Work

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

Year: 2026

Type: article

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.

Source: Nonlinearity

DOI: 10.1088/1361-6544/ae4cec

Authors: Elena Ambrogi, Qingyou He, Delphine Salort

Institutions: Centre National de la Recherche Scientifique, Sorbonne Université, Laboratoire Jacques-Louis Lions, University of Bologna

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,325

Funders: European Commission, Agence Nationale de la Recherche