Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage—and the highly nonlinear impact of that voltage back on the currents themselves—into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

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.

The partial tripolar electrode configuration is a relatively novel stimulation strategies that can generate more spatially focused electric fields than the commonly used monopolar configuration. Focused stimulation strategies should improve spectral resolution in cochlear implant users, but may also be more sensitive to local irregularities in the electrode-neuron interface. In this study, we develop a practical computer model of cochlear implant stimulation that can simulate neural activation in a simplified cochlear geometry and we relate the resulting patterns of neural activity to basic psychophysical measures. We examine how two types of local irregularities in the electrode-neuron interface, variations in spiral ganglion nerve density and electrode position within the scala tympani, affect the simulated neural activation patterns and how these patterns change with electrode configuration. The model shows that higher partial tripolar fractions activate more spatially restricted populations of neurons at all current levels and require higher current levels to excite a given number of neurons. We find that threshold levels are more sensitive at high partial tripolar fractions to both types of irregularities, but these effects are not independent. In particular, at close electrode-neuron distances, activation is typically more spatially localized which leads to a greater influence of neural dead regions.