A Fractional-Order Discrete-Time FitzHugh–Nagumo Model With Multiple Time Delays

Abstract

This paper introduces a novel fractional-order (FO) discrete-time FitzHugh–Nagumo model (FHNM) that incorporates multiple time delays to capture the combined effects of memory and transmission latencies inherent in neuronal dynamics. The model is formulated using the Caputo nabla difference operator, providing a rigorous mathematical framework that preserves the hereditary properties of fractional calculus while enabling direct numerical simulation. We establish sufficient conditions for the uniqueness of solutions through contraction arguments and boundedness assumptions. Furthermore, by constructing an appropriate Lyapunov functional (LF), we derive a criterion for local Mittag–Leffler stability (MLS) of the equilibrium point (EP), a fractional generalization of exponential stability particularly suited for systems with long-range memory. Comprehensive numerical simulations verify these theoretical predictions using realistic parameter values, demonstrating robust convergence to the EP across wide parameter ranges. An extensive sensitivity analysis explores the effects of FO, time delays, model parameters, and numerical implementation choices, identifying critical bifurcation thresholds where oscillations emerge via Hopf bifurcations. The results confirm the theoretical stability conditions and highlight the practical utility of the proposed framework for studying neuronal excitability under the influence of memory and delayed coupling. This work bridges the gap between continuous biophysical models and discrete data-driven analyses, with potential applications in computational neuroscience, neurological disorder research, and neuromorphic computing.