Unified Colored Noise Approximation Research Articles

Multimodality in systems driven by Ornstein-Uhlenbeck noise.

The presence of noise in nonlinear dynamical systems can significantly change their properties. Here, we study the properties of a noise perturbed motion in a single-well potential of |x|n (n>0) type. We explore under what conditions the action of the Ornstein-Uhlenbeck noise induces bimodality of stationary states in static, single-well, power-law potentials. In particular, we inspect the transition from unimodality (n⩽2) to bimodality (n>2). Results of numerical simulations are compared with estimates obtained from the unified colored-noise approximation. Furthermore, we explore the role of a harmonic addition to the general single-well power-law potentials showing its constructive or destructive role.

Delay Segmented Tristable Stochastic Resonance System Driven by Non-Gaussian Colored Noise and Its Application in Bearing Fault Detection

Abstract This study investigates the Delayed Segmented Tristable Stochastic Resonance (DSTSR) system under the influence of additive non-Gaussian colored noise. The research employs an improved segmented tristable potential function, wherein the equilibrium points and barrier heights can be independently controlled by parameters. Simultaneously, the segmented function on both sides reduces the restrictions of higher-order terms on the walls of the potential wells. The equivalent Langevin equation for the DSTSR system is obtained using the path integral method, the unified colored noise approximation method, and the small-delay approximation. Subsequently, the theoretical expressions for the steady-state probability density, mean first passage time (MFPT), and Signal-to-Noise Ratio (SNR) are derived from the resulting equations, and the impact of variations in system parameters on these performance metrics is discussed. Additionally, Monte Carlo simulations for MFPT are conducted to verify the accuracy of the theoretical derivations. Combining the results from the theoretical section and the impact of parameters on system performance, the article employs an adaptive genetic algorithm to optimize system parameters. This algorithm is then applied to simulation experiments and bearing fault detection. In the simulation experiments, the DSTSR system is compared with other systems. The results indicate that the DSTSR system exhibits the highest SNR improvement. Furthermore, in bearing fault detection under non-Gaussian colored noise, the DSTSR system shows higher spectral amplitude and SNR at the fault frequency compared to the tristable stochastic resonance system and the segmented tristable stochastic resonance system without time delay feedback. This suggests that stochastic resonance can effectively detect weak signals in non-Gaussian non-white noise scenarios, and the introduction of time delay contributes to the occurrence of stochastic resonance to a certain extent.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Logical stochastic resonance in a cross-bifurcation non-smooth system

This paper investigates logical stochastic resonance (LSR) in a cross-bifurcation non-smooth system driven by Gaussian colored noise. In this system, a bifurcation parameter triggers a transition between monostability, bistability and tristability. By using Novikov’s theorem and the unified colored noise approximation method, the approximate Fokker–Planck equation is obtained. Then we derive the generalized potential function and the transition rates to analyze the LSR phenomenon using numerical simulations. We simulate the logic operation of the system in the bistable and tristable regions respectively. We assess the impact of Gaussian colored noise on the LSR and discover that the reliability of the logic response depends on the noise strength and the bifurcation parameter. Furthermore, it is found that the bistable region has a more extensive parameter range to produce reliable logic operation compared with the tristable region, since the tristable region is more sensitive to noise than the bistable one.

Tumor state transitions driven by Gaussian and non-Gaussian noises

Tumor state transitions between the excited (high-concentration) and nonexcited (low-concentration) basins under the Gaussian white noise and non-Gaussian colored noise are investigated via the most probable steady states (MPSS) and the first escape probability (FEP)-based stochastic basin of attraction (SBA), respectively. Reducing the non-Gaussian colored noise and then utilizing the unified colored noise approximation (UCNA), the Markov system is derived. The extremal controlling equation of stationary probability density function (SPDF) is derived to analyze the impacts of noise on transitions in terms of MPSS. The existence of the ‘color’ of the non-Gaussian colored noise induces the reappearance of the uncorrelated additive white noise parameter that had vanished from the extremal controlling equation, completely reversing the inability of the uncorrelated additive Gaussian white noise to operate on transitions. The FEP-dependent SBA characterizing the excited basin stability is performed to further analyze the role of noise on the likelihood of escaping to the nonexcited state. Results show that the cross-correlated noises play a dual role in regulating SBA. The increased SBA indicating more difficulty to escape to the nonexcited state reflects a worse therapeutic effect. Therefore, enhancing the negatively correlated noise intensities and augmenting the non-Gaussian noise correlation time is essential for destabilizing the excited basin and achieving optimal therapeutic efficacy.

