

Preprint 11/2009
Moment analysis of the Hodgkin-Huxley system with additive noise
Henry Tuckwell and Jürgen Jost
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Submission date: 19. Feb. 2009
published in: Physica / A, 388 (2009) 19, p. 4115-4125
DOI number (of the published article): 10.1016/j.physa.2009.06.029
Bibtex
Keywords and phrases: Hodgkin-Huxley model, stochastic processes, Moment equations
Abstract:
We consider a classical space-clamped Hodgkin-Huxley (HH) model neuron stimulated by a current which has
a mean together with additive Gaussian white noise of amplitude
.
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 4th 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, when rhythmic spiking
is inhibited by noise in certain parameter ranges, 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
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 basin of attraction of the
of the limit cycle and the speed of the process.