurious eect
called aliasing (Jenkins and Watts, 1968 Weaver, 1989). The greater this excess is, the
greater the errors and the less adequate is the sampling rate for representing the signal.
From the complex coecients of eq. 5, the periodogram 1 can be obtained as:
1 Due to the high complexity of the EEG signal, it will be convenient to consider it as a realization
of a stochastic process (Lopes da Silva, 1993a Dummermuth and Molinari, 1987). However, it should
be stressed that this is a mathematical and not a biophysical model. On the other hand, in section
x5, the EEG will be considered as a realization of a non-linear deterministic \chaotic" system. The
deterministic/stochastic nature of the EEG will be discussed in sections x5.7 and x7.1.3. Regarding the
EEG as a random signal, the periodograms can be interpreted as estimators of the power spectrum.
14

I xx (k) =jX(k)j 2 = X(k) X (k) (9)
where denotes complex conjugation. The sample cross spectrum can be similarly
dened by making the product of eq. 9 between two time series x(n) andy(n) as follows
I xy (k) =X(k) Y (k) (10)
where X(k) and Y (k) are the Fourier Transforms of x(t) and y(t) respectively. The
sample cross spectrum gives a measure of the linear correlation between two signals for
the dierent frequencies f k , dened as in eq. 7. Since it is a complex measure, it can be
described by an amplitude and a phase. A useful measure of the amplitude is obtained
with its square value, normalized in the following way:
; 2 xy(k) =
jI xy (k)j 2
I xx (k) I yy (k)
(11)
Equation 11 is known as the squared coherence function, and it will give values
tending to 1 for highly linearly correlated signals and to 0 for non correlated ones.
The counterpart of the coherence is the phase function, dened as
= arctan ;=I xy(k)

2.3.1 Frequency bands
Since EEG data is non-stationary and contains many artifacts, the rst step when
making a Fourier analysis is to choose several representative data samples, generally of
1or2 seconds, free of artifacts (Nuwer et al., 1994).
Before computing the Fourier Transforms, eachepochismultiplied by an appropriate
windowing function (normally a Hanning window is used) in order to avoid border
problems (leakage) (Jenkins and Watts, 1968 Dumermuth and Molinari, 1987). Then,
the periodograms of the epochs are computed and averaged.
As stated in sec. x1.1.1,itisvery useful to group the frequencies in dierent bands.
Several quantitative parameters were dened from these frequency bands, as for example
the relative power between them (being the most used the alpha/theta ratio), reactivity
(ratio between eyes closed/eyes open alpha activity), asymmetry index ( (left power -
rightpower)/(left power + rightpower) ), etc. (Nuwer et al., 1994). Moreover, statistical
techniques can be used in order to establish normal ranges and their deviation with
several pathologies (John et al., 1987).
2.3.2 Topographical mapping
The information of the spectrograms of the dierent electrodes can be arranged in
topographical maps (Gotman, 1990a Gevins, 1987 Lehmann, 1987 Lopes da Silva,
1993a). Usually, these algorithms use linear or quadratic interpolations between the 3
or 4 nearest recording sites. Since the use of single electrode references can distort the
maps near the reference site (Lehmann 1971, 1987), one critical point in these plots is
the election of the reference. Several suggestions were proposed in order to avoid this
distortion, among them the use of averaged references and also the use of the average
of derivatives of the EEG activity (Lehmann, 1987 Lopes da Silva, 1993b). One nal
issue to be considered is how to project a three dimensional head into a two dimensional
map.
The use of topographic plots started more than 30 years ago by implementing different
techniques (Walter et al., 1966 Lehman, 1971), but it was after the introduction
of color topographic maps by Duy et al. (1979) that they became widely accepted,
and started to be used in several medical centers with the appearance of commercial
systems.
With these plots is easy to visualize asymmetries and to localize frequency band
activities. Furthermore, the topographic maps complement the quantitative parameters
described in the previous sections in the characterization of normality and in the study
and diagnosis of several pathologies (Duy, 1986 Maurer and Dierks, 1991 Pfurtscheller
16

17
Figure 5: Topographical mapping of the dierent EEG frequency bands from a normal EEG recording taken with eyes closed.

and Lopes da Silva, 1988 Garcia and Quian Quiroga, 1998).
Figure 5 shows the topographic mapping of an EEG recording of a normal subject
with eyes closed. Five dierent frequency bands are plotted. As expected, the power
is homogeneously distributed in all frequency bands, except in alpha, where there is a
simetrical increase in the posterior locations. This increase reects the presence of the
normal spontaneous alpha activity described in section 1.1.1.
2.3.3 Frequency analysis of evoked responses
Considering the proposed relation between EEG and ERP described in section x1.3,
the physiological functions of the frequency bands can be analyzed by studying their
response to dierent types of stimulation. Basar (1980) described a combined analysis
procedure for achieving this goal. The procedure relies on the study of enhancements of
dierent frequency bands as a response to stimulation. After a Fourier analysis of the
pre- and post-stimulus EEG, frequency ranges are determined and bands are separated
by using a selective ltering.
details see Basar, 1980):
I will summarize here some of these results (for more
Delta enhancements were correlated with cognitive processes like signal matching
or decision making by using auditory and visual oddball paradigms (Stampfer and
Basar, 1985 Basar-Eroglu et al., 1992 Schurmann et al., 1995 Demiralp et al.,
1999).
Theta enhancements were correlated with cognitive tasks like focused attention
and signal detection by using omitted stimulus paradigms in free moving cats
(Basar-Eroglu et al., 1991) and in humans (Demiralp and Basar, 1992).
Alpha enhancements were correlated to primary sensory processing in several
experiments using cross-modality stimulation in cats and humans (Basar, 1980
Basar et al., 1992 Basar and Schurmann, 1996).
Gamma enhancements were correlated with binding of features (Gray and Singer,
1989 Gray et al., 1989) and they were suggested to be involved in several other
sensory and cognitive processes (Basar-Eroglu et al., 1996a,b Schurmann et al.,
1997).
2.3.4 Coherence
The coherence function gives a measure of the linear correlation between two EEG
electrodes. The main advantage over previous \cross-correlation" measures (see Gevins,
18

1987) is that coherence gives the correlation as a function of frequency, thus allowing the
study of spatial correlations between dierent frequency bands. There is an extensive
literature about the use of coherence in EEGs. This section will only show some examples
and for a more complete review I suggest the work of Lopes da Silva (1993a).
Lopes da Silva et al. (1973a) computed thalamo-cortical and cortico-cortical coherences
for checking the validity of the assumption of a thalamic pacemaker of cortical
alpha rhythms, as proposed by Andersen and Andersson (1968). They found that
cortico-cortical coherences were higher than the thalamo-cortical ones, thus stressing
the importance of cortical mechanisms in the spread of alpha rhythms, in disagreement
with the thalamic pacemaker theory.
Storm van Leeuwen et al. (1978) used coherence analysis for dierentiating alpha
and murhythms. It is interesting to remark that since these frequencies are very similar,
this is a very dicult task to achieve with normal spectrograms.
Other interesting application of coherence is for the denition of characteristic \lengths
of synchrony" between neuronal populations. This is achieved by establishing lengths
with signicant coherence. Following this approach, after studying 189 children, Thatcher
et al. (1986) proposed a \two-compartmental model" stating that two separate sources
of EEG coherences are present: the rst one determined by short length axonal connections
and the second one by long distance connections.
Bullock and coworkers (1989, 1995a, 1995b) criticize the use of scalp electrodes for
measuring spatial structure of synchrony and they studied the spatial structure of coherence
with intracranial electrodes. In contradiction with previous reports of population
synchrony of the order of several centimeters, they reported that coherence varied in
the order of a few millimeters, thus showing a microstructure of the cell populations.
Furthermore, due to this small range structure, they criticize the view of the EEG as a
linear superposition of independent oscillators.
Several attempts were performed in order to analyze epileptic seizures with coherence
analysis. Gerch and Goddard (1970) used coherence for identifying the location of an
epileptic focus in a cat brain. By seeing a signicant decrease of coherence between two
channels after the subtraction of the eect of a third one (this method called partial
coherence), they deduce that the third one was driving the other two, thus being the
focus of the seizure.
Brazier (1972), introduced the measurement of time delays between channels (see
eq. 12) in order to follow routes of propagation and establish the origin of seizure activity.
Gotman (1983, 1987) proposed a modication of this algorithm by calculating the time
delay from the slope of a straight linettedto the phase spectrum when the coherence
is signicant. He showed that time delays most often indicate a lead from the side of
19

onset in dierent type of seizures in animals and humans. However, a straight line is
not always possible to t in the phase spectrum and coherence values are often very low
in order to extract any meaningful measure from the phase spectrum.
2.4 Conclusion
In this chapter the basic background and several applications of Fourier Transform to
EEG analysis were described. One important application is the comparison between the
power of dierent frequency bands and their topological distribution used to discriminate
between normal subjects and pathological cases. This analysis is already adapted to
many commercial systems and used in several medical centers. A word of caution
should be mentioned at this point. Although quantitative parameters can be easily
extracted from the EEG in an automated way, the visual inspection of the recordings
should not be left aside. This helps to avoid misinterpretations due to nonstationarity,
artifacts or others. At this respect, topographic mapping and quantitative values should
be considered as a complement and not as a replacement of the visual inspection of the
EEG. Fourier Transform can be used for studying the functions and sources of dierent
brain oscillations by analyzing EEGs or ERPs. Some examples of how the measure
of coherence can lead to interesting physiological interpretations were also presented.
Measures of connectivity or synchrony between neural populations can be established
and tested in several states and pathologies. Since an epileptic seizure is an abnormal
synchronization of cell groups, coherence appears as an ideal tool for measuring it.
Several works tried to use time delays calculated from phase dierences as a method for
source localization. However, the measure of time delays is dicult since it depends on
having high coherence values, this being not the usual case.
Despite of the fact that Fourier Transform proved to be very useful in EEG analysis,
I would like to mention here two main disadvantages. First, Fourier Transform
requires stationarity of the signal, the EEGs being highly non-stationary. Second, with
Fourier Transform the time evolution of the frequency patterns is lost. The methods
to be presented in the following sections are developed in order to overcome these two
limitations.
20

3 Gabor Transform (Short Time Fourier Transform)
3.1 Introduction
Since the Fourier Transform is based in comparing the signal with complex sinusoids that
extend through the whole time domain, its main disadvantage is the lack of information
about the time evolution of the frequencies. Then, if an alteration occurs at some time
boundary, the whole Fourier spectrum will be aected, thus also needing the requirement
of stationarity.
In many occasions, signals have time varying features that can not be resolved with
the Fourier Transform. A traditional example is a chirp signal (i.e. a signal characterized
by awell dened but steadily rising frequency). In the case of a chirp, Fourier analysis
can not dene the instantaneous frequency because it integrates over the whole time,
thus giving a broad frequency spectrum.
This is partially resolved by using the Gabor Transform, also called Short-Time
Fourier Transform. With this approach, Fourier Transform is applied to time-evolving
windows of a few seconds of data smoothed with an appropriate function (Cohen, 1995
Chui, 1992 Qian and Chen, 1996). Then, the evolution of the frequencies can be
followed and the stationarity requirement is partially satised by considering the signals
to be stationary in the order of the window length (Lopes da Silva, 1993a). In other
words, the procedure consists in breaking the signal in small time segments and then in
applying a Fourier Transform to the successive segments.
One application of Gabor Transform is to the analysis of Tonic-Clonic (Grand Mal)
seizures (see sec. x1.1.2). As we will see in this chapter, these seizures have interesting
time evolving frequency characteristics that can not be resolved with methods suchasthe
visual inspection of the EEG or the Fourier Transform (Quian Quiroga et al., 1997b
Blanco et al., 1998b). Moreover, since it is dicult to extract any information from
the traditional time-frequency graphic representation of the Gabor Transform, called
spectrogram, I will introduce three parameters, the relative intensity ratio and the mean
and maximum band frequencies, dened from the Gabor Transform, in order to obtain
quantitative measures of the dynamics of epileptic seizures.
After a brief theoretical outline, I will describe two dierent type of applications.
First, the application to scalp recorded tonic-clonic seizures (Quian Quiroga et al.,
1997b) and second, the application to intracranially recorded tonic-clonic seizures (Blanco
et al., 1995b,1996a).
21

3.2 Theoretical background
The Gabor transform of a signal x(t) is dened as follows:
G D (f t) =
Z 1
;1
x(t 0 ) g D(t 0 ; t) e ;i2ft0 dt 0 (14)
where denotes complex conjugation. Note that G D (f t) is the same as the Fourier
Transform (eq. 2) but with the introduction of the window g D(t 0 ; t) of wide D and
center in t. Furthermore, in analogy with the Fourier Transform, the Gabor Transform
can be considered as an inner product between the signal and the complex sinusoidal
functions e ;i2ft0
modulated by the window g D (compare with eq. 3), i.e.
G D (f t) = (15)
The window function g D is introduced in order to localize the Fourier Transform of
the signal at time t. For this reason, the window function mustbepeaked around t and
falling o rapidly, thus emphasizing the signal in the rst case and suppressing it for
distant times. Several window functions can be used for achieving this, among them,
Gabor (1946) proposed the use of a Gaussian function:
g (t) =
1=4
e
;
2 t2 (16)
Since the Fourier Transform of a Gaussian is again a Gaussian, this function allows a
simultaneous localization in time and frequency (see appendix xA). It should be noted
that the Gaussian is described in function of a positive constant and the window
function in eq. 14 was dened in function of the window length D. This is because
Gaussian functions do not have compact support (i.e. they are not 0 outside a certain
boundary). However, they approach asymptotically to 0 with a rate determined by
the parameter . In consequence, the Gaussian function can be truncated to have a
length D, by assuming that in the borders their values are nearly 0 due to the use of a
reasonable decay . Then, in the case of Gaussian windows the Gabor Transform can
be explicitly dened as a function of the parameter
G D (f t) ! G D(f t) (17)
Let us return now to the general case of any arbitrary window function and consider
the inverse transformation. Taking the inverse Fourier Transform of eq. 14 and after
some algebraic manipulation we obtain:
x(t) = 1
k g D k
Z 1 Z 1
;1
;1
G D (f t 0 ) g D (t ; t 0 ) e i2ft df dt 0 (18)
22

where k g D k= R 1
;1 jg D(t 0 )j 2 dt 0 . Eq. 18 is the inverse Gabor Transform and implies that
the original signal can be completely reconstructed from the coecients G D (f t).
Gabor Transform is highly redundant because it gives a time-frequency map from
every time value of the original signal. In order to decrease redundancy, asampled Gabor
Transform can be dened by taking discrete values of time and frequency, i.e.:
G D (f t) ! G D (mF nT ) (19)
where F and T represent the frequency and time sampling steps. Small F steps are
obtained by using large windows, and small T steps are obtained by using high overlapping
between successive windows. Depending on the resolution required, a proper
choice of F and T will decrease the redundancy and save computational time, but the
price to pay is that the reconstruction will be no longer straightforward as with eq. 18
(Qian and Chen, 1996).
In the following, for convenience I will keep the notation of the continuous Gabor
Transform dened in eq. 14. The spectrum can be dened as
I(f t) =jG D (f t)j 2 = G D(f t) G D (f t) (20)
and a time-frequency representation of the signal can be obtained by using a recursive
algorithm that slides the time window and plots the energy (I) as a function of the
frequency and time. These plots, called spectrograms (see g. 8), give an elegant visual
description of the time evolution of the dierent frequencies, but it is dicult to extract
from them any quantitative measure.
In order to quantify this information, I will dene the band power spectral intensity
for each one of the EEG traditional frequency bands i (i = :::) as:
I (i) (t) =
Z f
(i) max
f (i) min
I(f t) df i = ::: (21)
where ( f (i) min f (i) max ) are the frequency limits for the band i. The division and grouping
of the spectrum in frequency bands is not arbitrary, since as showed in section x1.1.1
and section x2.3.3, EEG bands are correlated with dierent sources and functions of the
brain.
Obviously the total power spectral intensity will be
I T (t) = X i
I (i) (t) i = ::: (22)
and we can dene the band relative intensity ratio (RIR) foreach frequency band i as
23

