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

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,
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