A voltage-conductance kinetic system from neuroscience: probabilistic reformulation and exponential ergodicity

Autor:	Dou, Xu'an, Kong, Fanhao, Xu, Weijun, Zhou, Zhennan
Rok vydání:	2023
Předmět:	Mathematics - Probability Mathematics - Analysis of PDEs Quantitative Biology - Neurons and Cognition 35B40 35Q84 35Q92 37A25 92B20
Druh dokumentu:	Working Paper
Popis:	The voltage-conductance kinetic equation for an ensemble of neurons has been studied by many scientists and mathematicians, while its rigorous analysis is still at a premature stage. In this work, we obtain for the first time the exponential convergence to the steady state of this kinetic model in the linear setting. Our proof is based on a probabilistic reformulation, which allows us to investigate microscopic trajectories and bypass the difficulties raised by the special velocity field and boundary conditions in the macroscopic equation. We construct an associated stochastic process, for which proving the minorization condition becomes tractable, and the exponential ergodicity is then proved using Harris' theorem. Comment: 23 pages, 3 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2305.03992 Zobrazit plný text záznamu View this record from Arxiv