Synchronization in large directed networks of coupled phase oscillators

We study the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks. We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider the case of networks with mixed positive-negative coupling strengths. We compare our theory with numerical simulations and find good agreement.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Synchronization in a neural network of phase oscillators with time delayed coupling

We investigate a neural network model designed as a system of the central oscillator and peripheral oscillators interacting with a time delay in a phase-locking loop. The delay corresponds to the finite velocity of signal propagation along nerve fibers. We study the synchronization under various values of . It is shown that under some conditions for a finite delay time there exist a multitude of synchronization frequencies in contrast to the case without delay where one has at most one solution. The criteria for the existence of multiple solutions and their stability are found. The asymptotic behavior under increasing connection strengths is analyzed.Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 8, pp. 945–960, August, 1994.     

Synchronization speed of identical oscillators on community networks

Synchronizability of complex oscillators networks has attracted much research interest in recent years. In contrast, in this paper we investigate numerically the synchronization speed, rather than the synchronizability or synchronization stability, of identical oscillators on complex networks with communities. A new weighted community network model is employed here, in which the community strength could be tunable by one parameter δ. The results showed that the synchronization speed of identical oscillators on community networks could reach a maximal value when δ is around 0.1. We argue that this is induced by the competition between the community partition and the scale-free property of the networks. Moreover, we have given the corresponding analysis through the second least eigenvalue λ₂ of the Laplacian matrix of the network which supports the previous result that the synchronization speed is determined by the value of λ₂.     

Synchronization in a multiplex network of gene oscillators

Study of the synchronization in the network of gene oscillator network has vital importance in understanding of rhythmicity of molecular and cellular activities. In this paper, we analyze a network of linearly coupled genetic oscillators in a multiplex structure. The coupling strength values are changed and the coupling range is considered to be fixed, but different in the two layers. The analyses are done in two cases of periodic and chaotic oscillations. By computing the statistical measures, the interlayer and intralayer synchronization states are studied. The results show that the layer with higher coupling range has more enhanced synchrony and is less affected by the turbulent behavior of the other layer. On the other hand, the layer with lower coupling range approaches synchronization by strengthening the interlayer and intralayer couplings. The interlayer synchronization is also achieved in high coupling strength values.     

Frequency clustering of coupled phase oscillators on small-world networks

We analyze the phenomenon of frequency clustering in a system of coupled phase oscillators. The oscillators, which in the absence of coupling have uniformly distributed natural frequencies, are coupled through a small-world network, built according to the Watts-Strogatz model. We study the time evolution and determine variations in the transient times depending on the disorder of the network and on the coupling strength. We investigate the effects of fluctuations in the average frequencies, and discuss the definition of the threshold for synchronization. We characterize the structure of clusters and the distribution of cluster sizes in the synchronization transition, and define suitable order parameters to describe the aggregation of the oscillators as the network disorder and the coupling strength change. The non-monotonic behavior observed in some order parameters is related to fluctuations in the mean frequencies.     

Synchronization of plant circadian oscillators with a phase delay effect of the vein network

Synchronization phenomena in coupled circadian oscillators of plant leaves were investigated experimentally using bioluminescence technology for a clock gene. Analyzing the phase of circadian oscillation, the phase-wave propagations and the phase delay caused by the vein network were observed. We describe these phase dynamics using a two-layer model with coupled Stuart-Landau equations. Global synchronization of circadian oscillators in the leaf is also investigated.     

Synchronization in the network of chaotic microwave oscillators

Time scale synchronization in networks of chaotic microwave oscillators with the different topologies of the links between nodes has been studied. As a node element of the network the one-dimensional distributed model of the low-voltage vircator has been used. To characterize the degree of synchronization in the whole network the synchronization index has been introduced. The transition to the synchronous regime is shown to take place via cluster time scale synchronization. Meanwhile, the spectral structure of the output signals is complicated sufficiently which allows using such devices in a number of practical applications.     

Synchronization in small-world systems

We quantify the dynamical implications of the small-world phenomenon by considering the generic synchronization of oscillator networks of arbitrary topology. The linear stability of the synchronous state is linked to an algebraic condition of the Laplacian matrix of the network. Through numerics and analysis, we show how the addition of random shortcuts translates into improved network synchronizability. Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes. However, the small-world property does not guarantee synchronizability: the synchronization threshold lies within the boundaries, but linked to the end of the small-world region.     

Complete and phase synchronization in a heterogeneous small-world neuronal network

Synchronous firing of neurons is thought to be important for information communication in neuronal networks. This paper investigates the complete and phase synchronization in a heterogeneous small-world chaotic Hindmarsh--Rose neuronal network. The effects of various network parameters on synchronization behaviour are discussed with some biological explanations. Complete synchronization of small-world neuronal networks is studied theoretically by the master stability function method. It is shown that the coupling strength necessary for complete or phase synchronization decreases with the neuron number, the node degree and the connection density are increased. The effect of heterogeneity of neuronal networks is also considered and it is found that the network heterogeneity has an adverse effect on synchrony.     

Chaotic phase synchronization in a modular neuronal network of small-world subnetworks

We investigate the onset of chaotic phase synchronization of bursting oscillators in a modular neuronal network of small-world subnetworks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that this bursting synchronization transition can be induced not only by the variations of inter- and intra-coupling strengths but also by changing the probability of random links between different subnetworks. We also analyze the effect of external chaotic phase synchronization of bursting behavior in this clustered network by an external time-periodic signal applied to a single neuron. Simulation results demonstrate a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this synchronization region increases with the signal amplitude and the number of driven neurons but decreases rapidly with the network size. Considering that the synchronization of bursting neurons is thought to play a key role in some pathological conditions, the presented results could have important implications for the role of externally applied driving signal in controlling bursting activity in neuronal ensembles.     

Data synchronization in a network of coupled phase oscillators

We devised a new method of data mining for a large-scale database. In the method, a network of locally coupled phase oscillators subject to Kuramoto's model substitutes for given multivariate data to generate major features through phase locking of the oscillators, i.e., phase transition of the data set. We applied the method to the national database of care needs certification for the Japanese public long-term care insurance program, and found three major patterns in the aging process of the frail elderly. This work revealed the latent utility of Kuramoto's model for data processing.     

耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

Multistable attractors in a network of phase oscillators with three-body interactions

Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with a three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in an asymptotic state can vary continuously within some range depending on the initial phase pattern.     

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.     

Synchronization with an arbitrary phase shift in a pair of synaptically coupled neural oscillators

The phase dynamics of a pair of spiking neural oscillators coupled by a unidirectional nonlinear connection has been studied. The synchronization effect with the controlled relative phase of spikes has been obtained for various coupling strengths and depolarization parameters. It has been found that the phase value is determined by the difference between the depolarization levels of neurons and is independent of the synaptic coupling strength. The synchronization mechanism has been studied by means of the construction and analysis of one-dimensional phase maps. The phase locking effect for spikes has been interpreted in application to the synaptic plasticity in neurobiology.     

Continuous opinion model in small-world directed networks

In the compromise model of continuous opinions proposed by Deffuant et al., the states of two agents in a network can start to converge if they are neighbors and if their opinions are sufficiently close to each other, below a given threshold of tolerance ?. In directed networks, if agent i is a neighbor of agent j,j need not be a neighbor of i. In Watts-Strogatz networks we performed simulations to find the averaged number of final opinions 〈F〉 and their distribution as a function of ? and of the network structural disorder. In directed networks 〈F〉 exhibits a rich structure, being larger than in undirected networks for higher values of ?, and smaller for lower values of ?.