Figure 6: Procedure for calculating the Gabor Transform and the RIR.
RIR (i) (t) = I (i) (t)
I T (t)
100 (23)
Figure 6 illustrates the iterative procedure for calculating the RIR. First, a window
centered in a time T 1 is applied to the data (step A) and the Gabor Transform of this
windowed data is calculated by using eq. 14 (step B). Then, the RIR is calculated and
plotted (step C). Finally, the window is displaced a xed time and the procedure is
repeated, obtaining a time representation of the RIR.
For further quantitative analysis, we dene the band mean weight frequency as
=
h Z f
(i) max
f (i) min
24
I(f t) f df
i
=I (i) (t) (24)

and the band maximum peak frequency (f (i)
M (t)) of the i-band, as the frequency value for
which I(f t) is maximum in the interval ( f (i) min f (i) max )ata time t, i.e.
I(f M t) >I(f t)
8 f 6= f M (f (i)
min f(i) max) (25)
Uncertainty Principle
One critical limitation appears when windowing the data due to the Uncertainty
Principle (Cohen, 1995 Chui, 1992 Qian and Chen, 1996 Kaiser, 1994). If the window
is too narrow, the frequency resolution will be poor, and if the window is to wide, the
time localization will be not so precise. Or in another words, sharp localizations in
time and frequency are mutually exclusive because a frequency can not be calculated
instantaneously. If we denote by t the time uncertainty (time duration) and by f the
uncertainty in the frequencies (frequency bandwidth), in the case of normalized signals
the Uncertainty Principle can be expressed as follows (see appendix xA for a proof):
t f 1
(26)
4
This limitation becomes important when the signal has transient components localized
in time as in the case of EEGs or ERPs. Gabor (1946) suggested a Gaussian
function as the smoothing function, owing to its good localization in time and frequency.
In fact, with a Gaussian function eq. 64 is an equality and furthermore, the equality also
holds for the mother function of the Gabor Transform, g D e i2ft (see appendix xA).
3.3 Application to intracranially recorded tonic-clonic seizures
3.3.1 Methods and Materials
Seizure EEG recordings were obtained from a 21 years old male patient. A nine hours
recording was performed with 12 depth electrodes ( each electrode having 5 to 15 contacts
) placed in the epileptogenic zone and propagating brain areas. Each signal was
amplied and ltered using a 1;40Hz band-pass lter. A 4 pole Butterworth lter was
used as low-pass lter and as an anti-aliasing scheme. After 10 bits A/D conversion the
EEG data was written continuously onto a hard drive with a sampling rate of 256Hz
per channel. Selected data sets of ictal and interictal activity were stored for subsequent
o-line analysis. We established the necessary frequency resolution in f = 0:25Hz
in order to have enough frequency values for calculating the mean and maximum band
frequencies and the time resolution was set to t = 0:25sec. This was done by using
a Gaussian window of D = 4sec width 2 with slide displacement steps of 0:25sec. The
2 from eq. 8, f = 1 = 1
= 1 Hz =0:25Hz
N 4256Hz(1=256Hz) 4
25

decay constant of the Gaussian was set to = 2=D 2 (see eq. 16), what makes their
values at the borders less than 5% than the ones of the peak, thus avoiding leakage
problems 3 . Finally, the frequency band limits were dened in the following way: delta
(0:5 ; 3:5Hz), theta (3:5 ; 7:5Hz), alpha (7:5 ; 12:5Hz ), beta-1 (12:5 ; 18Hz) and
beta-2 (18 ; 30Hz).
3.3.2 Results
Figure 7 shows 64 sec. of the EEG signal corresponding to one depth electrode in the left
hippocampus. The epileptic seizure starts about second 10, and nishes about second
54.
The spectrogram corresponding to the previous EEG recording is shown in g. 8.
We canobserve from this plot that after second 10, when the seizure starts, there is an
increase of the frequencies between 5 ; 20Hz, in agreement with what it can be seen
by visual inspection of the EEG. Furthermore, there is an abrupt increase of the low
frequencies (less than 3Hz) in the post-seizure stage, clearly correlated with the slow
oscillations dominating the last part of the recording in g. 7. Although this plot gives a
very elegant representation of the signal, it is dicult to extract more information from
it.
Figure 9 shows the relative intensity ratio (RIR), corresponding to the same EEG
recording. Looking at the time evolution of the RIR a good agreement with the signal
morphology can be also established, but in this case the evolution of the frequency bands
can be followed with more detail.
The main feature seen in this plot is a clear decrease of the delta band during the
seizure, with a dominance of alpha and some contribution of theta. In this case, higher
frequencies ( 1 and 2 ) do not give an important contribution during the seizure.
In gure 10 we show the time evolution of the mean and the maximum frequency
(f (i)
M (t) and see eq. 24 and eq. 25) for the dierent bands considered in this
work.
The most interesting result is the similarity between the mean and the maximum
frequency in the delta band between seconds 28 and 40. This reects the presence of a
well dened peak in this band, although, as seen in g. 12 its intensity at this time is
relatively low.
3 for simplicity the normalization constant ;
1=4 of eq. 16 was arbitrarily set to 1.
26

Figure 7: Intracraneal seizure recording. Seizure starts at about second 10 and ends at
about second 54.
27

Figure 8: Spectrogram of the previous EEG recording.
Figure 9: RIR of the previous seizure recording.
28

Figure 10: Mean (soft line) and maximum (line with breaks) band frequencies corresponding
to the same seizure recording.
29

3.3.3 Discussion
The RIR and the mean and maximum band frequencies give quantitative information
dicult to obtain from the visual inspection of the EEG recording or from the spectrogram.
Moreover, with the RIR the evolution of the dierent frequency bands can be
followed accurately.
The evolution of the frequency bands can be more interesting than the following
of single peaks, since the rst ones can be related with processes and sources of the
brain, as described in sec. x1.1.1. Moreover, the access to quantitative values allows
the statistical study of interesting features with a large sample of seizures, as it will be
shown in sec. x3.4.
The main result was that during the seizure there was an abrupt decrease of delta
activity until the ending of the seizure when it raised up again. The seizure was mostly
dominated by alpha oscillations, result that will be statistically veried in sec. x3.4 with
the analysis of several scalp seizure recordings.
Although during the seizure delta activity did not reach high amplitudes in comparison
with the ones of alpha and theta, it was the only one who showed a signicant
similarity between the mean and maximum frequencies, thus showing the presence of a
well dened peak. It can be conjectured that during the seizure delta oscillations act as
a latent pacemaker obscured by the higher amplitude of the alpha and theta ones, until
the ending of the seizure when it recruits more cell assemblies and completely dominates.
3.4 Application to scalp recorded tonic-clonic seizures
3.4.1 Methods and Materials
Subjects and data recording
Twenty tonic-clonic seizures from eight epileptic patients were analyzed. Scalp electrodes
with linked earlobes references were applied following the 10 ; 20 international
system (see sec. x1.1).
Each signal was digitized at 409:6Hz 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, the signal was digitally ltered with a 1;50Hz bandwidth lter and stored, after
decimation, at 102:4Hz inaPChard drive.
Gabor Transform and data processing
Analysis for each event included one minute of EEG before the seizure onset and two
minutes containing the ictal and post-ictal phases. All 3 minutes were analyzed from
the right central (C4) electrode, choosing this electrode after visual inspection of the
30

EEG as the one with least amount of artifacts.
The RIR was calculated as described in g. 6 by applying a Gaussian window with a
width of D =2:5sec: and slide displacement steps of 1:25sec: (half-overlapping windows).
The selection of D = 2:5sec: allows a frequency resolution f = 0:4Hz, which was
considered enough for obtaining a good measure of the RIR. The constant of the
Gaussian function (see eq. 16) was set as in section x3.3.1. Traditional frequency bands
were set to 1 ; 3:5, 3:5 ; 7:5, and 7:5 ; 12:5Hz respectively. Given that beta 1 and beta
2 are closely related to muscle artifact those bands were excluded from the analysis.
Plateau criteria
Clear decrements in delta activity were seen during the seizures. In order to quantify
this observation, the mean relative intensity ratio (MRIR) for the delta band was
calculated in the pre-ictal phase and compared with the value in the low intensity zones
(plateaus) observed during the seizure. Pre-ictal MRIR was dened as the mean value
of the RIR in the minute before the seizure, discarding those areas contaminated by
artifacts (dened by visual inspection of the EEG). In the ictal phase, plateaus were
dened according to the following criteria:
1. Plateaus must last at least 10sec in order to avoid local variations
2. MRIR(ictal) < 0:3 MRIR(pre ; ictal) and
3. Standard error of the plateau MRIR (SEM) must be less than 1toconrm low
variance.
Although the choice of these criteria is arbitrary, no plateaus were identied in the
pre-ictal phase, strongly indicating that the appearance of a plateau in the ictal phase
reects a dynamical change rather than a statistical phenomenon.
3.4.2 Results
As an example of the analysis performed I will describe the results of one of the seizures
and then I will summarize the global ndings.
Figure 11 discloses three minutes of EEG data of one typical seizure and the RIR
of this signal is shown in gure 12. Pre-ictal phase is characterized by asignalof50V
amplitude with a dominance of delta rhythms (pre-ictal delta MRIR 50%). Seizure
starts at second 80 (marked with an S in both graphs) with a discharge of slow waves
superposed by fast ones with lower amplitude. This discharge lasts approximately 8
seconds, has a mean amplitude of 100V , and, as seen in gure 12, produces a marked
rise in delta band, which reaches 90% of the RIR. Afterwards, seizure spreads making
31

the analysis of the EEG more complicated due to muscle artifacts however, it is possible
to establish the beginning of the clonic phase at around second 123 and the end of the
seizure at second 155 (marked with an E in both graphs) where there is an abrupt decay
of the signal.
Although it is dicult to extract any information from the EEG during the seizure,
in gure 12 we can follow its frequency pattern. From second 90, delta activity decreases
abruptly to values lower than 10% of the RIR, and theta and alpha bands alternatively
dominate. We also observe that the starting of the clonic phase is correlated with a
rise of theta frequencies and after second 140, when clonic discharges became more
separated, delta activity rises up again until the end of the seizure, also maintaining
this behavior in the post-ictal phase.
We can conclude from this example that the seizure was dominated by alpha and
theta rhythms with a corresponding abrupt decrease of delta activity. By applying
the criteria explained in the previous section, a plateau of delta decrement was dened
between seconds 99 and 138, lasting 39 seconds, having very low variance (0.14) and
also having a very low ictal to pre-ictal delta MRIR (2%).
Considering the whole group, a stereotyped pattern was identied in 14=20 (70%) of
the seizures. This pattern was characterized by a signicant reduction in delta activity
during the seizures, implying that they were dominated by theta and alpha rhythms
until the ending of the seizure when delta activity started to rise again.
One critical point in our results is the possible distortion due to spatial propagation
of the seizure, duetothefactthat data of C4 electrodes was analyzed, and the sources
of the seizures were mostly in temporal locations. In order to overcome this, Gabor
Transform was also applied to T3 and T4 electrodes, obtaining similar results to the
ones reported with C4 electrodes.
3.4.3 Discussion
The most robust nding is that in 70% of the seizures a signicant reduction in delta
activity was observed by applying the plateau criteria. This was usually seen at the
beginning of the seizure, thus emphasizing the dominance of alpha and theta bands
during the initial phases of a tonic-clonic seizure. This result is in agreement with our
previous observations using Gabor Transform to analyze ictal patterns recorded with
depth electrodes (Blanco et al., 1995b, 1996a, 1996b, 1997). Although the selection of
the plateau criteria might have inuenced these results, slight changes in the denition
of the plateau showed no signicant variations. Also, results do not depend on the
patient studied, and owing to the great variation in age, antiepileptic drugs, and source
of the seizure between patients, we can conclude that the results are not dependent on
32

Figure 11: Scalp EEG of the right central channel during a Tonic-Clonic seizure
Figure 12: RIR corresponding to the previous seizure
33

these factors.
A rise in delta activity was observed towards the end of the seizures. This pattern
was also seen with depth recordings, thus making unlikely the possibility of being generated
by artifacts. Neuronal fatigue, decrement in neuronal ring and preponderance of
inhibitory mechanisms are critical factors in the mechanisms underlying this observation
which prompts further research.
These results were obtained by avoiding muscle artifacts with Gabor Transform, thus
supporting their utility for the evaluation of tonic-clonic seizures, something dicult to
assess in scalp recorded EEGs. As Niedermeyer (1993c) pointed out:
\Muscle activity rapidly obscures the recording the vertex deviation, however,
may remain artifact-free due to the lack of underlying muscles. Informative
Grand Mal recordings can be secured only from patients with muscle
relaxation from curarization 4 and articial respiration".
Gotman et al. (1981) avoid this problem by the use of lters. However, he pointed
out that ltering the signal has several disadvantages. On the one hand it is impossible
to separate brain and muscle activity in the EEG, and further, it is well known
that ltering high frequencies also aects the morphology of the low ones. Gastaut and
Broughton (1972) instead, described a frequency pattern during a tonic-clonic seizure
of a curarizated patient. In the rst seconds after the seizure onset, they found an
epileptic \recruiting rhythm" of 10Hz associated with the tonic phase that lasted approximately
10 seconds later, as the seizure ended, there was a progressive increase of
the lower frequencies (5 ; 6Hz) associated with the clonic phase. Our ndings were
similar to the ones highlighted by Gastaut and Broughton, but recruiting rhythms were
observed to be also in the theta range or, as seen in most cases, uctuating between
alpha and theta. Darcey and Williamson (1985) also obtained similar patterns by analyzing
seizures recorded with depth electrodes. They reported an activity characterized
by 10Hz at the onset of the seizure, frequency that declined as the seizure ended. The
results presented here diers from these previous attempts in that a quantication of
the Gabor Transform was performed with the RIR, thus allowing the analysis of the
frequency content of the seizure. One important point to note is that these results were
obtained with scalp recordings and without the need for curare or any ltering method.
4 Curare is a drug that blocks neuromuscular transmission, thus inhibiting the muscular artifacts
characteristic of the Grand Mal seizures
34

3.5 Conclusion
In this chapter I described the use of quantitative parameters dened from the Gabor
Transform applied to the analysis of EEG signals. These parameters give an accurate
description of the time evolution of EEG rhythms, something dicult to perform from
the spectrograms. Spectrograms are more suitable for the analysis of long recordings
in which only large scale variations, of the order of minutes or hours, are relevant (see
examples of spectrogram analysis in Lopes da Silva, 1993b Gotman, 1990a). In this
context, spectrograms are for example very useful for discriminating between dierent
sleep stages, due to the slow variation of the frequency patterns. However, as showed in
sec. x3.3.2, spectrograms are not so suitable for the analysis epileptic seizures, in which
frequency patterns change in the order of seconds.
On the other hand, the quantication by means of the RIR allowed the performance
of a statistical analysis of the frequency content of Grand Mal seizures, by studying 20
scalp seizure recordings. This analysis showed marked decreases of the delta band during
the seizure due to the dominance of alpha and theta band at this stage. Since intracranial
recordings are nearly free of artifacts, the fact that the same pattern was seen in both
situations reinforces the idea that the results obtained with scalp electrodes were not
an spurious eect of muscle activity, orofthe procedure achieved for their elimination.
The introduction of the mean and maximum band frequencies allowed the study of
the time evolution of frequency patterns within each frequency band. It is interesting
to note that with this analysis it was possible to establish the presence of a well dened
delta oscillation during the seizure, in spite of the fact that its energy was very low and
remained latent until the ending of the seizure, when it reached very high amplitudes
and dominated the EEG.
Other interesting point to note is that although the grouping in frequency bands
implies a loose of frequency resolution, it can be more useful than the study of single
frequencies or peaks, due to the relations between frequency bands and functions or
sources of the brain.
The use of the present time-frequency analysis, together with the clinical patient
history and the visual assessment of the EEG, can contribute to the identication of the
source of epileptic seizures and to a better understanding of its dynamic.
Certainly, Gabor Transform is not intended to replace conventional EEG analysis,
but rather to complement it and also to provide further insights into the underlying
mechanisms of ictal patterns. In this context, RIR allows a fast interpretation of several
minutes of frequency variations in a single display, something dicult to perform with
traditional scalp EEG.
35

4 Wavelet Transform
4.1 Introduction
Fourier Transform consists in making a correlation between the signal to be analyzed
and complex sinusoids of dierent frequencies (see upper part of g. 13). As stated in
chapter x2, Fourier Transform gives no information about time and requires stationarity
of the signal. By \windowing" the complex sinusoidal mother functions of the Fourier
Transform, a time evolution of the frequencies can be obtained just sliding the windows
throughout the signal. This procedure, called the Gabor Transform, consists in correlating
the original signal with modulated complex sinusoids as showed in the middle part of
g. 13 (for details see section x3.2). Gabor Transform gives an optimal time-frequency
representation, but one critical limitation appears when windowing the data due to the
Uncertainty Principle (see x3.2 and xA Cohen, 1995 Strang and Nguyen, 1996 Chui,
1992). If the window is too narrow, the frequency resolution will be poor, and if the
window is too wide, the time localization will be not so precise. Data involving slow
processes will require wide windows and on the other hand, for data with fast transients
(high frequency components) a narrow window will be more suitable. Then, owing to
its xed window size, Gabor Transform is not suitable for analyzing signals involving
dierent range of frequencies.
Grossmann and Morlet (1984) introduced the Wavelet Transform in order to overcome
this problem. The main advantage of wavelets is that they have avarying window
size, being wide for slow frequencies and narrow for the fast ones (see lower part of
g. 13), thus leading to an optimal time-frequency resolution in all the frequency ranges
(Chui, 1992 Strang and Nguyen, 1996 Mallat, 1989). Furthermore, owing to the fact
that windows are adapted to the transients of each scale, wavelets lack of the requirement
of stationarity.
This chapter starts with a brief theoretical background of the Wavelet Transform.
Details of a straightforward implementation called the multiresolution analysis and an
alternative decomposition of time signals called the Wavelet Packets analysis will be
given. In section x4.4, I will show the results obtained by applying trigonometric wavelet
packets (Serrano, 1996) to the study of tonic-clonic seizures (Blanco et al., 1998a). In
sections x4.5 and x4.6, I will analyze two dierent types of event related potentials. By
using the multiresolution decomposition method, in the rst case I will study the alpha
band responses (Quian Quiroga and Schurmann, 1998, 1999) and in the second case the
ones corresponding to the gamma band (Sakowicz et al., 1999). Finally, I will discuss
advantages of Wavelets in the study of brain signals.
36

Original signal
FOURIER TRANSFORM
GABOR TRANSFORM
WAVELET TRANSFORM
Figure 13: Frequency and time-frequency methods. Fourier Transform is obtained by
correlating the original signal with complex sinusoids of dierent frequencies (upper
part). In Gabor Transform, the signal is correlated with modulated sinusoidal functions
that slides upon the time axis, thus giving a time-frequency representation (middle
part). Wavelets give an alternative time-scale representation but due to their varying
window size, a better resolution for each scale is achieved (bottom part). Furthermore,
the function to be correlated with the original signal can be chosen depending on the
application (e.g. in the graph quadratic B-Splines wavelets are shown).
37

