Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

In contrast to the previous studies which have dealt with stochastic resonance induced by random transitions of system motion between two coexisting limit cycle attractors in the FitzHugh–Nagumo (FHN) neuron model after Hopf bifurcation and which have dealt with the phenomenon of stochastic resonance induced by external noise when the model with periodic input has only one attractor before Hopf bifurcation, in this paper we have focused our attention on stochastic resonance (SR) induced by a novel transition behavior, the transitions of motion of the model among one attractor on the left side of bifurcation point and two attractors on the right side of bifurcation point under the perturbation of noise. The results of research show: since one bifurcation of transition from one to two limit cycle attractors and the other bifurcation of transition from two to one limit cycle attractors occur in turn besides Hopf bifurcation, the novel transitions of motion of the model occur when bifurcation parameter is perturbed by weak internal noise; the bifurcation point of the model may stochastically slightly shift to the left or right when FHN neuron model is perturbed by external Gaussian distributed white noise, and then the novel transitions of system motion also occur under the perturbation of external noise; the novel transitions could induce SR alone, and when the novel transitions of motion of the model and the traditional transitions between two coexisting limit cycle attractors after bifurcation occur in the same process the SR also may occur with complicated behaviors types; the mechanism of SR induced by external noise when FHN neuron model with periodic input has only one attractor before Hopf bifurcation is related to this kind of novel transition mentioned above.     

Multifractal detrended fluctuation analysis of analog random multiplicative processes

We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function.     

Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises

In this paper, we first present some conditions for the upper semicontinuity of perturbed random attractors to a limiting random attractor. Then we apply this result to establish the upper semicontinuity of random attractors for the first order stochastic lattice differential equations with random coupled coefficients and multiplicative/additive white noises in the weighted space of infinite sequences as the coefficient of the white noise term tends to zero.     

Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh–Nagumo neural model

We study the dynamics of neurons via a bistable modified stochastic FitzHugh–Nagumo model having two stable fixed points separated by one unstable fixed point. Due to the ability of a neuron to detect and enhance weak information transmission, we show numerically that starting from the resting potential, we get firing activities (spiking) when operating slightly beyond the supercritical Hopf bifurcation. For real biological systems which are sometimes embedded in the complex environment, we observe that a gradual increase or decrease noise intensities did not result in a gradual change of the membrane potential distribution thanks to noise induced transition phenomena. We shown analytically that for zero correlation between two sine Wiener noises, additive noise has no effect on the transition between monostable and bistable phase on the neural model. We adapted a general expression of the signal-to-noise ratio for a general two-state theory extended in the asymmetric case and non-Gaussian noises in our model to study the influence of noise strength in stochastic resonance. Our investigation revealed that in the evolution of excitable system, neurons may use noises to their advantage by enhancing their sensitivity near a preferred phase to detect external stimuli or affect the efficiency and rate of information processing.     

加性二值噪声激励下Duffing系统的随机分岔

研究了Duffing系统在加性二值噪声作用下的随机分岔现象.首先,根据二值噪声的统计特性,推导得到二值噪声状态间的跃迁概率,据此对二值噪声进行了数值模拟.其次,利用四阶Runge-Kutta(龙格-库塔)数值算法得到该系统位移和速率的稳态联合概率密度及位移的稳态概率密度.然后,通过对位移稳态概率密度单双峰结构变化的研究,发现加性二值噪声的状态和强度能够诱导系统产生随机分岔现象.最后,观察到随着系统非对称参数的逐渐变化,系统同样产生了随机分岔现象.     

Existence of multiple limit cycles in Chen system

In this paper, the existence of multiple limit cycles for Chen system are investigated. By using the method of computing the singular point quantities, the simple and explicit parametric conditions can be determined to the number and stability of multiple limit cycles from Hopf bifurcation. Especially, at least 4 limit cycles can be obtained for the Chen system as a three-dimensional perturbed system.     

