Moment analysis of the Hodgkin–Huxley system with additive noise

We consider a classical space-clamped Hodgkin–Huxley (HH) model neuron stimulated by a current which has a mean μ together with additive Gaussian white noise of amplitude σ. A system of 14 deterministic first-order nonlinear differential equations is derived for the first- and second-order moments (means, variances and covariances) of the voltage, V, and the subsidiary variables n, m and h. The system of equations is integrated numerically with a fourth-order Runge–Kutta method. As long as the variances as determined by these deterministic equations remain small, the latter accurately approximate the first- and second-order moments of the stochastic Hodgkin–Huxley system describing spiking neurons. On the other hand, for certain values of μ, when rhythmic spiking is inhibited by larger amplitude noise, the solutions of the moment equation strongly overestimate the moments of the voltage. A more refined analysis of the nature of such irregularities leads to precise insights about the effects of noise on the Hodgkin–Huxley system. For suitable values of μ which enable rhythmic spiking, we analyze, by numerical examples from both simulation and solutions of the moment equations, the three factors which tend to promote its cessation, namely, the increasing variance, the nature and shape of the basins of attraction of the limit cycle and stable equilibrium point and the speed of the process.