A Fokker-Planck equation with superlinear drift at infinity for Integrate-and-Fire model

TL;DR

This paper extends the classical Noisy Integrate-and-Fire model by incorporating a superlinear drift in the Fokker-Planck equation, establishing well-posedness, boundary conditions, entropy dissipation, and exponential convergence to equilibrium.

Contribution

It introduces a novel framework on the full line for the model with superlinear drift, proving key properties and convergence results.

Findings

Well-posedness of the extended model established

Boundary condition at infinity rigorously defined

Exponential convergence to the stationary state proven

Abstract

The Integrate-and-Fire model is a Fokker-Planck equation arising in neuroscience. It describes the evolution of the probability density of the neuronal membrane potential and fitting has shown that the inclusion of a em superlinear drift provides the most realistic description. To make sense of this, we propose to set the equation on the full line, the neural activity being described by the flux at infinity. This framework serves as a model extension of the classical Noisy Integrate-and-Fire model, with a fixed firing potential. We first establish the well-posedness of the solution, establish the boundary condition at infinity which is the major difficulty. Then, state rigorously the entropy dissipation property. Finally, using Doeblin's method, we prove the exponential convergence of the solution toward the unique stationary state in full generality.