On the exponential convergence rate for a non-gradient Fokker-Planck equation in Computational Neuroscience

TL;DR

This paper proves the exponential convergence rate of a non-gradient Fokker-Planck equation with Robin boundary conditions, relevant to neuronal population models, extending previous numerical findings with rigorous analysis.

Contribution

It provides a rigorous proof of exponential convergence for a non-gradient Fokker-Planck equation with Robin boundary conditions, applicable to computational neuroscience.

Findings

Established exponential convergence rate analytically

Validated previous numerical observations

Applicable to neuron population models

Abstract

This paper concerns the proof of the exponential rate of convergence of the solution of a Fokker-Planck equation, with a drift term not being the gradient of a potential function and endowed by Robin type boundary conditions. This kind of problem arises, for example, in the study of interacting neurons populations. Previous studies have numerically shown that, after a small period of time, the solution of the evolution problem exponentially converges to the stable state of the equation.