Active particles driven by competing spatially dependent self-propulsion and external force

We investigate how the competing presence of a nonuniform motility landscape and an external confining field affects the properties of active particles. We employ the active Ornstein-Uhlenbeck particle (AOUP) model with a periodic swim-velocity profile to derive analytical approximations for the steady-state probability distribution of position and velocity, encompassing both the Unified Colored Noise Approximation and the theory of potential-free active particles with spatially dependent swim velocity recently developed. We test the theory by confining an active particle in a harmonic trap, which gives rise to interesting properties, such as a transition from a unimodal to a bimodal (and, eventually multimodal) spatial density, induced by decreasing the spatial period of the self propulsion. Correspondingly, the velocity distribution shows pronounced deviations from the Gaussian shape, even displaying a bimodal profile in the high-motility regions. We thus show that the interplay of two relatively simple physical fields can be employed to generate complex emerging behavior.

Transition and basin stability in a stochastic tumor growth model with immunization

The phenomenon of noise-induced transition driven by the correlated Gaussian and non-Gaussian colored noises is investigated by the first escape probability (FEP) and the mean first exit time (MFET). To derive the Markovian approximation of the original tumor growth model and obtain the analytical expressions of the FEP and MFET, we reduce the non-Gaussian colored noise and then expand the unified colored noise approximation (UCNA). Additionally, the stochastic basin of attraction (SBA), a recent geometric concept based on the FEP and MFET, is introduced to provide further insight into the effects of noisy fluctuations on the basin stability of a certain domain. A higher FEP or shorter MFET in the high tumor population region B facilitates the transition from B to the low tumor population region Bc, which indicates the weaker stability of domain B. Our main results demonstrate that (i) the transitions from B to Bc can be induced by both the Gaussian and non-Gaussian noise sources; (ii) the stronger noise intensity, especially the non-Gaussian noise intensity, with a larger deviation parameter and immune coefficient improves the FEP, shortens the MFET, and hence benefits the transitions. However, the enlargement of the correlation between noises strengthens the basin stability of domain B and impedes the transitions; (iii) the size of SBA expands due to the larger cross-correlated intensity. In contrast, the enhancements of noise intensities with a larger departure degree reduce the size of SBA, which weakens the basin stability and is less in favor of tumor treatment. Furthermore, the Monte Carlo simulations of the original system are employed to verify the feasibility and accuracy of the analytical predictions.

Most probable trajectories in the delayed tumor growth model excited by a multiplicative non-Gaussian noise

This paper investigates the transition phenomenon in the delayed tumor growth model under a multiplicative non-Gaussian colored noise based on the most probable trajectories. Firstly, the unified colored noise approximation and the small delay approximation are utilized to approximate the model in this paper. We then detect the effects of the time-delay, the correlation time and the noise intensity of non-Gaussian colored noise, and find that they can all promote the switch of most probable trajectories from the tumor state to the tumor-free state, namely, the most probable transition time decreases gradually. Conversely, these parameters suppress the transition of most probable trajectories from the tumor-free state to the tumor state, that is, the most probable transition time will increase successively. By analyzing the dynamics under these parameters, it is found that the noise intensity has the greatest influence on the transition behavior. Finally, it can be observed that the sum of the most probable transition time from the tumor-free state to the tumor state and from the tumor state to the tumor-free state is equal to the total observation time.

Stochastic resonance in asymmetric bistable coupled network systems driven by Gaussian colored noise

