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

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.