Stochastic Analysis and Numerical Simulation of Piecewise Fractional Hindmarsh-Rose Neuron Models via Generalized Polynomial Chaos

This paper presents a rigorous stochastic framework for the analysis and numerical simulation of chaotic neuronal dynamics governed by piecewise fractional differential operators. We consider a modified Hindmarsh–Rose system in which the memory effect is described by a piecewise-defined fractional derivative with uncertain order. The resulting model captures switching memory regimes and incorporates uncertainty in the fractional dynamics through a random fractional order of the form [Formula: see text] where [Formula: see text] is a prescribed random variable. To efficiently quantify the impact of uncertainty, we employ a generalized polynomial chaos (gPC) expansion, which transforms the stochastic fractional system into a coupled deterministic system for the modal coefficients. This reformulation enables the use of a predictor–corrector scheme specifically adapted to fractional operators, ensuring an accurate treatment of the inherent nonlocal memory effects. Within this framework, key statistical quantities, such as the mean and variance, are directly reconstructed from the expansion coefficients, thereby eliminating the need for costly sampling procedures. Building upon this numerical strategy, a detailed analysis is conducted to investigate the influence of both the fractional order and the switching time on the system dynamics. The results indicate that decreasing the fractional order intensifies memory effects, leading to larger oscillation amplitudes and more complex attractor structures, whereas increasing the switching time enhances stability and reduces uncertainty propagation. Furthermore, comparisons with Monte Carlo simulations demonstrate that the proposed approach achieves higher accuracy while significantly reducing computational cost. The developed framework thus provides a robust and efficient tool for the study of nonlinear stochastic fractional systems and offers deeper insight into the interplay between memory effects and uncertainty in chaotic neuronal dynamics.