On the Voltage-Conductance kinetic equation

TL;DR

This paper analyzes a nonlinear Voltage-Conductance kinetic equation from neuroscience, proving existence of solutions and exponential stability of steady states using ultracontractivity properties.

Contribution

It establishes the existence of solutions and linear exponential stability for the Voltage-Conductance kinetic equation, extending previous estimates with new analytical techniques.

Findings

Existence of solutions in weighted $L^\infty$ space.

Linear exponential stability of the steady state.

Use of ultracontractivity properties in analysis.

Abstract

We consider the nonlinear Voltage-Conductance kinetic equation arising in neuroscience. We establish the existence of solutions in a weighted $L^\infty$ framework in a weak interaction regime. We also prove the linear asymptotic exponential stability of the steady state making constructive a recent estimate of Xu'an Dou et al. (2023). Both results are based in a fundamental way on some ultracontractivity property of theflow associated to the linear (possibly time dependent) Voltage-Conductance kinetic equation.