Propagation and enhancement of the noise-induced signal in a coupled cell system

The response of an array of unidirectional coupled cells to local external noise is investigated. The cells are all tuned near the Hopf bifurcation point for calcium oscillation. It is found that when the first cell of the chain is perturbed by noise, stochastic calcium oscillation can be induced and propagates along the chain with considerable enhancement and a rather regular signal output is obtained at the end of the chain. It indicates that noise can result in the internal order, which is likely to show the rhythm of life in life systems. It is demonstrated that the occurrence of such a phenomenon depends on the coexistence of three factors: being close to the Hopf bifurcation point, unidirectional coupling and optimal coupling strength. It is important to note that this phenomenon is quite different from the noise enhanced signal propagation, where a larger coupling strength is always more favorable. Our result may find important applications in calcium signaling processes in vivo.     

噪声激励下水平扫视眼动系统的随机分岔

研究了眼动系统在神经噪声作用下的随机分岔现象.首先,基于水平眼动系统模型,用加性的Gauss(高斯)白噪声模拟神经系统中的噪声,建立眼动系统的随机动力学模型.其次,利用数值算法得到眼球运动位移的Poincaré分岔图和系统在不同参数下的位移和速度的稳态联合概率密度以及位移的稳态概率密度.研究发现:噪声强度和抑制性神经元的作用强度都能诱导产生随机P分岔现象,使得位移的稳态概率密度出现峰的个数从1到3的转换,间歇性眼球震颤产生.此外,还发现当抑制性神经元的作用强度增大到一定值时,稳态概率密度始终呈现单峰结构.该结论对此类疾病的治疗有一定的指导作用.     

白噪声参激Hopf分叉系统的两次分叉研究

本文研究了白噪声参数激励下的Hopf分叉系统的两次分叉行为．明确了由于噪声的介入而使得系统的分叉类型产生了实质性的改变并导致了分叉点的漂移．     

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.     

Impact of two types of time delays and cross-correlated multiplicative and additive noises on stability and stochastic resonance for a metapopulation system

In this paper, we investigate the stability and the shift between the extinction state and the stable one of a large density and the stochastic resonance (SR) for a metapopulation system subjected to two types of time delay terms, cross-correlation noises and multiplicative signal. By using the fast descent method and the method of small delay approximation, the expressions of the effective potential function and the signal-to-noise ratio (SNR) are obtained. We denote by Q the intensity of the multiplicative noise, and M the intensity of the additive noise, θ and τ the two time delay terms introduced into the metapopulation system. Our main results show some facts that time delay θ and the strength of correlation noise λ can restrain the development of the metapopulation, while the other term of time delay τ can accelerate the expansion of the population from the extinction state to the large stable one. We discover that it is possible to enhance the signal-to-noise ratio by adjusting the intensities of the multiplicative, additive noises and the time delays of the stochastic metapopulation system     

Stochastic stability and bifurcation in a macroeconomic model

On the basis of the work of Goodwin and Puu, a new business cycle model subject to a stochastically parametric excitation is derived in this paper. At first, we reduce the model to a one-dimensional diffusion process by applying the stochastic averaging method of quasi-nonintegrable Hamiltonian system. Secondly, we utilize the methods of Lyapunov exponent and boundary classification associated with diffusion process respectively to analyze the stochastic stability of the trivial solution of system. The numerical results obtained illustrate that the trivial solution of system must be globally stable if it is locally stable in the state space. Thirdly, we explore the stochastic Hopf bifurcation of the business cycle model according to the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simply way on the potential applications of stochastic stability and bifurcation analysis.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

First-passage failure of linear oscillator with non-classical inelastic impact

The first-passage failure of linear oscillator with inelastic impact subjected to the additive and multiplicative random noises is investigated. The impact is described by the non-classical inelastic impact model, which is essentially different from the traditional impact model and can provide the whole information of the impact process. First of all, the impact force in the motion equation is replaced by the quasi-linear damping and nonlinear stiffness terms. Then, the stochastic averaging is adopted and the averaged Itô stochastic deferential equation of the total system energy is derived. Last, by solving the established backward Kolmogorov equation and Pontryagin equation from the averaged Itô equation numerically, the conditional reliability, the conditional probability density function (PDF) and the mean time of first-passage failure can be obtained. The comparison between the analytical results and those from Monte-Carlo simulation reveals the proposed procedure is effective. The influences of some system parameters are discussed in detail.     

