An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

Abstract

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.

Keywords:

• FitzHugh–Nagumo
• nonstandard finite difference
• equilibrium point
• kinematic model

2010 Mathematics Subject Classifications:

• 65M06
• 65M08
• 65L12
• 97N40

Acknowledgments

Thanks are addressed to the anonymous reviewers whose suggestions have contributed to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the support of South African DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences $\left(M^3{B}^2\right)$ of the University of Pretoria. MC and AR also acknowledge the support of National Research Foundation of South Africa under Grant Numbers 93476 and 95864, respectively.