In this work studied is the synergistic effect of asymmetric bistable coupled network systems under the action of Gaussian colored noise and periodic signal. The system is a network model consisting of a large number of oscillators. The interaction and change between individuals produce complex nonlinear behavior patterns. For further research, firstly, the original N-dimensional system is reduced and approximated by using the mean field theory, the unified colored noise approximation theory and the equivalent nonlinearization method. Secondly, the Langevin equation of simplified model is obtained through the slaving principle by using the two-state model theory to derive the theoretical expression of signal-to-noise ratio. It is found that the system produces the phenomenon of scale stochastic resonance. Finally, the effects of Gaussian color noise parameters, system parameters and periodic signal parameters on the stochastic resonance behavior of asymmetric coupled network systems are discussed. The results show that the increase of Gaussian colored noise correlation time and noise intensity can promote the scale stochastic resonance phenomenon; selecting appropriate coupling coefficient can achieve the optimal stochastic resonance effect. And the stochastic resonance phenomenon of the system driven by the Gaussian colored noise and the Gaussian white noise, respectively, are analyzed and compared with each other. Research result shows that Gaussian colored noise is more conducive to enhancing stochastic resonance phenomenon.

Correlated escape of active particles across a potential barrier.

We study the dynamics of one-dimensional active particles confined in a double-well potential, focusing on the escape properties of the system, such as the mean escape time from a well. We first consider a single-particle both in near and far-from-equilibrium regimes by varying the persistence time of the active force and the swim velocity. A non-monotonic behavior of the mean escape time is observed with the persistence time of the activity, revealing the existence of an optimal choice of the parameters favoring the escape process. For small persistence times, a Kramers-like formula with an effective potential obtained within the unified colored noise approximation is shown to hold. Instead, for large persistence times, we developed a simple theoretical argument based on the first passage theory, which explains the linear dependence of the escape time with the persistence of the active force. In the second part of the work, we consider the escape on two active particles mutually repelling. Interestingly, the subtle interplay of active and repulsive forces may lead to a correlation between particles, favoring the simultaneous jump across the barrier. This mechanism cannot be observed in the escape process of two passive particles. Finally, we find that in the small persistence regime, the repulsion favors the escape, such as in passive systems, in agreement with our theoretical predictions, while for large persistence times, the repulsive and active forces produce an effective attraction, which hinders the barrier crossing.

Stochastic resonance of an asymmetric tristable system driven by cross-correlated Ornstein–Uhlenbeck noise

Stochastic resonance of an asymmetric tristable system driven by a periodic forcing and cross-correlated Ornstein–Uhlenbeck (O–U) noise is investigated. An approximate Fokker-Planck Equation for an asymmetric tristable system driven by a periodic forcing and cross-correlated O–U noise is derived in the adiabatic limit by using the unified colored-noise approximation method and the stochastic equivalent rules. Afterwards, the expression for spectral amplification (SA), the stationary probability density function (PDF) and the mean first passage time (MFPT) are derived. By considering the influence of asymmetric potential function and cross-correlated noise, the phenomenon of noise enhancing stability (NES) and stochastic resonance (SR) can be found. It can also be found that the response of the tristable system can be effectively improved by properly selecting the relevant noise intensity and the asymmetric constant. This proves the excellent performance of the system, and provides a good theoretical support for practical engineering application.

Noise and delay enhanced stability in tumor-immune responses to chemotherapy system

The stability of tumor-immune responses to chemotherapy system with Gaussian colored noises and time delay is discussed by means of the maximum Lyapunov exponent. The unified colored noise approximation of multidimensional stochastic dynamic system with Gaussian colored and white noises is extended to the general case for the Gaussian colored noises having different self-correlation times and the correlated noises. We derive the analytic formula of the maximum Lyapunov exponent of system as a function of intensities and correlation times of Gaussian colored noises as well as delay, these parameters could change the stability of system between asymptotically stable and unstable, among, the system appears the noise and delay enhanced stability, which is detected by the maximum Lyapunov exponent.

The stability analysis of tumor-immune responses to chemotherapy system driven by Gaussian colored noises

Abstract The influence of Gaussian colored noises on the stability of tumor-immune responses to chemotherapy system is analyzed. By means of the unified colored noise approximation of multidimensional stochastic dynamic system, the system is changed into the stochastic system with Gaussian white noises. I derive the analytic formula of the maximum Lyapunov exponent of system as a function of intensities and correlation times of Gaussian colored noises, then I find the noise-enhanced stability phenomenon, which is detected by the maximum Lyapunov exponent. Moreover, the correlation time τ1 of ξ1(t) influences the stability of steady state becomes unstable if τ1 → 0.15, which also increases as the correlation time τ2 of ξ2(t) increases. These indicate that, once tumor grows, by controlling the stochastic effect in process of chemotherapy, such as Gaussian colored noises, it is possible to control the stability of the tumor, which is very important for treatment. Especially, one could regulate the correlation time τ1 for this purpose.

