Noise-induced Oscillations Research Articles

Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment.

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.

System size expansion using Feynman rules and diagrams

Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the chemical master equation that allows us to calculate multi-time correlation functions which are accurate to any desired order in van Kampenʼs system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.

Effect of Noise Correlation on Noise-Induced Oscillation Frequency in the Photosensitive Belousov–Zhabotinsky Reaction in a Continuous Stirred Tank Reactor

We report on the experimental study of noise-induced oscillations in the photosensitive Ru(bpy)3(2+)-catalyzed Belousov-Zhabotinsky reaction in a continuous stirred tank reactor (CSTR). In the absence of deterministic oscillations and any external periodic forcing, oscillations appear when the system is perturbed by stochastic fluctuations in light irradiation with sufficiently high amplitude in the vicinity of the bifurcation point. The frequency distribution of the noise-induced oscillations is strongly affected by noise correlation. There is a shift of the noise-induced oscillation frequency toward higher frequencies for an intermediate range of the noise correlation exponent, indicating the occurrence of coherence resonance. Our findings indicate that, in principle, noise correlation can be used to direct chemical reactions toward certain behavior.

How reliable is the linear noise approximation of gene regulatory networks?

BackgroundThe linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems.ResultsWe implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations.ConclusionThe software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing computationally efficient and quantitative studies of intrinsic noise in gene regulatory networks.

Dynamics, control and information in delay-coupled systems: an overview

Dynamics, control and information in delay-coupled systems: an overview

Effects of bursty protein production on the noisy oscillatory properties of downstream pathways

Experiments show that proteins are translated in sharp bursts; similar bursty phenomena have been observed for protein import into compartments. Here we investigate the effect of burstiness in protein expression and import on the stochastic properties of downstream pathways. We consider two identical pathways with equal mean input rates, except in one pathway proteins are input one at a time and in the other proteins are input in bursts. Deterministically the dynamics of these two pathways are indistinguishable. However the stochastic behavior falls in three categories: (i) both pathways display or do not display noise-induced oscillations; (ii) the non-bursty input pathway displays noise-induced oscillations whereas the bursty one does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify systems involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production.

Signatures of nonlinearity in single cell noise-induced oscillations

A class of theoretical models seeks to explain rhythmic single cell data by postulating that they are generated by intrinsic noise in biochemical systems whose deterministic models exhibit only damped oscillations. The main features of such noise-induced oscillations are quantified by the power spectrum which measures the dependence of the oscillatory signal's power with frequency. In this paper we derive an approximate closed-form expression for the power spectrum of any monostable biochemical system close to a Hopf bifurcation, where noise-induced oscillations are most pronounced. Unlike the commonly used linear noise approximation which is valid in the macroscopic limit of large volumes, our theory is valid over a wide range of volumes and hence affords a more suitable description of single cell noise-induced oscillations. Our theory predicts that the spectra have three universal features: (i) a dominant peak at some frequency, (ii) a smaller peak at twice the frequency of the dominant peak and (iii) a peak at zero frequency. Of these, the linear noise approximation predicts only the first feature while the remaining two stem from the combination of intrinsic noise and nonlinearity in the law of mass action. The theoretical expressions are shown to accurately match the power spectra determined from stochastic simulations of mitotic and circadian oscillators. Furthermore it is shown how recently acquired single cell rhythmic fibroblast data displays all the features predicted by our theory and that the experimental spectrum is well described by our theory but not by the conventional linear noise approximation.

Phase description of stochastic oscillations.

We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows us to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine isophases of observed irregular oscillations, providing a basis for a refined phase description in data analysis.