4.2 Theoretical Background
4.2.1 Continuous Wavelet Transform
A wavelet family ab is a set of elemental functions generated by dilations and translations
of a unique admissible mother wavelet (t):
ab(t) = jaj ;1=2
t ; b
where a b 2 R, a 6= 0 are the scale and translation parameters respectively.
a
(27)
As a
increases the wavelet becomes more narrow and by varying b, the mother wavelet is
displaced in time. Thus, the wavelet family gives a unique pattern and its replicas at
dierent scales and with variable localization in time.
The continuous wavelet transform of a signal X(t) 2 L 2 (R) (nite energy signals) is
dened as the correlation between the signal and the wavelet functions ab , i.e.
(W X)(a b) = jaj ;1=2 Z 1
;1
X(t) ( t ; b
a ) dt = < X(t) ab > (28)
where denotes complex conjugation. Then, the dierent correlations
< X(t) ab > indicates how precisely the wavelet function locally ts the signal at
every scale a. Since the correlation is made with dierent scales of a single function,
instead of with complex sinusoids characterized by their frequencies, wavelets give a
time-scale representation.
4.2.2 Dyadic Wavelet Transform
The continuous Wavelet Transform maps a signal of one independent variable t onto
a function of two independent variables a b. This procedure is redundant and not
ecient for algorithm implementations. In consequence, it is more practical to dene
the Wavelet Transform only at discrete scales a and discrete times b. One way toachieve
this, is by choosing the discrete set of parameters fa j = 2 j b jk = 2 j kg, with j k 2 Z.
Replacing in eq. 27 we obtain the discrete wavelet family
jk(t) = 2 ;j=2 ( 2 ;j t ; k ) j k 2 Z (29)
that forms a basis of the Hilbert space L 2 (R), and whose correlation with the signal is
called Dyadic Wavelet Transform.
38

4.2.3 Multiresolution Analysis
Contracted versions of the wavelet function will match high frequency components of
the original signal and on the other hand, dilated versions will match low frequency
oscillations. By correlating the original signal with wavelet functions of dierent sizes
we can obtain the details of the signal in dierent scale levels. Then, the information
given by the dyadic Wavelet Transform can be organized according to a hierarchical
scheme called multiresolution analysis (Mallat, 1989 Chui, 1992).
If we denote by W j the subspaces of L 2 generated by the wavelets jk for each level
j, the space L 2 can be decomposed as a direct sum of the subspaces W j ,
L 2 = X j2Z
W j (30)
Let us dene the closed subspaces
V j = W j+1 W j+2 ::: j 2Z (31)
The subspaces V j are a multiresolution approximation of L 2 and they are generated by
scalings and translations of a single function jk called the scaling function. Then, for
the subspaces V j we have the orthogonal complementary subspaces W j , namely:
V j;1 = V j W j j 2Z (32)
Let us suppose we have a discrete signal X(n), which we will denote as x 0 , with nite
energy and without loss of generality, let us suppose that the sampling rate is t =1.
Then, we can successively decompose it with the following recursive scheme
x j;1 ( n ) = x j ( n ) r j ( n ) (33)
where the terms x j (n) 2 V j give the coarser representation of the signal and r j (n) 2 W j
give the details for each scale j =0 1 N. Then, for any resolution level N > 0, we
can write the decomposition of the signal as
x N (k) ( 2 ;N n ; k ) +
NX
X
C j (k) jk (n) (34)
X(n) = X k
j=1
k
where () is the scaling function, C j (k) are the wavelet coecients, and the sequence
fx N (k)g represents the coarser signal at the resolution level N. The second term is
the wavelet expansion. The wavelet coecients C j (k) can be interpreted as the local
residual errors between successive signal approximations at scales j ; 1 and j, and
39

j (n) = X k
C j (k) jk (n) (35)
is the detail signal at scale j.
Figure 14 shows as an example the multiresolution decomposition method applied
to an event-related potential and the corresponding reconstruction achieved by using
the inverse transform. The method starts by correlating the signal with shifted versions
(i.e. thus giving the time evolution) of a contracted wavelet function, the coecients
obtained thus giving the detail of the high frequency components. The remaining part
will be a coarser version of the original signal that can be obtained by correlating the
signal with the scaling function, which is orthogonal to the wavelet function. Then, the
wavelet function is dilated and from the coarser signal the procedure is repeated, thus
giving a decomposition of the signal in dierent scale levels, or what it is analog, in
dierent frequency bands. This method gives a decomposition of the signal that can be
implemented with very ecient algorithms due to the fact that it can be achieved just
by applying simple lters, as showed by Mallat (1989, see table 1 for an example). The
lower levels give the details corresponding to the high frequency components and the
higher levels the ones corresponding the the low frequencies. As pointed out in sec. x4.1,
the levels related with higher frequencies have more coecients that the lower ones, due
to the varying window size of the Wavelet Transform.
4.2.4 B-Splines wavelets
An important point to be discussed is howtochoose the mother functions to be compared
with the signal. In principle, the wavelet function should have a certain shape that we
would like to localize in the original signal. However, due to mathematical restrictions,
not every function can be used as a wavelet. Then, one criterion for choosing the wavelet
function is that \it looks similar" to the patterns of the original signal. In this respect,
B-Spline functions seem suitable for decomposing ERPs (see bottom part of gure 13).
B-Splines are piecewise polynomial functions of a certain degree that form a base in
L 2 (R) (see e.g. Chui, 1992). Filter coecients corresponding to quadratic B-Splines are
shown in table 1. We can remark the following properties that makes them very suitable
for the analysis of ERP (Unser et al., 1992, 1993 Unser, 1997 Chui, 1992 Strang and
Nguyen, 1996):
1. Smoothness: the smooth behavior of B-Splines is very important in order to avoid
border eects when making the correlation between the original signal and a
wavelet function with abrupt patterns.
40

Original signal
-15
0
15
-1sec
0
1sec
Multiresolution Decomposition
Multiresolution Reconstruction
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
-15
0
15
-15
0
15
-1sec
0 1sec
-1sec 0
1sec
Figure 14: Multiresolution decomposition method. The signal is decomposed in scale
levels each one representing the activity in dierent frequency bands. The wavelet coef-
cients show how closely the signal matches locally the dierent dilated versions of the
wavelet mother function (in this case a quadratic B-Spline). Furthermore, by applying
the inverse transform, the signal can be reconstructed from the wavelet coecients for
each scale level.
41

k h(k) g(k) p 2 (k) q 2 (k)
-10 0.00157 -0.00388
-9 0.01909 -0.03416
-8 -0.00503 0.03416
-7 -0.0444 0.07933
-6 0.01165 -0.02096
-5 0.10328 -0.18408
-4 -0.02593 0.04977 0.00208
-3 -0.24373 0.42390 -0.06040
-2 0.03398 -0.14034 0.25 0.30625
-1 0.65523 -0.90044 0.75 -0.63125
0 0.65523 0.90044 0.75 0.63125
1 0.03398 0.14034 0.25 -0.30625
2 -0.24373 -0.42390 -0.06040
3 -0.02593 -0.04977 -0.00208
4 0.10328 0.18408
5 0.01165 0.02096
6 -0.0444 -0.07933
7 -0.00503 -0.00901
8 0.01909 0.03416
9 0.00157 0.00388
Table 1: Filter coecients for quadratic B-Splines. h and g are the decomposition lters
and p 2 , q 2 are the reconstruction ones (from Ademoglu et al, 1997 see also Strang and
Nguyen, 1996).
2. Time-frequency resolution: it was shown that B-Spline functions have nearly optimal
time-frequency resolution (i.e. nearly the maximum allowed by the uncertainty
principle Unser et al., 1992).
3. Compact support: this means that the wavelet function does not extend to innity.
This fact is important in order not to include in a certain wavelet coecient
correlations with points far in the past or in the future.
4. Semi-orthogonality: the use of B-Splines as mother functions when applying the
multiresolution decomposition guarantees orthogonalitybetween the dierent scales.
42

4.2.5 Wavelet Packets
In the approach described before, only the successive signals x j (n) are decomposed, but
in many cases, it is also interesting to decompose the details r j (n). If we decompose
both x j (n) and r j (n), then the original signal can be represented in dierent ways as
combinations of x j (n) and r j (n) ofdierent levels j.
In this work, a decomposition called trigonometric wavelet packets was used (Serrano,
1996). The main idea is to decompose the components r j (n) in portions.
We dene any portion or local signal as
r (ml)
j ( n ) =
l+2 m ;1
X
k=l
C j (k) jk ( n ) (36)
where the parameters m and l are chosen for r (ml)
j (n) to cover the full time interval
2 ;j l n 2 ;j (l +2 m ), which is a relative long interval of length 2 m;j . Note that we
dened the local wavelet packet with 2 m basic functions jk (n) fork = l l+2 m ; 1.
Now, we dene the set of fundamental frequencies
! mh = +2h=2 m (37)
with 0 h 2 m;1 and associated Fourier matrix M (m) given by
M (m)
dk
= 2 ;m=2 8><
>:
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
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 (ml)
j (n)
as a 2 m dimensional vector obtained from
for 1 d 2 m .
(ml)
jd
( n ) =
l+2 m ;1
X
k=l
(38)
M (m)
dk jk ( n ) (39)
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

(ml)
j ( n ) =
2 m X
d=1
and the corresponding coecients are easily computed as
where 1 d 2 m .
D (ml)
j ( d ) =
D (ml)
j ( d ) (ml)
jd
( n ) : (40)
l+2 m ;1
X
k=l
M (m)
dk
C j ( k ) (41)
The trigonometric wavelet packets (ml)
jd
(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
(ml)
transform ^
jd
(!) is centered at the fundamental frequency ! mh , when d = 2h or
(ml)
d =2h +1. Moreover, ^ jd (!) =0on the other fundamental frequencies.
In other words, the coecients fD (ml)
j
spectrum
for the local signal r (ml)
j
of coecients fC j (k)D (ml)
j
(d)g can be considered as the discrete Fourier
(n). Summing up, we can resume in the double set
(d)g the time-scale-frequency information of the local signal
r (ml)
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 il i )
j (n), according the structure of the
signal,
r j ( n ) = X m i
r (m il 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 signals
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 applications
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 amultiresolution 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 multiresolution
decomposition compared with the continuous Wavelet Transform but with a
44

marked decrease in computational time. Other works also used this approach for automatic
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 selectiveaveraging of the single trials, thus obtaining a better
denition of the P300. Basar et al. (1999) reported the utility ofWavelet 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 eventrelated
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

ain activity during an epileptic seizure, we eliminated B 1 and B 2 bands, both containing
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 corresponding
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 and7:2Hz, 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

3
2
7.2 Hz 7.6 Hz
3
2
1
1
0
0
3
2
0 20 40 60 80
0 20 40 60 80
3
3
3
8.0 Hz 8.4 Hz 8.8 Hz 9.2 Hz
2
2
2
1
1
1
1
0
0
0
0
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
3
2
3
3
3
9.6 Hz 10 Hz 10.4 Hz 10.8 Hz
2
2
2
48
1
1
1
1
0
0
0
0
3
2
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
3
11.2 Hz 11.6 Hz 12.0 Hz 12.4 Hz
3
2
2
2
3
1
1
1
1
0
0
0
0
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
Figure 16: Wavelet packets for the scale level B3.

3
2
3
3
3
3.2 Hz 3.6 Hz 4.0 Hz 4.4 Hz
2
2
2
1
1
1
1
0
0
0
0
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
3
2
3
3
4.8 Hz 5.2 Hz 5.6 Hz 6.0 Hz
2
2
2
3
1
1
1
1
0
0
0
0
49
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
0 20 40 60 80
3
6.4 Hz
2
1
0
0 20 40 60 80
Figure 17: Wavelet packets for the scale level B4.

4.4.3 Discussion
The high time-frequency resolution achieved with Wavelet Packets allowed a very detailed
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 3 ; 4Hz,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 potentials
4.5.1 Introduction
EEG alpha rhythms can be dened as oscillations between 8 and 13 Hz, with an amplitude
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 meanings.
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 different
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 of1:5sec: 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 decomposition,
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 computations
(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 responses
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 comparison
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 selected
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 jk 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

sweep #1 sweep #2 sweep #3 sweep #4 sweep #5
-50
-50
-50
-50
-50
Original
sweep
0
50
0
50
0
50
0
50
0
50
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
53
Digital
filtering
-50
-50
0
0
-50
-50
-50
0
0
0
50
50
50
50
50
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
-150
-150
-150
-150
-150
Wavelet
coeff.
0
0
0
0
0
150
150
150
150
150
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
-50
-50
-50
-50
-50
Wavelet
reconstr.
0
50
0
50
0
50
0
50
0
50
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
Figure 18: Examples of the better performance of the Wavelet Transform in comparison with an \ideal" digital ltering for ve
single sweeps.

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 transientmarked with an arrow. In fact in this last case, the transient
is due to the cognitive P300wave 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 peakappears 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
F3
F4
F3
F4
F3
F4
µV -15
µV
-15
µV-15
Cz
0
Cz
0
Cz
0
55
P3
15
-1sec 0 1sec
P4
P3
15
-1sec 0 1sec
P4
P3
15
-1sec 0 1sec
P4
O1
O2
O1
O2
O1
O2
Figure 19: Grand average of the wide-band frequency responses.

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 electrodes
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 <
0:05).
56

VEP non-target target
F3
F4
F3
F4
F3
F4
µV
-15
µV
-15
µV
-15
Cz
0
Cz
0
Cz
0
57
15
-1sec 0 1sec
15
-1sec 0 1sec
15
-1sec 0 1sec
P3
P4
P3
P4
P3
P4
O1
O2
O1
O2
O1
O2
Figure 20: Alpha responses of one typical subject.

VEP non-target target
F3
F4
F3
F4
F3
F4
µV
-5
µV
-5
µV
-5
Cz
0
Cz
0
Cz
0
5
-1sec 0 1sec
5
-1sec 0 1sec
5
-1sec 0 1sec
58
P3
P4
P3
P4
P3
P4
O1
O2
O1
O2
O1
O2
Figure 21: Grand average (over N = 10 subjects) of the alpha responses.

Electrode F3 F4 Cz P3 P4 O1 O2
Delay 246.03 290.86 248.20 209.20 209.27 172.37 181.03
SEM 21.09 23.35 22.76 20.51 22.10 18.42 17.45
F3 XXX - - - - < 0.05 < 0.05
F4 XXX - < 0.01 < 0.01 < 0.01 < 0.01
Cz XXX - - < 0.05 < 0.05
P3 XXX - - -
P4 XXX - -
O1 XXX -
O2
XXX
Table 3: Multiple factor ANOVA comparison of the time delays of the maximum alpha
band wavelet components for the factor electrode. Note: SEM means standard error of
the mean, - means no signicance, delays in ms.
4.5.4 Discussion
Resonance theory
In the present work, occipital maximum enhancements were located between 100 and
200ms, strongly stressing the importance of the alpha band in the generation of the P-
100 peak and the following N-200 negative rebound. This is in agreement with the results
of Stampfer and Basar (1985), who, by digitally ltering auditory evoked potentials in
humans, showed that sensory evoked responses were generated and shaped mostly by
alpha and theta oscillations. This result is directly related with the resonance theory.
In principle, if the alpha enhancements were found only at the time of appearance of
the P100-N200 peaks, it can be argued that they were just a result of the morphology
of these peaks. However, the prolonged alpha response observed in some of the subjects
showed that this is not the case, and that it is more plausible to think about enhanced
spontaneous oscillations as a response to the stimulus, according to the resonance theory
presented in section x1.3.
Functional correlates of event-related alpha oscillations
Post-stimulus amplitude increases of alpha band were observed in all electrodes,
being signicantly higher in the occipital ones. Furthermore, anterior responses were
delayed in comparison with the posterior ones. Frontal maximum responses appeared
between 50 and 100ms after the occipital ones, this delay being statistically signicant.
Consequently, a primary response was reached at occipital locations, later propagating
to parietal, central and frontal electrodes. Then, owing to the anatomy of the visual
59

pathway (Mason and Kandel, 1991 Shepherd, 1988), the occipital localization and the
short latency of this response points towards a relation between event-related alpha
oscillations and the rst stages of sensory processing. Moreover, the idea of sensory
processing is reinforced by the independence of the response from the type of stimulus
used. Since the 1940s, the event-related alpha response was interpreted as the reactiveness
of the brain to sensory stimuli. A sensory alpha response was also observed
in several intracortical structures of the cat brain upon auditory and visual stimulation
(for a review of these works see Basar et al., 1997).
However, this process should be not considered as the only one related with the alpha
band. For example, in some subjects alpha responses were prolonged upon TARGET
stimuli in posterior locations. Spontaneous alpha activity, low amplitude long-latency
responses and low phase locking could be responsible of the lack of statistical signicance
when considering the whole group of subjects. Consequently, it could be conjectured
that other processes related with a cognitive function, due to the long latency and the
appearance only upon target stimuli, were present in some of the subjects.
The cognitive function of alpha band was already pointed out in previous works
(Stampfer and Basar, 1985 Kolev and Schurmann, 1992 Basar et al., 1992 Klimesch,
1997 Maltseva and Masloboev, 1997 and Petsche et al., 1997). Further experiments
applying dierent types of tasks should be performed in order to learn more about
cognitive related alpha responses.
Sources of event-related alpha oscillations
Another interesting question about alpha rhythms deals with its sources of generation.
Alpha responses were best visible in occipital electrodes, but enhancements were
also present in other locations with some time delay. Owing to the high dierences in
the delays between occipital and anterior electrodes, the conjecture of a single generator
and dispersion by volume conduction must be ruled out. A more plausible idea is to
think about several generators activated at dierent times and with dierent strength
depending on the stimuli and paradigm used. Several previous ndings support a multiple
alpha generator theory. Alpha rhythms were described mainly in the thalamus
and cortex (Adrian, 1941 Andersen and Andersson, 1968 Lopes da Silva and Storm
van Leewen, 1977), but alpha activity was also found in other structures like the brain
stem, cerebellum and limbic system, (for a review see Basar et al., 1997). Further evidence
showing that alpha rhythms cannot be explained through generators only in the
thalamus or cortex are the experiments performed on the cerebral ganglion of aplysia
and with the isolated ganglia of Helix pomata (Schutt et al., 1992 Schutt and Basar,
1992). These studies provided evidence that 10Hz activity can be recorded in vitro in
such small neural populations, each consisting only of approximately 2000 neurons.
60

