Accurate Langevin approaches to simulate Markovian channel dynamics

A fast Markovian method for modeling channel noise in neurons

Channel noise results from rapid transitions of protein channels from closed to open state and is generally considered as the most dominant source of electrical noise causing membrane-potential fluctuations even in the absence of synaptic inputs. The simulation of a realistic channel noise remains a source of possible error. Although the Markovian method is considered as the golden standard for appropriate description of channel noise, its computation time increasing exponentially with the number of channels, it is poorly suitable to simulate realistic features. We describe here a novel algorithm at discrete time unit for simulating ion channel noise based on Markov chains (MC). Although this new algorithm refers to a Monte-Carlo process, it only needs few random numbers whatever the number of channels involved. Our fast MC (FMC) model does not exhibit the drawbacks due to approximations based on stochastic differential equations and the values of spike jitter are comparable to those obtained with the true Markovian method. In fact, we show here, that these drawbacks can be highlighted in the approximation based on stochastic differential equation methods even for a high number of channels (standard deviation of the 5th spike is about two-fold larger than that of MCF or true Markovian method for 5000 sodium channels). The FMC model appears therefore as the most accurate method to simulate channel noise with a fast execution time that does not depend on the channel number.

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

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⊤=D. 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 × 28 noise coefficient matrix S. 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 S 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 L1 and L∞ 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 S with minimal loss of accuracy under both voltage- and current-clamp conditions.

Chemical Langevin equation: A path-integral view of Gillespie's derivation

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path-integral description of the CME and show how applying Gillespie's two conditions to it directly leads to a path-integral equivalent to the CLE. We compare this approach to the path-integral equivalent of a large system size derivation and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems and are readily generalizable.

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

Strong diffusion formulation of Markov chain ensembles and its optimal weaker reductions

Two self-contained diffusion formulations, in the form of coupled stochastic differential equations, are developed for the temporal evolution of state densities over an ensemble of Markov chains evolving independently under a common transition rate matrix. Our first formulation derives from Kurtz's strong approximation theorem of density-dependent Markov jump processes [Stoch.

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Their Appl. 6, 223 (1978)STOPB70304-414910.1016/0304-4149(78)90020-0] and, therefore, strongly converges with an error bound of the order of lnN/N for ensemble size N. The second formulation eliminates some fluctuation variables, and correspondingly some noise terms, within the governing equations of the strong formulation, with the objective of achieving a simpler analytic formulation and a faster computation algorithm when the transition rates are constant or slowly varying. There, the reduction of the structural complexity is optimal in the sense that the elimination of any given set of variables takes place with the lowest attainable increase in the error bound. The resultant formulations are supported by numerical simulations.

Interpretation of amperometric kinetics of content release during contacts of vesicles with a lipid membrane

The exocytotic pathway of secretion of molecules from cells includes transport by vesicles, tether-mediated fusion of vesicles with the plasma membrane accompanied by pore formation, and diffusion-mediated release of their contents via a pore to the outside. In related basic biophysical studies, vesicle-content release is tracked by measuring corresponding amperometric spikes. Although experiments of this type have a long history, the understanding of the underlying physics is still elusive. The present study elucidates the likely contribution of line energy, membrane tension and bending, osmotic pressure, hydration forces, and tethers to the potential energy for fusion-related pore formation and evolution. The overdamped Langevin equation is used to describe the pore dynamics, which are in turn employed to calculate the kinetics of content release and to interpret the shape of amperometric spikes.

Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system

Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.