Additive Noise Intensity Research Articles

The mean first-passage time for piecewise nonlinear system driven by colored correlated additive and multiplicative colored noises

In this paper, We have studied the effects of intensity and correlation time of noises on the mean first-passage time in a picecewise nonlinear system driven by multiplicative and additive colored noises with colored cross-correlation. We derived the expression of the mean first-passage time (MFPT) by applying the unified colored approximation method and the steepest-descent approximation. Results show that the MFPT of the system exhibits a mono-peak structure and the “resonance” phenomena enhance with the increase of multiplicative noise intensity. The value of the peak decreases with increasing additive noise intensity and the correlation between the additive and multiplicative noises. However, the MFPT of the system increases with the increase of additive noise intensity. That is, the effects of the additive noise and the multiplicative noise on MFPT are different. Moreover, the negative and passive correlations play different roles in the MFPT.

Doubly stochastic coherence in complex neuronal networks

A system composed of coupled FitzHugh-Nagumo neurons with various topological structures is investigated under the co-presence of two independently additive and multiplicative Gaussian white noises, in which particular attention is paid to the neuronal networks spiking regularity. As the additive noise intensity and the multiplicative noise intensity are simultaneously adjusted to optimal values, the temporal periodicity of the output of the system reaches the maximum, indicating the occurrence of doubly stochastic coherence. The network topology randomness exerts different influences on the temporal coherence of the spiking oscillation for dissimilar coupling strength regimes. At a small coupling strength, the spiking regularity shows nearly no difference in the regular, small-world, and completely random networks. At an intermediate coupling strength, the temporal periodicity in a small-world neuronal network can be improved slightly by adding a small fraction of long-range connections. At a large coupling strength, the dynamical behavior of the neurons completely loses the resonance property with regard to the additive noise intensity or the multiplicative noise intensity, and the spiking regularity decreases considerably with the increase of the network topology randomness. The network topology randomness plays more of a depressed role than a favorable role in improving the temporal coherence of the spiking oscillation in the neuronal network research study.

Effect of Time-Delay in a Time-Delayed Mono-Stable System Subject to White Noise and Dichotomous Noise

The stochastic resonance (SR) for a time-delayed mono-stable system, driven by periodic square-wave signal, a dichotomous noise and a static force as well as multiplicative and additive noise is considered. It is found that the output signal-to-noise ratio (SNR) increases monotonically with the increase of the delayed time. The SNR varies non-monotonically with the intensities of the dichotomous noise and multiplicative and additive noise as well as the system parameter and the amplitude of the input square-wave signal.

THE BISTABLE POTENTIAL: AN ARCHETYPE FOR CLASSICAL AND QUANTUM SYSTEMS

In this work we analyze the transient dynamics of three different classical and quantum systems. First, we consider a classical Brownian particle moving in an asymmetric bistable potential, subject to a multiplicative and additive noise source. We investigate the role of these two noise sources on the life time of the metastable state. A nonmonotonic behavior of the lifetime as a function of both additive and multiplicative noise intensities is found, revealing the phenomenon of noise enhanced stability. Afterward, by using a Lotka–Volterra model, the dynamics of two competing species in the presence of Lévy noise sources is analyzed. Quasiperiodic oscillations and stochastic resonance phenomenon in the dynamics of the competing species are found. Finally the dynamics of a quantum particle subject to an asymmetric bistable potential and interacting with a thermal reservoir is investigated. We use the Caldeira–Leggett model and the approach of the Feynman–Vernon functional in discrete variable representation. We obtain the time evolution of the population distributions in energy eigenstates of the particle, for different values of the coupling strength with the thermal bath.

Stochastic Resonance-Like and Resonance Suppression-Like Phenomena in a Bistable System with Time Delay and Additive Noise

Stochastic resonance (SR)-like and resonance suppression (RS)-like phenomena in a time-delayed bistable system driven by additive white noise are investigated by means of stochastic simulations of the power spectrum, the quality factor of the power spectrum, and the mean first-passage time (MFPT) of the system. The calculative results indicate that: (i) as the system is driven by a small periodic signal, the quality factor as a function delay time exhibits a maximal value at smaller noise intensities, i.e., an SR-like phenomenon. With the increment in additive noise intensity, the extremum gradually disappears and the quality factor decreases monotonously with delay time. (ii) As the additive noise intensity is smaller, the curve of the MFPT with respect to delay time displays a peak, i.e., an RS-like phenomenon. At higher levels of noise, however, the non-monotonic behavior is lost.