In conclusion, our results point towards a distributed alpha system with functional
relation to sensory processing and possibly to further processes.
4.6 Application to gamma responses of bisensory event-related
potentials
4.6.1 Introduction
During the last ten years gamma-band phenomena gained increasing popularity since
the reports of Gray and Singer (1989). Measuring the spike activity of cellular units
and multi-units of the cat's striate cortex they found burst activities to be in congruence
with oscillatory LFPs (local eld potentials) in orientation-specic columns. Later
ndings (Eckhorn et al., 1988, Gray etal. 1989) led to the proposal of a mechanism suf-
cient to explain the formation of distant neuronal units to assemblies. Such assembly
formation has been suggested as the brain's solution to the problem of encoding. The
so-called \binding-theorem" provided a solid explanatory base in congruence with the
experimental data.
Although being very elegant in solving the problem to encode a quasi-unlimited
number of real object features and combinations in a limited number of neural elements
interest has, nevertheless, not swayed from the experimental results, which were not
directly related to theoretical questions. As, for example, the studies of Tiitinen et al.
(1993) and Pulvermuller et al. (1995) could demonstrate, the 40Hz oscillatory activity
is not restricted to sensory processing, but can be rather modulated or triggered by
cognitive processes as well.
4.6.2 Material and Methods
Event-related potentials were measured in 15 healthy subjects who had neither any
known neurological decit nor reported intake ofdrugsknown to aect the EEG. Electrodes
were placed according to the 10 ; 20 system in F3, F4, Cz, C3, C4, T3, T4, P3,
P4, O1 and O2 locations, referenced to linked earlobes. Data were amplied with a time
constant of 1:5sec: andalow-pass lter at 70Hz. For each single sweep, 1sec: pre- and
post-stimulus EEG were digitized with a sampling rate of 250Hz and stored in a hard
disk.
Subjects were instructed to view and listen passively to the stimuli. A recording
session consisted of3parts:
1. Registration of 120 sweeps covering 2s time-windows with application of unimodal
stimulation presenting a 2kHzsinusoidal tone-step binaurally.
61

2. Registration of 120 sweeps covering 2s time-windows with application of unimodal
stimulation using a rectangular light-step centered in the visual eld at a distance
of 1.5 m.
3. Registration of 120 sweeps covering 2s time-windows with application of multimodal
stimulation. Stimuli of (1) and (2) were applied simultaneously.
After visual inspection of the data, 30 sweeps free of artifacts were selected for each
modality for future analysis. A Wavelet Transform was applied to each single sweep
using a quadratic B-Spline function as mother wavelet. After a ve octave wavelet
decomposition, the components of the gamma band 31 ; 62Hz were analyzed. For each
subject the gamma components of the 30 single sweeps were averaged. Results for each
subject were averaged obtaining a \grand average".
Statistical analysis
The wavelet coecients processed were the ones obtained after averaging the responses
of the 30 single trials for each electrode and subject. Then, wavelet coecients
were rectied and the maximum coecient was computed in a time window of 250ms
post-stimulation. A multiple factor ANOVA (MANOVA) with repeated measures test
was applied in order to compare dierences between modalities and electrodes. Furthermore,
maximum wavelet components pre- and post-stimulation were compared with
t-tests in order to analyze statistical signicance of the amplitude enhancements for
each electrode and modality.
4.6.3 Results
Gamma wavelet coecients of one typical subject are shown in gure 22. In central
locations gamma amplitude enhancements are more pronounced upon bimodal stimulation.
In this case, pre-stimulus EEG amplitude is less than 5 (in arbitrary units) and
post-stimulus enhancements reach values up to 12. Further increases are seen in right
frontal and right occipital electrodes, in the latter case appearing with some delay.
The grand average of all the subjects is shown in gure 23. With auditory stimulation
enhancements are visible in the central electrode and more poorly dened in the occipital
ones. Visual stimulation shows enhancement onlyinthecentral electrode. On the other
hand, bimodal stimulation evokes signicantly higher enhancements than the other two
modalities (p < 0:01 see also g. 24) and they are distributed all along the surface
EEG, being more pronounced in the central location. Furthermore, bimodal responses
are clearly not a linear superposition of the auditory and visual ones, this fact being
very clear in central electrodes and specially in the frontal ones, where enhancements
occur only upon bimodal stimulation.
62

63
Figure 22: Gamma responses for atypical subject.

Figure 23: Grand average (over N =15subjects) of the gamma responses.
64

Figure 24: T-test comparison of the pre- and post-stimulus gamma amplitudes.
Finally, in gure 24 comparisons between maximum components pre- and poststimulation
are shown for each electrode and modality. Enhancements are clearly higher
upon bimodal stimulation in right frontal, posterior and central electrodes, where they
reach high signicance (p

information that two dierent stimuli are manifestations of the same process. However,
the possibility thatanintensity rise on joined-polymodal conditions leads to behavioral
changes in terms of attention and arousal functions can not be discarded and deserves
further studies.
Less data is available on the role of 40 Hz-oscillatory processes in bisensory integration.
Sheer and Schrock (1986) conducted their studies on focused attention by looking
for modications in 40 Hz-SSRs (steady-state responses, seesec.x1.2) driven by means of
simultaneously applied click-and-ash stimuli, from which one or the other modality had
to be ignored. With the stimulus trains being not in phase, they found the 40 Hz-SSRs
of the non-focused stimulus modality reduced. Other modes of intermodal perturbation
of the 40 Hz-SSRs have been tested by Rohrbaugh et al. (1990). Both salient \foreground"
visual and auditory stimuli implied a latency and amplitude decrease in the
40Hz auditory steady-state responses, stimuli of the same modality thereby dominating
in eect. Thus both exemplary studies have discriminative functions in common, which
are modiers of the 40Hz surface EEG.
4.7 Conclusion
Wavelet decomposition proved to be a very useful tool for analyzing brain signals. I
would like to remark two advantages of Wavelet Transform over Fourier based methods.
First, Wavelet Transform lacks of the requirement of stationarity, this being crucial for
avoiding spurious results when analyzing brain signals, already known to be highly nonstationary.
Second, owing to the varying window sizeoftheWavelet Transform, a better
time-frequency resolution can be achieved when the signal has patterns involving dierent
scales. This is particularly important inthe case of event-related potentials, where
the relevant response is limited to a fraction of a second, a good time resolution thus
being crucial for making any physiological interpretation. In section 4.5.2, I showed
with some selected sweeps how the multiresolution decomposition implemented with
B-Splines functions leads to a better resolution of the event-related oscillations in comparison
with a conventional ideal lter used in several previous works. Moreover, due
to the fact that the multiresolution decomposition method is implemented as a ltering
scheme, it can be seen as a way to construct lters with an optimal time-frequency resolution.
This exemplies and complements the theoretical description of the advantages
of wavelets introduced in section 4.2. Furthermore, the multiresolution decomposition
is a way of data reduction, thus giving relevant (i.e. non-redundant) coecients that
allows a straightforward implementation of statistical tests.
Alpha responses to pattern visual event-related potentials were related with primary
sensory processing, having several generators. Furthermore, the analysis of discrete
66

wavelet coecients allowed an easy design and implementation of statistical tests. In
this context, the resolution achieved with Wavelet Transform was very important for
obtaining signicance of the results.
The frequency dynamics during the seizures was already described with Gabor Transform
(sec. x3.4). However, with wavelets packets it was possible to follow with better
accuracy the time evolution of the frequency peaks. Further advantages can be obtained
when comparing frequency patterns of dierent channels in order to obtain information
about the sources of the seizures. In this case, the time resolution of wavelets can be
crucial due to the fact that the seizure spread, from the focus to other locations, can
take place in a few milliseconds.
Wavelet Transform, due to its varying window size, is more suitable for analyzing
signals involving dierent ranges of frequencies. In fact, as showed with alpha and
gamma responses to ERPs, frequency behaviors can be resolved up to fractions of a
second. On the other hand, with the election of a adequate window, Gabor Transform
is more suitable for the analysis of signals with a more limited frequency content as shown
in the previous chapter with Grand Mal seizures, in which the interesting activity was
limited to the lower frequencies of the EEG (up to 12:5Hz).
67

5 Deterministic Chaos
5.1 Introduction
In the previous chapters I described the application of several time-frequency methods
to the analysis of brain signals. Since these methods are linear, a completely dierent
approach canbeachieved by applying the concepts of non-linear dynamics, also known
as Deterministic Chaos theory.
Chaotic systems have an apparently noisy behavior but are in fact ruled by deterministic
laws. They are characterized by their sensibility on initial conditions. That
means, similar initial conditions give completely dierent outcomes after some time.
Since chaotic signals look like noise and furthermore, since they also have a broadband
frequency spectrum, linear approaches as the ones described in the previous sections
are sometimes not suitable for their study. Several methods were developed in order
to calculate the degree of determinism (or random nature), complexity, chaoticity, etc.
of these signals. Among these, the Correlation Dimension, Lyapunov Exponents and
Kolmogorov Entropy have been the most popular. In this chapter I will give a mathematical
background of these methods and then I will describe their application to the
study of EEG signals.
The chapter is organized as follows. After a brief denition of the basic concepts
related with chaos, the mathematical background for the calculation of the Correlation
Dimension and Lyapunov Exponents will be described. In section x5.2.4, I will remark
some problems in the selection of computational parameters. One of these problems is
the stationarity of the signal to be studied. In Section x5.3, I will describe a criterion
based on weak stationarity (Blanco et al., 1995a). In section x5.4, I will give a brief
review of previous results of chaos analysis of EEG signals. In sections x5.5 and x5.6,
I will show the application of Chaos methods to scalp normal EEG recordings and to
intracranially recorded Grand Mal seizures (Blanco et al., 1995a,b, 1996a). Finally in
section x5.7, I will discuss about the applicability andresults of chaos analysis to EEG
signals.
5.2 Theoretical Background
5.2.1 Basic concepts
Phase space: It is a common practice in physics to represent the evolution of a system
in phase spaces. For example, in mechanics,itisvery useful to represent the movement
of a system as a pendulum by plotting the position vs. the velocity or more generally,
68

the position vs. the linear momentum. In general, the phase space is identied with a
topological manifold. An n-dimensional phase space is spanned by a set of n-dimensional
\embedding vectors", each one dening a point in the phase space, thus representing
the instantaneous state of the system. The sequence of such states over the time scale
denes a curve in the phase space, called trajectory.
Attractor: In some cases, trajectories of dissipative dynamical systems (systems with
a volume contraction in the phase space) converge with increasing time to a bounded
subset of the phase space. This bounded region to which all suciently close trajectories
(trajectories lying in the basin of attraction) converges asymptotically is called the
attractor (Shuster, 1988). According to their topology, several types of attractors can
be distinguished (see Abraham and Shaw 1983 Shuster, 1988 Ott et al, 1994 Basar
and Quian Quiroga, 1998):
1. Fixed Point: Trajectories in phase space tend to a pointatrest. Atypical example
is a damped pendulum that has come to rest after some time.
2. Limit cycle: The attractor is a closed (one dimensional) curve in the phase space
representing a periodic motion. The standard example is a damped periodically
driven oscillator. circumstances
3. Torus: The attractor is a two-dimensional toroidal surface. This type of attractor
represent a quasiperiodic motion where two incommensurable frequencies correspond
to the movement around and along the torus.
4. Strange attractor: The main property of strange attractors is their sensitive dependence
on initial conditions. Points that are initially close in the phase space,
become exponentially separated after some time. All known strange attractors
have a non-integer dimension. Signals corresponding to strange attractors have a
random appearance.
5.2.2 Correlation Dimension
The Correlation Dimension (D 2 ) has become the most widely used quantitative parameter
to describe attractors. It is a measure of complexity ofthe system related with its
number of degrees of freedom, or in a more intuitive way with its topological dimension.
It is already a common practice to calculate D 2 and to investigate how itchanges upon
dierent . Furthermore, since in principle D 2 converges to nites values for deterministic
systems and do not converge in the case of a random signal, D 2 isagoodparameterfor
evaluating the deterministic or noisy inherent nature of a system. As we will see later
69

in this chapter, the question of a deterministic or noisy nature of EEG signals keep the
attention of many research groups during the last years.
5.2.3 Calculation of the Correlation Dimension
I will illustrate briey a proposal made by Grassberger and Procaccia (1983) for computing
the correlation dimension D 2 . First of all, a phase space must be constructed, this
space being spanned by a set of embedding vectors. In the case of univariate signals, following
a proposal made by Takens (1981) called the \time shift method", n-dimensional
vectors are constructed in the following way:
~x = fx(t)x(t + ):::x(t +(n ; 1) )g (43)
where is a xed time increment and n is the embedding dimension. Every instantaneous
state of the system is therefore represented by thevector ~x which denes apoint in the
phase-space. Once the phase space is constructed, the correlation integral as a function
of variable distances R is dened as:
C(R) = lim
N!1
1 X
(R ;j~x
N 2 i ; ~x j j) (44)
i6=j
where N is the number of data points and is the Heavyside function. Then C(R)
is a measure of the probability that two arbitrary points ~x i , ~x j will be separated by
a distance less than R. Theiler (1986) made a correction to this method in order to
avoid spurious temporal correlations. He proposed that the vectors to be compared
when calculating the correlation integral, should be distanced at least W data points
(ji ; jj >W), where W is a measure of the temporal correlation of the signal (e.g. the
rst zero of the autocorrelation function.)
In the case the attractor is a simple curve in the phase space, the number of pair of
vectors whose distance in less than a certain radius R will be proportional to R 1 . In
the case the attractor is a two dimensional surface, C(R) R 2 and for a xed point,
C(R) R 0 . Generalizing we can write the following relation:
C(R) R D 2
(45)
then, if the number of data and the embedding dimension are suciently large we obtain
D 2 = lim
R!0
log C(R)
log R
The main point isthatC(R) behaves as a power of R for small R. By plotting log C(R)
versus log R, D 2 can be calculated from the slope of the curve. For an attractor with
70
(46)

unknown topology it is necessary to calculate C(R) for several embedding dimensions 6 .
Then, for deterministic signals, the correlation dimension D 2 should converge towards
a saturation value, for some n>n o .
Singular Value Decomposition (SVD)
In many occasions the amount ofnoisein a signal makes impossible the calculation
of D 2 . One way of reducing the noise from the signal is by using the SVD method.
The main idea is to form a matrix from the embedding vectors (eq. 43) and then, after
calculating the eigenvectors and eigenvalues, to make a reconstruction of the signal
by rotating the matrix to a reduced base composed by the eigenvectors considered
signicant. Since the eigenvectors with relative small eigenvalues are supposed to be
dominated by noise, this phase space reduction is a way of eliminating noise from the
signal.
5.2.4 Problems arising when calculating the Correlation Dimension
In order to make the phase space reconstruction described in eq. 43 an appropriate time
lag and an appropriate embedding dimension n should be chosen. Since an unfortunate
election could lead to wrong results, several criteria were proposed (Blanco et al., 1996a
Abarbanel et al., 1993).
Concerning the time lag, if it is to small, the components in eq. 43 will have nearly the
same value and they will not properly span the phase space (redundancy). In the extreme
case, the attractor will be constricted to a diagonal line and its topology will be lost. On
the other hand, if the time lag is very high, the components of each embedding vector
will become totally unrelated to each other, thus becoming meaningless (irrelevance).
Several methods were developed for estimating the optimal time lag, among them, the
rst zero of the autocorrelation function has been the most used. This criterion assures
linear independence of the components. Some variations were also proposed as for
example the value of a certain decay of the autocorrelation function. Furthermore, for
taking into account non linear eects, criteria based on non-linear correlations were
dened (average of mutual information) (Abarbanel et al., 1993 Pjin, 1990). Other
approach was given by Rosenstein et al. (1994), who proposed a geometrical-based
method. The main idea is to look for a time lag that gives an optimal expansion of the
embedding vectors with respect to the diagonal in the phase space.
The election of a proper n was also subtle to discussion. If n is too small, the attractor
will be not completely unfolded and on the other hand, if n is too high, calculations
6 In fact, following Takens's proposal the dimension n of the embedding phase-space should be chosen
at least twice the dimension of the attractor.
71

will be dominated by noise. One rst criteria is to take n > 2D 2 (Takens, 1981). In
this case, the attractor will be completely unfolded but this criterion assumes a previous
estimation of D 2 . One solution is then to repeat the calculations for dierent sets of
embedding dimensions and in case of nding convergence, then to cheek if this criterion
is fullled. However, this is sometimes dicult to implement because the calculations
do not depend solely on or on n, on the contrary, they depend on the combination
of both (Broomhead and King, 1986 Mees et al., 1987 Albano et al., 1988 Palus and
Dvorak, 1992). Then, an estimation of a minimum n would be helpful. In this respect,
Kennel et al. (1992) proposed the false nearest neighbors method. The main idea is to
calculate if for a certain n nearest neighbors in the phase space still remain close for a
dimension n +1. If this is not the case, then the attractor was not completely unfolded
and the embedding dimension must be higher. The procedure is repeated for increasing
embedding dimensions until neighbors remain close.
Another important problem arises when choosing the linear region in the plots of
log C(R) vs. log R and also in how to calculate the slope. Up to the moment there is
no unique solution to this question and dierent groups have dierent approaches for
obtaining the values of D 2 from the log C(R) vs. log R plots. Some researchers prefer to
show directly the plots (Pjin et al., 1997) or to limit the calculation to the correlation
integral (eq. 44).
5.2.5 Lyapunov Exponents and Kolmogorov Entropy
Another useful tool for characterizing the attractor are the Lyapunov exponents. Lyapunov
exponents provide a quantitative indication of the level of chaos of a system.
They measure the mean exponential divergence of initially close phase space trajectories
with time. As more rapidly two trajectories diverges for a certain period of time,
more chaotic is the system and more sensitive to initial conditions.
Let us consider a small spherical hypervolume in the phase space. After a short
time, as trajectories evolve, the sphere will have an ellipsoid shape with its axes deformed
according the Lyapunov exponents. If the system is known to be dissipative, the volume
in the phase space will tend to contract and the sum of the Lyapunov exponents will be
negative. The longest axis of the ellipsoid will correspond to the most unstable direction,
determined by the largest Lyapunov exponent. Usually only this exponent is computed.
If it is positive, trajectories will diverge otherwise, they will get closer reaching a non
chaotic attractor. Following this argument, a necessary condition for a system to be
chaotic is that at least one of the exponents (the largest one) is positive. Lyapunov
exponents also give an indication of the period of time in which predictions are possible
and this is strongly related with the concept of information theory and entropy. In fact,
72

