Existence and stability of traveling waves in a biologically constrained model of seizure wave propagation

Abstract

Epilepsy -- the condition of recurrent, unprovoked seizures -- manifests in brain voltage activity with characteristic spatio-temporal patterns. One of the patterns typically observed during a seizure is a traveling wave. To characterize these waves, we analyze high-density local field potential (LFP) data recorded in vivo from human cortex during a seizure from three patients. We show that traveling wave patterns emerge in the LFP with consistent quantitative features. Using a mean-field approach we model the neuronal population activity observed in the LFP and obtain explicit traveling wave solutions for this model. We then employ the LFP data to constrain the model and obtain parameter configurations that support traveling wave solutions with features consistent with the observed LFP waves. In particular, our model formulation is able to capture the "reverberation" of the activity following the traveling wave that was found in the clinical data. We obtain biologically reasonable parameter estimates for two important features: the timescales of the model and the extent of the connectivity. In this way, we link the observed LFP waves during seizure to proposed biological mechanisms. We also study the linear stability of the traveling wave solutions by constructing an Evans function. We find for some parameters the existence of two waves: one wave is slow and narrow whereas the other wave is fast and wide. Moreover, the fast and wide wave has speed and width consistent with the observed LFP waves. We numerically analyze the Evans function to determine stability (instability) of the fast (slow) wave.