Metastable states and quasicycles in a stochastic Wilson-Cowan\ud model of neuronal population dynamics

We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N -> infinity, where N determines the size of each population, the dynamics is described by deterministic Wilson–Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady–state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N, we calculate the exponentially small rate of noise–induced transitions between the resulting metastable states using a Wentzel–Kramers–Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory/inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles)

Chemical master versus chemical langevin for first-order reaction networks

Markov jump processes are widely used to model interacting species in circumstances where discreteness and stochasticity are relevant. Such models have been particularly successful in computational cell biology, and in this case, the interactions are typically rst-order. The Chemical Langevin Equation is a stochastic dierential equation that can be regarded as an approximation to the underlying jump process. In particular, the Chemical Langevin Equation allows simulations to be performed more eectively. In this work, we obtain expressions for the rst and second moments of the Chemical Langevin Equation for a generic rst-order reaction network. Moreover, we show that these moments exactly match those of the under-lying jump process. Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation. Our work assumes that a unique solution exists for the Chemical Langevin Equation. We also show that the moment matching re- sult extends to the case where a gene regulation model of Raser and O'Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin equations. We nish with numerical experiments on a dimerization model that involves second order reactions, showing that the two regimes continue to give similar results

Mirror symmetry breaking as a problem in dynamical critical phenomena

The critical properties of the Frank model of spontaneous chiral synthesis are discussed by applying results from the field theoretic renormalization group (RG). The long time and long wavelength features of this microscopic reaction scheme belong to the same universality class as multi-colored directed percolation processes. Thus, the following RG fixed points (FP) govern the critical dynamics of the Frank model for d<4: one unstable FP that corresponds to complete decoupling between the two enantiomers, a saddle-point that corresponds to symmetric interspecies coupling, and two stable FPs that individually correspond to unidirectional couplings between the two chiral molecules. These latter two FPs are associated with the breakdown of mirror or chiral symmetry. In this simplified model of molecular synthesis, homochirality is a natural consequence of the intrinsic reaction noise in the critical regime, which corresponds to extremely dilute chemical systems.Comment: 9 pages, 3 figure

Coarse-graining of overdamped Langevin dynamics via the Mori-Zwanzig formalism

The Mori–Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the importance of memory effects in determining the correct temporal behaviour of such systems

Chemical master equation and Langevin regimes for a gene transcription model

Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie's algorithm) allows for practical simulations to be performed. However, for large networks and/or fast reactions, such computations can be prohibitatively expensive. The Chemical Langevin regime replaces the massive ordinary dierential equation system with a small stochastic dierential equation system that is more amenable to computation. Although the transition from Chemical Master Equation to Chemical Langevin Equation can be justied rigorously in the large system size limit, there is very little guidance available about how closely the two models match for a xed system. Here, we consider a transcription model from the recent literature and show that it is possible to compare rst and second moments in the two stochastic settings. To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito's Lemma. We nd that there is a perfect match|both modelling regimes give the same means, variances and correlations for all components in the system. The model that we analyse involves 'unimolecular reactions', and we nish with some numerical simulations involving dimerization to show that the means and variances in the two regimes can also be close when more general 'bimolecular reactions' are involved

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms

Biochemical reactions can happen on different time scales and also the abundance of species in these reactions can be very different from each other. Classical approaches, such as deterministic or stochastic approach, fail to account for or to exploit this multi-scale nature, respectively. In this paper, we propose a jump-diffusion approximation for multi-scale Markov jump processes that couples the two modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into fast and slow sets, where the fast set is simulated by a stochastic differential equation and the slow set is modeled by a discrete chain. The error bound leads to a very efficient dynamic partitioning algorithm which has been implemented for several multi-scale reaction systems. The gain in computational efficiency is illustrated by a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics.Comment: 32 pages, 7 figure