  We consider the nonlinear Voltage-Conductance kinetic equation arising in
neuroscience. We establish the existence of solutions in a weighted $L^\infty$
framework in a weak interaction regime. We also prove the linear asymptotic
exponential stability of the steady state making constructive a recent estimate
of Xu'an Dou et al. (2023). Both results are based in a fundamental way on some
ultracontractivity property of theflow associated to the linear (possibly time
dependent) Voltage-Conductance kinetic equation.
