Solution moment stability in stochastic differential delay equations

Mackey, Michael C.; Nechaeva, I. G.

doi:10.1103/physreve.52.3366

Cited by 105 publications

(77 citation statements)

References 39 publications

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“…Equation (1) is understood under the Stratonovich interpretation of stochastic calculus [3]. This interpretation is appropriate when white noise can be considered as the limiting case of the colored noise actually existing in the system [4]. This observation has been confirmed experimentally [5].…”

Section: Introductionmentioning

confidence: 58%

Correlation times in stochastic equations with delayed feedback and multiplicative noise

2011

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We obtain the characteristic correlation time associated with a model stochastic differential equation that includes the normal form of a pitchfork bifurcation and delayed feedback. In particular, the validity of the common assumption of statistical independence between the state at time t and that at t − τ , where τ is the delay time, is examined. We find that the correlation time diverges at the model's bifurcation line, thus signaling a sharp bifurcation threshold, and the failure of statistical independence near threshold. We determine the correlation time both by numerical integration of the governing equation, and analytically in the limit of small τ . The correlation time T diverges as T ∼ a −1 , where a is the control parameter so that a = 0 is the bifurcation threshold. The small-τ expansion correctly predicts the location of the bifurcation threshold, but there are systematic deviations in the magnitude of the correlation time.

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Section: Introductionmentioning

confidence: 58%

Correlation times in stochastic equations with delayed feedback and multiplicative noise

2011

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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).…”

mentioning

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). The limit of short delay time has been used to show how a univariate nondelayed stochastic differential equation can be used to approximate the original system (34), and, more recently, a noise-driven bistable system with delayed feedback was reduced to a two-state model with delayed transition rates to demonstrate the phenomenon of coherence resonance (35).…”

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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.

515

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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“…(B1) stands for a recurrent coupling given by the sigmoidal function of G(x); other notations follow those of Eqs. (1) and (2). After some manipulations, we get DEs for means, variance and covarinces given by…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In recent years, linear SDDEs of Langevin equation are beginning to gain much attention [1]- [6]. The parameter range for the stationary solutions of the Langevin equation has been examined with the use of the step by step method [1], the moment mothod [2] and the Fokker-Planck equation (FPE) method [3] [4].…”

Section: Introductionmentioning

confidence: 99%

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

Hasegawa

2004

Phys. Rev. E

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Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semi-analytical augmented moment method (AMM) which has been proposed in a recent paper [H. Hasegawa, E-print: cond-mat/0311021]. For N -unit FN neuron ensembles, AMM transforms original 2N -dimensional stochastic delay differential equations (SDDEs) to infinite-dimensional deterministic DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level m in the level-m AMM (AMMm), yielding 8(m + 1)-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the fluctuation-induced enhancement at the transition between nonoscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.

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Solution moment stability in stochastic differential delay equations

Cited by 105 publications

References 39 publications

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Delay-induced stochastic oscillations in gene regulation

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

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