Numerical simulation of motor neural system transmission models driven by multiplicative white noise: mean-square stability and convergence analysis

Abstract

The signal transmission of motor neural system is inherently subject to random perturbations. Understanding the regulatory role of such noise in neural information encoding remains a central challenge in computational neuroscience and applied mathematics. This paper develops a proper orthogonal decomposition (POD) based reduced-order fully implicit finite difference method for the stochastic FitzHugh-Nagumo (FHN) neural system transmission model driven by multiplicative white noise. A fully implicit finite difference scheme is constructed to discretize the stochastic reaction-diffusion system, which overcomes stability constraints in traditional semi-implicit methods. We rigorously prove that both the full-order and reduced-order schemes are unconditionally mean-square stable without any restrictions on temporal and spatial step sizes. By employing a multi-sample joint snapshot strategy, the POD technique is used to extract dominant dynamic modes and project the high-dimensional system onto a low-dimensional space, while preserving the statistical structure of multiplicative noise. Comprehensive numerical experiments are carried out on one-dimensional, two-dimensional and three-dimensional stochastic FHN models. Theoretical analysis and numerical results confirm that the proposed method maintains high approximation accuracy comparable to the full-order scheme, achieves a speed-up factor of more than 10 times, and remains stable even under strong multiplicative noise. This work provides an efficient, stable, and reliable computational framework for large-scale and long-time simulations of stochastic motor neural dynamics, and can be extended to a broader class of semilinear stochastic partial differential equations.