A comprehensive numerical investigation of a coupled mathematical model of neuronal excitability

TL;DR

This paper conducts a detailed numerical study of the FitzHugh-Nagumo model, analyzing stability, convergence, and robustness of various numerical schemes to better understand neuronal excitability.

Contribution

It introduces an alternative numerical approach using Taylor polynomials and difference schemes, with a focus on stability and robustness for the FitzHugh-Nagumo model.

Findings

Numerical schemes are stable and convergent for the model.

Analytical and numerical stability conditions are consistent.

The proposed methods effectively simulate neuronal excitability.

Abstract

Being an example for a relaxation oscillator, the FitzHugh-Nagumo model has been widely employed for describing the generation of action potentials. In this paper, we begin with a biological interpretation of what the subsequent mathematical and numerical analyses of the model entail. The interaction between action potential variable and recovery variable is then revisited through linear stability analysis around the equilibrium and local stability conditions are determined. Analytical results are compared with numerical simulations. The study aims to show an alternative approach regarding Taylor polynomials and constructed difference scheme which play a key role in the numerical approach for the problem. The robustness of the schemes is investigated in terms of convergency and stability of the techniques. This systematic approach by the combination of numerical techniques provides beneficial results which are uniquely designed for the FitzHugh-Nagumo model. We describe the matrix representations with the collocation points. Then the method is applied in order to acquire a system of nonlinear algebraic equations. On the other hand, we apply finite difference scheme and its stability is also performed. Moreover, the numerical simulations are shown. Consequently, a comprehensive investigation of the related model is examined.