Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron

Abstract

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

Diffusion approximation-based simulation of stochastic ion channels: which method to use?

“…Previously, Goldwyn et al (2011) also allowed the variables to freely change, but instead used a steady state approximation on the voltage to maintain the RSR constraint. However, such an approximation can be rather inaccurate (Dangerfield et al, 2012; Orio and Soudry, 2012; Huang et al, 2013a), even when the number of channels is relatively high.…”

“…This is done using projection into a simplex (Chen and Ye, 2011). Huang et al (2013a) proposed a different method. When the boundary constraint is broken, the variables should first be truncated.…”

“…Under this common theoretical approach, each implementation differs in how it handles various numerical difficulties—such as bounding of state variables to [0,1]. Here we review and test a set of the most recently published DA implementations (Goldwyn et al, 2011; Linaro et al, 2011; Dangerfield et al, 2012; Orio and Soudry, 2012; Schmandt and Galán, 2012; Güler, 2013; Huang et al, 2013a), comparing all of them in a set of numerical simulations that assess numerical accuracy and computational efficiency on three different models: (1) the original Hodgkin and Huxley model, (2) a model with faster sodium channels, and (3) a multi-compartmental model inspired in granular cells. We conclude that for a low number of channels (usually below 1000 per simulated compartment) one should use MC—which is the fastest and most accurate method.…”

The impact of channel and external synaptic noises on spatial and temporal coherence in neuronal networks

“…Recently, Goldwyn et al proposed a more accurate channel-based approach to the Markov-chain model by properly representing the structure and dynamics of ion channels [32]. An improved channel-based SDE methods was also developed by Huang et al by implementing a truncation and restoration procedure [33]. In this paper, we choose to model the ion channel kinetics by a Markov chain.…”

Molecular Na-channel excitability from statistical physics

“…Most of these approaches 31 incorporate fluctuations in some of the elements of the 32 HH theory, for instance by using Langevin [7][8][9][10] or mas-33 ter equations [11] for the conductance equations or noise 34 terms in the equation for the membrane charge [12]. A 35 more microscopic approach used Langevin equations for 36 the ions with Poisson equation for the potential membrane 37 [13] in order to obtain the effects of fluctuations on the 38 membrane conductivity.…”