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

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