  The random transitions of ion channels between conducting and non-conducting
states generate a source of internal fluctuations in a neuron, known as channel
noise. The standard method for modeling fluctuations in the states of ion
channels uses continuous-time Markov chains nonlinearly coupled to a
differential equation for voltage. Beginning with the work of Fox and Lu, 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 preserve 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
effect 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, and the subunit-based approaches do not
correctly account for this structure. Our study therefore confirms and
elucidates the findings of previous numerical investigations of subunit-based
SDE models. Moreover, it presents the first 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.
