Mathematical Model of a Main Rhythm in Limbic Seizures

Abstract

While synchronization in the brain neural networks has been studied, the emergency of the main oscillation frequency and its evolution at different normal and pathological states remains poorly investigated. We propose a new concept of the formation of a main frequency in limbic epilepsy. The idea is that the main frequency is not a result of the activity of a single cell, but is formed due to collective dynamics in a ring of model neurons connected with delay. The individual cells are in an excitable mode providing no self-oscillations without coupling. We considered the ring of a different number of Hodgkin–Huxley neurons connected with synapses with time delay. We have shown that the proposed circuit can generate oscillatory activity with frequencies close to those experimentally observed. The frequency can be varied by changing the number of model neurons, time delay in synapses and coupling strength. The linear dependence of the oscillation period on both coupling delay and the number of neurons in the ring was hypothesized and checked by means of fitting the values obtained from the numerical experiments to the empirical formula for a constant value of coupling coefficient.

Keywords

• Hodgkin–Huxley equations
• neural networks
• time delay system
• collective dynamics
• central pattern generator
• approximation