Collective Dynamics of Two-Mode Stochastic Oscillators

Physica A: Statistical Mechanics and its Applications, 2003

We model the collective clapping of spectators by globally coupled two-mode stochastic oscillators. All distinct experimentally observable clapping modes are successfully reproduced. Surprisingly, it is found that in an extended region of the parameter space the periodicity of the collective output is strongly enhanced by the considered coupling. The model o ers a realistic way to generate periodic dynamics by coupling largely stochastic units.

2008

It is known that an ensemble of two-mode pulse-emitting stochastic oscillators globally coupled through their collective output will synchronize for some controllable parameter values. These oscillators proved to be appropriate for modeling the dynamics of rhythmic applause or several other synchronization phenomena in biological systems. Within this work we generalize this two-mode oscillator model to the case of several modes and investigate the effect on their synchronization. Computer simulation proves that synchronization appears for several modes as well and the system undergoes a phase transition-like phenomena while changing a controllable parameter. However, if the number of modes is increased synchronization will be present for smaller and smaller parameter range.

2006

We examine microscopic mechanisms for coupling stochastic oscillators so that they display similar and correlated temporal variations. Unlike oscillatory motion in deterministic dynamical systems, complete synchronization of stochastic oscillators does not occur, but appropriately defined oscillator phase variables coincide. This is illustrated in model chemical systems and genetic networks that produce oscillations in the dynamical variables, and we show that suitable coupling of different networks can result in their phase synchronization.

Commun Nonlinear Sci Numer Simulat, 2019

We investigated the phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three parameters that define: a) the beginning of activation of an element α, b) the duration of the activation δ and c) the strength of the coupling g. We observed a rich diversity of different types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed the canard in-phase solutions and the chaotic anti-phase activity, which corresponds to a strange attractor that appears via the cascade of period doubling bifurcations of limit cycles. In addition, we investigated an interesting phenomenon when two different chaotic attractive regimes corresponding for two different types of chaotic anti-phase activity merge in a single strange attractor. As a result, a new type of chaotic anti-phase regime appears by explosion from the collision of these two strange attractors. We also provided the detailed study of bifurcations which lead to the transitions between all these regimes. We detected on the (α, δ) parameter plane regions that correspond to the above-mentioned regimes. We also showed numerically the existence of bistability regions where various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specified regions corresponding to coexisting various types of sequential activity.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021

The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed design of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitude death states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude desynchronized domain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to death state are characterized using an average temporal interaction approximation, which agrees with the numerical results in temporal interaction. A first order phase transition behavior may change into a second order transition in spatial dynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possible abrupt first order like transition is completely non-existent in the case of temporal dynamic interaction. Besides the study on periodic Stuart-Landau systems, we present results for paradigmatic chaotic model of Rössler oscillators and Mac-arthur ecological model. Population biology of ecological networks, person to person communication networks, brain functional networks, possibility of outbreaks and spreading of disease through human contact networks, to name but a few examples which attest to the importance of researches based on temporal interaction approach. Studies based on representating several complex systems as time-varying networks of dynamical units have been shown to be extremely beneficial in understanding real life processes. Surprisingly, in all the previous studies on time-varying interaction, death state receives little attention in a network of coupled oscillators. In addition, only a few studies on dynamic interaction have considered the proximity of the individual systems' trajectories in the context of their interaction. In this paper, we propose a simple yet effective dynamic interaction scheme among nonlinear oscillators, which is capable of relaxing the collective oscillatory dynamics towards the dynamical equilibrium under appropriate choices of parameters. The dynamics of coupled oscillators can show fascinating complex behaviors including various dynamical phenomena. A qualitative explanation of the numerical observation is validated through linear stability analysis and interestingly, a linear stability analysis is persued even when the system is time-dependent. An elaborate study is contemplated to reveal the influences of our proposed dynamic interaction in terms of all the network parameters.

Physical review research, 2021

We derive a unified amplitude-phase decomposition for both noisy limit cycles and quasicycles; in the latter case, the oscillatory motion has no deterministic counterpart. We extend a previous amplitude-phase decomposition approach using the stochastic averaging method (SAM) for quasicycles by taking into account nonlinear terms up to order 3. We further take into account the case of coupled networks where each isolated network can be in a quasi-or noisy limit-cycle regime. The method is illustrated on two models which exhibit a deterministic supercritical Hopf bifurcation: the Stochastic Wilson-Cowan model of neural rhythms, and the Stochastic Stuart-Landau model in physics. At the level of a single oscillatory module, the amplitude process of each of these models decouples from the phase process to the lowest order, allowing a Fokker-Planck estimate of the amplitude probability density. The peak of this density captures well the transition between the two regimes. The model describes accurately the effect of Gaussian white noise as well as of correlated noise. Bursting epochs in the limit-cycle regime are in fact favored by noise with shorter correlation time or stronger intensity. Quasicycle and noisy limit-cycle dynamics are associated with, respectively, Rayleigh-type and Gaussian-like amplitude densities. This provides an additional tool to distinguish quasicycle from limit-cycle origins of bursty rhythms. The case of multiple oscillatory modules with excitatory all-to-all delayed coupling results in a system of stochastic coupled amplitude-phase equations that keeps all the biophysical parameters of the initial networks and again works across the Hopf bifurcation. The theory is illustrated for small heterogeneous networks of oscillatory modules. Numerical simulations of the amplitude-phase dynamics obtained through the SAM are in good agreement with those of the original oscillatory networks. In the deterministic and nearly identical oscillators limits, the stochastic Stuart-Landau model leads to the stochastic Kuramoto model of interacting phases. The approach can be tailored to networks with different frequency, topology, and stochastic inputs, thus providing a general and flexible framework to analyze noisy oscillations continuously across the underlying deterministic bifurcation.

Applied Physics Express, 2008

We have investigated cooperative dynamics of an artificial stochastic resonant system, which is a recurrent ring connection of neuron-like signal transducers (NST) based on stochastic resonance (SR), using electronic circuit experiments. The ring showed quasi-periodic, tunable oscillation driven by only noise. An oscillation coherently amplified by noise demonstrated that SR may lead to unusual oscillation features. Furthermore, we found that the ring showed synchronized oscillation in a chain network composed of multiple rings. Our results suggest that basic functions (oscillation and synchronization) that may be used in the central pattern generator of biological system are induced by collective integration of the NST element.

Physical Review E, 2009

Oscillations are ubiquitous phenomena in biological systems. Conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops. Some specific proteins function as transcription factors, which in turn negatively regulate the expression of the genes that encode these "clock proteins". These loops may lead to rhythmic changes in gene expression in a cell. In the case of multi-cellular tissue, collective oscillation is often due to synchronization of these cells, which manifest themselves as autonomous oscillators. In contrast, we propose here a different scenario for the occurrence of collective oscillation in a group of non-oscillatory cells. Neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in present system. By adopting a spatially inhomogeneous active factor, we observe and analyze a coupling-induced oscillation, inherent to the phenomenon of wave propagation due to intracellular communication.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011

We investigate the behavior of the order parameter describing the collective dynamics of a large set of driven, globally coupled excitable units. We derive conditions on the parameters of the system that allow to bound the degree of synchrony of its solutions. We describe a regime where time dependent nonsynchronous dynamics occurs and, yet, the average activity displays low dimensional, temporally complex behavior.

Physical Review E, 2021