Fractional-order FitzHugh–Nagumo dynamics: From single-neuron stability bifurcations to synchronization in small-world networks

Abstract

This study pioneers a unified theoretical framework for fractional-order (FO) FitzHugh–Nagumo (FHN) neurodynamics, uncovering novel order-dependent and topology-order synergetic regulation mechanisms. By integrating FO stability theory with an extended master stability function approach, we achieve three pivotal breakthroughs: First, we identify a novel order-dependent stability bifurcation in single FOFHN neurons, where a reduced order enhances nodal stability via the intrinsic memory effects of FO calculus. Second, FOFHN networks exhibit counterintuitive non-monotonic synchronization bifurcations, which reveal the dual regulatory role of memory effects: while FO memory effects stabilize individual neurons, they can either facilitate or impair synchronous behavior at the network scale. Third, we discover unique nonlinear interactions between small-world topology and fractional order that generate distinct network synchronization patterns, where optimal synchronization arises from the balanced interplay of fractional order, topological structure, and nodal dynamical properties. This work bridges critical gaps in cross-scale FO neurodynamics, offering fundamental new insights into memory-dependent neuronal dynamics and establishing practical design principles for the modeling and control of FO neuronal networks.