Stochastic resonance in coupled weakly-damped bistable oscillators subjected to additive and multiplicative noises

With coupled weakly-damped periodically driven bistable oscillators subjected to additive and multiplicative noises under concern, the objective of this paper is to check to what extent the resonant point predicted by the Gaussian distribution assumption can approximate the simulated one. The investigation based on the dynamical mean-field approximation and the direct simulation demonstrates that the predicted resonant point and the simulated one are basically coincident for the case of pure additive noise, but for the case including multiplicative noise the situation becomes somewhat complex. Specifically speaking, when stochastic resonance (SR) is observed by changing the additive noise intensity, the predicted resonant point is lower than the simulated one; nevertheless, when SR is observed by changing the multiplicative noise intensity, the predicted resonant point is higher than the simulated one. Our observations imply that the Gaussian distribution assumption can not exactly describe the actual situation, but it is useful to some extent in predicting the low-frequency stochastic resonance of the coupled weakly-damped bistable oscillator.

Stochastic resonance in a bistable system with global delay and two noises

The global time delay is introduced into a bistable system driven by two noises and a periodic signal. The signal power amplification factor η is employed to characterise stochastic resonance (SR) of the system. Numerical simulation results indicate the following. (1) For a periodic signal with low frequency (Ω), the typical behaviour of the SR is lowered monotonically by increasing the delay time τ; for moderate Ω, τ enhances the SR behaviour and then weakens it, with a critical value at which the SR is optimum. (2) Multiplicative noise intensity D and additive noise intensity α have different influences on the SR performance, viz D enhances the SR monotonically while α enforces the SR initially and then restrains it; (3) correlation intensity λ between the two noises always weakens the SR behaviour of the system.

RELAXATION PHENOMENA IN CLASSICAL AND QUANTUM SYSTEMS

classical and quantum systems are investigated. First, the role of multiplicative and additive noise in a classical metastable system is analyzed. The mean lifetime of the metastable state shows a nonmonotonic behavior with a maximum as a function of both the additive and multiplicative noise intensities. In the second system, the simultaneous action of thermal and non-Gaussian noise on the dynamics of an overdamped point Josephson junction is studied. The eect of a Levy noise generated by a Cauchy‐Lorentz distribution on the mean lifetime of the superconductive metastable state, in the presence of a periodic driving, is investigated. We find resonant activation and noise enhanced stability in the presence of Levy noise. Finally, the time evolution of a quantum particle moving in a metastable potential and interacting with a thermal reservoir is analyzed. Within the Caldeira‐Legget model and the Feynman‐Vernon functional approach, we obtain the time evolution of the population distributions in the position eigenstates of the particle, for dierent values of the thermal bath coupling strength.

Stochastic resonance in FHN neural system driven by non-Gaussian noise

Stochastic resonance (SR) is studied in the FitzHugh-Nagumo (FHN) neural system subject to multiplicative non-Gaussian noise, additive Gaussian white noise and a periodic signal. Using the path integral approach and the two-state theory, the expression of the signal-to-noise ratio (SNR) is derived. The simulation results show that conventional SR and double SR occur in the FHN neural model under different values of system parameters. The effects of the additive and multiplicative noise intensities on SNR are different. Moreover, the addition of non-Gaussian noise is conductive to the enhancement of the response to the output signal of the FHN neural system.

Effect of noises on the stability of a metapopulation

The Levins model subjected to the noise is employed to study the stability of a metapopulation. The analytic expressions of the stationary probability distribution function and the mean extinction time of the metapopulation are obtained according to the Fokker-Planck Equation. The results show that for the case of no correlation between the additive noise and the multiplicative noise (=0, is the intensity of correlation between multiplicative and additive noise), the increase of the additive noise intensity weakens the stability of a metapopulation; for the case of 0, enhances the stability of a metapopulation. For -(c-e-D)2/(4cD)1, can induce the resonance restrain phenomenon. Meantime, there exists a critical value of D. When D is lower than the critical value, the stability of the system is enhanced.

