Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

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 [Formula: see text] and suggested a Langevin formulation with noise coefficient matrix [Formula: see text] such that SS[Formula: see text]. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 [Formula: see text] 28 noise coefficient matrix [Formula: see text]. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix [Formula: see text] that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) 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; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the [Formula: see text] and [Formula: see text] metric-space norms; and (5) 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 [Formula: see text] with minimal loss of accuracy under both voltage- and current-clamp conditions.