Stochastic shielding and edge importance for Markov chains with timescale separation

Abstract

Nerve cells produce electrical impulses (“spikes”) through the coordinated opening and closing of ion channels. Markov processes with voltage-dependent transition rates capture the stochasticity of spike generation at the cost of complex, time-consuming simulations. Schmandt and Galán introduced a novel method, based on the stochastic shielding approximation, as a fast, accurate method for generating approximate sample paths with excellent first and second moment agreement to exact stochastic simulations. We previously analyzed the mathematical basis for the method’s remarkable accuracy, and showed that for models with a Gaussian noise approximation, the stationary variance of the occupancy at each vertex in the ion channel state graph could be written as a sum of distinct contributions from each edge in the graph. We extend this analysis to arbitrary discrete population models with first-order kinetics. The resulting decomposition allows us to rank the “importance” of each edge’s contribution to the variance of the current under stationary conditions. In most cases, transitions between open (conducting) and closed (non-conducting) states make the greatest contributions to the variance, but there are exceptions. In a 5-state model of the nicotinic acetylcholine receptor, at low agonist concentration, a pair of “hidden” transitions (between two closed states) makes a greater contribution to the variance than any of the open-closed transitions. We exhaustively investigate this “edge importance reversal” phenomenon in simplified 3-state models, and obtain an exact formula for the contribution of each edge to the variance of the open state. Two conditions contribute to reversals: the opening rate should be faster than all other rates in the system, and the closed state leading to the opening rate should be sparsely occupied. When edge importance reversal occurs, current fluctuations are dominated by a slow noise component arising from the hidden transitions.

Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

“…For sufficiently large number of channels, Schmidt and Thomas (2014) and Schmidt et al (2018) showed that under voltage clamp, equations 3.1 and 3.2 can be approximated by a multidimensional Ornstein-Uhlenbeck (OU) process (or Langevin equation) in the form 5…”

“…Note that reciprocal edges have identical R k due to detailed balance. Under voltage clamp, the largest value of R k for the HH channels always corresponds to directly observable transitions, that is, edges k such that |c ζ k | > 0, although this condition need not hold in general (Schmidt, Galán, & Thomas, 2018).…”

“…In this article, we advocate for a class of models combining the best features of conductance-based Langevin models with the recently developed stochastic shielding approximation (Schmandt & Galán, 2012;Schmidt & Thomas, 2014;Schmidt et al, 2018). We propose a Langevin model with a 14-dimensional state space, representing the voltage, five states of the K + channel, and eight states of the Na + -channel; and a 28-dimensional representation of the driving noise: one independent gaussian noise term for each directed edge in the channel-state transition graph.…”

“…However, while detailed balance holds for the HH model under stationary voltage clamp, this condition is violated during active spiking. Finally, the stochastic shielding approximation (Schmandt & Galán, 2012;Schmidt & Thomas, 2014;Schmidt et al, 2018) does not have a natural formulation in the representation associated with an n × n noise coefficient matrix S; in the cases of rectangular S matrices used in Orio and Soudry (2012) and Dangerfield et al (2012) stochastic shielding can only be applied to reciprocal pairs of edges. We elaborate on these points in section 6.…”

Linear Noise Approximation of Intensity-Driven Signal Transduction Channels

“…Just as we assume the population of independent transmitters is large enough to justify a Gaussian approximation for the input concentration signal, we also assume a large population of N 1 independent receptors exposed to the same input signal. For large N , the fluctuations in number of transitions per time among states, relative to the mean number of transitions per unit time, scale as 1/ √ N [15]. In general, we may consider a variety of scaling relations between M , the number of sources, and N , the number of receptors.…”