sine-Wiener Noises Research Articles

Semiconductor laser systems excited by multiplicative Ornstein-Uhlenbeck noise and additive sine-Wiener noise in relation to real and imaginary parts.

In this paper we study a semiconductor laser system excited by multiplicative Ornstein-Uhlenbeck (OU) noise and additive sine-Wiener (SW) noise with a related real part and imaginary part. The two-dimensional Langevin equation(LE) with cross-correlating complex OU noise and complex SW noise is equivalent to the six-dimensional LE. Based on functional methods and spatial dimension extension methods, the six-dimensional Fokker-Planck equation(FPE) is achieved. Through Taylor expansion approximation and linear transformation, the dimensionality of FPE is reduced, and the exact expression of the probability distribution (SPD) in steady state is obtained. The impacts of noise parameters on laser intensity on SPD are further analyzed. Moreover, based on the reduced FPE, the information entropy of the system is obtained. This calculation is useful for us to understand further the influence of system parameters and noise correlation coefficient on the system. The main work lies in obtaining an alternative method for deriving a FPE corresponding to the two-dimensional stochastic semiconductor laser model, which is more applicable than methods in previous literature. An equivalent expression for cross-correlated OU noise and cross-correlated SW noise as well as a method for approximate solutions of a high-dimensional FPE are also provided.

Autonomous learning of nonlocal stochastic neuron dynamics.

Neuronal dynamics is driven by externally imposed or internally generated random excitations/noise, and is often described by systems of random or stochastic ordinary differential equations. Such systems admit a distribution of solutions, which is (partially) characterized by the single-time joint probability density function (PDF) of system states. It can be used to calculate such information-theoretic quantities as the mutual information between the stochastic stimulus and various internal states of the neuron (e.g., membrane potential), as well as various spiking statistics. When random excitations are modeled as Gaussian white noise, the joint PDF of neuron states satisfies exactly a Fokker-Planck equation. However, most biologically plausible noise sources are correlated (colored). In this case, the resulting PDF equations require a closure approximation. We propose two methods for closing such equations: a modified nonlocal large-eddy-diffusivity closure and a data-driven closure relying on sparse regression to learn relevant features. The closures are tested for the stochastic non-spiking leaky integrate-and-fire and FitzHugh-Nagumo (FHN) neurons driven by sine-Wiener noise. Mutual information and total correlation between the random stimulus and the internal states of the neuron are calculated for the FHN neuron.

Investigation on the properties of sine-Wiener noise and its induced escape in the particular limit case D → ∞

Sine-Wiener (SW) noise is increasingly adopted in realistic stochastic modeling for its bounded nature. However, many features of the SW noise are still unexplored. In this paper, firstly, the properties of the SW noise and its integral process are explored as the parameter D in the SW noise tends to be infinite. It is found that although the distribution of the SW noise is quite different from Gaussian white noise, the integral process of the SW noise shows many similarities with the Wiener process. Inspired by the Wiener process, which uses the diffusion coefficient to denote the intensity of the Gaussian noise, a quantity is put forward to characterize the SW noise’s intensity. Then we apply the SW noise to a one-dimensional double-well potential system and the Maier–Stein system to investigate the escape behaviors. A more interesting result is observed that the mean first exit time also follows the well-known Arrhenius law as in the case of the Gaussian noise, and the quasi-potential and the exit location distributions are very close to the results of the Gaussian noise.

An equivalent analytical method to deal with cross-correlated exponential type noises in the nonlinear dynamic system

An equivalent analytical method to deal with cross-correlated exponential type noises in the nonlinear stochastic dynamical system is presented, which are equivalent to a linear combination of multiplicative white noises. This equivalent analytical method solves a recent conundrum which arose in the study of cross-correlated sine Wiener noises and provides a simple and effective method to deal with the related stochastic dynamical problem of exponential type noises. Applying this method, we investigate the FitzHugh-Nagumo dynamic system with a periodic signal and driven by cross-correlated sine-wiener noises, make steady-state analysis, and point out stochastic resonance phenomena induced by cross-correlated sine-wiener noises. We show the effects of the noise parameters on the stationary probability density and the Signal-to-Noise Ratio of the stochastic dynamical system. The cross-correlation time τ and the cross-correlation strength λ enhance the stochastic resonance excited by the additive sine Wiener noise η(t), in contrast, and inhibit the stochastic resonance excited by the multiplicative sine Wiener noise ε(t).

Properties of bounded stochastic processes employed in biophysics

Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated within a solid mathematical ground. Four families are object of investigation: the Sine-Wiener (SW), the Doering–Cai–Lin (DCL), the Tsallis–Stariolo–Borland (TSB), and the Kessler–Sørensen (KS) families. We address mathematical questions on existence and uniqueness of the processes defined through Stochastic Differential Equations, which often conceal non-obvious behavior, and we explore the behavior of the solutions near the boundaries of the state space. The expression of the time-dependent probability density of the Sine-Wiener noise is provided in closed form, and a close connection with the Doering–Cai–Lin noise is shown. Further relationships among the different families are explored, pathwise and in distribution. Finally, we illustrate an analogy between the Kessler–Sørensen family and Bessel processes, which allows to relate the respective local times at the boundaries.

Sine-Wiener bounded noise-induced logical stochastic resonance in a two-well potential system

Abstract Some noisy nonlinear system can be used to realize reliable logic operation based on the mechanism of logical stochastic resonance (LSR). However, most previous studies focus mainly on Gaussian noise-driven system. In this paper, the effect of non-Gaussian sine-Wiener (SW) bounded noise on the reliability and agility of logic system is explored based on a SW noise-driven two-potential well system. The success probability P of obtaining desired reliable logic operation increases quickly, reaches a maximum of P = 1, and then decrease with the increase of noise intensity or self-correlation time of SW noise, showing the occurrence of LSR. Furthermore, with increasing self-correlation time of SW noise, the optimal window of noise intensity moves toward left and becomes narrower. For too long self-correlation time, SW noise cannot induce LSR. The optimal parameter regions of SW noise are strongly dependent on bias b of logic system. Therefore, adjusting bias b can realize the control of noise and let noise produce constructive effect. Additionally, the agility of reliable logic gate can be improved by properly increasing noise intensity. Taken together, the results presented here are beneficial to the design of new logic devices based on LSR.

Effects of sine-Wiener noise on signal propagation in a randomly connected neural network

We investigate the effects of sine-Wiener (SW)-noise on signal propagation in a randomly connected neural network based on Izhikevich neuron model in detail, in which the axonal conduction delays of synapses, the linkage probability between neurons and the ratio between excitatory and inhibitory neurons of the network are set similarly with the mammalian neocortex. It is found that the SW-noise can enhance the propagation of weak signal in the network. Besides the parameters of SW-noise, the characteristic parameters of the network also play important roles in signal propagation. Furthermore, it is found that the neural network has its sensitive frequency that can optimally enhance the propagation of weak signal when the signal’s frequency is close to the network’s sensitive frequency. In summary, the results here suggest that the SW-noise with suitable self-correlation time and intensity can facilitate the propagation of weak signal in the randomly connected neural network.

Switch process induced by the sine-Wiener noises in the gene transcriptional regulatory system

Switch process induced by the sine-Wiener noises in the gene transcriptional regulatory system were investigated by numerical simulations. Results show that (i) Only multiplicative noise which is the fluctuation of degradation reaction rate can induce a switch process (on → off) as the intensity of multiplicative noise increases for the case of uncorrelated noises; (ii) For the case of correlated noises, the switch process can be induced not only by intensity of multiplicative noise but also by intensity of additive noise which is the fluctuation of synthesis reaction rate and the cross-correlation intensity between the multiplicative and the additive noise can also induce the switch process; (iii) A successive switch process (on → off → on) occurs accompanied by a noise intensity increase when the correlation intensity takes larger value.

Delay-induced synchronization transition in a small-world neuronal network of FitzHugh–Nagumo neurons subjected to sine-Wiener bounded noise

Noise and delay are ubiquitous in brain and they have significant effects on neuronal network synchronization and even brain functions. Based on a small-world neuronal network of delayed FitzHugh–Nagumo (FHN) neurons subjected to sine-Wiener (SW) bounded noise, the effects of delay and SW noise on synchronization and synchronization transition are numerically investigated by calculating a synchronization measure R and plotting spatiotemporal patterns. The phenomenon of delay-induced synchronization transition is observed as delay [Formula: see text] is increased. And large self-correlation time and strength of SW noise can increase the number of delay-induced synchronization transition. In addition, delay-induced synchronization transition is robust against the change of topology structure of neuronal network and this phenomenon becomes much easier to see for small nearest neighbors k in the small-world network. Since synchronization transition may imply functional switch, our results may have important implications, and inspire future studies.