On the Relationship Between Solutions of Stochastic and Random Differential Inclusions

Some results on the relationship of the solutions of a stochastic differential inclusion and the corresponding random differential inclusion obtained after a change of variable are proved. As a consequence, we obtain the pullback convergence of the solutions of the stochastic inclusion to a compact random set. The cases of a reaction–diffusion inclusion perturbed by additive and multiplicative noises are considered.     

Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics

In this paper, we consider the growth of densities of two kinds of typical HAB algae: diatom and dianoflagellate on some coasts of China’s mainland. Since there exist many random factors that cause the change of the algae densities, we shall develop a new nonlinear dynamical model with stochastic excitations on the algae densities. Applying a stochastic averaging method on the model, we obtain a two-dimensional diffusion process of averaged amplitude and phase. Then we investigate the stability and the Hopf bifurcation of the stochastic system with FPK (Fokker Planck–Kolmogorov) theory and obtain the stationary transition probability density of the process. We obtain the critical values of parameters for the occurrences of Hopf bifurcation in terms of probability. We also investigate numerically the effects of various parameters on the stationary transition probability density of the occurrences of Hopf bifurcation. The numerical results are in good correlation with the analysis. We draw the conclusion that if the Hopf bifurcation occurs with a radius large enough, i.e., if the densities of the HAB algae reach a high value, the HAB will take place with comparatively high probability.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Stochastic stability and state shifts for a time-delayed cancer growth system subjected to correlated multiplicative and additive noises

In the present paper, we investigate the stationary probability distribution(SPD) and the mean treatment time of a time-delayed cancer growth system induced by cross-correlated intrinsic and extrinsic noises. Our main results show that the resonant-like phenomenon of the mean first-passage time (MFPT) appears in the tumor cell growth model due to the interaction of all kinds of noises and time delay. Due to the existence of the resonant-like peak value, by increasing the intensity of multiplicative noise and time delay, it is possible to restrain effectively the development of the cancer cells and enhance the stability of the system. During the process of controlling the diffusion of the tumor cells, it contributes to inhibiting the development of cancer by increasing the cross-correlated noise strength and weakening the additive noise intensity and time delay. Meanwhile, the proper multiplicative noise intensity is conducive to the process of inhibition. Conversely, in the process of exterminating cancer cells of a large density, it can exert positive effects on eliminating the tumor cells by increasing noises intensities and the value of time delay.     

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.     

Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer–van der Pol system via Chebyshev polynomial approximation

In this paper, we analyzed stochastic chaos and Hopf bifurcation of stochastic Bonhoeffer–van der Pol (SBVP for short) system with bounded random parameter of an arch-like probability density function. The modifier ‘stochastic’ here implies dependent on some random parameter. In order to study the dynamical behavior of the SBVP system, Chebyshev polynomial approximation is applied to transform the SBVP system into its equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. Thus, we can further explore the nonlinear phenomena in SBVP system. Stochastic chaos and Hopf bifurcation analyzed here are by and large similar to those in the deterministic mean-parameter Bonhoeffer–van der Pol system (DM–BVP for short) but there are also some featuring differences between them shown by numerical results. For example, in the SBVP system the parameter interval matching chaotic responses diffuses into a wider one, which further grows wider with increasing of intensity of the random variable. The shapes of limit cycles in the SBVP system are some different from that in the DM–BVP system, and the sizes of limit cycles become smaller with the increasing of intensity of the random variable. And some biological explanations are given.