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 letter, 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.

Noise source importance in linear stochastic models of biological systems that grow, shrink, wander, or persist

“…Noise source importance was originally introduced to identify unimportant noise sources that could be ignored as a form of model reduction [1,2]. In some cases, such as in the Hodgkin-Huxley model for neuron firing, this approach can lead to efficient, yet accurate, approximate simulation [3][4][5]. To illustrate this application we will ask, which binding events can be ignored when simulating the inositol triphosphate channel responsible for calcium induced calcium release?…”

“…The stochastic shielding approximation reduces a reaction network by dropping noise from unimportant edges [1, 2, 7]. Example applications to neural dynamics are discussed in [3–5, 8–10].…”

Memristive Hodgkin-Huxley Spiking Neuron Model for Reproducing Neuron Behaviors

“…The space-clamped Hodgkin-Huxley model effectively inhibits the production of spikes under the injection of the noisy synaptic input (Tuckwell and Ditlevsen, 2016 ). The Langevin is combined with the Hodgkin-Huxley system performs accurate interspike interval (ISI) and realizes the accuracy minimal loss (Pu and Thomas, 2020 ). The Berger-Levy theory is introduced to the Hodgkin-Huxley model, demonstrate that the information communication between neurons is related to the presynaptic firing rate and the synchronization (Ghavami et al, 2018 ).…”

Resolving Molecular Contributions of Ion Channel Noise to Interspike Interval Variability through Stochastic Shielding

“…In our previous work [47], we numerically estimated the contribution of each directed edge in the transition graph (Fig. 2A,B) to the variance of ISIs.…”