the sum of the positive exponents (i.e. the ones giving the rate of expansion of the
volume), equals the Kolmogorov entropy (Pesin, 1977).
K 2 = X >0
i (47)
5.2.6 Calculating Lyapunov Exponents
Wolf Method
Wolfetal. (1985) proposed an algorithm for calculating the largest Lyapunov exponent.
First, the phase space reconstruction is made (eq.43) and the nearest neighbor is
searched for one of the rst embedding vectors. A restriction must be made when searching
for the neighbor: it must be suciently separated in time in order not to compute
as nearest neighbors successive vectors of the same trajectory. Without considering this
correction, Lyapunov exponents could be spurious due to temporal correlation of the
neighbors. Once the neighbor and the initial distance (L) is determined, the system is
evolved forward some xed time (evolution time) and the new distance (L 0 ) is calculated.
This evolution is repeated, calculating the successive distances, until the separation is
greater than a certain threshold. Then a new vector (replacement vector) is searched as
close as possible to the rst one, having approximately the same orientation of the rst
neighbor. Finally, Lyapunov exponents can be estimated using the following formula:
L 1 =
1
(t k ; t 0 )
kX
i=1
ln L0 (t i )
L(t i;1 )
(48)
where k is the number of time propagation steps.
Rosenstein Method
Rosenstein et al. (1993) developed another algorithm for the calculation of the largest
Lyapunov Exponent in short and noisy time series. As before, the rst step is to make
a phase space reconstruction. Then the nearest neighbor for each embedding vector
is found. After this, the system is evolved some xed time and the largest Lyapunov
Exponent can be estimated as the mean rate of separation of the neighbors.
Assuming that the separation is determined by the largest Lyapunov Exponent (),
then at a time t the distance will be:
d(t) C e t (49)
where C is the initial separation. Taking the natural logarithm of both sides we obtain:
73

ln d(t) ln C + t (50)
This gives a set of parallel lines for the dierent embedding dimensions, and the
largest Lyapunov Exponent can be calculated as the mean slope, averaging all the
embedding vectors.
Lyapunov exponents are very sensible to the election of the time lag, the embedding
dimension and especially to the election of the evolution time. If the evolution time is too
short, neighbor vectors will not evolve enough in order to obtain relevant information.
If the evolution time is too large, vectors will \jump" to other trajectories thus giving
unreliable results.
5.3 Stationarity
Due to the fact that Chaos methods require stationary signals and EEGs are known to
be highly non stationary, I will briey discuss this topic and I will propose a criterion of
stationarity. In principle, non-stationarity means that characteristics of the time series,
such as the mean, variance or power spectra, change with time. More technically, if we
have a time series of discrete observed values fx 1 x 2 :::x N g, stationarity means that
the joint probability distribution function f 12 (x 1 x 2 ) depends only on the time dierences
jt 1 ; t 2 j and not on the absolute values t 1 and t 2 (Jenkins and Watts, 1968). Statistical
tests of stationarity have revealed a variety of results in EEGs, and estimates of
stationary epochs range from some seconds to several minutes (Lopes da Silva, 1993a).
However, whether or not the same data segment is considered stationary, depends on
the problem being studied and the type of analysis to be performed.
In the case of EEGs, due to the large amount of data needed for the application of
the non linear dynamic methods, strict stationarity is almost impossible to achieve. This
problem has brought agreatvariety of results exposed by dierent authors (Basar, 1990
Basar and Bullock, 1989). A less restrictive requirement, called \weak stationarity" of
order n, is that the moments up to some order n are fairly stable with time. If the
probability distribution of a signal is Gaussian, it can be completely characterized by its
mean ( m ), its variance ( 2 ) and its autocorrelation function. In this case, second order
stationarity ( n = 2 ), plus an assumption of normality, is enough to assure complete
stationarity (Jenkins and Watts, 1968). Consequently, in order to check for stationarity
of the EEG segments to be used for analysis, the following procedure was used (Blanco
et al., 1995a).
1. The total EEG time series was divided in bins with a xed number of data. The
election of the bin length depends on the type of data and on the analysis to be
74

performed. On one hand, must be large enough in order to give a reliable statistic
and on the other hand if it is too large, it will not be possible to observe fast
changes.
2. The mean and variance for each bin were calculated, and zones in which their
values do not have signicant changes (e.g. less than 20%) were selected.
3. Finally, the corresponding histogram for this zone was constructed, and the normality
of the obtained distribution was veried.
I would like to remark that although the above criterion is arbitrary and does not
prove stationarity 7 , it was very useful as a rst approximation in order to select EEG
data segments to work with, or to reject others with clear non stationary behavior.
However, I would like to stress that in general elaborated statistical criteria can be
hardly fullled by EEG signals. For a more complete discussion about stationarity
related with the application of Chaos methods, see for example Schreiber (1997).
Eckman et al. (1987) presented another approach to the problem of stationarity by
using the so called recurrence plots. Recurrence plots are based on distance calculations
in the phase space. After a phase space reconstruction, the Euclidean distance between
the embedding vectors (lets say N in total) is calculated. Then, an NxN plot is performed
in the following way: for each pair of embedding vectors (i j) (i represented in
the horizontal axis and j in the vertical axis) a point is plotted if its distance is less
than a certain value r. Stationary signals will be characterized by homogeneous plots
and non-stationary signals will show inhomogeneities due to varying features of the embedding
vectors in the phase space. As an example, Fig. 25 shows the recurrence plot
of an stationary one minute EEG signal (see Blanco et al., 1995a).
Recurrence plots give an elegant representation of the stationarity of the signal, but
in many occasions their interpretation is subjective and further analysis is required.
5.4 Short review of Chaos analysis of EEG signals
5.4.1 Correlation Dimension
Since the pioneering work of Babloyantz and coworkers (Babloyantz, 1985), several
groups started to study the dynamics of the brain activity by reconstructing the trajectory
of the system in the phase space and by calculating parameters as the Correlation
Dimension (D 2 ). Low D 2 values correspond to simple systems and high D 2 values to
7 Autocorrelation function is not checked, and the choice of what is considered as stable mean and
variances is in fact arbitrary.
75

Figure 25: Recurrence Plot of a stationary EEG signal. Horizontal axis (i) andvertical
axis (j) represent the 3200 embedding vectors obtained from the signal after phase space
reconstruction. A point is plotted in the position (i j) if the distance between vectors
i and j is less than a certain value r.
more complex ones this value tending to innity, in theory, in the case of noise. D 2
proved to be very useful for characterizing the brain dynamics in dierent sleep states.
It was found that D 2 decreases in deep sleep stages, thus reecting a synchronization of
the EEG (Babloyantz, 1986, Roschke and Aldenho, 1991 Achermann et al., 1994a,b
Pradhan et al., 1995). D 2 was also used for characterizing the brain dynamics doing
mental tasks (Lutzenberger et al., 1995 Molle et al., 1996). Furthermore, D 2 was compared
between normal subjects and patients with dierent pathologies, with the general
result that decreases in D 2 are related with abnormal synchronizations of the EEG, as
demonstrated in epilepsy (Blanco et al., 1996a Babloyantz and Destexhe, 1986 Pijn
et al., 1991 Iasemidis et al., 1990 Lehnertz and Elger, 1995, 1998) and other pathologies
as Alzheimer, dementia, Parkinson, depression, schizophrenia or Creutzfeld-Jakob
coma. (Pritchard et al., 1994, Besthorn et al., 1995 Stam et al., 1995, Roschke et al.,
1994 Gallez and Babloyantz, 1991). For a more detailed review of dierent application
of chaos analysis in brain signals I suggest the works of Basar (1989), Pritchard and
76

Duke (1992) and Elbert et al. (1994) and Basar and Quian Quiroga (1998).
The Correlation Dimension was also used for characterizing the nature of EEG signals.
In principle, converging D 2 values point towards a non-linear deterministic nature
and diverging D 2 values would stress the interpretation of EEG signals as noise. First
results showed converging values of D 2 in several situations. However, Osborne and
Provenzale (1989) showed that ltered noise can also give nite D 2 values. Pijn et al.
(1991) proposed the use of surrogate tests in order to validate the results obtained with
D 2 . In brief, random (surrogate) signals are constructed from the original one with the
same linear characteristics (frequency spectrum) and then, converging D 2 values of the
original signal should be considered valid, only if the ones of the surrogates diverge. By
applying this procedure, they found that values of the original and surrogate data dier
in the case of epileptic seizures in the case of the normal EEG, this analysis showing
that it is indistinguishable from Gaussian noise. Achermann et al. (1994a) also reported
no dierences between EEG in sleep stages and noise.
5.4.2 Lyapunov Exponents
Although the determination of the existence of a positive Lyapunov exponent could be a
signofchaos, publications about Lyapunov exponents are rare in comparison to the ones
with Correlation Dimension. I will report here some important ndings with Lyapunov
exponents in epilepsy, sleep stages and in dierent pathologies.
Epilepsy
One of the rst attempts to apply Lyapunov exponents to EEG data was done by
Babloyantz and Destexhe (1986) using the Wolf method for the evaluation of a short
epileptic \Petit Mal" seizure. They obtained a value of =2:9 0:6, concluding that
although the attractor has a global stability during an epileptic seizure (due to a very
low Correlation Dimension), the presence of a positiveLyapunov exponent shows a great
sensitivity to initial conditions, giving a rich response to external outputs.
Frank et al. (1990) studied a longer \Grand Mal" epileptic seizure. They proposed a
modied version of the Wolf method, choosing in a dierent way the replacementvectors
and making multiple passes through the time series. They point out that Lyapunov
exponents are sensible to the evolution time and to the embedding dimension, reporting
a value of =1 0:2 estimated across dierent embedding dimensions.
Iasemidis and Sackellares (1991) also used a modied Wolf algorithm and analyzed
seizures recorded with subdural electrodes. They observed a drop in the Lyapunov
Exponents during seizures, with greater values (implying a more chaotic state) postictally
than ictally or pre-ictally. Furthermore, they found a phase-locking of the focal
77

sites minutes before the starting of the seizure, with a progressive phase entrainment of
the nonfocal ones. They propose this phase-locking as a method for assigning degrees
of participation of each focal site and for classifying their importance in the developing
of the seizure. Krystal and Weiner (1991), using the same algorithm, obtained similar
results in electroconvulsive therapy seizures.
Sleep
Some groups also had success in evaluating Lyapunov exponents during dierent
sleep stages. Babloyantz (1988) reported positive Lyapunov exponents during deep
sleep, obtaining a value of = 0:4 ; 0:8 for stage II and a value of = 0:3 ; 0:6 for
stage IV.
Principe and Lo (1991) reported a greater value of = 2:1 for sleep stage II, but
they remark that an accurate value is impossible to obtain because of the complexity of
the signal, its time varying nature and the sensibility of the results with the election of
the parameters for the calculations.
Roschke et al. (1993), following the modication of the Wolf algorithm proposed by
Frank et al. (1990) calculated the Lyapunov exponent of recordings from 15 healthy
male subjects in sleep stages I, II, III, IV and REM. They found in all cases positive
values, thus stating that EEG signals are neither quasiperiodic waves, nor simple noise.
They also report a decrement in the Lyapunov exponents as sleep becomes slower.
Roschke et al. (1994) studied dierences in Correlation Dimension and Lyapunov
exponents in sleep recordings of depressive and schizophrenic patients compared with
healthy controls. They mainly found alterations during slow sleep in depression, and
during REM sleep in schizophrenia.
Other studies
Gallez and Babloyantz (1991) analyzed the complete Lyapunov spectrum in awake
\eyes closed" state (alpha waves), deep sleep (stage IV) and Creutzfeld-Jakob coma.
They found in all cases studied at least two positive Lyapunov exponents, increasing
this numberuptothreeinthecaseofalphawaves, implying that alpha waves correspond
to a more complex system than the one present during sleep.
Stam et al. (1995) studied 13 Parkinson and 9 demented patients against 9 healthy
subjects by using the Correlation Dimension, the Lyapunov Exponents and the Kolmogorov
entropy calculated from a spatial reconstruction of the embedding vectors
(multichanneling). They report a value of = 6:17 for the control subjects, a similar
value of =6:12 (but with lower Correlation Dimension) for Parkinson patients and a
signicant lower value of =4:84 for demented patients.
Wallenstein and Nash (1991) implemented the Wolf algorithm with avarying prop-
78

agation time. They applied this procedure to study ERPs in central and parietal
locations, nding lower values in non-target than in target stimulus, without signicant
dierences between electrodes. In the central location (Cz) they report a value of
=0:397 0:18 for non-target stimulus and of =0:794 0:44 for the target ones.
Although there is a great variance in the results of dierent groups, there is a general
agreement that EEG signals have at least one positive Lyapunov exponent, implying
that EEGs (in case of having a deterministic origin) reect a chaotic activity.
5.5 Application to scalp recorded EEGs
5.5.1 Material and Methods
EEG recordings of 6 subjects were studied. Recordings were performed in a \no-task"
awake state with eyes closed. Three of these subjects had normal EEG recordings ( N1,
N2, N3 ) and other three had abnormal ones (A1,A2,A3). The main abnormalities
of these last ones were: slow waves, increases in theta rhythms, very low alpha reactivity
and important decrease in the alpha/theta ratio.
EEGs were digitized using a 8 bits analog-to-digital (A/D) converter. 20 electrodes
were disposed according to the 10 ; 20 system with earlobe references. The data was
sampled with a frequency of 256 Hz, and ltered with a high-pass lter at 0.5 Hz and
alow-pass lter at 32 Hz. We chose for our analysis the central electrodes because they
were the ones less contaminated by artifacts.
5.5.2 Results and Discussion
The objective of this section is to establish some criteria for the analysis of EEG signals
previous to any calculation with Chaos methods. This is done in order to avoid spurious
results due to unfortunate selections of the segments of data to be analyzed.
The rst step is to choose data segments without artifacts. Then, after selecting
proper zones free of artifacts or with very few interruptions, the stationarity of the data
can be veried by applying the criteria described in sec. x5.3. In this case, the length of
the data bins was 1000 data points (about 4 seconds of digitized EEG signal).
In Fig. 26 and Fig. 27 the mean and variance for the time series denoted by N1
are showed. In this case, mean and variance were considered stable when their changes
were in a range of 20% (see sec. 5.3). From these gures it can be concluded that the
stationarity criterion previously proposed is satised between bins 9 and 26. Note that
in this case, the variance does not give any restriction about selected bins to be used in
the following analysis.
79

Figure 26: Mean values for bins of 1000 data points for the EEG data N1.
Figure 27: Standard deviation for bins of 1000 data points for the EEG data N1.
Fig. 28 shows that the histogram for this selected segment ofN1 has approximately
a normal distribution (at least, no higher order moments seem to be relevant).
The algorithm for the calculation of Correlation Dimension assumed stationarity and
noise free time series. Since EEGs are usually contaminated by noise it is convenient
prior to the calculation of the Correlation Dimension to apply noise reduction techniques
as the Singular Value Decomposition (SVD) (Albano et al., 1988 Broomhead and King,
1986)(see sec. x5.2.3). The SVD has the advantage of reducing the embedding dimension
of the system and of partially eliminating the noiseofthesignal.
In Table 4 the total length of the series, the stationary portions and the results of
the calculations of D 2 and are presented. The time lag was chosen as the rst zero
of the autocorrelation function and the embedding dimension by satisfying the Takens
80

Figure 28: Histogram for the EEG data N1 in the selected zone (see text).
EEG N 0 N d D e D 2 1
N1 34000 15000 13 5:5 ; 6:0 0:26
N2 32000 13000 13 5:0 ; 6:0 0:23
N3 35000 14000 13 4:5 ; 5:5 0:25
A1 35000 15000 13 4:5 ; 5:5 0:28
A2 34000 14000 13 5:0 ; 5:5 0:26
A3 36000 15000 13 4:5 ; 5:0 0:27
Table 4: Correlation Dimension (D 2 ) and maximum Lyapunov exponent ( 1 ) for the
dierent EEG signals (N means normal recordings and A abnormal ones). N 0
is the
total length of the EEG recordings, N d the length employed for the calculations and D e
the embedding dimension.
81

criterion (see sec. x5.2.4). D 2 varied between 4:5 ; 6 without a clear dierence between
normal and pathological EEG recordings. Maximum Lyapunov exponents were positive
in all the cases giving an evidence of chaotic activity (in the case of signals having a
deterministic origin).
5.6 Application to intracranially recorded tonic-clonic seizures
5.6.1 Material and Methods
The subject and data analyzed was described in sec. x3.3. The EEG corresponds to a
9-hour intracranial recording from a 21 years old patient with tonic-clonic seizures.
5.6.2 Results and Discussion
Figure 7 shows the EEG signal corresponding to a segment of 64 sec from one depth
electrode in the left hippocampus. As seen in the gure, the epileptic seizure starts
about second 10 and nishes about second 54.
In order to study changes in the EEG behavior the signal was divided in intervals of
8 sec as follows: ( i ) pre-seizure (0 ; 8sec) ( ii ) starting of the seizure (10 ; 18sec) (
iii ) full development of the seizure (21 ; 29sec) and (29 ; 37sec) ( iv ) ending of the
seizure (37 ; 44sec) and (44 ; 52sec).
For these intervals, stationaritywas tested following the criterion described in sec. x5.3.
The length of the bins used was of 512 data. In the second interval (10 ; 18sec) the
criterion of stationarity was not satised due to the fast changes in the EEG morphology.
The time delay and the minimum embedding dimension were estimated with the
geometrical method introduced by Rosestein and the False Nearest Neighbor method,
respectively (see sect. x5.2.4).
Fig. 29 displays the two dimensional projection of the phase portraits for the selected
intervals. Each EEG segment was characterized by its Correlation Dimensions and
maximum Lyapunov exponents.
Table 5 shows the stationary zones, optimal time lags , minimum embedding dimensions,
Correlation Dimension and maximum Lyapunov exponent for the dierent
EEG intervals. These last two parameters were obtained as an average among the corresponding
values obtained for all the considered embedding dimensions. These values
are in good agreement with those given in the literature in background activity and in
epileptic seizures (see sec. x5.4).
The seizure is characterized by a drop in the value of the maximum Lyapunov exponent.
A similar situation is observed for the Correlation Dimension. From the values
82

