Statistical Model for Excitation and Hypersynchronization in the Small Neural Populations

Abstract

The mathematical modeling of epileptic seizures appearing in small neural populations can follow a few alternative ways: modeling of individual cells and their interaction vs. modeling groups and clusters on neurons. The purpose of this work is invention of a novel continuous (population-based) model for the appearance of the hyper-synchronized firing cells of the epileptiform type. In the same time, we use here the master equations based on the transition probabilities among different states of the cell excitation and hyper-synchronization. We developed an ODE model combining the dynamical equations for different sub-populations (unexcited, excited, and, as our novelty, hypersynchronized). Our model may serve as a simple but powerful tool to analyze the appearance and development of epileptiform dynamics in artificial neural networks. It can cover different cases of microepilepsy, and also may open the gate for studying drug-resistant epilepsy regime. Our dynamical set can be extended with the control inputs mimicking the external perturbations of the neural clusters with the electrical or optogenetic signals. In this case, the set of control algorithms can be applied to detect and suppress the epileptiform dynamics. Thus, the dynamic processes of epilepsy in small neural populations do not demand necessary the development of detailed models for individual neurons. Even the ‘averaged’ dynamical set for the unexcited, excited and hypersynchronized sub-populations can serve as an efficient tool for investigation and numerical simulations of microscopic seizures.