The validity of quasi steady-state approximations in discrete stochastic simulations

In biochemical networks, reactions often occur on disparate timescales and can be characterized as either "fast" or "slow." The quasi-steady state approximation (QSSA) utilizes timescale separation to project models of biochemical networks onto lower-dimensional slow manifolds. As a result, fast elementary reactions are not modeled explicitly, and their effect is captured by non-elementary reaction rate functions (e.g. Hill functions). The accuracy of the QSSA applied to deterministic systems depends on how well timescales are separated. Recently, it has been proposed to use the non-elementary rate functions obtained via the deterministic QSSA to define propensity functions in stochastic simulations of biochemical networks. In this approach, termed the stochastic QSSA, fast reactions that are part of non-elementary reactions are not simulated, greatly reducing computation time. However, it is unclear when the stochastic QSSA provides an accurate approximation of the original stochastic simulation. We show that, unlike the deterministic QSSA, the validity of the stochastic QSSA does not follow from timescale separation alone, but also depends on the sensitivity of the non-elementary reaction rate functions to changes in the slow species. The stochastic QSSA becomes more accurate when this sensitivity is small. Different types of QSSAs result in non-elementary functions with different sensitivities, and the total QSSA results in less sensitive functions than the standard or the pre-factor QSSA. We prove that, as a result, the stochastic QSSA becomes more accurate when non-elementary reaction functions are obtained using the total QSSA. Our work provides a novel condition for the validity of the QSSA in stochastic simulations of biochemical reaction networks with disparate timescales.Comment: 21 pages, 4 figure

A perturbation analysis of spontaneous action potential initiation by stochastic ion channels

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris- Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently-developed quasi-stationary perturbation method. Using the fact that the activating ionic channel’s random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT

Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models

Markov Population Model is a commonly used framework to describe stochastic systems. Their exact analysis is unfeasible in most cases because of the state space explosion. Approximations are usually sought, often with the goal of reducing the number of variables. Among them, the mean field limit and the quasi-equilibrium approximations stand out. We view them as techniques that are rooted in independent basic principles. At the basis of the mean field limit is the law of large numbers. The principle of the quasi-equilibrium reduction is the separation of temporal scales. It is common practice to apply both limits to an MPM yielding a fully reduced model. Although the two limits should be viewed as completely independent options, they are applied almost invariably in a fixed sequence: MF limit first, QE-reduction second. We present a framework that makes explicit the distinction of the two reductions, and allows an arbitrary order of their application. By inverting the sequence, we show that the double limit does not commute in general: the mean field limit of a time-scale reduced model is not the same as the time-scale reduced limit of a mean field model. An example is provided to demonstrate this phenomenon. Sufficient conditions for the two operations to be freely exchangeable are also provided

Metastability in a stochastic neural network modeled as a velocity jump Markov process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorprated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks