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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
11/2009

Moment analysis of the Hodgkin-Huxley system with additive noise

Henry Tuckwell and Jürgen Jost

Abstract

We consider a classical space-clamped Hodgkin-Huxley (HH) model neuron stimulated by a current which has a mean $\mu$ together with additive Gaussian white noise of amplitude $\sigma$.

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 $\mu$ 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.

Received:
Feb 19, 2009
Published:
Feb 23, 2009
Keywords:
Hodgkin-Huxley model, stochastic processes, Moment equations

Related publications

inJournal
2009 Repository Open Access
Henry C. Tuckwell and Jürgen Jost

Moment analysis of the Hodgkin-Huxley system with additive noise

In: Physica / A, 388 (2009) 19, pp. 4115-4125