Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States

Patricio Orio,Daniel Soudry,Michele Giugliano

doi:10.1371/journal.pone.0036670

Patricio Orio, Daniel Soudry + Show 1 more

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https://doi.org/10.1371/journal.pone.0036670

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Abstract

BackgroundThe phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies.Main ContributionsWe derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable – allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.

Highlights

Noise and variability are present throughout the nervous system, from sensory systems to the motor output and perhaps more importantly in the higher brain areas [1]
Very recently is was described [35] and mathematically proven [32,33,40] that when the gating particles are considered to be coupled or ‘tied’ in groups, the resulting conductance fluctuations have statistics that cannot be adequately reproduced with an uncoupled Diffusion Approximation (DA) algorithm
Numerical Efficiency Following the practical approach of this work, we numerically evaluated the computational cost of three different algorithms: Markov Chains (MC), our DA algorithm and the uncoupled version of DA

Summary

Introduction

Noise and variability are present throughout the nervous system, from sensory systems to the motor output and perhaps more importantly in the higher brain areas [1]. The effects of channel noise on neuronal excitability are to a large extent studied with the use of mathematical models, either by constructing and analyzing models with stochastic channels [15,16,17,18] or by introducing a noisy conductances in dynamic clamp experiments [19,20]. It is of interest, to develop and analyze numerical models that faithfully reproduce the stochastic nature of ion channels. A steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies

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