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arXiv · 2005.10598

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

Shusen Pu·Peter J. Thomas
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Abstract

Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor $D$ and suggested a Langevin formulation with noise coefficient matrix $S$ such that $SS^\intercal=D$. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this paper, we present a natural 14-dimensional dynamics for the HH system in which each \emph{directed} edge in the ion channel state transition graph acts as an independent noise source, leading to a $14\times 28$ noise coefficient matrix $S$. We show that (i) the corresponding 14D system of ordinary differential \rev{equations} is consistent with the classical 4D representation of the HH system; (ii) the 14D representation leads to a noise coefficient matrix $S$ that can be obtained cheaply on each timestep, without requiring a matrix decomposition; (iii) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (iv) our 14D representation (and those equivalent to it) give the most accurate interspike-interval distribution, not only with respect to moments but under both the $L_1$ and $L_\infty$ metric-space norms; and (v) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies $S$ with minimal loss of accuracy under both voltage- and current-clamp conditions.

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Shusen Pu, Peter J. Thomas. 2020-05-21. Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics. https://doi.org/10.1162/neco_a_01312

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