Spike trains in a stochastic Hodgkin-Huxley system.

We consider a standard Hodgkin-Huxley model neuron with a Gaussian white noise input current with drift parameter mu and variance parameter sigma(2). Partial differential equations of second order are obtained for the first two moments of the time taken to spike from (any) initial state, as functions of the initial values. The analytical theory for a 2-component (V,m) approximation is also considered. Let mu(c) (approximately 4.15) be the critical value of mu for firing when noise is absent. Large sample simulation results are obtained for mumu(c), for many values of sigma between 0 and 25. For the time to spike, the 2-component approximation is accurate for all sigma when mu=10, for sigma>7 when mu=5 and only when sigma>15 when mu=2. When mu>mu(c), most paths show similar behavior and the moments exhibit smoothly changing behavior as sigma increases. Thus there are a different number of regimes depending on the magnitude of mu relative to mu(c): one when mu is small and when mu is large; but three when mu is close to and above mu(c). Both for the Hodgkin-Huxley (HH) system and the 2-component approximation, and regardless of the value of mu, the CV tends to about 1.3 at the largest value (25) of sigma considered. We also discuss in detail the problem of determining the interspike interval and give an accurate method for estimating this random variable by decomposing the interval into stochastic and almost deterministic components.