Steady-state fluctuations of a genetic feedback loop with fluctuating rate parameters using the unified colored noise approximation

A common model of stochastic auto-regulatory gene expression describes promoter switching via cooperative protein binding, effective protein production in the active state and dilution of proteins. Here we consider an extension of this model whereby colored noise with a short correlation time is added to the reaction rate parameters—we show that when the size and timescale of the noise is appropriately chosen it accounts for fast reactions that are not explicitly modeled, e.g., in models with no mRNA description, fluctuations in the protein production rate can account for rapid multiple stages of nuclear mRNA processing which precede translation in eukaryotes. We show how the unified colored noise approximation can be used to derive expressions for the protein number distribution that is in good agreement with stochastic simulations. We find that even when the noise in the rate parameters is small, the protein distributions predicted by our model can be significantly different than models assuming constant reaction rates.

Statistical field theory and effective action method for scalar active matter

We employ Statistical Field Theory techniques for coarse-graining the steady-state properties of Active Ornstein-Uhlenbeck particles. The computation is carried on in the framework of the Unified Colored Noise approximation that allows an effective equilibrium picture. We thus develop a mean-field theory that allows to describe in a unified framework the phenomenology of scalar Active Matter. In particular, we are able to describe through spontaneous symmetry breaking mechanism two peculiar features of Active Systems that are (i) The accumulation of active particles at the boundaries of a confining container, and (ii) Motility-Induced Phase Separation (MIPS). \textcolor{black}{We develop a mean-field theory for steric interacting active particles undergoing to MIPS and for Active Lennard-Jones (ALJ) fluids.} \textcolor{black}{Within this framework}, we discuss the universality class of MIPS and ALJ \textcolor{black}{showing that it falls into Ising universality class.} We \textcolor{black}{thus} compute analytically the critical line $T_c(\tau)$ for both models. In the case of MIPS, $T_c(\tau)$ gives rise to a reentrant phase diagram compatible with an inverse transition from liquid to gas as the strength of the noise decreases. \textcolor{black}{However, in the case of particles interacting through anisotropic potentials, } the field theory acquires a $\varphi^3$ term that, \textcolor{black}{in general, cannot be canceled performing the expansion around the critical point.} In this case, the \textcolor{black}{Ising} critical point might \textcolor{black}{be replaced} by a first-order phase transition \textcolor{black}{region}.

The unified colored noise approximation of multidimensional stochastic dynamic system

The unified colored noise approximation of multidimensional stochastic dynamic system driven by correlated Gaussian colored noises is developed. For general multidimensional stochastic dynamic system driven by Gaussian colored noises, which maybe correlate with each other and have small or large correlation times, it is transformed into multidimensional stochastic dynamic system driven by white noises. By merging method of white noises, the multiple diffusion terms of each equation of system is merged into one diffusion term, then the unified colored noise approximation for multidimensional stochastic dynamic system is established. In order to facilitate application, the specific derivation formulas of the unified colored noise approximation for common one-dimensional, two-dimensional, and three-dimensional stochastic dynamic systems are also given respectively, among, one also proves that it includes the unified colored noise approximation for one-dimensional stochastic dynamic system previously created.

Estimating the nonlinear effects of an ecological system driven by Ornstein-Uhlenbeck noise

• An effective FPE for a dynamical system driven by multiple O-U noises is derived. • A Levins model driven by O-U noise and cross-correlated noise is constructed. • Noise enhanced stability phenomena is observed. In this paper, a simple ecological model driven by Ornstein-Uhlenbeck noise and cross-correlated noise is constructed based on the classical Levins model. Initially, an approximate Fokker-Planck Equation for a general dynamical system driven by multiple sources of Ornstein-Uhlenbeck and cross-correlated noise is obtained by using the unified colored-noise approximation method and the stochastic equivalent rules. Then, the effects of Ornstein-Uhlenbeck noise on the dynamical characteristics of the modified Levins model are investigated. The theoretical dynamical quantities, including the steady-state probability distribution function and the mean extinction time, respectively, are analyzed. It is found that the external noises can lead to a decrease in the probability of an occupied patch, while, the internal noise, the cross-correlated noise and the self-correlation time of the external and internal noises can cause an increase in the probability of an occupied patch. Further, the impact of the noise intensities on the mean extinction time is analyzed. It is especially worthy to note that as the intensities of the internal noise and the cross-correlated noise increase, a noise-enhanced stability phenomenon is induced. Moreover, the mean extinction time increases as the self-correlated time and the intensity of the cross-correlated noise increase, yet the mean extinction time decreases as the intensity of the internal noise increases. The validity of theoretical method is checked by numerical simulation.

