Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system.

Anqi Ling,Yueheng Lan,Jianwei Shuai,Yandong Huang

doi:10.1038/srep22662

Abstract

Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.

Highlights

Hodgkin and Huxley first proposed a classical way to deterministically characterize neuronal dynamics based on a quantitative analysis of experimental results[1]
The stochastic kinetics of ion channels could be defined as a Markov chain, with discrete phase space states where each state in the chain represents a particular configuration of the ion channel
As discussed in the Methods section, the propagation of the action potential along an axon is described by a stochastic version of the HH equation, which could be transformed to a partial differential equation (PDE) for the generating function of the channel system

Summary

Introduction

Hodgkin and Huxley first proposed a classical way to deterministically characterize neuronal dynamics based on a quantitative analysis of experimental results[1]. The random transition of an ion channel from one state to another just depends on its current state in the Markov assumption, which can be exactly simulated by the Gillespie algorithm[28] This algorithm tracks the number of channels in each state at each time point on one trajectory, and many trajectories are computed for well converged statistics. As reviewed by Huang et al, despite for small channel numbers, currently proposed Langevin approaches cannot accurately replicate the statistical properties of the Markov HH model even at large channel numbers, calling for a better approach to the stochastic HH dynamics[35]

Methods

Results

Conclusion