  We consider a stochastic Hodgkin-Huxley model driven by a periodic signal as
model for the membrane potential of a pyramidal neuron. The associated five
dimensional diffusion process is a time inhomogeneous highly degenerate
diffusion for which the weak Hoermander condition holds only locally. Using a
technique which is based on estimates of the Fourier transform, inspired by
Fournier 2008, Bally 2007 and De Marco 2011, we show that the process admits
locally a strictly positive continuous transition density. Moreover, we show
that the presence of noise enables the stochastic system to imitate any
possible deterministic spiking behavior, i.e. mixtures of regularly spiking and
non-spiking time periods are possible features of the stochastic model. This is
a fundamental difference between stochastic and deterministic Hodgkin-Huxley
models.
