  We formulate simple criteria for positive Harris recurrence of strongly
degenerate stochastic differential equations with smooth coefficients when the
drift depends on time and space and is periodic in the time argument. There is
no time dependence in the diffusion coefficient. Our criteria rely on control
systems and the support theorem, existence of an attainable inner point of full
weak Hoermander dimension and of some Lyapunov function. Positive Harris
recurrence enables us to prove limit theorems for such diffusions.
  As an application, we consider a stochastic Hodgkin-Huxley model for a
spiking neuron including its dendritic input. The latter carries some
deterministic periodic signal coded in its drift coefficient and is the only
source of noise for the whole system. This amounts to a 5d SDE driven by 1d
Brownian motion for which we can prove positive Harris recurrence using our
criteria. This approach provides us with laws of large numbers which allow to
describe the spiking activity of the neuron in the long run.
