Analysis of Nonlinear Dynamics in Epilepsy using a Koopman Operator Framework

Abstract

Epilepsy is a disorder characterized by recurrent seizures, and seizure prediction using electroencephalogram (EEG) and electrocorticogram (ECoG) signals is challenging. Conventional signal processing methods, which assume stationarity and linearity for the system considered, are inefficient at capturing the complex temporal structures of these signals; therefore, this study focused on dynamic mode decomposition (DMD), which has recently been used in fluid dynamics to extract the spatiotemporal dynamic features of non-stationary and nonlinear signals.

This study applied an advanced data-driven nonlinear time series analysis method that combines time-delay embedding, diffusion mapping, and Koopman operator analysis, to the nonlinear dynamics of epileptic ECoG data. We analyzed 5 min of ECoG data from an 11-year-old boy with refractory Rolandic epilepsy. The collected data were embedded into a high-dimensional time-delayed coordinate space, and then its dimensions were reduced by diffusion mapping. The Koopman operator was estimated using extended DMD (eDMD), yielding its eigenvalues, modes, and eigenfunctions that represent the underlying dynamics of brain activity.

We identified 11 of 32 Koopman eigenfunctions as significantly correlated with the occurrence of epileptic spikes, representing the temporal features of the ECoG data. We also found six Koopman modes that were significantly correlated with the spatial pattern of epileptic spike propagation, capturing the spatial features of the ECoG data.

Understanding the spatiotemporal brain dynamics underlying epileptic EEG and ECoG signals may provide new clues to seizure prediction in epilepsy.