Are Correlation Dimension and Lyapunov Exponents Useful Tools for Prediction Of Epileptic Seizures?

Epilepsy, a common neurological disorder, manifests with seizures that occur suddenly, cyclically but aperiodically, features that enhance its disabling power. This chapter reviews the results of correlation dimension analysis from an extensive clinical database-over 2,000 total hours of continuous ECoG from 20 subjects with epilepsy. It examines the sensitivity of the estimated dimension to properties such as the signal amplitude and autocorrelation and describes the effect of the embedding (necessary for estimating dimension from time series. Following this, it discusses the filtering method and performs surrogate data analysis. Based on this understanding, it elucidates that the correlation dimension has no predictive power for seizures. Furthermore, it describes the fundamental relation between the fractal dimension and the Lyapunov exponents, as given by the Kaplan-Yorke formula, and argues for the inability of Lyapunov exponents to predict seizures. It then reviews the control studies based on non-stationary dynamical systems modeled by discrete-time maps and by continuous-time flows, for which the behavior of the Lyapunov exponents is known a priori, to show that the exponents have no predictive or even detective power for characteristic system changes-even when only an exceedingly small amount of noise is present. Finally, it discusses the supporting results from analysis of clinical ECoG data.