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The random transitions of ion channels between conducting and non-conductingstates generate a source of internal fluctuations in a neuron, known as channelnoise. The standard method for modeling fluctuations in the states of ionchannels uses continuous-time Markov chains nonlinearly coupled to adifferential equation for voltage. Beginning with the work of Fox and Lu, therehave been attempts to generate simpler models that use stochastic differentialequation (SDEs) to approximate the stochastic spiking activity produced byMarkov chain models. Recent numerical investigations, however, have raiseddoubts that SDE models can preserve the stochastic dynamics of Markov chainmodels.  We analyze three SDE models that have been proposed as approximations to theMarkov chain model: one that describes the states of the ion channels and twothat describe the states of the ion channel subunits. We show that the formerchannel-based approach can capture the distribution of channel noise and itseffect on spiking in a Hodgkin-Huxley neuron model to a degree not previouslydemonstrated, but the latter two subunit-based approaches cannot. Our analysisprovides intuitive and mathematical explanations for why this is the case: thetemporal correlation in the channel noise is determined by the combinatorics ofbundling subunits into channels, and the subunit-based approaches do notcorrectly account for this structure. Our study therefore confirms andelucidates the findings of previous numerical investigations of subunit-basedSDE models. Moreover, it presents the first evidence that Markov chain modelsof the nonlinear, stochastic dynamics of neural membranes can be accuratelyapproximated by SDEs. This finding opens a door to future modeling work usingSDE techniques to further illuminate the effects of ion channel fluctuations onelectrically active cells.

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