Exploring a noisy van der Pol type oscillator with a stochastic approach

Yuan, Ruoshi; Wang, Xinan; Ma, Yuanlin; Yuan, Bo; Ao, Ping

doi:10.1103/physreve.87.062109

Cited by 30 publications

(37 citation statements)

References 39 publications

(63 reference statements)

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“…There is a limit cycle that attracts all trajectories except the unstable fixed point at the origin (Figure 1(a)), and the global invariant set is composed of the limit cycle and the fixed point. The system is a standard example of a nonlinear oscillator and has been studied extensively, including with stochastic forcing [24,12,43]. Here we find nearly tight bounds on averages of x 2 + y 2 both with and without noise.…”

Section: Introductionmentioning

confidence: 62%

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Fantuzzi¹,

Goluskin²,

Huang³

et al. 2016

SIAM J. Appl. Dyn. Syst.

106

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Abstract. We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.

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Section: Introductionmentioning

confidence: 62%

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Fantuzzi¹,

Goluskin²,

Huang³

et al. 2016

SIAM J. Appl. Dyn. Syst.

106

View full text Add to dashboard Cite

show abstract

“…The accuracy of the deterministic picture deteriorates as we scale down the size of our system so that at some point, we must account for the fluctuating molecular nature of the oscillator (24,25). As a first-order approximation, we may describe the dynamics of phase and amplitude as being driven by a Gaussian white noise ηðtÞ,…”

Section: Phenomenological Model Of Dephasingmentioning

confidence: 99%

On the dephasing of genetic oscillators

Potoyan

Wolynes

2014

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

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The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We show that unlike ordinary chemical oscillators, the dichotomic molecular noise of gene state switching in gene oscillators affects the stochastic dephasing in a way that may not always be captured by phenomenological limit cycle-based models. Through simulations of a realistic model of the NFκB/IκB network, we also illustrate the dephasing phenomena that are important for reconciling single-cell and population-based experiments on gene oscillators. C yclic rhythms are a common feature of many self-organized systems (1) manifesting themselves in myriad forms in biology, ranging from subcellular biochemical oscillations to cell division and on to the familiar predator-prey cycles of ecology. An oscillatory response of a gene regulatory circuit, whether transient or self-sustained, can provide several advantages over a temporally monotonic response (2, 3). The ability of copies of a system to synchronize may lead to dramatic noise reduction and greater precision for timing in assemblies. On the subcellular level, rhythmic dynamics span time scales from a few seconds, as in the calcium oscillations, to days, as in the circadian rhythms, or years for cicada cycles (1, 4-6). Ultradian genetic oscillations, which take place on an intermediate scale from minutes to a few hours, are medically important. A singularly important case is the Nuclear Factor Kappa B (NFκB) gene network, which organizes the mammalian cell's response to various types of external stress and plays a role in regulating inflammation levels in populations of cells. The response of the NFκB circuit to continuous external stimulation has been studied by Hoffmann et al. (7) who observed damped oscillatory dynamics for NFκB, which has been linked to the presence or absence of particular isoforms of the inhibitor IκB. On the other hand, experiments carried out on individual cells have detected more sustained NFκB/IκB oscillations that either are completely self-sustained (8, 9) or damp at a much slower rate (9) than found for the population, depending on the duration of external stimulation. It follows that some type of averaging takes place, but the physical mechanisms and the stochastic aspects of this population averaging are not fully understood. Many aspects of the NFκB oscillatory dynamics may be rationalized using deterministic mass action rate equations (10, 11), but how stochastic self-sustained oscillations average out at a cell population level remains unclear.In this work, we provide a conceptual framework for understanding stochastic averaging as a result of "dephasing" of genetic oscillators. By dephasing, one essentially means the loss of common phase or coherence of oscillations in populations of mRNA or protein by-products of gene activation, caused by the stochastic molecular events in the course of an oscillator's operation that occur at di...

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“…Here x = 0, 1, .... No special boundary conditions are required for this system, as J 1 (x, t) and J 2 (x, t) at the boundary x = 0 take the values specified by Eq. (19). The single-reactional velocity v k (x, t) ∈ R can be written as:…”

Section: The Birth and Death Processmentioning

confidence: 99%

“…These models either generate time-evolving landscapes of probabilities over different microstates [9][10][11][12], or generate trajectories along which the systems travel [8,13]. Vector fields of probability flux and probability velocity are also of significant interest, as they can further characterize time-varying properties of the reaction systems, including that of the non-equilibrium steady states [14][15][16][17][18][19]. For example, determining the probability flux can help to infer the mechanism of dynamic switching among different attractors [20,21].…”

mentioning

confidence: 99%

Discrete flux and velocity fields of probability and their global maps in reaction systems

Terebus

Liang

2018

The Journal of Chemical Physics

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Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate, and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model. Keywords: Stochastic biochemical reaction networks, discrete flux and velocity fields of probability INTRODUCTIONBiochemical reactions in cells are intrinsically stochastic [1][2][3][4]. When the concentrations of participating molecules are small or the differences in reaction rates are large, stochastic effects become prominent [3,[5][6][7]. Many stochastic models have been developed to gain understanding of these reaction systems [8][9][10][11][12]. These models either generate time-evolving landscapes of probabilities over different microstates [9][10][11][12], or generate trajectories along which the systems travel [8,13]. Vector fields of probability flux and probability velocity are also of significant interest, as they can further characterize time-varying properties of the reaction systems, including that of the non-equilibrium steady states [14][15][16][17][18][19]. For example, determining the probability flux can help to infer the mechanism of dynamic switching among different attractors [20,21]. Quantifying the probability flux can also help to characterize the departure of non-equilibrium reaction systems from detailed balance [16,22,23], and can help to identify barriers and checkpoints between different stable cellular states [24]. Computing probability fluxes and velocity fields has found ...

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Exploring a noisy van der Pol type oscillator with a stochastic approach

Cited by 30 publications

References 39 publications

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

On the dephasing of genetic oscillators

Discrete flux and velocity fields of probability and their global maps in reaction systems

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