Bounded noise-induced coherence resonance in a single Rulkov neuron

Based on the chaotic Rulkov model, we investigate the impacts of bounded noise on firing activity of the neural system. First, the chaotic neural model and bounded noise are introduced and analyzed. Then, different firing patterns of the neural model driven by bounded noise are present. Finally, taking the coherence parameter R to measure the regularity of firing behavior, it is demonstrated that the regularity of neural membrane potential could achieve optimal by adjusting either the intensity of bounded noise or its frequency to an appropriate value, which means that the phenomenon of coherence resonance occurs. Moreover, it is found that coherence resonance can also be induced by another kind of bounded noise, called sine-Wiener noise. The results of this paper are beneficial to explore the functional role of noise on neural system.

Enhancement of temporal regularity and degradation of spatial synchronization induced by cross-correlated sine-Wiener noises in regular and small-world neuronal networks

Most previous studies assumed noise to be unbounded Gaussian. Here the effects of non-Gaussian cross-correlated sine-Wiener (CCSW) bounded noises on temporal regularity and spatial synchronization of coupled neurons are investigated in regular and small-world networks of the FitzHugh–Nagumo (FHN) neurons. For regular neuronal networks, as cross-correlation time of CCSW noises is increased, CCSW noises-induced temporal coherent resonance occurs at appropriately cross-correlation time of CCSW noises, and spatial synchronization in neuronal network is always impaired. In addition, the increasing degree of correlation of CCSW noises may enhance or impair temporal regularity, which is dependent on cross-correlation time. On the other hand, the increasing degree of correlation of CCSW noises destroys spatial synchronization or has no effect on spatial synchronization. For small world neuronal network, CCSW noises-induced enhancement of temporal regularity and degradation of spatial synchronization are still observed. Furthermore, the increasing rewiring probability make spatiotemporal pattern more order in time and in space. If delay is introduced into small world network, the aforementioned CCSW noises-induced phenomena still exist. Meanwhile, delay let spatiotemporal pattern more disorder.

Impact of cross-correlated sine-Wiener noises in the gene transcriptional regulatory system.

We studied fluctuation-induced switching processes in the gene transcriptional regulatory system under cross-correlated sine-Wiener (CCSW) noises. It is numerically demonstrated that the increase of the multiplicative noise intensity A and cross-correlation time τ in CCSW noises can reduce the concentration of the TF-A monomer and switch to an "off" state. In addition, when the cross-correlation time τ is small, the increase of the additive noise intensity B leads to a switch of the process from "off"→"on". Simultaneously, the increase of the cross-correlation intensity λ of CCSW noises contributes to maintaining the current state. When the cross-correlation time is large, the high concentration state has two peaks and the stationary probability distribution presents a three-peak structure.

Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises

The sine-Wiener noise, as one new type of bounded noise and a natural tool to model fluctuations in dynamical systems, has been applied to problems in a variety of areas, especially in biomolecular networks and neural models. In this paper, by virtue of the Novikov theorem, Fox’s approach, and the ansatz of Hanggi, an approximate Fokker–Planck equation is derived for an one-dimensional Langevin-type equation with cross-correlated sine-Wiener noise. Meanwhile, the dynamical characters of a bistable system driven by cross-correlated sine-Wiener noise are investigated by applying the approximate theoretical method. For the bistable system, the cross-correlation intensity $$\lambda $$ can induce the reentrance-like phase transition, while the other noise intensities and the self-correlation time, except for the self-correlation time of additive bounded noise, can induce the first-order-like phase transition. The transition from the stable state to another one can be accelerated by $$\alpha $$ (additive bounded noise intensity), $$\tau _1$$ (the self-correlation time of the multiplicative bounded noise), and $$\tau _2$$ (the self-correlation time of the additive bounded noise) and can be restrained with $$\lambda $$ and $$\tau _3$$ (self-correlation time of the cross-correlation bounded noise). It is interesting that the noise-enhanced stability phenomenon is observed with D (multiplicative bounded noise intensity) varying for the positive correlation ($$\lambda >0$$) and is enhanced as $$\lambda $$ increases. The numerical results are in basic agreement with the theoretical predictions.

Weak periodic signal detection by sine-Wiener-noise-induced resonance in the FitzHugh-Nagumo neuron.

Based on the FitzHugh-Nagumo (FHN) neuron model subjected to sine-Wiener (SW) noise, impacts of SW noise on weak periodic signal detection are investigated by calculating response measure Q for characterizing synchronization between the input signal and the output temporal activities of the neuron. It is numerically demonstrated that the response measure Q can achieve the optimal value under appropriate and moderate intensity or correlation time of SW noise, suggesting the occurrence of SW-noise-induced stochastic resonance. Furthermore, the optimal value of Q is sensitive to correlation time. Consequently, the correlation time of SW noise has a great influence on the performance of signal detection in the FHN neuron.

