  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.
