Modeling adaptive memory in neural dynamics: a dual nabla Fractional-Order FitzHugh–Nagumo framework

Mathematical modeling of neural dynamics has been greatly advanced by fractional calculus (FCs), which incorporates memory and hereditary effects often observed in biological systems. This study introduces a novel discrete fractional FitzHugh-Nagumo (FHN) model, extending the classical excitable system through the use of a dual nabla Caputo (DNC) operator. The model allows the fractional order (FO) to vary in time, thereby capturing adaptive memory effects and time-modulated dynamical behaviors more realistically. A rigorous stability analysis is conducted, establishing sufficient conditions for local Mittag-Leffler stability (LMLS) of equilibrium points (EPs) and proving global Mittag-Leffler stability (GMLS) under specific parameter constraints. An efficient finite-difference numerical scheme is implemented to validate the theoretical results, demonstrating the model’s ability to simulate complex spatiotemporal patterns, modulated wave propagation, and convergence behaviors. The proposed framework offers a unified mathematical structure that generalizes both integer-order and constant-order fractional FHN models, and its fully discrete formulation is directly amenable to implementation on digital and neuromorphic hardware, bridging advanced FCs with practical bio-inspired engineering applications.