Molecular noise induces concentration oscillations in chemical systems with stable node steady states

It is well known that internal or molecular noise induces concentration oscillations in chemical systems whose deterministic models exhibit damped oscillations. In this article we show, using the linear-noise approximation of the chemical master equation, that noise can also induce oscillations in systems whose deterministic descriptions admit no damped oscillations, i.e., systems with a stable node. This non-intuitive phenomenon is remarkable since, unlike noise-induced oscillations in systems with damped deterministic oscillations, it cannot be explained by noise excitation of the deterministic resonant frequency of the system. We here prove the following general properties of stable-node noise-induced oscillations for systems with two species: (i) the upper bound of their frequency is given by the geometric mean of the real eigenvalues of the Jacobian of the system, (ii) the upper bound of the Q-factor of the oscillations is inversely proportional to the distance between the real eigenvalues of the Jacobian, and (iii) these oscillations are not necessarily exhibited by all interacting chemical species in the system. The existence and properties of stable-node oscillations are verified by stochastic simulations of the Brusselator, a cascade Brusselator reaction system, and two other simple chemical systems involving auto-catalysis and trimerization. It is also shown how external noise induces stable node oscillations with different properties than those stimulated by internal noise.

Delay-managed tradeoff in the molecular dynamics of the segmentation clock

The molecular segmentation clock is a complex regulatory network that governs the periodic somite segmentation in vertebrate embryos. Underlying the rhythm of the segmentation clock is a single-cell level pace-making circuit, where inevitable molecular noise and time delay impose normal operating constraints to the pace-making. However, how the molecular mechanisms of the core circuit of the segmentation clock coordinate the operating constraints and maintain the rhythmic nature of the developmental process remains poorly understood. To address this question, we construct two biologically-motivated mathematical models with multiple clock protein transcription binding sites, with differential or rate-limited decay rates for protein monomers and dimers. We demonstrate that the rate-limited decay significantly enlarges the parameter space of noise-induced and delay-induced oscillations. Interestingly, focusing on the stochastic characters of noise-induced and delay-induced oscillations in terms of phase coherence and phase averaged amplitude noise in the polar coordinate, we find that there is a delay-managed tradeoff between phase coherence and phase averaged amplitude noise. In particular, the model with both multiple binding sites and rate-limited decay can show regular tunability as the delay increases. Our results indicate that transcriptional and post-translational mechanisms constrain the combined effects of noise and delay on the molecular dynamics of the segmentation clock.

Non-linear corrections to the time-covariance function derived from a multi-state chemical master equation

The time-covariance function captures the dynamics of biochemical fluctuations and contains important information about the underlying kinetic rate parameters. Intrinsic fluctuations in biochemical reaction networks are typically modelled using a master equation formalism. In general, the equation cannot be solved exactly and approximation methods are required. For small fluctuations close to equilibrium, a linearisation of the dynamics provides a very good description of the relaxation of the time-covariance function. As the number of molecules in the system decrease, deviations from the linear theory appear. Carrying out a systematic perturbation expansion of the master equation to capture these effects results in formidable algebra; however, symbolic mathematics packages considerably expedite the computation. The authors demonstrate that non-linear effects can reveal features of the underlying dynamics, such as reaction stoichiometry, not available in linearised theory. Furthermore, in models that exhibit noise-induced oscillations, non-linear corrections result in a shift in the base frequency along with the appearance of a secondary harmonic.

Demographic noise and piecewise deterministic Markov processes

We explore a class of hybrid (piecewise deterministic) systems characterized by a large number of individuals inhabiting an environment whose state is described by a set of continuous variables. We use analytical and numerical methods from nonequilibrium statistical mechanics to study the influence that intrinsic noise has on the qualitative behavior of the system. We discuss the application of these concepts to the case of semiarid ecosystems. Using a system-size expansion we calculate the power spectrum of the fluctuations in the system. This predicts the existence of noise-induced oscillations.

The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

BackgroundIt is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions.ResultsWe here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations.ConclusionsA new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.

