Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

Noise-induced critical breakdown of phase lockings in a forced van der Pol oscillator

The effect of noise on the period of the van der Pol oscillator forced by a sinusoidal signal is analyzed in detail. The dynamics of the forced nonlinear oscillator with noise is described by the evolution of probability densities by a Markov operator. The oscillator shows a 1 : 1 phase-locked oscillation whose period is equal to the period of the sinusoidal forcing when the forcing is sufficiently strong. Noise modulates the oscillator's period as expected, but we show that this modulation is not uniform; the period of the oscillator changes drastically at a certain strength of noise. Thus this noise-induced breakdown of the phase locking can be considered as a stochastic bifurcation. The breakdown of stochastic phase-locking or the stochastic bifurcation is also analyzed from the viewpoint of the spectra of the Markov operator.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Duffing系统随机相位抑制混沌与随机共振并存现象的机理研究

针对随机相位作用的Duffing混沌系统, 研究了随机相位强度变化时系统混沌动力学的演化行为及伴随的随机共振现象. 结合Lyapunov指数、庞加莱截面、相图、时间历程图、功率谱等工具, 发现当噪声强度增大时, 系统存在从混沌状态转化为有序状态的过程, 即存在噪声抑制混沌的现象, 且在这一过程中, 系统亦存在随机共振现象, 而且随机共振曲线上最优的噪声强度恰为噪声抑制混沌的参数临界点. 通过含随机相位周期力的平均效应分析并结合系统的分岔图, 探讨了噪声对混沌运动演化的作用机理, 解释了在此过程中随机共振的形成机理, 论证了噪声抑制混沌与随机共振的相互关系.     

Random dynamics of the Morris-Lecar neural model

Determining the response characteristics of neurons to fluctuating noise-like inputs similar to realistic stimuli is essential for understanding neuronal coding. This study addresses this issue by providing a random dynamical system analysis of the Morris-Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris-Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. In both the transitions from excitable to oscillating regimes are associated with different bifurcation scenarios. This work examines how random perturbations affect these two bifurcation scenarios. It is first numerically shown that the Morris-Lecar model driven by white Gaussian noise current tends to have a unique stationary distribution in the phase space. Numerical evaluations also reveal quantitative and qualitative changes in this distribution in the vicinity of the bifurcations of the deterministic system. However, these changes notwithstanding, our numerical simulations show that the Lyapunov exponents of the system remain negative in these parameter regions, indicating that no dynamical stochastic bifurcations take place. Moreover, our numerical simulations confirm that, regardless of the asymptotic dynamics of the deterministic system, the random Morris-Lecar model stabilizes at a unique stationary stochastic process. In terms of random dynamical system theory, our analysis shows that additive noise destroys the above-mentioned bifurcation sequences that characterize class I and class II regimes in the Morris-Lecar model. The interpretation of this result in terms of neuronal coding is that, despite the differences in the deterministic dynamics of class I and class II membranes, their responses to noise-like stimuli present a reliable feature.     

Stochastic phase resetting of stimulus-locked responses of two coupled oscillators: transient response clustering,synchronization, and desynchronization

Transient phase dynamics, synchronization, and desynchronization which are stimulus-locked (i.e., tightly time-locked to a repetitively administered stimulus) are studied in two coupled phase oscillators in the presence of noise. The presented method makes it possible to detect such processes in numerical and experimental signals. The time resolution is enormous, since it is only restricted by the sampling rate. Stochastic stimulus locking of the phases or the n:m phase difference at a particular time t relative to stimulus onset is defined by the presence of one or more prominent peaks in the cross-trial distribution of the phases or the n:m phase difference at time t relative to stimulus onset in an ensemble of poststimulus responses. The oscillators' coupling may cause a transient cross-trial response clustering of the poststimulus responses. In particular, the mechanism by which intrinsic noise induces symmetric antiphase cross-trial response clustering in coupled detuned oscillators is a stochastic resonance. Unlike the presented approach, both cross-trial averaging (where an ensemble of poststimulus responses is simply averaged) and cross-trial cross correlation (CTCC) lead to severe misinterpretations: Triggered averaging cannot distinguish a cross-trial response clustering or decorrelation from a mean amplitude decrease of the single responses. CTCC not only depends on the oscillators' phase difference but also on their phases and, thus, inevitably displays "artificial" oscillations that are not related to synchronization or desynchronization.     

