Average Synchronization and Temporal Order in a Noisy Neuronal Network with Coupling Delay

Average synchronization and temporal order characterized by the rate of firing are studied in a spatially extended network system with the coupling time delay, which is locally modelled by a two-dimensional Rulkov map neuron. It is shown that there exists an optimal noise level, where average synchronization and temporal order are maximum irrespective of the coupling time delay. Furthermore, it is found that temporal order is weakened when the coupling time delay appears. However, the coupling time delay has a twofold effect on average synchronization, one associated with its increase, the other with its decrease. This clearly manifests that random perturbations and time delay play a complementary role in synchronization and temporal order.     

Synchronization of Complex Network with Drivingly Coupled Scheme

We present a network model with a new coupled scheme which is the generalization of drive-response systems called a drivingly coupled network. The synchronization of the network is investigated by numerical simulations based on Lorenz systems. By calculating the largest transversal Lyapunov exponents of such network, the stable and unstable regions of synchronous state for eigenvalues in such network can be obtained and many kinds of drivingly coupled arrays based on Lorenz systems such as all-to-all, star-shape, ring-shape and chain-shape networks are considered.     

Synchronization stability of general complex dynamical networks with time-varying delays

The synchronization problem of some general complex dynamical networks with time-varying delays is investigated. Both time-varying delays in the network couplings and time-varying delays in the dynamical nodes are considered. The novel delay-dependent criteria in terms of linear matrix inequalities (LMI) are derived based on free-weighting matrices technique and appropriate Lyapunov functional proposed recently. Numerical examples are given to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Synchronization of complex dynamical networks with nonidentical nodes

This Letter investigates the synchronization problem of a complex network with nonidentical nodes, and proposes two effective control schemes to synchronize the network onto any smooth goal dynamics. By applying open-loop control to all nodes and placing adaptive feedback injections on a small fraction of network nodes, a low-dimensional sufficient condition is derived to guarantee the global synchronization of the complex network with nonidentical nodes. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network composed of nonidentical nodes, and an upper bound of impulsive intervals is estimated to ensure the global stability of the synchronization process. Numerical simulations are given to verify the theoretical results.     

Synchronization analysis of delayed complex networks with time-varying couplings

In this paper, a new method is presented to analyze the linear stability of the synchronized state in arbitrarily coupled complex dynamical systems with time delays. The coupling configurations are not restricted to the symmetric and irreducible connections or the non-negative off-diagonal links. The stability criteria are obtained by using Lyapunov-Krasovskii functional method and subspace projection method. These criteria reveal the relationship between coupling matrices and stability of the dynamical networks.     

Projective Synchronization in Time-Delayed Chaotic Systems

For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.     

Global Synchronization of General Delayed Dynamical Networks

Global synchronization of general delayed dynamical networks with linear coupling are investigated. A sufficient condition for the global synchronization is obtained by using the linear matrix inequality and introducing a reference state. This condition is simply given based on the maximum nonzero eigenvalue of the network coupling matrix. Moreover, we show how to construct the coupling matrix to guarantee global synchronization of network, which is very convenient to use. A two-dimension system with delay as a dynamical node in network with global coupling is finally presented to verify the theoretical results of the proposed global synchronization scheme.     

Synchronization of Coupled Oscillators on Newman-Watts Small-World Networks

We investigate the collection behaviour of coupled phase oscillators on Newman-Watts small-world networks in one and two dimensions. Each component of the network is assumed as an oscillator and each interacts with the others following the Kuramoto model We then study the onset of global synchronization of phases and frequencies based on dynamic simulations and finite-size scaling. Both the phase and frequency synchronization are observed to emerge in the presence of a tiny fraction of shortcuts and enhanced with the increases of nearest neighbours and lattice dimensions.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

Different Types of Synchronization in Time-Delayed Systems

We investigate different types of synchronization between two unidirectionally nonlinearly coupled identical delay- differential systems related to optical bistable or hybrid optical bistable devices. This system can represent some kinds of delay-differential models, i.e. Ikeda model, Vall~e model, sine-square model, Mackey Glass model, and so on. We find existence and sufficient stability conditions by theoretical analysis and test the correctness by＂ numerical simulations. Lag, complete and anticipating synchronization are observed, respectively. It is found that the time-delay system can be divided into two parts~ one is the instant term and the other is the delay term. Synchronization between two identical chaotic systems can be derived by adding a coupled term to the delay term in the driven system.     

Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme

Encoding information by firing patterns is one of the basic neural functions, and synchronization is important collective behaviour of a group of coupled neurons. Taking account of two schemes for encoding information （that is, rate coding and temporal coding）, rhythm synchronization of coupled neurons is studied. There are two types of rhythm synchronization of neurons： spike and burst synchronizations. Firstly, it is shown that the spike synchronization is equivalent to the phase synchronization for coupled neurons. Secondly, the similarity function of the slow variables of neurons, which have relevant to the bursting process, is proposed to judge the burst synchronization. It is also found that the burst synchronization can be achieved more easily than the spike synchronization, whatever the firing patterns of the neurons are. Hence the temporal encoding scheme, which is closely related to both the spike and burst synchronizations, is more comprehensive than the rate coding scheme in essence.     

Synchronization in Oscillator Networks with Nonlinear Coupling

Synchronization in coupled oscillator networks has attracted much attention from many fields of science and engineering. In this paper, it is firstly proved that the oscifiator network with nonlinear coupling is also eventually dissipative under the hypothesis of eventual dissipation of the uncoupled oscillators. And the dynamics of the network is analyzed in its absorbing domain by combining two methods developed recently. Suufficient conditions for synchronization in the oscillator networks with nonlinear coupling are obtained. The two methods are combined effectively and the results embody the respective merits of the two methods. Numerical simulations confirm the validity of the results.     

Chaos Synchronization Criterion and Its Optimizations for a Nonlinear Transducer System via Linear State Error Feedback Control

Global chaos synchronization of two identical nonlinear transducer systems is investigated via linear state error feedback control. The sufficient criterion for global chaos synchronization is derived firstly by the Gerschgorin disc theorem and the stability theory of linear time-varied systems. Then this sufficient criterion is further optimized in the sense of reducing the lower bounds of the coupling coefficients with two methods, one based on Gerschgorin disc theorem itself and the other based on Lyapunov direct method. Finally, two optimized criteria are compared theoretically.     

Synchronization in a class of weighted complex networks with coupling delays

It is commonly accepted that realistic networks can display not only a complex topological structure, but also a heterogeneous distribution of connection weights. In addition, time delay is inevitable because the information spreading through a complex network is characterized by the finite speeds of signal transmission over a distance. Weighted complex networks with coupling delays have been gaining increasing attention in various fields of science and engineering. Some of the topics of most concern in the field of weighted complex networks are finding how the synchronizability depends on various parameters of the network including the coupling strength, weight distribution and delay. On the basis of the theory of asymptotic stability of linear time-delay systems with complex coefficients, the synchronization stability of weighted complex dynamical networks with coupling delays is investigated, and simple criteria are obtained for both delay-independent and delay-dependent stabilities of the synchronization state. Finally, an example is given as an illustration testing the theoretical results.     

Phase synchronization on scale-free networks with community structure

In this Letter, we propose a growing network model that can generate scale-free networks with a tunable community strength. The community strength, C, is directly measured by the ratio of the number of external edges to that of the internal ones; a smaller C   corresponds to a stronger community structure. By using the Kuramoto model, we investigated the phase synchronization on this network and found an abnormal region (C?0.002$C?0.002$), in which the network has even worse synchronizability than the unconnected case (C=0$C=0$). On the other hand, the community effect will vanish when C exceeds 0.1. Between these two extreme regions, a stronger community structure will hinder global synchronization.     

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 complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.     

Detection of Mechanism of Noise-Induced Synchronization between Two Identical Uncoupled Neurons

We investigate the noise-induced synchronization between two identical uncoupled Hodgkin-Huxley neurons with sinusoidal stimulations. The numerical results confirm that the value of critical noise intensity for synchronizing two systems is much less than the magnitude of mean size of the attractor in the original system, and the deterministic feature of the attractor in the original system remains unchanged. This finding is significantly different from the previous work [Phys. Rev. E 67 （2003） 027201] in which the value of the critical noise intensity for synchronizing two systems was found to be roughly equal to the magnitude of mean size of the attractor in the original system, and at this intensity, the noise swamps the qualitative structure of the attractor in the original deterministic systems to synchronize to their stochastic dynamics. Further investigation shows that the critical noise intensity for synchronizing two neurons induced by noise may be related to the structure of interspike intervals of the original systems.     

Stochastic switching and dynamical freezing in nonlinear spin systems

We consider the controlled switching of individual spins in a nonlinear, interacting spin chain by means of external magnetic fields. We show analytically and by full numerical simulations that stochastic switching is achievable when the driving fields are such that the underlying semi-classical dynamics is chaotic. On the basis of random matrix theory and the geometry of quantum evolution we confirm the quantum case to follow qualitatively the semi-classical behavior.