Figure 29: Attractors corresponding to the selected zones (see text) during a Grand Mal
seizure.
83

EEG segment D e (min) D 2 1
0 ; 8 sec 0:0195 8 4:30 4:6
21 ; 29 sec 0:0156 13 2:60 4:0
29 ; 37 sec 0:0156 10 2:50 4:0
37 ; 44 sec 0:0156 11 2:05 3:5
44 ; 52 sec 0:0156 13 2:15 3:0
Table 5: Optimal value of time lag (insec ), minimum embedding dimension D (min)
e ,
Correlation Dimension (D 2 ), and maximum Lyapunov exponent ( 1 ) for the stationary
EEG segments (Data-set size: N = 2048 see text).
showed in table 5, it can be concluded that the dynamical behavior was more regular
during the seizure.
Summarizing, we found a good evidence that during the epileptic seizure there is
a transition from a complex to a simple dynamical behavior. Furthermore, this result
yields insights with respect to the theory of how epileptic seizures occur (i.e.
synchronization of the disordered background EEG activity).
as a
5.7 Conclusion
Deterministic chaos theory gives an alternative new type of analysis compared with
the ones given by the traditional methods. However, the application of chaos methods,
should be done with care in order to avoid misleading results. In this way a right election
of parameters required for the calculations is critical.
Chaos analysis has several prerequisites, some of them making impossible its application
to the study of EEG signals. One of these prerequisites is the availability of
long data recordings but on the other hand, data must be stationary. Consequently,
chaos analysis cannot be applied in nonstationary short data recordings as in the case of
event-related potentials. In these types of series, the dynamics of the system is changing
completely in fractions of a second due to the eect of the stimulus and therefore, it has
no sense to dene an attractor and to calculate any metric invariant from it.
Since methods for calculation of D 2 and 1 were developed for ideal signals (i.e.
innite, noise-free and stationary), then, due to the complex structure of the EEGs,
results should be considered of relative relevance and, as suggested by Pritchard and
Duke (1992), is more realistic to talk about Dimensionality instead of Correlation Dimension.
By using this view, several groups reported changes in D 2 and 1 values in
dierent brain states and pathologies. Nevertheless, these measures seem not to be suit-
84

able for automatic detection processes or on-line analysis (Jansen, 1991). Furthermore,
other methods turned out to be more suitable for these purposes (see Pjin et al., 1997
Gotman, 1990b).
Since it was reported that ltered noise could lead to converging low D 2 values,
conclusions based on absolute values of D 2 should be revised and validated. In this
respect, the use of surrogate tests (Theiler et. al., 1992) was proposed. Furthermore,
absolute results should be stable with respect to dierent election of the parameters
necessary for calculations. This type of approach was mainly used for discriminating
between deterministic chaos vs. random behavior of the EEGs. In this respect, as I will
discuss in detail in section x7, the failure to validate converging low dimensional values
of D 2 with surrogate tests (then proving a low dimensional chaotic nature) implies that
methods of Chaos are not suitable for answering this question rather than implying a
random nature of the EEG signals.
I would like to remark that I discussed the most used approach of Chaos analysis
to EEG signals, namely, by means of the Correlation Dimension, Lyapunov exponents
or Kolmogorov entropy. Although their results are not so promising as believed before,
Chaos analysis is not limited to these invariants and alternative non linear approaches as
for example the study of chaotic synchronization (Arnhold et al., 1999 Quian Quiroga
et al., 1999c) could lead to interesting results.
85

6 Wavelet-entropy
6.1 Introduction
One very interesting question related with brain activity deals with its nature. Should
brain signals be considered as noisy activity or as a deterministic phenomenon with
some degree of order? and in the latter case, how can we quantify this degree of order?
Several approaches were performed in order to shed light on this topic, especially
after the introduction of the non-linear dynamics (Chaos) theory to the study of EEGs
(see section x5).
Another approach to this question is to consider the entropy. The concept of thermodynamic
entropy is well known in physics as a measure of the order of a system (for
a more detailed physical description see Feynman et al., 1964). Left side of g. 30 shows
schematically this idea. In the upper graph (situation A), the molecules of a certain gas
are isolated in the left side of a recipient because the division was just removed. After
some time (situation B), molecules are subject to a free expansion then lling the whole
recipient. Situation A corresponds to a more ordered state because the molecules are
restricted to the left side therefore, situation A is associated with a lower entropy than
situation B.
Furthermore, for studying the nature of EEG signals it is very important to consider
how these signals behave in several situations. Among these, the study of event-related
potentials (i.e. the reaction of the ongoing EEG to an external or internal stimulation)
could shed light on this question. As suggested by Basar (1980), ERPs are due
to selective enhancements or synchronizations of the spontaneous brain oscillations of
the ongoing EEG, thus giving a transition from a disordered to an ordered state (see
section 1.3). Then, a method for measuring the entropy appears as an ideal tool for
quantifying the \ordering" of the EEG signals upon stimulation.
Recently, a measure of entropy dened from the Fourier power spectrum, the spectral
entropy (Powell and Percival, 1979), started to be applied to the study of brain signals
(Inouye et al., 1991, 1993). An ordered activity (i.e. a sinusoidal signal) is manifested
as a narrow peak in the frequency domain, thus having low entropy (see situation A in
the right sideofg.30).On the other hand, random activity has a wide band response
in the frequency domain, reected in a high entropy value (situation B in the right side
of g. 30). However, since the spectral entropy is based on the Fourier Transform, it
has some disadvantages. As stated in previous chapters, Fourier Transform does not
take into account the time evolution of the frequency patterns and furthermore, Fourier
Transform requires stationarity of the signals and EEGs are highly non-stationary (Lopes
da Silva, 1993a Mpitsos, 1989 Blanco et al., 1995a).
86

ENTROPY
Thermodynamics
Signal analysis
Situation A
Order
=
low entropy
Fourier
Situation B
Disorder
=
high entropy
Fourier
Figure 30: Entropy in thermodynamics and in signal analysis.
These disadvantages of the Fourier Transform are partially resolved by using the
Gabor Transform (Powell and Percival already dened a time evolving entropy from
the Gabor Transform by using a Hanning window). Nevertheless, as already discused in
previous chapters, the selection of the window size is critical due to the UncertaintyPrinciple
and this limitation becomes important when the signal has transient components
localized in time as in ERPs.
Wavelets haveavarying window size, thus allowing a better time-frequency resolution
for all the scales. Consequently, a further improvement to the spectral entropy is to
dene the entropy from the Wavelet transform. In the latter case, the time evolution of
the frequency patterns can be followed with an optimal time-frequency resolution.
In this chapter, I will rst introduce the denition of the Wavelet-entropy and then I
will describe its application to the study of ERPs (Quian Quiroga et al., 1999a,b Rosso
et al., 1998).
87

6.2 Theoretical Background
Although the denition of this time evolving entropy can be done from any timefrequency
representation, as Gabor Transform or dierent type of wavelets, I will describe
the method from the wavelet coecients C ij (where i denotes time and j are the
dierent scales) obtained after applying the multiresolution decomposition method (see
sec. x4.2.3).
Once the coecients C ij are known, the energy for each time i and level j can be
calculated as
E ij = C 2 ij (51)
Since the number of components for each resolution level is dierent, I will redene
the energy by calculating, for each level, its mean value in successive time windows
(t = 128ms) denoted by the index k which will now give the time evolution. Then,
the energy will be:
E kj = 1 N
i 0
X+t
i=i 0
E ij (52)
where i 0 is the starting value of the time window (i 0 =1 1+t 1+2t : : :) and N is
the number of components in the time window for each resolution level. For every time
window k, the total energy can be calculated as:
E k = X j
E kj (53)
and we can dene the quantity
p kj = E kj
E k
(54)
as a probability distribution associated with the scale level j. Clearly, for each time
window k, P j p kj = 1 and then, following the denition of entropy given by Shannon
(1948), the time varying Wavelet-entropy can be dened as (for further details see Blanco
et al., 1998a):
WS k = ; X j
p kj log 2
p kj (55)
88

6.3 Application to visual event-related potentials
6.3.1 Methods and Materials
In 9 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 achecker reversal.
2. Oddball paradigm (NON-TARGET/TARGET stimuli): subjects were watching
the same pattern as above. Two dierent 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.
In both cases, 200 stimuli were presented and the duration of each stimulus was
1 second. 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. 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 decomposition,
the components of the following bands were obtained: 62 ; 125Hz,31; 62Hz
(gamma), 16 ; 31Hz (beta), 8 ; 16Hz (alpha), 4 ; 8Hz (theta) and the residues in the
0:5 ; 4Hz band (delta).
For each subject the results of the wavelet decomposition of the 30 single sweeps
were averaged, obtaining the mean components (C ij ). Finally, from this coecients
the Wavelet-entropy was calculated as described in the previous section. In this case,
since the WS is calculated from the averaged wavelet coecients, only phase-locked
oscillations will contribute to it, the others being cancelled (for the same data, an analysis
of phase-locking in the alpha band was done in Quian Quiroga et.al., 1999d).
89

Signal WS
X 1 0:15
X 2 0:36
X 3 0:53
Table 6: Wavelet entropy for a sinusoidal signal (X 1 ), for a random signal (X 3 ) and for
a sinusoidal signal with noise (X 2 = 1 (X 2 1 + X 3 )).
Statistical analysis
Statistical analysis was performed by computing the mean entropy values in the
second pre- and post-stimulus. Signicance of entropy decreases were calculated for
each electrode and stimuli type by comparing minimum pre- and post-stimulation with
t-test comparisons.
6.3.2 Results
The Wavelet entropy (WS) was tested with 3 dierent time series. A: a pure sinusoidal
signal with a frequency of 12 Hz (X 1 ) B: a random signal (X 3 ) and C: a signal composed
of the sum of the two previous ones (X 2 ). The random signal was obtained from a white
noise generator. Table 6 shows the results of the WS for the three signals. As expected,
the WS is higher for the noisy signal (broad-band spectrum) and minimum for the
sinusoidal one (narrow-band spectrum).
The event-related responses of one typical subject are shown in gure 31. Only the
central and left electrodes are shown, having the right ones similar behavior. Left side
of the gure corresponds to VEP, center to NON-TARGET and right sidetoTARGET
stimuli. The P100 response is clearly visible upon all stimuli types at about 100 ms,
best dened in occipital locations. In the case of TARGET stimulation, a marked
positive peak appears between 400 and 500 ms, according to the expected cognitive
P300 response.
Figure 32 shows the results of the band and total energies for the same subject.
Only the lower frequency bands are plotted since the higher ones showed no relevant
contribution to the total energy. All stimuli types show alpha and theta band increases in
occipital locations at 100-200 ms, correlated with the P100-N200 complex. Only upon
TARGET stimulation an increase in the posterior locations is visualized in the delta
band at 500-600 ms, this increase being related with the cognitive response(P300).
Results of the WS for the same subject are shown in gure 33. Decreases in the
entropy are more signicant in posterior sites upon TARGET stimulation, strongly
90

VEP
Non Target
Target
F3
-20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
10
20
Cz
-20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
10
20
91
P3
-20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
10
20
O1
-20
-10
0
10
20
-20
-10
0
10
20
-1sec. 0 1sec. -1sec. 0 1sec. -1sec. 0 1sec.
-20
-10
0
10
20
Figure 31: Evoked responses of one typical subject (y-axis values in V ).

500
VEP NON-TARGET TARGET
500
500
Alpha
Theta
Delta
Total
F3
F3
0
500
-1.0 -0.5 0.0 0.5 1.0
0
500
-1.0 -0.5 0.0 0.5 1.0
0
500
-1.0 -0.5 0.0 0.5 1.0
Cz
Cz
0
0
0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
92
1000
1000
1000
500
P3
500
P3
500
0
0
0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
1000
1000
1000
500
O1
500
O1
500
0
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Figure 32: Total and band energy for the recordings of the previous gure.

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Figure 33: Wavelet Entropy corresponding to the same recording.

correlated with the denition of the P300 response. There are no decreases of entropy
correlated with the P100 response. This is consistent with the distribution of energy
described in g. 32, since the P100 peaks have a wide-band frequency composition
corresponding to alpha and theta oscillations. On the other hand, the P300 response
corresponded only to delta oscillations, thus having a lower entropy. It is very interesting
to note that the results of the entropy are not directly related with the ones of the energy.
This can be clearly seen, for example, by analyzing the TARGET response of electrode
O1, where there is a high energy increase at about 200ms without signicant changes in
the entropy for this time.
Considering the whole group of subjects, the grand average of the WS is shown in
gure 34. As pointed out with the typical subject, posterior decreases are observable
upon TARGET stimulation at about 600 ms, these decreases being clearly correlated
with the denition of the P300 response. On the contrary, P100 peaks produced no
relevant decreases in the entropy since they corresponded to a wider range of frequencies.
In order to verify statistically the decreases of entropy correlated with the P300
responses, pre- and post-stimulus mean entropy values were compared by using t-tests.
As shown in gure 35, entropies are signicantly decreased in posterior electrodes upon
TARGET stimulation. Then, entropy decreases respond to a dynamical process related
with the stimulation, rather than to random uctuations.
6.3.3 Discussion
Results of the calculation of the WS in dierent digitally generated signals conrmed
the expected correlation between the WS and the complexity of them, being high for
the random signal and low for the sinusoidal one. This shows that WS could be used as
a measure for describing the behavior of EEG signals, since a noisy activity will have
a broadband spectrum and consequently a high entropy, and on the other side, more
ordered activity as a sinusoidal signal will have lower entropy. Then, WS gives a new
approach to the measure and quantication of the order of a system, its physiological interpretation
being very interesting due to its relation with tuning of cell groups involved
in the generation of the EEG signal.
Entropy in signal analysis
The concept of entropy emerged last century as a useful state function applied to
the study of the thermodynamic of gases (Feynman, 1964). Furthermore, entropy was
related with the order of a system and its applications rapidly expanded to several disciplines
due to its very interesting meanings. One important milestone was the introduction
of the theory of communication (Shannon, 1948 see also Feynman, 1996) relating
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Figure 34: Grand average of the Wavelet Entropy.