有界随机噪声激励下软弹簧Duffing振子的安全盆分叉

研究了软弹簧Duffing振子在有界随机噪声激励下，系统安全盆的侵蚀现象，并提出了随机安全盆分叉的概念. 计算表明，由于随机扰动的影响，系统的随机安全盆分叉点发生了偏移. 关键词： Duffing振子 有界随机噪声 安全盆 分叉     

Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.     

Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron

Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure.     

Intrinsic noise and division cycle effects on an abstract biological oscillator

Oscillatory dynamics are common in biological pathways, emerging from the coupling of positive and negative feedback loops. Due to the small numbers of molecules typically contained in cellular volumes, stochastic effects may play an important role in system behavior. Thus, for moderate noise strengths, stochasticity has been shown to enhance signal-to-noise ratios or even induce oscillations in a class of phenomena referred to as "stochastic resonance" and "coherence resonance," respectively. Furthermore, the biological oscillators are subject to influences from the division cycle of the cell. In this paper we consider a biologically relevant oscillator and investigate the effect of intrinsic noise as well as division cycle which encompasses the processes of growth, DNA duplication, and cell division. We first construct a minimal reaction network which can oscillate in the presence of large or negligible timescale separation. We then derive corresponding deterministic and stochastic models and compare their dynamical behaviors with respect to (i) the extent of the parameter space where each model can exhibit oscillatory behavior and (ii) the oscillation characteristics, namely, the amplitude and the period. We further incorporate division cycle effects on both models and investigate the effect of growth rate on system behavior. Our results show that in the presence but not in the absence of large timescale separation, coherence resonance effects result in extending the oscillatory region and lowering the period for the stochastic model. When the division cycle is taken into account, the oscillatory region of the deterministic model is shown to extend or shrink for moderate or high growth rates, respectively. Further, under the influence of the division cycle, the stochastic model can oscillate for parameter sets for which the deterministic model does not. The division cycle is also found to be able to resonate with the oscillator, thereby enhancing oscillation robustness. The results of this study can give valuable insight into the complex interplay between oscillatory intracellular dynamics and various noise sources, stemming from gene expression, cell growth, and division.     

Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given.     

基于Laguerre多项式逼近法的随机双势阱Duffing系统的分岔和混沌研究

应用Laguerre正交多项式逼近法研究了含有随机参数的双势阱Duffing系统的分岔和混沌行为.系统参数为指数分布随机变量的非线性动力系统首先被转化为等价的确定性扩阶系统，然后通过数值方法求得其响应.数值模拟结果的比较表明，含有随机参数的双势阱Duffing系统保持着与确定性系统相类似的倍周期分岔和混沌行为，但是由于随机因素的影响，在局部小区域内随机参数系统的动力学行为会发生突变. 关键词： 双势阱Duffing系统 指数分布概率密度函数 Laguerre多项式逼近 随机分岔     

Self-induced stochastic resonance in excitable systems

The effect of small-amplitude noise on excitable systems with strong time-scale separation is analyzed. It is found that vanishingly small random perturbations of the fast excitatory variable may result in the onset of a deterministic limit cycle behavior, absent without noise. The mechanism, termed self-induced stochastic resonance, combines a stochastic resonance-type phenomenon with an intrinsic mechanism of reset, and no periodic drive of the system is required. Self-induced stochastic resonance is different from other types of noise-induced coherent behaviors in that it arises away from bifurcation thresholds, in a parameter regime where the zero-noise (deterministic) dynamics does not display a limit cycle nor even its precursor. The period of the limit cycle created by the noise has a non-trivial dependence on the noise amplitude and the time-scale ratio between fast excitatory variables and slow recovery variables. It is argued that self-induced stochastic resonance may offer one possible scenario of how noise can robustly control the function of biological systems.     

