Modelling of double reset spiking neuron and Fitzhugh–Nagumo equations with coupling kernel functions

Creation of a computationally efficient and biologically plausible neuron and neural network models is an exceptionally important occupation, the results of which are much needed for the neuroscience in general or for the neurophysiology in partial. Besides, the applying neural network technological areas are always looking for novel improvements of the methodological approaches, algorithms, computational facilities.In this paper, a new (in some aspects) hybrid integrate-and-fire (HIF) neuron model, based on N  -shaped Na+Na+ and K+K+ ionic current–voltage characteristics and a double discrete reset mechanism, is proposed. In the results of computational experiments, the spike series from a single spiking to the burst and chaotic burst trains are demonstrated.In the other part of the paper, analogous FitzHugh–Nagumo neuron model as a generalized neural networks system with an expansion to an inclusion of a coupling kernel function has been considered. The mean field approximation of neuronal potentials and recovery currents inside a neuron cluster was used. The biologically more realistic nonlinear N-shaped Na+Na+ characteristic and kernel functions (one - or two-dimensional) were applied. A possibility to present the nonlinear integral differential equations with kernel functions under the Fourier transformation by partial differential equations was used that allowed us to overcome the analytical and numerical modeling difficulties. The equivalence of two kinds of solutions was additionally confirmed based on the error evaluation. The approach of the equivalent partial differential equations was successfully employed to solve the system with the heterogeneous synaptic functions. The analytical studies are corroborated by many numerical modeling experiments.