Breakup of Spiral Wave and Order-Disorder Spatial Pattern Transition Induced by Spatially Uniform Cross-Correlated Sine-Wiener Noises in a Regular Network of Hodgkin-Huxley Neurons

Spatially uniform cross-correlated sine-Wiener (CCSW) noises are imposed on each Hodgkin-Huxley (HH) neuron in a two-dimensional (2D) regular network. Noise-induced spiral wave destruction and order-disorder spatial pattern transition can be observed by adjusting cross-correlation time or cross-correlation intensity of CCSW noises. The sudden change of the curve of synchronization factor R and the time series of average membrane voltage F can be used to semiquantitatively assess this spatial pattern transition and spiral wave destruction induced by CCSW noises, respectively. In addition, moderate cross-correlation time and strong cross-correlation intensity of CCSW noises are both detrimental to survival of spiral wave. Comparing with non-cross-correlated sine-Wiener noises, CCSW noises can destroy organized spiral wave with lesser noise amplitude.

Detection of sub-threshold periodic signal by multiplicative and additive cross-correlated sine-Wiener noises in the FitzHugh–Nagumo neuron

We study the effects of multiplicative and additive cross-correlated sine-Wiener (CCSW) noises on the performance of sub-threshold periodic signal detection in the FitzHugh–Nagumo (FHN) neuron by calculating Fourier coefficients Q for measuring synchronization between sub-threshold input signal and the response of system. CCSW noises-induced transitions of electrical activity in the FHN neuron model can be observed. Moreover, the performance of sub-threshold periodic signal detection is achieved at moderate noise strength, cross-correlation time and cross-correlation strength of CCSW noises, which indicate the occurrence of CCSW noises-induced stochastic resonance. Furthermore, the performance of sub-threshold signal detection is strongly sensitive to cross-correlation time of CCSW noises. Therefore, the performance can be effectively controlled by regulating cross-correlation time of CCSW noises. These results provide a possible mechanism for amplifying or detecting the sub-threshold signal in the nervous system.

Coherence resonance induced by cross-correlated sine-Wiener noises in the FitzHugh–Nagumo neurons

We study coherence resonance (CR) in the FitzHugh–Nagumo (FHN) neurons under cross-correlated sine-Wiener (CCSW) noises. It is numerically demonstrated that the reciprocal coefficient of variance of inter-spike intervals (R) increases with increasing amplitude or correlation time of CCSW noises, reaches the maximum at proper amplitude or correlation time, and then decreases, suggesting the appearance of CR phenomenon. In addition, the occurrence of CR is sensitive to a parameter range of amplitudes and correlation times of CCSW noises. Thus, CR can be controlled by regulating the amplitudes and correlation times of CCSW noises in the FHN excitable systems.

Phase transitions induced by time-delay and different noises

The effects of global and local time-delay on a genotype selection model driven by colored or sine-Wiener noise are investigated. The stationary probability distributions are derived by simulating the Langevin Equation directly. It is found that compared to the colored noise, the bounded noise can enhance transitions, though they have the same mean and stationary correlation. Local time-delay and the global one have opposite role in the system with different noises. When time-delay is global, neither the colored noise nor the sine-Wiener noise can induce phase transitions.

The effect of sine-Wiener noises on transition in a genotype selection model with time delays

A genotype selection system interplay with sine-Wiener noises and time delays is investigated. Stationary probability distribution function is obtained by numerical simulations. Results show that the multiplicative bounded noise can facilitate the gene separation, while the additive bounded noise suppresses the gene separation. Besides, local time delays α and β, being in gene transformation and gene heredity progress respectively, play opposite roles in the gene selection process. What is more interesting is that there is no transition during the process of gene select when time delays α = β (i.e., the system is subjected to global time delay).

Spatio-temporal sine-Wiener bounded noise and its effect on Ginzburg–Landau model

In this work we introduce a spatio-temporal bounded noise derived by the sine-Wiener noise and by the spatially colored unbounded noise proposed by Garcia-Ojalvo, Sancho, and Ramirez-Piscina (GSR noise). We characterize the behavior of the equilibrium distribution of this novel noise by showing its dependence on both the temporal and the spatial autocorrelation lengths. In particular, we show that the distribution experiences a stochastic transition from bimodality to trimodality. Then, we employ the noise here defined to study the emergence of phase transitions in the real Ginzburg–Landau model. Various phenomena are evidenced by means of numerical simulations, among which reentrant transitions, as well as differences in the response of the system to “equivalent” GSR additive noise perturbations.