Chimera states for coupled oscillators

Arrays of identical oscillators can display a remarkable spatiotemporal pattern in which phase-locked oscillators coexist with drifting ones. Discovered two years ago, such "chimera states" are believed to be impossible for locally or globally coupled systems; they are peculiar to the intermediate case of nonlocal coupling. Here we present an exact solution for this state, for a ring of phase oscillators coupled by a cosine kernel. We show that the stable chimera state bifurcates from a spatially modulated drift state, and dies in a saddle-node bifurcation with an unstable chimera state.     

脉冲激励下环形耦合Duffing振子间的瞬态同步突变现象

研究非周期信号激励下Duffing振子动力学行为变化特征时,发现处于倍周期分岔的环形耦合Duffing振子系统,在一定的参数条件下,脉冲信号能引起其中一个振子与其他振子运动轨迹间出现短暂失同步的现象即瞬态同步突变现象.利用这种现象可以快速检测出强噪声背景中的微弱脉冲信号,从而扩展了现有的Duffing振子对非周期信号的检测范围及应用领域. 关键词： 瞬态同步突变 微弱信号检测 脉冲信号 Duffing振子     

谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌

研究了软弹簧Duffing振子在确定性谐和外力和有界随机噪声联合作用下，系统安全盆的侵蚀和混沌现象.推导出系统的随机Melnikov过程，根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值，然后用数值模拟方法计算了系统的安全盆分叉点.结果表明，由于随机扰动的影响，系统的安全盆分叉点发生了偏移，并且使得混沌容易发生. 关键词： Duffing振子 安全盆 分叉 混沌     

Peculiarities of the transitions to synchronization in coupled systems with amplitude death

The paper presents the results of the study of the sequences of bifurcation leading to the synchronization and amplitude death in a system of two dissipatively coupled self-sustained oscillators with inertial nonlinearity. Two types of synchronizations tongues have been identified. In one of them phase locking regions exist where the synchronization is achieved by the saddle-node bifurcation and regions where the transition to synchronization leads through Neimark-Sacker bifurcation. In the second type of the tongues there are only phase locking regions. It has been shown that for a weak non-identity of the system parameters, the first type tongues merge together. The transition between the synchronization tongues can occur without bifurcations, i.e., transition between the synchronized regimes with different periods of oscillations can occur gradually.     

A semi-analytic method for computing the long-time order parameter dynamics in mean-field coupled overdamped oscillators with colored noises

The order parameter dynamics of a mean-field model is frequently investigated in macroscopic cumulant dynamics, from which a bifurcation can be predicted qualitatively. In this Letter, for quantitatively investigating the long-time order parameter dynamics, a semi-analytic method is proposed based on approximate nonlinear Fokker-Planck equations. Applying the new method to the mean-field model of periodically driven overdamped bistable oscillators with colored noise, we exhibit the bifurcation behavior and the nonlinear stochastic resonance of the order parameter by tuning noise intensity or coupling coefficient, and the accuracy of the new method are verified by direct simulation. Our observations disclose some new properties about the order parameter dynamics of the mean-field model. For example, the periodic signal shifts the critical coupling coefficient to a larger value, while the nonzero correlation time of the colored noise shifts it to a lower value. Our observation also discloses that there is no quantitatively corresponding relation between the resonant peak and the critical bifurcation parameter of the Gaussian moment system.     

Effects of demographic noise on the synchronization of a metapopulation in a fluctuating environment

We use the theory of noise-induced phase synchronization to analyze the effects of demographic noise on the synchronization of a metapopulation of predator-prey systems within a fluctuating environment (Moran effect). Treating each local predator-prey population as a stochastic urn model, we derive a Langevin equation for the stochastic dynamics of the metapopulation. Assuming each local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive the steady-state probability density for pairwise phase differences between oscillators, which is then used to determine the degree of synchronization of the metapopulation.