Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons

Joshua H. Goldwyn, Nikita S. Imennov, Michael Famulare, and Eric Shea-Brown
Phys. Rev. E 83, 041908 – Published 11 April 2011; Erratum Phys. Rev. E 83, 049902 (2011)

Abstract

The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. E 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can capture the stochastic dynamics of Markov chain models.

We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effects on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case. The temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, but the subunit-based approaches do not correctly account for this structure. Our study confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.

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  • Received 21 September 2010
  • Publisher error corrected 14 April 2011

DOI:https://doi.org/10.1103/PhysRevE.83.041908

©2011 American Physical Society

Corrections

14 April 2011

Erratum

Publisher’s Note: Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons [Phys. Rev. E 83, 041908 (2011)]

Joshua H. Goldwyn, Nikita S. Imennov, Michael Famulare, and Eric Shea-Brown
Phys. Rev. E 83, 049902 (2011)

Authors & Affiliations

Joshua H. Goldwyn1, Nikita S. Imennov2, Michael Famulare3, and Eric Shea-Brown1,4

  • 1University of Washington, Department of Applied Mathematics, P.O. Box 352420, Seattle, Washington 98195 USA
  • 2University of Washington, Department of Bioengineering, P.O. Box 355061, Seattle, Washington 98195 USA
  • 3University of Washington, Department of Physics, P.O. Box 351560, Seattle, Washington 98195 USA
  • 4University of Washington, Department of Applied Mathematics and University of Washington, Program in Neurobiology and Behavior, P.O. Box 357270, Seattle, Washington 98195 USA

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Issue

Vol. 83, Iss. 4 — April 2011

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