Time delay induced transition of gene switch and stochastic resonance in a genetic transcriptional regulatory model

BackgroundNoise, nonlinear interactions, positive and negative feedbacks within signaling pathways, time delays, protein oligomerization, and crosstalk between different pathways are main characters in the regulatory of gene expression. However, only a single noise source or only delay time in the deterministic model is considered in the gene transcriptional regulatory system in previous researches. The combined effects of correlated noise and time delays on the gene regulatory model still remain not to be fully understood.ResultsThe roles of time delay on gene switch and stochastic resonance are systematically explored based on a famous gene transcriptional regulatory model subject to correlated noise. Two cases, including linear time delay appearing in the degradation process (case I) and nonlinear time delay appearing in the synthesis process (case II) are considered, respectively. For case I: Our theoretical results show that time delay can induce gene switch, i.e., the TF-A monomer concentration shifts from the high concentration state to the low concentration state ("on"→"off"). With increasing the time delay, the transition from "on" to "off" state can be further accelerated. Moreover, it is found that the stochastic resonance can be enhanced by both the time delay and correlated noise intensity. However, the additive noise original from the synthesis rate restrains the stochastic resonance. It is also very interesting that a resonance bi-peaks structure appears under large additive noise intensity. The theoretical results by using small-delay time-approximation approach are consistent well with our numerical simulation. For case II: Our numerical simulation results show that time delay can also induce the gene switch, however different with case I, the TF-A monomer concentration shifts from the low concentration state to the high concentration state ("off"→"on"). With increasing time delay, the transition from "on" to "off" state can be further enhanced. Moreover, it is found that the stochastic resonance can be weaken by the time delay.ConclusionsThe stochastic delay dynamic approach can identify key physiological control parameters to which the behavior of special genetic regulatory systems is particularly sensitive. Such parameters might provide targets for pharmacological intervention. Thus, it would be highly interesting to investigate if similar experimental techniques could be used to bring out the delay-induced switch and stochastic resonance in the stochastic gene transcriptional regulatory process.

Stochastic resonance on a modular neuronal network of small-world subnetworks with a subthreshold pacemaker

We study the phenomenon of stochastic resonance on a modular neuronal network consisting of several small-world subnetworks with a subthreshold periodic pacemaker. Numerical results show that the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the intensity of additive spatiotemporal noise. This effect of pacemaker-driven stochastic resonance of the system depends extensively on the local and the global network structure, such as the intra- and inter-coupling strengths, rewiring probability of individual small-world subnetwork, the number of links between different subnetworks, and the number of subnetworks. All these parameters play a key role in determining the ability of the network to enhance the noise-induced outreach of the localized subthreshold pacemaker, and only they bounded to a rather sharp interval of values warrant the emergence of the pronounced stochastic resonance phenomenon. Considering the rather important role of pacemakers in real-life, the presented results could have important implications for many biological processes that rely on an effective pacemaker for their proper functioning.

Combined effects of asymmetry and noise correlation on the noise-enhanced stability phenomenon in a bistable system

We investigate the effects of asymmetry and noise correlation on the noise-enhanced stability (NES) phenomenon in an asymmetric bistable system driven by cross-correlated noises. The expressions for the average escape time from the left stable state TL and from the right stable state TR are derived. The results indicate that TL and TR, as a function of the multiplicative noise intensity D, show non-monotonic behavior with the presence of a maximum value in the cases of positive and negative correlation between noises, respectively. This is the distinguishing characteristic of the NES phenomenon. However, the additive noise cannot produce the NES effect. The NES effect for both states is enhanced as the cross-correlation strength between noises increases, but is weakened as the additive noise intensity increases, whereas the NES effect is enhanced for the left state and weakened for the right state as the asymmetry parameter increases. Moreover, the validity of the analytical approximation is confirmed by stochastic simulations.

The relaxation time of processes in a FitzHugh–Nagumo neural system with time delay

In this paper, we study the relaxation time (RT) of the steady-state correlation function in a FitzHugh–Nagumo neural system under the presence of multiplicative and additive white noises and time delay. The noise correlation parameter λ can produce a critical behavior in the RT as functions of the multiplicative noise intensity D, the additive noise intensity Q and the time delay τ. That is, the RT decreases as the noise intensities D and Q increase, and increases as the time delay τ increases below the critical value of λ. However, above the critical value, the RT first increases, reaches a maximum, and then decreases as D, Q and τ increase, i.e. a noise intensity D or Q and a time delay τ exist, at which the time scales of the relaxation process are at their largest. In addition, the additive noise intensity Q can also produce a critical behavior in the RT as a function of λ. The noise correlation parameter λ first increases the RT of processes, then decreases it below the critical value of Q. Above the critical value, λ increases it.

