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We formulate simple criteria for positive Harris recurrence of stronglydegenerate stochastic differential equations with smooth coefficients when thedrift depends on time and space and is periodic in the time argument. There isno time dependence in the diffusion coefficient. Our criteria rely on controlsystems and the support theorem, existence of an attainable inner point of fullweak Hoermander dimension and of some Lyapunov function. Positive Harrisrecurrence enables us to prove limit theorems for such diffusions.  As an application, we consider a stochastic Hodgkin-Huxley model for aspiking neuron including its dendritic input. The latter carries somedeterministic periodic signal coded in its drift coefficient and is the onlysource of noise for the whole system. This amounts to a 5d SDE driven by 1dBrownian motion for which we can prove positive Harris recurrence using ourcriteria. This approach provides us with laws of large numbers which allow todescribe the spiking activity of the neuron in the long run.

Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model