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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • Abeer Al-Nana,
  • Iqbal M. Batiha

摘要

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.