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

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