A polynomial differential quadrature-based numerical scheme to simulate the nerve pulse propagation in the spatial Fitzhugh-Nagumo model

Abstract

Nonlinear dynamics connect the neurons that form the brain, and thus, the complex information is produced and transported. The function of the neurons and the problem of understanding the dynamics of the brain has been the research area of mathematical neuroscience. In this study, the modelling and simulation of the propagation of the electric field based Action Potential (AP) on the Two Dimensional (2-D) field of axon network, whose matrix consists of 128×128 electrically coupled neurons were done using nonlinear Spatial FitzHugh Nagumo (SFN) equations. SFN equations are a particular class of Partial Differential Equation’s (PDE’s) exhibiting travelling wave behaviour occurred in neuron systems. The motivation of this paper is to evaluate the SFN equation, which is a special kind of the time-dependent nonlinear reaction-diffusion problem governing neuron dynamics numerically in 2-D space addressed by investigating the Polynomial-based Differential Quadrature Method (PDQM) having Chebyshev-Gauss-Lobatto quadrature points. The solution occurs as elliptical spiral waves induced by electrical stimulation. Thus, the neuronal system behaviour and the interaction with the specific type of Boundary Conditions (BC’s) are predicted. The space derivatives are discretised through PDQM. In this way, the problem is reduced into a system of first-order non-linear differential equations. Hereafter the time derivatives of the SFN equation are solved through the Finite Difference Method (FDM). The various dynamical behaviour that governs the travelling wave pattern regarding the Initial Condition’s (IC’s), BC’s and way of stimulation of the neuron model examined in details. Numerical results indicated that the proposed PDQM provide reliable, fast, and efficient solutions.