Nonlinear stability for a two-dimensional Fokker-Planck equation with partial diffusion in neuroscience

Abstract The primary objective of this paper is to investigate the large-time asymptotic behaviour of a two-dimensional nonlinear Fokker–Planck equation, characterized by partial diffusion and pulse dynamics, arising in mathematical neuroscience. We prove that, within a weakly nonlinear regime, the solution to the evolution problem converges exponentially fast toward a unique stationary state. As a prerequisite for this stability analysis, we first prove the existence, uniqueness, and regularity of weak solutions to the corresponding stationary problem under weak connectivity. To address the analytical challenges introduced by the nonlinear terms in the long-term dynamics, our methodology combines the Doeblin–Harris framework for non-autonomous linear equations with perturbation theory. These theoretical findings are further supported by numerical simulations illustrating the convergence properties of the model.