1.0
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and the Lorenz equations (a model of Rayleigh-Benard convection). By changing the
control parameters of the models, they showed increases of the spectral entropy upon
the disordering of the systems in their route to chaoticity.
Inouye and coworkers (1991, 1993) introduced the spectral entropy to the analysis
of EEG signals. From the spectral entropy they dened an irregularity indexand they
reported in rest EEGs, more irregularityinanterior than in occipital areas. Furthermore,
they reported a greater degree of EEG desynchronization during mental arithmetic in
comparison with rest EEG. Stam and coworkers (Stam et al., 1993) instead, dened an
accelaration spectrum entropy from the second derivative of the signal and they reported
dierences of this measure in dementia, Parkinson disease (Jelles et al., 1995) and during
mental activation (Thomeer et al., 1994).
Blanco et al. (1998a) proposed a further improvement by dening the entropy from
the Wavelet Transform due to its advantages over the Fourier transform. Among these,
the most important one is that the time evolution of the entropy can be followed. Since
the entropy is dened from the time-frequency representation of the signal, the nearly
optimal time-frequency resolution of the Wavelet Transform is crucial for having an
accurate measure. This is particularly important in the case of event-related potentials,
in which the relevant response is limited to a fraction of a second. Furthermore, Wavelet
Transform lacks of the requirement of stationarity.
I showed the application of the Wavelet-entropy (WS) to the study of ERPs. WS
proved to be a very useful tool for characterizing the event-related responses, furthermore,
the information obtained with the WS probed not to be trivially related with the
energy and consequently with the amplitude of the signal. This means that with this
method, new information can be accessed with an approach dierent from the traditional
analysis of amplitude and delays of the event-related responses.
WS as a measure of resonance in the brain
Following the resonance theory, the ERP can be seen as an evoked synchronization,
frequency stabilization, frequency selective enhancement and/or phase reordering of the
ongoing EEG, and it should not be interpreted as an additive component to a noisy
background EEG (see sec. x1.3).
WS appears as an ideal tool for measuring the synchronization of the EEG oscillations
upon stimulation. In this context, synchronization means that the group of cells
involved in the generation of the response react to the stimulation tuned in frequency.
That means, they produce a narrow band in the frequency domain and consequently
they are correlated with a decrease in the entropy. Basar (1980) already mentioned
the importance of the entropy for understanding the relation between the pre-stimulus
ongoing EEG and the event-related responses by making an analogy with the concept
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of magnetization in physics.
WS is a natural measure of the resonance phenomena (see section x1.3). Resonance is
related with a decrease of entropy, since only some of the spontaneous oscillations of the
ongoing EEG will be enhanced, thus giving a more ordered frequency distribution than
the broadband spectrum of the EEG. This relation was showed with the simulations
described in table 1, in whose noisy (broadband) signals had higher entropy than the
ones corresponding to ordered (narrowband) behaviors.
Study of the P100-N200 complex
In agreement with previous ndings (overview in Regan, 1989), the P100 peak was
best dened in occipital locations. Furthermore, its task independence (at least the one
related with the TARGET stimuli) points towards a relation between this response and
primary sensory rocessing.
According to the resonance theory, it is very interesting to analyze the frequency
characteristics of the dierent evoked peaks in order to understand the response mechanisms
involved in its generation. The distribution of energy in the frequency domain
showed that the P100-N200 responses corresponded to a wide range of frequencies mostly
in the alpha and theta bands. These responses did not produce any signicant decreases
of entropy compared with the one of the normal ongoing EEG due to its wide-band frequency
distribution of the involved generators.
The involvement of alpha and theta oscillation in the generation of these peaks was
already described by Stampfer and Basar (1985), who obtain a similar result by digitally
ltering auditory evoked potential in humans.
Study of the P300 response
According to previous evidence (Regan, 1989 Polich and Kok, 1995 Basar-Eroglu et
al., 1992), the P300 positive deection was observed upon TARGET stimuli, best dened
in parietal and occipital electrodes at about 500-600 ms. These responses, traditionally
related with a cognitive process, showed a signicant decrease in the WS since they
involved only delta generators. Although the correlation between the P300 and the
entropy decreases was clear due to their common appearance only upon TARGET stimuli
and their posterior localization, a dierence of about 100-200 ms was observed between
them. This is because after the P300 there is a positive slow rebound in the delta range
(see g. 3). Consequently, although the minimum of the P300 is at about 400-500 ms,
the P300 complex can be viewed as a delta wave extending up to larger latencies. A
similar observation was made by Stampfer and Basar (1985), who after ltering in the
delta band the TARGET stimuli of an oddball paradigm in auditory ERP, described
the presence of a negative deection at about 600 ms as a continuation of the positive
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P300 deection.
Due to the relation between the P300 and the decreases in the entropy, it can be
tentatively assumed that the cognitive (P300) response involves a higher degree of order
than the one related with the ongoing EEG and the one related with the P100 response.
From this, we can postulate that the cognitive response is possibly related with a tuning
of the EEG oscillations, thus giving a narrower frequency response reected in a low
entropy value. However, this conjecture should be veried with more subjects and with
dierent experiments. Furthermore, since in this case the WS was dened from the
mean wavelet coecients (after averaging the coecients of the single trials) we can
not discard the possibility of other oscillations being present in a non phase-locked way
(therefore their contribution being cancelled when averaging).
The importance of delta oscillations in the generation of the P300 was reported
in several previous works. Basar-Eroglu et al. (1992) by using an auditory oddball
paradigm in 10 subjects, pointed out the importance of delta oscillators in the generation
of the P300, suggesting that they are related mainly with decision making and matching.
Schurmann et al. (1995), also remarked the importance of delta oscillations in the
generation of the P300, nding also delta enhancement uponTARGET responses in the
single trials. Furthermore, they obtain better averaged responses by selectively averaging
single trials with an enhanced delta response. A similar result was recently reported by
Demiralp et al. (1999), who after applying a wavelet multirresolution decomposition to
evoked responses, used the delta coecients as discriminators between good and bad
single trials.
6.4 Conclusions
Wavelet-entropy proved to be a very useful tool for characterizing the event-related
responses. With this method a new approach to the measure of order/noise of a system
can be achieved 8 . Up to now, this type of descriptions were merely based on Chaos
analysis. However, these methods have several prerequisites, some of them making their
application to the study of EEG signals impossible. One of these prerequisites is the
data length. Chaos analysis cannot be applied to short data recordings as in the case of
event-related potentials. In this type of signals, the dynamics of the system is changing
completely in fractions of a second duetotheeectof the stimulus, and then it has no
sense to dene an attractor and to calculate parameters as the Correlation Dimension,
8
We should remark that high WS values does not necesarily mean noise since for example chaotic
deterministic systems also have a broadband spectra. However, analysis of how the EEG gets tuned in
frequency after stimulation is directly related with the question of the deterministic/random nature of
it as it will be further discussed in sec.7.1.3.
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Lyapunov Exponent or Kolmogorov Entropy from them.
Previous methods for measuring the spectral entropy from the Fourier Transform are
also not suitable for the analysis of ERPs. This is because the Fourier Transform requires
stationarity of the signal, and can not describe the time evolution of the frequencies.
In this context, the denition of a time-varying entropy allowed the study of the time
evolution of the \ordering" of the ongoing EEG due to stimulation, thus making possible
the correlation between changes in entropy of the signal and sensory/cognitive processes
of the ERPs.
The information obtained with the Wavelet-Entropy turned out not to be trivially
related with the energy. This means that with this method, new information can be
accessed with a dierent approach than the one obtained by making the traditional
analysis of amplitude of the evoked responses.
WS appears as a natural measure of order in EEG signals. This is very interesting
in order to study synchronizations upon dierent stimuli as in the case of ERPs. Taking
into account the resonance theory, WS can measure the degree of synchronicity of the
cell groups involved in the dierent responses.
Further implementations of the method are in progress in order to optimize the
results by having more frequency bands. This would allow a more accurate denition of
the entropy and moreover will allow the denition of the entropy for dierent frequency
bands. However, we would like to remark that in the case of ERPs this is not so easy
to achieve since a high time resolution is needed, thus limiting the frequency resolution
due to the uncertainty principle.
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7 General Discussion
In this chapter I will rst discuss how the results described in this thesis are related
with several questions of neurophysiology, so far still unresolved with the traditional
approaches. The joining of evidence obtained with the described methods sheds lighton
these topics and allows the conjecture of physiological mechanisms. In the second part of
this chapter I will compare these methods, stressing their advantages and disadvantages
when applied to the study of EEG signals.
7.1 Physiological considerations
7.1.1 Dynamics of Grand Mal seizures
Chaos analysis of epileptic seizures leads to the general result that during seizures a
transition from a complex system to a simpler one takes place.
On the other hand, by using the RIR dened from the Gabor Transform, I showed
and quantied a well dened frequency behavior during seizures. Grand Mal seizures
were dominated by alpha and theta frequencies. Delta oscillations decreased during them
and had an abrupt increase correlated with the clonic phase. With Wavelet Packets,
I showed with a better resolution the temporal evolution of these frequency patterns
and it was possible to establish that the low frequency activity (3 ; 4Hz) related with
the rhythmic contractions of the clonic phase, was in fact originated by the \slowing"
of higher frequencies (at least 8 ; 9Hz). Then, during Grand Mal seizures there is
a clear frequency dynamics: some seconds after the starting of the seizure alpha and
theta activity dominates, these oscillations later becoming slower and when they are
in the limit of the delta band (about 3 ; 4Hz) the clonic phase of the seizure starts
and delta activity has an abrupt increase dominating the EEG recording. Moreover,
it is reasonable to conjecture that the violent contractions of the clonic phase are the
response to brain oscillations that are generated in higher frequencies, but owing to
the fact that muscles cannot react so fast, muscle activity is then limited to a tonic
contraction (muscular tension) until brain oscillations become slower and muscles are
capable of contracting in resonance with them.
The frequency pattern described is in agreement with studies in animals and with
computer simulations. Furthermore, it would be very interesting to investigate possible
causes of this behavior. Neuronal fatigue is one of the most plausible explanations.
The ring of a neuron is produced as a response to excitatory inputs of neighboring
neurons. This connection is done mainly by means of synaptical processes generated
by neurotransmitters produced in the neurons. During an epileptic seizure there is an
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abnormal ring of the neurons, reected in high amplitude paroxysms as spikes or more
generally in a marked increase of the overall amplitude of the EEG. Then, if the neurons
are not capable of generating the necessary amount of neurotransmitters for sustaining
this uncommon intensive ring, synaptical connections will decrease and excitatory input
can not be transmitted. This process, known as neuronal fatigue, produces a change
in the oscillatory behavior of cell groups, probably leading to the described \slowing"
of the frequency patterns during Grand Mal seizures. It can be also conjectured that
this frequency dynamics is due to a recruitment of GABA inhibitory channels, variation
in the neurotransmitter balance, etc. Results in this respect should be validated and
compared with analysis in intracranial recordings.
I also showed with the mean and maximum band frequencies, dened from the
Gabor Transform, the presence of a low amplitude but well dened peak in the delta
band (1 ; 3Hz) during a seizure (referred to as a latent pacemaker). Then, based on the
frequency dynamics previously described, it is possible to conjecture that oscillations in
the range of the alpha or theta band characteristic of Grand Mal seizures, once they start
to oscillate in the range of the delta band (after the \slowing" process), they provoke
an abrupt increase of the delta activity duetoaresonance phenomenon with the latent
delta pacemaker. This resonance and massive synchronization would be responsible of
the starting of the clonic phase of the seizure.
7.1.2 Event-related responses
Alpha responses to visual event-related potentials were studied with the Wavelet Transform.
Since the sources and functions of alpha oscillations are still subject to discussion,
the aim of this approach was to obtain more information about this topic by using a
new and powerful method, namely wavelets. In this context, I showed that stimulation
leads to alpha enhancements visualized upon all electrodes with some signicant delays
between the anterior and posterior locations. Enhancements were signicantly higher in
occipital locations, had short latency and were statistically independent of the stimulus
type (one of them requiring a cognitive process), thus pointing towards a relation between
alpha oscillations and sensory (i.e. \pre-cognitive") processing. Furthermore, the
delay dierences between responses in dierent electrodes imply that event-related alpha
oscillations have several sources, thus ruling out the hypothesis of a unique generator
and propagation by volume conduction. It is interesting to remark that the resolution
of the Wavelet Transform was crucial for achieving statistical signicance of the results.
However, it is important tomention that \alpha oscillations" is a concept that include
several type of dierent processes and the previous interpretation should not be taken
as exclusive or general description of them.
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Wavelet analysis was also applied to study the gamma responses to bimodal stimulation
(simultaneous auditory and visual stimulation). This analysis showed a signicant
enhancement of the responses upon bimodal stimulus in comparison with unimodal ones
(auditory and visual separately). Then, it was possible to conjecture a relation between
gamma oscillations and a fast process responsible of carrying the information that the
two sensory perceptions of a bimodal stimulation correspond in fact to the same stimulus.
Finally, a very interesting analysis of the event-related responses was accessed by
the Wavelet-entropy. By seeing the ERP as a synchronization or selective enhancement
of some of the ongoing EEG oscillations, the WS appears as a natural method for
obtaining a quantitative measure of the ordering of the EEG spontaneous oscillations
due to stimulation. Following this view, the P300 response, traditionally related with
cognitive processes, seems to be related with a \tuned" response of EEG oscillations. On
the other hand, P100 responses were related with oscillations not \tuned" in frequency,
thus having a wider frequency composition.
7.1.3 Are EEG signals chaos or noise?
First reports of Chaos analysis on EEG signals showed convergence of the Correlation
Dimension (D 2 ) to small values, thus claiming that EEG dynamics correspond to low
dimensional deterministic chaos. After the nding that ltered noise can also have
convergent low dimensional D 2 values, other works stressed the necessity of validation
of the metric estimates by means of surrogates tests. In this direction, it was stated
that the nature of EEG signals is indistinguishable from noise.
In order to deal with this dispute, in the following I will discuss about noise as
an inherent property of the system under study, and not about noise produced by the
surrounding or by limitations of the measuring systems (ampliers, etc.). In principle
we can state that a system is random when we cannot predict its outcome, but in fact
the concept of noise is an idealization since it depends on our ability for analyzing the
system (I am excluding in this discussion problems related with quantum mechanics
and the uncertainty principle). For example, if from the spin, the initial impulse, the
mechanical laws, etc., we can calculate the evolution of a coin ipped in the air, then
its outcome will not be random.
In the past, the dynamics of air convections of the atmosphere, for example, was considered
random, until Lorenz (1969) showed that it can be modeled by three dierential
equations, thus being deterministic. Chaos theory has grown very fast, developing new
methods that showed how many systems, in former times considered noise, have in fact
a deterministic chaotic nature. One of the most used methods for distinguishing deter-
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ministic chaos and noise is the D 2 . However, this method was in principle developed for
stationary, noise-free and long data sets. Several eorts were made in order to adapt
EEG signals to these requirements with dierent results, among them, singular value
decomposition, alternative phase space reconstructions (multichanneling), ltering, etc.
But, if it is impossible to distinguish EEG signals from noise with \chaos methods", is
it because the EEG corresponds to a noisy phenomenon, or is it because the available
methods are not suitable for analyzing the nature of the EEGs? It is like the question
of the ipped coin. Is this phenomenon random, or is it that we don't have the ability,
or a suitable method to study it?
Probably neurophysiological evidence could give more straightforward evidence to
resolve this dispute. In fact, EEG alpha oscillations are blockade with eyes opening
external or internal stimulation lead to evoked responses that are reproducible upon
similar conditions pre-stimulus EEG inuences evoked responses (Basar, 1980) EEG
activity can be synchronized in sleep or in pathologies as epilepsy. In summary, EEG
patterns have reproducible clear changes upon dierent conditions.
Since all these experiments point towards a deterministic origin of the EEG, then,
in my opinion, the question about a noisy origin of EEGs based on results obtained
with methods as the D 2 should be restricted to a discussion of the possibilities of these
methods and not to a discussion about the nature of the signal.
7.2 Comparison of the methods
In principle, there is not a \best method" for quantitative analysis of EEG signals. The
selection of the adequate method will depend on the type of signal to be studied and on
the questions expected to be answered.
7.2.1 Fourier Transform vs. Gabor Transform
Fourier Transform gives a representation in the frequency domain. This allows the visualization
of periodicities that would be dicult to observe from the EEG, especially when
several rhythms occur simultaneously. Frequencies are grouped in bands, their total or
relative power and their topographical distribution constituting the main quantitative
method of analysis of the EEG. This procedure was adapted to several commercial systems
and has been used as a diagnostical tool in medical centers. It is important to
remark that the grouping in frequency bands is not arbitrary because it was shown that
these bands can be related to dierent functions, sources and pathologies.
However, Fourier Transform gives no information about the time of occurrence of
the frequency patterns. Moreover, transients localized in time in the original signal will
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aect the whole spectrum and for this reason, in order to avoid spurious eects in the
Fourier spectrum, signals must be stationary. This is particularly important in the case
of EEG signals due to the presence of artifacts. Artifacts are alterations (usually of high
amplitude) of the ongoing EEG due to causes not related with the brain activity (e.g.
blinking, head movements, etc.) and they give spurious eects in the Fourier spectrum
that can lead to misinterpretations. In the case of analyzing background EEGs, this
is partially resolved by selecting small \artifact free" segments of data, later averaging
the spectrum of the selected segments. However, this widely used procedure is very
subjective because it requires the decision of what should be considered an appropriate
segment to be analyzed (i.e. which segments are representative of the whole EEG?).
An easy and intuitive way to obtain a time evolution of the frequency patterns is by
making the Fourier spectrum of successive segments (\windows") of data, then plotting
them as a function of time. This procedure is called the Short Time Fourier Transform
or Gabor Transform and the plots obtained are called spectograms. The problem of
stationarity ispartially resolved by taking short windows. Spectograms give an elegant
representation of the signal and they are suitable for visualizing large scale frequency
variations (i.e. of the order of minutes or hours) as for example for studying sleep stages.
However, as I showed in section x3, they are not suitable for analyzing epileptic seizures,
in which the frequency patterns change in the order of seconds.
In this context, the introduction of the band relative intensity ratio (RIR) and the
mean and maximum band frequencies, allowed a more detailed study of the frequency
behavior during Grand Mal epileptic seizures, as already summarized in section x7.1.1.
Furthermore, with these quantitative parameters it was possible to make a statistical
analysis of the frequency behavior in several scalp recorded seizures. Other interesting
point to mention is that although in scalp recordings of Grand Mal seizures muscle
activity obscures completely the EEG, it was possible to study the frequency patterns
by leaving aside the high frequency components related with muscle artifacts, thus
obtaining a very interesting quantitative frequency pattern that keeps \hidden" with
the traditional analysis of EEG recordings.
7.2.2 Gabor Transform vs. Wavelet Transform
Gabor Transform gives an optimal representation of the EEG in the time-frequency
domain. However, one critical limitation arises when choosing the size of the window to
be applied due to the Uncertainty Principle. If the window is too narrow, the frequency
resolution will be poor, and if the window is too wide, the time localization will be
not so precise. In fact, frequencies can not be resolved instantaneously. Then, for slow
processes a wide window will be necessary and for data involving fast processes, a narrow
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window will be more suitable. Due to its xed window size, Gabor Transform is not
optimal for analyzing signals having dierent ranges of frequencies.
The main advantage of the Wavelet Transform is that the size of the window isvariable,
being wide when studying low frequencies and narrow when studying the high ones.
Then, the time-frequency resolution is automatically adapted (see appendix xA.3), thus
being an optimal method for analyzing signals involving dierent ranges of frequencies.
Furthermore, due to their adapted window size, wavelets lacks of the requirement of
stationarity.
Wavelet Transform consists in making a correlation between the original signal and
scaled versions of the same \mother function". This mother function can be chosen
between a wide range of options each one having dierent characteristics that can be
more or less appropriate depending on the signal to be analyzed. At this respect, B-
Spline functions were very suitable for analyzing EEG signals due to their compact
support and smoothness.
Successive correlations of the signal to be studied with scaled versions of the wavelet
function (and their complementary function) can be arranged in a hierarchical scheme
allowing the decomposition of the signal in dierent scales (frequency bands). This
method, the multiresolution decomposition, allowed the study of the alpha responses
to visual event-related potentials and the study of the gamma responses upon bimodal
stimulation.
The time-frequency resolution of wavelets was crucial for making physiological interpretations
of the event-related responses (see section x7.1.2) because these results were
based in statistically signicant dierences in the amplitudes and time delays of the
responses, dierences that in the latter case were in the order of 100 ms and would have
been very dicult to resolve with other methods like the Gabor Transform. Furthermore,
the access to discrete coecients in the case of the Wavelet Transform allows a
very easy implementation of statistical tests.
As showed with alpha and gamma responses to ERPs, with the Wavelet Transform
frequency behaviors can be resolved up to fractions of a second. On the other hand, with
the election of an adequate window, Gabor Transform is more suitable for the analysis
of signals with a more limited frequency content as shown with Grand Mal seizures, in
which the interesting activity was limited to the lower frequencies of the EEG (up to
12:5Hz see section x3.4).
Although with Gabor Transform the general characteristics of the frequency dynamics
during the Grand Mal seizures was already visible (see section x7.1.1), by using an
alternative decomposition of the signal based on the Wavelet Transform, the Wavelet
Packets, it was possible to study with a better resolution the evolution of the frequency
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peaks.
7.2.3 Wavelet Transform vs. conventional digital ltering
The multiresolution decomposition based on the Wavelet Transform has several advantages
over conventional digital ltering based on the Fourier Transform (ideal lters).
Moreover, due to the fact that the multiresolution decomposition method is implemented
as a ltering scheme, it can be seen as a way to construct lters with an optimal
time-frequency resolution.
An important pointtobementioned is that the multiresolution decomposition has a
powerful mathematical background in fact, it can be seen as a sequence of correlations
between the original signal and dilatations and translations of a single \mother function"
(and its complementary function, see section x4.2.3). Furthermore, the optimal mother
function to be applied to a certain signal can be chosen based on its mathematical
properties or just based on visual features that can be more or less suitable for the
analysis of a certain signal (see sec. 4.2.4).
Since wavelets have a varying window size adapted to each frequency range, the
\ltering" of some frequency bands does not aect the morphology of the others. For
example, it is well known that when ltering with Fourier based lters the high frequencies
of the EEG, the morphology of the low frequencies is also aected (e.g. in the
case of a recording of an epileptic seizure, the shape of the spikes can be modied, thus
obscuring important details). Moreover, in the case of ERPs, the use of a method based
in wavelets avoids unwanted eects as \ringing" (i.e. the spurious appearance of an
stimulus related amplitude enhancement previous to stimulation). In general, it can be
stated that the Fourier based ltering gives a more smooth signal than the one obtained
by using the multiresolution decomposition due to the nearly optimal time-frequency
resolution of the Wavelet Transform for every scale. In order to exemplify these advantages,
in section 4.5.2 I showed with some selected sweeps how the multiresolution
decomposition implemented with B-Spline functions leads to a better resolution of the
event-related responses in comparison with an \ideal lter".
Another interesting point is that the multiresolution decomposition gives discrete
coecients, thus making very easy the design and implementation of statistical tests.
Similar type of analysis are more dicult to implement from digitally ltered signals.
In this respect, the straightforward implementation of statistical tests plus the high
time-frequency resolution of wavelets could be critical for obtaining signicant results
as showed for example in the analysis of ERPs (see sec. x4.5 and x4.6).
107

