Spike trains in a stochastic Hodgkin-Huxley system

Abstract

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 mu<mu(c) and mu>mu(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), sigma must be large to induce firing so paths are always erratic. As the noise increases, the coefficient of variation (CV) has a well-defined minimum, and then climbs steadily over the range considered. If mu is just above mu(c), when the noise is small, paths are close to deterministic and the standard deviation and CV of the time to spike are small. As sigma increases, some very erratic paths (some almost oscillatory) appear, making the mean, standard deviation and CV of the spike time very large. These erratic paths start to have a large influence, so all three statistics have very pronounced maxima at intermediate sigma. 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.