Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Hinch, Robert; Chapman, S. Jonathan

doi:10.1017/s0956792505006194

Cited by 47 publications

(64 citation statements)

References 31 publications

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“…In the case of a single neural population, it is possible to calculate the prefactor in the expression for the lifetime τ of a metastable state by carrying out an asymptotic expansion of the original master equation (2.13) based on a WKB approximation [41,18,28]. However, this asymptotic analysis becomes considerably more difficult in the case of multiple interacting populations or when parameters of the network become time dependent.…”

Section: Hamiltonian Dynamics and Rare Event Statisticsmentioning

confidence: 99%

Stochastic Neural Field Theory and the System-Size Expansion

Bressloff¹

2010

SIAM J. Appl. Math.

152

232

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Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N → ∞) we recover standard activity-based or voltage-based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N ) can be truncated to form a closed system of equations for the first-and second-order moments. Taking a continuum limit of the moment equations while keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the mean-field dynamics; such an analysis is not possible using the system-size expansion since the latter cannot accurately determine exponentially small transitions.Key words. neural field theory, master equations, stochastic processes, system-size expansion, path integrals AMS subject classifications. Primary, 92; Secondary, 60 DOI. 10.1137/090756971 1. Introduction. Continuum models of neural tissue have been very popular in analyzing the spatiotemporal dynamics of large-scale cortical activity (reviewed in [24,35,7,16]). Such models take the form of integrodifferential equations in which the integration kernel represents the spatial distribution of synaptic connections between different neural populations, and the macroscopic state variables represent populationaveraged firing rates. Following seminal work in the 1970s by Wilson, Cowan, and Amari [72,73,2], many analytical and numerical results have been obtained regarding the existence and stability of spatially structured solutions to continuum neural field equations. These include electroencephalogram (EEG) rhythms [55,46,67], geometric visual hallucinations [25,26,70,8], stationary pulses or bumps [2,62,50,51,30], traveling fronts and pulses [27,61,6,74,17,32], spatially localized oscillations or breathers [31,33,59], and spiral waves [44,49,71,48].One potential limitation of neural field models is that they only take into account mean firing rates and thus do not include any information about higher-order statistics such as correlations between firing activity at different cortical locations. It is well known that the spike trains of individual cortical neurons in vivo tend to be very noisy, having interspike interval distributions that are close to Poisson [68]. However, it is far less clear how noise at the single cell...

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Section: Hamiltonian Dynamics and Rare Event Statisticsmentioning

confidence: 99%

Stochastic Neural Field Theory and the System-Size Expansion

Bressloff¹

2010

SIAM J. Appl. Math.

152

232

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“…In the first half of Figs. 11(b,d) we plot (X(t), Y (t)) (20) (solid lines) and the 95% confidence intervals (dashed lines) of ρ ± (x) and q ± (y). In the second half, instead of the average quantities we plot sampled values (X, Y ) from Q ± (x, y) [depending on the value of S(t)].…”

Section: Regime 2: ǫ ≪ τS ∼ τYmentioning

confidence: 99%

“…Then the reduced low-dimensional model consists of a discrete jump process between states j, where the switching rates are calculated from the transition rates of the original process X(t). 18,20,21 In higherdimensional systems where analytical approximations are not possible, computational techniques can be applied to determine the state space of the metastable variables and to sample the transition probabilities. A recent survey of such techniques applied to the simulation of rare events in molecular dynamics (such as conformation changes) can be found in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…If τ y ≪ τ s , the reduced model is simply a discrete jump process. 20 If τ y ≫ τ s , the slow variables evolve on a timescale which is even slower than the switches in x and the classical QSS approximation applies. In the intermediate regime in which τ y = O(τ s ) things become more interesting, as the evolution of y critically depends on the metastable behaviour of x.…”

Section: Introductionmentioning

confidence: 99%

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Model reduction for slow–fast stochastic systems with metastable behaviour

Bruna

Chapman

Smith

2014

The Journal of Chemical Physics

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The quasi-steady-state approximation (or stochastic averaging principle) is a useful tool in the study of multiscale stochastic systems, giving a practical method by which to reduce the number of degrees of freedom in a model. The method is extended here to slow-fast systems in which the fast variables exhibit metastable behaviour. The key parameter that determines the form of the reduced model is the ratio of the timescale for the switching of the fast variables between metastable states to the timescale for the evolution of the slow variables. The method is illustrated with two examples: one from biochemistry (a fast-species-mediated chemical switch coupled to a slower-varying species), and one from ecology (a predator-prey system). Numerical simulations of each model reduction are compared with those of the full system.

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“…the probability that there are x molecules of the chemical species X at time t in the system. The stationary solution of this infinite set of ODEs can be found in a closed form [10], i.e. one can find an exact formula for the stationary distribution plotted in Figure 3.1(b).…”

mentioning

confidence: 99%

Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

Erban¹,

Chapman²,

Kevrekidis³

et al. 2009

SIAM J. Appl. Math.

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Abstract.A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker-Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size.

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Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Cited by 47 publications

References 31 publications

Stochastic Neural Field Theory and the System-Size Expansion

Stochastic Neural Field Theory and the System-Size Expansion

Model reduction for slow–fast stochastic systems with metastable behaviour

Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

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