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Transition densities for stochastic Hodgkin-Huxley models

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.