Information Technology Reference
In-Depth Information
many normal physiological processes, such as the sleep-wake cycle and motor con-
trol, as well as pathological conditions such as epilepsy, Parkinson's disease and
other movement disorders, and sleep disorders such as sleep apnea and narcolepsy.
Traditionally, neurophysiologists analyze such signals using visual inspection or
through statistical analysis of linear signal properties such as the spectrogram and
coherence. More recently, investigators have begun to investigate the spatiotempo-
ral dynamical features of neurological signals. However, many of these techniques
have been applied to mathematical models or to dynamical systems that are much
less complex than the brain. Therefore, there is need to develop and evaluate math-
ematical techniques that provide robust results in higher dimensional, nonstationary
and noisy biological systems such as the brain. Although difficult or even impossi-
ble to prove, there has been little debate that brain activities should be modeled as
nonlinear systems. As a result, during the last decade, a variety of nonlinear time
series analysis techniques have been applied repeatedly to EEG recordings during
physiologic and pathologic conditions. Among those, the algorithms based on the
Lyapunov exponents appears promising for characterizing the spatiotemporal dy-
namics in electroencephalogram (EEGs) time series recorded from patients with
temporal lobe epilepsy. Nevertheless, there are many improvements can be made in
algorithms for finding Lyapunov exponents so that the estimation can be more ro-
bust, especially with respect to the presence of noise in the EEG. As the complexity
of the algorithms for finding Lyapunov exponents with the noise and nonstationarity
of EEG, this task requires development of novel techniques and numerous compu-
tational experiments.
This chapter is organized as follows. In the next section, we describe Lyapunov
exponents and discuss some of the algorithms for finding them. In Section 16.3,
an optimization model for estimating Lyapunov exponents is presented and a solu-
tion technique for the model is proposed. Brief descriptions of the models used for
computational experiments and a comparison of performance, including sensitivity
analysis, between the proposed algorithm and the two well-known algorithms are
given in Sections 16.4 and 16.5, respectively. The details of the numerical compu-
tations are also given in Section 16.5.
16.2 Lyapunov Exponents
Chaos is one type of behavior exhibited by nonlinear dynamical systems, which
are systems whose time evolution equations are nonlinear, that is, the dynamical
variables describing the properties of the systems (for example, position, veloc-
ity, acceleration, pressure) appear in the equations in a nonlinear form. There are
several techniques to measure chaos, depending on what one wants to character-
ize in the chaotic trajectory. Some of the techniques include: simple visual in-
spection of either the time series represented by a time plot of the trajectory, or
the bounded strange attractor reconstructed from the time series; spectral analy-
sis of the time series; Lyapunov exponents; and entropy analysis. Here, we focus