The Transition between Stochastic and Deterministic Behavior in an Excitable Gene Circuit

We explore the connection between a stochastic simulation model and an ordinary differential equations (ODEs) model of the dynamics of an excitable gene circuit that exhibits noise-induced oscillations. Near a bifurcation point in the ODE model, the stochastic simulation model yields behavior dramatically different from that predicted by the ODE model. We analyze how that behavior depends on the gene copy number and find very slow convergence to the large number limit near the bifurcation point. The implications for understanding the dynamics of gene circuits and other birth-death dynamical systems with small numbers of constituents are discussed.

Non-linear dynamics of biological systems

All biological systems can be classified as open, dissipative and non-linear. This review introduces the most typical phenomena associated with non-linearity, dissipation and openness in biological systems. Namely, damped oscillations, self-oscillations, synchronisation, chaotic and noise-induced oscillations are explained, and illustrated by examples from various biological systems. The link is made between the experimentally registered activities and their mathematical counterparts with the help of the qualitative theory of ordinary differential equations. We introduce the ideas of non-linear and oscillatory thinking proposed by Mandelstam in the 1930s, and show how they can be applied to biological systems. An emphasis is made on the intuitive explanation of mathematical concepts rather than mathematical rigour.

Noise-Induced Oscillations in the flow of Concentrated Suspensions

The flow of a concentrated suspension with an N-shaped dependence of the shear rate on the shear stress applied to the system is investigated. A mathematical model with two dynamic states, a steady-state and an self-oscillatory state, is examined. It is shown that, in the zone of steady-state flows possessing a determinate stability, even small fluctuations in the shear stress can give rise to large amplitude oscillations in the flow rate. These noise-induced oscillations are analysed using the technique of stochastic response functions. Theoretical results using a parametric description of their dispersion are presented for the stochastic flow oscillations observed in the model.

Synchronization of multi-frequency noise-induced oscillations

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.

Two-parameter coherent resonance behavior in catalytic oxidation of CO on platinum surface

We study the effects of two-parameter noises on rate oscillations during CO oxidation on platinum surface, in a parameter region sub-threshold to deterministic Hopf bifurcation. It is found that the performance of noise-induced oscillations, characterized by an effective signal-to-noise ratio, shows ridge shape in the DCO∼DO2 plane, where DCO and DO2 measure the strength of two different parameter noises. It is indicating that the ‘two-parameter coherent resonance’ phenomenon occurs. Stochastic normal form theory is employed to analyze the non-trivial effects of two-parameter noises and the simulation results are well reproduced.

Oscillation propagation in neural networks with different topologies

In light of the issue of oscillation propagation in neural networks, various topologies of FitzHugh-Nagumo neuron populations are investigated. External Gaussian white noise is injected into the first neuron only. Before the oscillation spreads to the other neurons in the network, some of the inherent stochasticity within the noise-induced oscillation of the first neuron is filtered out due to the neuron's nonlinear dynamics. Both the temporal and the spatial coherence of the evoked activity's propagation are analyzed in conjunction with the network topology randomness p, the coupling strength between neurons g, and the noise amplitude D. The temporal periodicity of the global neural network presents a typical coherence biresonance (CBR) characteristic with regard to the noise intensity. The network topology randomness exerts different influences on the resonance effects for different coupling strength regimes. At an intermediate coupling strength, the random shortcuts reinforce the interactions between the neurons, and then more stochasticity in the firings of the first neuron spreads within the network. Consequently, CBR is decreased with the increase of the network topology randomness. At a large coupling strength, the random shortcuts assist the nonlinearity in impairing the stochastic components, and consequently help to enhance the resonance effects, which differed significantly from previous related work. However, the degree of the spatial synchronization of the systems increases monotonically as the network topology randomness increases at any coupling strength.

The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics

The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The analysis captures the key elements supporting the nearly regular oscillations of the infected and susceptible populations, namely, the interaction of the deterministic and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic amplitude equation describing the envelope of the oscillations yields two criteria providing explicit parameter ranges where they can be observed. These conditions are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent without noise. In this context the criteria indicate how loss of immunity and other factors can lead to a significant increase in the parameter range for prevalence of the sustained oscillations, without any external driving forces. Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of the oscillations.