  In this paper we study the existence of densities for strongly degenerate
stochastic differential equations (SDEs) whose coefficients depend on time and
are not globally Lipschitz. In these models neither local ellipticity nor the
strong H\"ormander condition is satisfied. In this general setting we show that
continuous transition densities indeed exist in all neighborhoods of points
where the weak H\"ormander condition is satisfied. We also exhibit regions
where these densities remain positive. We then apply these results to
stochastic Hodgkin-Huxley models with periodic input as a first step towards
the study of ergodicity properties of such systems in the sense of [27]-[28].
