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

Xu'an Dou; Benoît Perthame; Delphine Salort; Zhennan Zhou

doi:10.3934/dcds.2022136

Article Contents

2023, Volume 43, Issue 3&4: 1366-1382. Doi: 10.3934/dcds.2022136

This issue Previous Article Parabolic methods for ultraspherical interpolation inequalities Next Article A new convergence proof for approximations of the Stefan problem

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

1.
Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China
2.
Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions, 75005, Paris, France
3.
Sorbonne Université, CNRS, Laboratoire de Biologie Computationnelle et Quantitative, 75005, Paris, France

^*Corresponding author: Benoît Perthame
^*Corresponding author: Benoît Perthame

In honour of Juan-Luis Vázquez's 75th birthday

Received: February 2022

Revised: August 2022

Early access: September 27, 2022

Published online: September 27, 2022

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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.

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Mathematics Subject Classification: Primary: 35B65, 35Q84; Secondary: 92B20.

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