Evolution of probability densities in stochastic coupled map lattices

Losson, Jérôme; Mackey, Michael C.

doi:10.1103/physreve.52.1403

Cited by 14 publications

(7 citation statements)

References 31 publications

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“…The behavior of stochastic delay-differential equations (SDDEs) has been extensively studied, and various approximation techniques have been developed and utilized (30)(31)(32)(33)(34)(35). For example, delayed differential equations have been reduced to coupled map lattices and perturbed by white noise, demonstrating how the phase space density reaches a limit cycle in the asymptotic regime (30).…”

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confidence: 99%

“…For example, delayed differential equations have been reduced to coupled map lattices and perturbed by white noise, demonstrating how the phase space density reaches a limit cycle in the asymptotic regime (30). The stability of the moment equations for linear SDDEs has been explored to elucidate the oscillatory properties of the first and second moments (31), and the master equation approach has been applied to a delayed random walker in an effort to demonstrate the effects of delay in an analytically tractable system (32,33).…”

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confidence: 99%

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Delay-induced stochastic oscillations in gene regulation

Bratsun

Volfson

Tsimring

et al. 2005

Proc. Natl. Acad. Sci. U.S.A.

513

487

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The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.master equation ͉ stochastic delay equations ͉ noise ͉ time delay ͉ systems biology T here is considerable experimental evidence that noise can play a major role in gene regulation (1-10). These fluctuations can arise from either intrinsic sources, which are related to the small numbers of reactant biomolecules, or extrinsic sources, which are attributable to a noisy cellular environment. Although the importance of fluctuations in gene regulation was stressed Ͼ30 years ago (11), recent experimental advances have renewed interest in the stochastic modeling of the biochemical reactions that underlie gene regulatory networks (12-16). The most typical approaches are the utilization of the Gillespie algorithms (17)(18)(19)(20), the direct analysis of the master equation, or the development of simplified descriptions based on the Fokker-Planck or Langevin equations (see ref. 21 for a review). A common thread in many of these approaches has been to consider intrinsic noise as the dominant source of variability in gene expression.One major difficulty often encountered in the analysis of gene regulatory networks is the vast separation of time scales between what are typically the fast reactions (dimerization, protein-DNA binding͞unbinding) and the slow reactions (transcription, translation, degradation). There have been many studies devoted to the development of reduced descriptions of these systems using the idea of quasiequilibrium for the fast processes compared with the slow dynamics (cf. ref. 22 and references therein). These approaches have thus far implicitly assumed that all...

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confidence: 99%

Delay-induced stochastic oscillations in gene regulation

Bratsun

Volfson

Tsimring

et al. 2005

Proc. Natl. Acad. Sci. U.S.A.

513

487

View full text Add to dashboard Cite

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“…Markov operators have a wide range of applications; for example, they can be applied to analyses of coupled map lattices [11,12], nonlinear oscillators driven by stochastic inputs [5,6,13,14], neuronal dynamics [15][16][17][18][19][20][21], and neural networks [22]. * yamanobe@med.hokudai.ac.jp Markov operators can be constructed using stochastic kernels, which are transition densities corresponding to the given stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

Stochastic phase transition operator

Yamanobe

2011

Phys. Rev. E

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In this study a Markov operator is introduced that represents the density evolution of an impulse-driven stochastic biological oscillator. The operator's stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the Fokker-Planck equation. The Markov operator is shown to successfully approximate the density evolution of the biological oscillator considered. The response of the oscillator to both periodic and time-varying impulses can be analyzed using the operator's transient and stationary properties. Furthermore, an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the Markov operators.

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“…The coupled map lattices applied thus far in ecology have, in general, been deterministic models of predator-prey interactions. Since even small perturbations can have profound effects on the dynamics of a coupled map lattice (Losson and Mackey, 1995), these deterministic models are open to the criticism that the patterns they produce, usually spiral waves, are simply an artefact of determinism. To address such criticism, Wilson et al (1993) and Bascompte et al (1997) have shown that individual-based models can produce the same types of spatial patterning as their deterministic counterparts.…”

Section: Discussionmentioning

confidence: 99%

Coupled map lattice approximations for spatially explicit individual-based models of ecology

Brännström

Sumpter

2005

Bulletin of Mathematical Biology

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Spatially explicit individual-based models are widely used in ecology but they are often difficult to treat analytically. Despite their intractability they often exhibit clear temporal and spatial patterning. We demonstrate how a spatially explicit individual-based model of scramble competition with local dispersal can be approximated by a stochastic coupled map lattice. The approximation disentangles the deterministic and stochastic element of local interaction and dispersal. We are thus able to understand the individual-based model through a simplified set of equations. In particular, we demonstrate that demographic noise leads to increased stability in the dynamics of locally dispersing single-species populations. The coupled map lattice approximation has general application to a range of spatially explicit individual-based models. It provides a new alternative to current approximation techniques, such as the method of moments and reaction-diffusion approximation, that captures both stochastic effects and large-scale patterning arising in individual-based models.

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Evolution of probability densities in stochastic coupled map lattices

Cited by 14 publications

References 31 publications

Delay-induced stochastic oscillations in gene regulation

Delay-induced stochastic oscillations in gene regulation

Stochastic phase transition operator

Coupled map lattice approximations for spatially explicit individual-based models of ecology

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