7.2.4 Chaos analysis vs. time-frequency methods (Gabor, Wavelets)
Since all the time-frequency methods described in this thesis are linear, due to the very
complex non linear nature of EEG signals, a non-linear analysis as the one performed
with the methods of Chaos theory could give new information, an alternative way of
visualization, or at least a new way ofquantication. Moreover, Chaos analysis can put
constraints to the system, thus helping in making models. It can also give a useful way
of data reduction, this for example being very useful for calculating cross-correlations
between dierent EEG channels. However, since these methods have several prerequisites
such as stationarity, small amount of noise, long data recordings, etc., they are very
dicult to implement in the case of EEG signals and their results should be taken as
relative values. In this respect, the validity of Chaos analysis for discriminating between
a random or deterministic nature of EEG signals is subject to several criticisms as I
described in section x7.1.3.
Up to now, parameters such as the D 2 , 1 , etc., characterized general properties
of the EEG, such as the complexity or chaoticity of the brain in dierent states and
pathologies. However, the conclusions reached with these methods are still very vague
in comparison with more precise conjectures about physiological processes accessed with
time-frequency analysis, as described for example in section x7.1.1 in the analysis of
epileptic seizures. In this respect, physiological interpretation from results obtained with
time-frequency methods is very useful due to the relation of dierent brain oscillations
with sources, functions and pathologies of the brain. Moreover, in many cases results
reported with Chaos methods are easily visualized in the EEG or with other more simple
methods. Then, it is reasonable to check if the results obtained with this approach
are in fact new, by comparing them with traditional methods as simple correlations,
Fourier analysis or the time-frequency methods described in this thesis. Despite all these
diculties, new methods based on Chaos theory should be implemented in the analysis
of EEG signals, this approach still being very promising for \visualizing" dynamical
properties of the EEG that keep obscure to the linear methods.
7.2.5 Wavelet-entropy vs. frequency analysis
Wavelet-entropy gives new information about EEG signals in comparison with the one
obtained by using frequency analysis or other standard methods. In fact, I showed
that WS is independent of the amplitude or energy of the signal. Instead of giving an
amplitude measurement, the WS shows how the energy is distributed in the dierent
frequency bands (see section x6.3.2). WS gives a measure of the \order" of the signal, i.e.
signals characterized by narrow peaks in the frequency domain are more ordered than
108

the ones having wide peaks (being the limit case a wide band spectrum corresponding
to noise or to a chaotic system). Then, the concept of entropy isvery interesting due to
the fact that it can be associated to frequency tuning of the neuronal groups.
Although in principle the entropy of the signal can be visualized from the Fourier
spectrum (i.e. just by looking how narrow or wide are the peaks), the WS allows the
following of its time evolution and furthermore, gives a reliable way of quantication.
Moreover, the advantage of dening the measure of entropy from the wavelet coecients
and not from other alternative time-frequency distribution as the Gabor Transform is
that the resolution of the Wavelet Transform is crucial for analyzing the evolution of
fast varying signals as in the case of event-related potentials.
7.2.6 Wavelet-Entropy vs. Chaos analysis
As stated in several parts of this thesis, ERPs can be considered as a selective enhancement,
synchronization or in another words, an ordering of the spontaneous EEG
oscillations. In this context, Wavelet Entropy appears as a natural and optimal method
for measuring this evoked ordering of the EEG signals.
Although using a completely dierent approach, and applied to dierenttypeofEEG
signals, Chaos analysis was in principle used for answering similar type of questions. Parameters
such asD 2 or 1 give a measure of the complexity, chaoticity, and by extension
give an idea of the order of the signal. In this context, converging low values of D 2 for
example, were used as a proof of a low dimensional, deterministic, ordered dynamic of
the EEG signals in several situations. However as I already mentioned, chaos methods
are very dicult to apply to EEG signals and furthermore, in many cases lead to pitfalls
and wrong results.
As discussed in section x7.1.3, it is very dicult to resolve disputes about the
nature of EEG signals by using these methods and it is more reasonable to consider
other physiological evidence as for example the response of the EEG to stimulation
(ERP). However, Chaos methods are limited to the analysis of long and stationary EEG
recordings, being impossible to implement them to the analysis of fast varying nonstationary
signals as ERPs. On the other hand, WS is applicable to ERPs and appears
as an ideal parameter for obtaining quantitative answers to these type of questions. In
particular, I showed in section x6.3.2 that decreases of entropy after stimulation were
correlated with an ordering of the brain rhythms involved in a cognitive process.
109

Conclusion
In this thesis I showed the application of several methods to the analysis of dierent
type of EEG signals. Due to the high complexity of EEGs, these methods needed
to be adapted or extended. Furthermore, it was possible to compare their abilities
in reaching results that allow interesting physiological interpretations. The methods
described complement the information obtained by the visual inspection of the EEG
carried by trained electroencephalographers and furthermore, they give aquantication
that allows the performance of statistical analysis. Moreover they can give an alternative
way of visualization and in the best case the access to information that keeps \hidden"
in the visual inspection of the EEG.
In this context, I showed a dynamics of the frequency patterns during Grand Mal
seizures. Seizures were dominated by alpha and theta rhythms, later becoming slower
with the starting of the clonic phase. Moreover, Chaos analysis showed a transition to
a simpler system during the seizures.
I also showed a distributed origin of event-related alpha oscillations, this ones being
related with primary sensory processing. It was possible to conjecture a relation between
gamma oscillations and a process responsible of carrying out the information that two
sensory perceptions of a bimodal stimulation correspond in fact to the same stimulus.
Responses to unexpected (TARGET) stimulation, traditionally related with cognitive
processing, showed a \tuning" in their frequency composition in comparison with the
ongoing EEG.
All these results give a valuable contribution in understanding the brain dynamics.
However, despite all the advances due to the development of new techniques and experiments,
is still very little what we know about this topic. The attempts to understand
the dynamics of the brain by analyzing the EEG is like trying to understand the conversations
occurring in a building by analyzing a sound recorded from far away. Dierent
stages and apartments are making dierent tasks, and we are not able to get inside
and see what is going on. The EEG, the sound recorded from far away, is still one of
our main tools to access to one of the most unknown and complex systems in nature,
one of the still elusive \treasures" of science. In this context, we must design clever
experiments and we are obliged to develop a great variety of methods and improve them
up to the limits of their abilities in order to learn more about the behavior of the brain.
And that is what makes the study of the brain so fascinating!
110

A
Time-frequency resolution and the Uncertainty
Principle
In this section after the introduction of some basic concepts from signal analysis, I will
show the proof of the Uncertainty Principle. Then, I will describe its application to the
Fourier, Gabor and Wavelet Transform, stressing the advantages of each method related
with time-frequency localization properties.
A.1 Preliminary concepts
Lets consider a normalized signal x(t) (i.e. x2 (t)dt = 1) with a corresponding
Fourier Transform X(!) 9 . The energy density can be written as jx(t)j 2 in the time
domain or as jX(!)j 2 in the frequency domain. Since the total energy or intensity should
be the same in both domains, we have the following relation (Parceval's theorem)
I =
Z 1
;1
jx(t)j 2 dt = 1
2
R 1
;1
Z 1
;1
jX(!)j 2 d! (56)
Considering the energy densities per time and frequency, the average time can be
dened as:
and the average frequency as:
=
Z 1
;1
t jx(t)j 2 dt (57)
=
Z 1
;1
! jX(!)j 2 d! (58)
Furthermore, the mean frequency can be calculated without the previous computation
of the Fourier spectrum, just by using the following relation (see demonstration
in Cohen, 1995 pp:11)
=
Z 1
;1
! jX(!)j 2 d! =
Z 1
;1
x (t) 1 i
d
x(t) dt (59)
dt
were denotes complex conjugation. From the energy densities we can also dene the
second order moments as:
=
Z 1
;1
t 2 jx(t)j 2 dt (60)
9 for simplicity I will use in this section the angular frequency ! =2f
111

Z 1
=
;1
! 2 jX(!)j 2 d! =
Z 1
;1
d
2
dt x(t)
dt (61)
where we used eq. 59. Finally, the time localization, or duration, can be dened by
means of the standard deviation as:
2 t =
Z 1
;1
and the frequency localization, or bandwidth, as:
2 ! =
Z 1
;1
(t; ) 2 jx(t)j 2 dt = ; 2 (62)
(!; ) 2 jX(!)j 2 d! =
A.2 Uncertainty Principle
Theorem
Proof
Z 1
;1
1
i
d
dt ;
For a normalized signal x(t), if p tx(t) ! 0 for jtj !1, then
t ! 1 2
or t f 1
4
x(t)
2
dt (63)
Due to the fact that the mean time and the mean frequency can be changed by
using a simple transformation (Cohen, 1995), lets assume that = 0 and =0.
Then, from eq. 62 and eq. 63 we obtain:
and therefore
2 ! =
Z 1
;1
2 t 2 ! =
2 t =
Z 1
;1
! 2 jX(!)j 2 d! =
Z 1
;1
t 2 jx(t)j 2 dt
(64)
t 2 jx(t)j 2 dt (65)
Z 1
;1
Z 1
;1
d
2
dt x(t)
d
2
dt x(t)
From the Schwartz inequality 10 , by taking f = tx and g = s 0 we obtain:
Z 1
;1
t 2 jx(t)j 2 dt
Z 1
;1
and resolving the right handintegrand
10 R 1
R
1
;1 jf(x)j2 dx
;1 jg(x)j2 dx
R
1
d
2
dt x(t)
;1 f (x)g(x) dx
112
dt
2
Z 1
;1
dt (66)
dt (67)
tx(t) d dt x(t) dt
2
(68)

Z 1
;1
tx(t) d dt x(t) dt = 1 2
Z 1
;1
t d dt x2 (t) dt = tx2 (t)
2
+1
;1
; 1 2
Z 1
;1
x 2 (t) dt = ; 1 2
replacing in eq. 68 and then in eq. 67, the uncertainty principle is proved.
A.3 Time-frequency resolution of the Fourier, Gabor and Wavelet
Transform
Fourier Transform is based in making a correlation between the original signal and
complex sinusoidal functions (see eq. 3). Since the Fourier Transform of these functions
is:
x(t) =e i! 0t
! X(!) =2 (! ; ! 0 ) (69)
then, the sinusoidal \mother functions" of the Fourier Transform are perfectly localized
in the frequency domain. However, they are not localized in time.
As stated in sec. x3.2, Gabor Transform consist in making a correlation between
the signal and amplitude modulated complex sinusoids, by using a windowing function.
Gabor (1946) suggested the use of a Gaussian function as window due to its simultaneous
localization in time and frequency domains. In fact, the Fourier Transform of a Gaussian
function is also a Gaussian:
r
x(t) =g (t) =e ;t2 ! X(!) =
e
; !2
4
(70)
Moreover, for a Gaussian function the inequality 68isanequality and ! t = 1 2 ,thus
having Gaussian functions the best time-frequency resolution allowed by the uncertainty
principle. In the case of the mother functions of the Gabor Transform, i.e. g e ;i!t
where g is the Gaussian windowing function, the equality in eq. 68 also holds, thus
having an optimal time-frequency resolution.
The frequency resolution (bandwidth) of a normalized Gaussian function can be
calculated by using eq. 63:
2 ! = 2
(71)
and in the case of the functions of the Gabor Transform, g e ;i!t , by applying eq. 59
and eq. 63 we obtain for one of these mother functions (! = ! 0 ) the same bandwidth
but localized in the frequency ! 0 , i.e.:
113

Figure 36: Time-frequency resolution of the Gabor Transform.
= ! 0 2 ! = 2
(72)
Then, in the case of Gabor Transform, the frequency resolution depends on the window
wide, corresponding to small values of (wide windows see eq. 16) a better frequency
resolution (but a worst time resolution see also discussion of the window sizein
sec. x3.2).
analyzing
The area determined by the frequency window multiplied by the time window is
called time-frequency window. Plotting of the time-frequency window is an easy way
to visualize the time-frequency resolution: its wide represents the time resolution, its
height represents the frequency resolution and its area represents the time-frequency
resolution, this last one having alower bound determined by the uncertainty principle.
In the case of the Gabor Transform, once the window is xed, the frequency resolution
is the same for all the frequencies (and therefore the time resolution too see g. 36).
Then, for low frequency signals, a wide window will be suitable and in the case of high
frequencies, a narrow window will do the job. However, due to its xed window size,
Gabor Transform is not suitable for analyzing signals with dierent ranges of frequencies.
On the other hand, in the case of the Wavelet Transform, the size of the window is
adapted, thus giving an optimal resolution for all frequencies (see g. 37).
The Wavelet Transform consists in making a correlation between the original signal
114

Figure 37: Time-frequency resolution of the Wavelet Transform.
and dilatations (contractions) of the same mother function (t)
t ; b
ab(t) = jaj ;1=2
a
were a denotes the dierent scales of dilatation (contraction) and b denotes the time
localization of the mother function. Let us denote by ~! and by ! the mean frequency
and the bandwidth of the mother function (t) and by ~!(a) and ! (a) the ones of the
scaled functions. Then, by applying eq. 59 and eq. 63 we obtain:
(73)
~!(a) = ~! a
2 !(a) = 1 a 2 2 ! (74)
Then, equation 74 shows the main feature of the Wavelet Transform. Wavelets provide
a time-frequency window which automatically narrows when studying high frequencies
and widens when analyzing low frequencies, but always keeping the area of the timefrequency
window constant, as shown in g. 37.
115

116

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Biographical sketch
21.03.1967 born in Buenos Aires Argentina
Nov. 1984
March 1993
nished the high school studies
nished the studies on Physics at the University of Buenos Aires
March 1991 second class teaching assistant at the Dept. of Physics - University
- March 1994 of Buenos Aires
March 1994 rst class teaching assistant at the Dept. of Physics - University
- April 1996 of Buenos Aires
March 1989 second class teaching assistant at the Ciclo Basico Comun -
- April 1993 University of Buenos Aires
April 1993 rst class teaching assistant at the Ciclo Basico Comun -
-May 1996 University of Buenos Aires
May/August given advanced course over Physics applied to Anesthesiology
1994 at the Hospital Italiano
April 1993
-May 1995
May 1995
-May 1996
June 1996
-May 1998
scholar at the department of Neurophysiology - Institute of
Neurological investigations (FLENI), Argentina
scholar at the department of Epilepsy - Institute of Neurological
investigations (FLENI), Argentina
guest scientist at the Institute of Physiology - Medical University
of Lubeck, Germany. Supported by the BMBF
June 1998 guest scientist at the Institute of Physiology - Medical University
- August 1998 of Lubeck, Germany. Supported by the Med. Univ. Lubeck
Nov. 1998 -
post-doc position at the Forschungszentrum Julich, Germany
129