Bounds and long term convergence for the voltage-conductance kinetic system arising in neuroscience

TL;DR

This paper establishes bounds and long-term convergence for a voltage-conductance kinetic system in neuroscience, addressing boundary condition challenges using advanced mathematical techniques.

Contribution

It provides the first L-infinity bounds and convergence results for the voltage-conductance equation with boundary degeneracy in neuronal network modeling.

Findings

Proved an L-infinity bound on the steady state solution

Established long-term convergence to the stationary state

Developed methods to handle boundary degeneracy in hypoelliptic equations

Abstract

The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently. We prove an L $\infty$ bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads us to use specific weights in the Alikakos iterations and adapt the relative entropy method.