Frequency and excitability disorder in epilepsy

Epilepsy and seizure disorders are conditions that affect millions of people worldwide. The onset of seizures, as well as the effect of treatment, undergo immense amounts of study, including approaches using theoretical and phenomenological models. We consider a model where the dynamics of the EEG or local field potential is described using a stochastic differential equation. The transition to the seizing phase is characterized by the coexistence of a stable attractive fixed point, marking normal background brain activity, and a stable limit cycle, which characterizes seizure-like dynamics. In this phase the transition between the two is activated by noise. Borrowing from neural field models, we promote this set of equations to a spatially continuous model, which allows for the use of field theory to study the transition and the correlation structure of the relevant variable. As the main focus, we consider spatial disorder both in the excitability and the frequency of spiking of the coarse-grained regions, as these inhomogeneities are very likely present in brains and neural cultures. We show how this spatial randomness affects the propensity of the system to seize, using both numerical and analytical tools. We hope that such an analysis would give insight into the onset of seizures, like the ictal wave, as well as provide insights into making normal brain activity more robust against perturbations.