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

Abstract

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.

arXiv comment19 pages, including 6 figures and 3 appendices

References in corpus (4)

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• The what and where of adding channel noise to the Hodgkin-Huxley equations
• Propagation of chaos in neural fields
• Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics
• Analysis of inverse stochastic resonance and the long-term firing of Hodgkin-Huxley neurons with Gaussian white noise
• Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States
• Channel noise induced stochastic facilitation in an auditory brainstem neuron model
• Exact simulation of the jump times of a class of Piecewise Deterministic Markov Processes
• Empirical quantification of predictive uncertainty due to model discrepancy by training with an ensemble of experimental designs: an application to ion channel kinetics
• Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics
• Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels
• Channel Noise Effects on Neural Synchronization
• Multiscale modeling via split-step methods in neural firing
• Lamperti Semi-Discrete method
• Effect of Channel Noise in Synchronization and Metabolic Energy Consumption in Unidirectionally Coupled Neurons: Drug Blocking of Sodium and Potassium Channels