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The voltage-conductance kinetic equation for an ensemble of neurons has beenstudied by many scientists and mathematicians, while its rigorous analysis isstill at a premature stage. In this work, we obtain for the first time theexponential convergence to the steady state of this kinetic model in the linearsetting. Our proof is based on a probabilistic reformulation, which allows usto investigate microscopic trajectories and bypass the difficulties raised bythe special velocity field and boundary conditions in the macroscopic equation.We construct an associated stochastic process, for which proving theminorization condition becomes tractable, and the exponential ergodicity isthen proved using Harris' theorem.

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