First escape probability and mean first exit time for a time-delayed ecosystem driven by non-Gaussian colored noise

Abstract The first escape probability (FEP) and the mean first exit time (MFET) are utilized to explore the stability of the high vegetation basin in an extended ecosystem with the two important features: the randomness and the delay. The more realistic non-Gaussian colored noise is chosen as a stochastic perturbation. In order to overcome the effects of the system inherent parameters, two novel basin stability indexes are proposed based on the FEP and the MFET. The unified colored noise approximation and the small delay approximation are exploited to further simplify the system and obtain the stability indexes. The results show that either increased herbivores or reduced rainfall can increase the FEP and decrease the MFET of the high vegetation ecosystem, and non-Gaussian colored noise disturbance also exacerbates the problem. Conversely, the time delay can control deterioration of the ecosystem. Furthermore, by comparing the theoretical analysis results and Monte Carlo simulation results of the original system, it is found that both the approximation methods and the process of solving the FEP and the MFET are feasible.

Asymmetric stochastic resonance under non-Gaussian colored noise and time-delayed feedback**Project supported by the National Natural Science Foundation of China (Grant No. 60551002) and the Natural Science Foundation of Hunan Province, China (Grant No. 2018JJ3680).

Based on adiabatic approximation theory, in this paper we study the asymmetric stochastic resonance system with time-delayed feedback driven by non-Gaussian colored noise. The analytical expressions of the mean first-passage time (MFPT) and output signal-to-noise ratio (SNR) are derived by using a path integral approach, unified colored-noise approximation (UCNA), and small delay approximation. The effects of time-delayed feedback and non-Gaussian colored noise on the output SNR are analyzed. Moreover, three types of asymmetric potential function characteristics are thoroughly discussed. And they are well-depth asymmetry (DASR), well-width asymmetry (WASR), and synchronous action of well-depth and well-width asymmetry (DWASR), respectively. The conclusion of this paper is that the time-delayed feedback can suppress SR, however, the non-Gaussian noise deviation parameter has the opposite effect. Moreover, the correlation time plays a significant role in improving SNR, and the SNR of asymmetric stochastic resonance is higher than that of symmetric stochastic resonance. Our experiments demonstrate that the appropriate parameters can make the asymmetric stochastic resonance perform better to detect weak signals than the symmetric stochastic resonance, in which no matter whether these signals have low frequency or high frequency, accompanied by strong or weak noise.

Stochastic Kinetics for a FitzHugh–Nagumo Neural System with Time Delay Driven by Non-Gaussian Noise and a Multiplicative Periodic Signal

In the present paper, the stability and the phenomena of stochastic resonance (SR) for a FitzHugh–Nagumo (FHN) system with time delay driven by a multiplicative non-Gaussian noise and an additive Gaussian white noise are investigated. By using the fast descent method, unified colored noise approximation and the two-state theory for the SR, the expressions for the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) are obtained. The research results show that the two noise intensities and time delay can always decrease the probability density at the two stable states and impair the stability of the neural system; while the noise correlation time [Formula: see text] can increase the probability density around both stable states and consolidate the stability of the neural system. Furthermore, the other noise correlation time [Formula: see text] can increase the probability at the resting state, but reduce that around the excited state. With respect to the SNR, it is discovered that the two noise strengths can both weaken the SR effect, while time delay [Formula: see text] and the departure parameter [Formula: see text] will always amplify the SR phenomenon. Moreover, the noise correlation time [Formula: see text] can motivate the SR effect, but not alter the peak value of the SNR. What’s most interesting is that the other noise correlation time [Formula: see text] can not only stimulate the SR phenomenon, but also results in the occurrence of two resonant peaks, whose heights are simultaneously improved because of the action of [Formula: see text].