Mean Field Modeling of the FitzHugh–Nagumo Neuronal Network Model with Kernel Functions and Time-Delayed Couplings

Abstract

In this paper, the nonlinear neural network FitzHugh–Nagumo model with an expansion by the excited neuronal kernel function has been investigated. The mean field approximation of neuronal potentials and recovery currents inside neuron ensembles was used. The biologically more realistic nonlinear sodium ionic current–voltage characteristic and kernel functions were applied. A possibility to present the nonlinear integral differential equations with kernel functions under the Fourier transformation by partial differential equations allows us to overcome the analytical and numerical modeling difficulties. An equivalence of two kinds solutions was confirmed basing on the errors analysis. The approach of the equivalent partial differential equations was successfully employed to solve the system with the heterogeneous synaptic functions as well as the FitzHugh–Nagumo nonlinear time-delayed differential equations in the case of the Hopf bifurcation and stability of stationary states. The analytical studies are corroborated by many numerical modeling experiments.The digital simulation at the transient and steady-state conditions was carried out by using finite difference technique. The comparison of the simulation results revealed that some of the calculated parameters, i.e. response and sensitivity is the same, while the others, i.e. half-time of the steady-state is significantly different for distinct models.