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 μ and variance parameter σ 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 μ c ( ≈ 4.15 ) be the critical value of μ for firing when noise is absent. Large sample simulation results are obtained for μ < μ c and μ > μ c , for many values of σ between 0 and 25. For the time to spike, the 2-component approximation is accurate for all σ when μ = 10 , for σ > 7 when μ = 5 and only when σ > 15 when μ = 2 . When μ < μ c , σ 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 μ is just above μ 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 σ 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 σ . When μ ≫ μ c , most paths show similar behavior and the moments exhibit smoothly changing behavior as σ increases. Thus there are a different number of regimes depending on the magnitude of μ relative to μ c : one when μ is small and when μ is large; but three when μ is close to and above μ c . Both for the Hodgkin–Huxley (HH) system and the 2-component approximation, and regardless of the value of μ , the CV tends to about 1.3 at the largest value (25) of σ 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.