On discrete FitzHugh–Nagumo reaction–diffusion model: Stability and simulations

This research paper focuses on the analysis of a discrete FitzHugh–Nagumo reaction–diffusion system. We begin by discretizing the FitzHugh–Nagumo reaction–diffusion model using the second-order and L1-difference approximations. Our study examines the local stability of the equilibrium points of the system. To identify conditions that ensure the global asymptotic stability of the steady-state solution, we employ various techniques, with a primary focus on the direct Lyapunov method. Theoretical results are supported by numerical simulations that demonstrate the practical validity of the asymptotic stability conclusions. Our findings provide new insights into the stability characteristics of discrete FitzHugh–Nagumo reaction–diffusion systems and contribute to the broader understanding of such systems in mathematical biology.