THE CRITICAL EFFECTS OF TIME DELAY AND NOISE CORRELATION ON STOCHASTIC RESONANCE IN AN ASYMMETRIC BISTABLE SYSTEM

We investigate the effects of time delay and noise correlation on the stochastic resonance induced by a multiplicative signal in an asymmetric bistable system. By the two-state theory and small delay approximation, the expression of the output signal-to-noise ratio (SNR) is obtained in the adiabatic limit. The results show that SNR as a function of the multiplicative noise intensity D shows a transition from two peaks to one peak with the decreasing of cross-correlation strength λ and the increasing of delay time τ. Moreover, there are the doubly critical phenomena for SNR versus λ and τ, and SNR versus D and α (additive noise intensity).

Effects of Time Delay on Stability of an Unstable State in a Bistable System with Correlated Noises

Effects of time delay on stability of an unstable state in a time-delayed bistable system are investigated. The analytic expression of the transition rate W(xu, τ) from unstable state xu to stable state x+ is derived. The numerical calculation results of W(xu, τ) indicate that W(xu, τ) decreases with the increasing multiplicative noise intensity, the additive noise intensity and the strength of correlations between the multiplicative and the additive noise increase, but W(xu, τ) increases with increasing delay time. Namely, the multiplicative noise, the additive noise and the correlations between the multiplicative and the additive noises enhance the stability of the unstable state in the time-delayed bistable system but the stability is weakened by time delay.

Bistability, Probability Transition Rate and First-Passage Time in an Autoactivating Positive-Feedback Loop

A hallmark of positive-feedback regulation is bistability, which gives rise to distinct cellular states with high and low expression levels, and that stochasticity in gene expression can cause random transitions between two states, yielding bimodal population distribution (Kaern et al., 2005, Nat Rev Genet 6: 451-464). In this paper, the probability transition rate and first-passage time in an autoactivating positive-feedback loop with bistability are investigated, where the gene expression is assumed to be disturbed by both additive and multiplicative external noises, the bimodality in the stochastic gene expression is due to the bistability, and the bistability determines that the potential of the Fokker-Planck equation has two potential wells. Our main goal is to illustrate how the probability transition rate and first-passage time are affected by the maximum transcriptional rate, the intensities of additive and multiplicative noises, and the correlation of additive and multiplicative noises. Our main results show that (i) the increase of the maximum transcription rate will be useful for maintaining a high gene expression level; (ii) the probability transition rate from one potential well to the other one will increase with the increase of the intensity of additive noise; (iii) the increase of multiplicative noise strength will increase the amount of probability in the left potential well; and (iv) positive (or negative) cross-correlation between additive and multiplicative noises will increase the amount of probability in the left (or right) potential well.

Diffusion of Active Particles Subject both to Additive and Multiplicative Noises

We consider a Langevin equation of active Brownian motion which contains a multiplicative as well as an additive noise term. We study the dependences of the effective diffusion coefficient Deff on both the additive and multiplicative noises. It is found that for fixed small additive noise intensity Deff varies non-monotonously with multiplicative noise intensity, with a minimum at a moderate value of multiplicative noise, and Deff increases monotonously, however, with the multiplicative noise intensity for relatively strong additive noise; for fixed multiplicative noise intensity Deff decreases with growing additive noise intensity until it approaches a constant. An explanation is also given of the different behavior of Deff as additive and multiplicative noises approach infinity, respectively.

Mean first-passage time and stochastic resonance in an asymmetric bistable system driven by non-Gaussian noise

In this paper, mean first-passage time (MFPT) and stochastic resonance (SR) are investigated in an asymmetric bistable system driven by multiplicative non-Gaussian noise and additive Gaussian white noise. Using the path integral approach and two-state theory, the expression of MFPT and the signal-to-noise ratio (SNR) are derived. The results show that the influences of the asymmetric coefficient on the MFPTs in two opposite directions are entirely different. SNR is a non-monotonic function of the additive noise intensity and asymmetric coefficient, therefore, an SR is found in this system. Whereas SNR is a monotonic function of the multiplicative noise intensity and no SR appears. This demonstrates that the effect of the multiplicative noise intensity on SNR is different from that of the additive noise intensity in the system.

Relaxation Time for an Optical Bistable System Subjected to Color Noises

The relaxation time T of an optical bistable system with cross-correlated color noises and small time delay is investigated. Using the Novikov theorem and Fox approach, the steady probability distribution is obtained. The expression of T is derived using the Stratonovich decoupling ansatz formalism. It is found that the relaxation time T increases with the increasing cross-correlation time τ0 between the two noises or the self-correlation time τ1 of the multiplicative noise, but decreases with the increasing self-correlation time τ2 of the additive noise. T decreases with the increasing correlation intensity λ or the multiplicative noise intensity Q, but increases with the increasing additive noise intensity D. There exists a peak in the curve of T versus delay time τ.