Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.     

Delay-induced diversity of firing behavior and ordered chaotic firing in adaptive neuronal networks

In this paper, we study the effect of time delay on the firing behavior and temporal coherence and synchronization in Newman–Watts thermosensitive neuron networks with adaptive coupling. At beginning, the firing exhibit disordered spiking in absence of time delay. As time delay is increased, the neurons exhibit diversity of firing behaviors including bursting with multiple spikes in a burst, spiking, bursting with four, three and two spikes, firing death, and bursting with increasing amplitude. The spiking is the most ordered, exhibiting coherence resonance (CR)-like behavior, and the firing synchronization becomes enhanced with the increase of time delay. As growth rate of coupling strength or network randomness increases, CR-like behavior shifts to smaller time delay and the synchronization of firing increases. These results show that time delay can induce diversity of firing behaviors in adaptive neuronal networks, and can order the chaotic firing by enhancing and optimizing the temporal coherence and enhancing the synchronization of firing. However, the phenomenon of firing death shows that time delay may inhibit the firing of adaptive neuronal networks. These findings provide new insight into the role of time delay in the firing activity of adaptive neuronal networks, and can help to better understand the complex firing phenomena in neural networks.     

Community structure in real-world networks from a non-parametrical synchronization-based dynamical approach

This work analyzes the problem of community structure in real-world networks based on the synchronization of nonidentical coupled chaotic Rössler oscillators each one characterized by a defined natural frequency, and coupled according to a predefined network topology. The interaction scheme contemplates an uniformly increasing coupling force to simulate a society in which the association between the agents grows in time. To enhance the stability of the correlated states that could emerge from the synchronization process, we propose a parameterless mechanism that adapts the characteristic frequencies of coupled oscillators according to a dynamic connectivity matrix deduced from correlated data. We show that the characteristic frequency vector that results from the adaptation mechanism reveals the underlying community structure present in the network.     

Impulsive synchronization of time delay bursting neuron systems with unidirectional coupling

In the article, impulsive synchronization of chaotic bursting in Hindmarsh–Rose neuron systems with time delay via partial state signal is investigated. Based on impulsive control theory of dynamical systems, the sufficient conditions on feedback strength and impulsive interval are established to guarantee the synchronization. Numerical simulations show the effectiveness of the proposed scheme. The obtained results may be helpful to understand dynamical mechanism of signal transduction in real neuronal activity. © 2014 Wiley Periodicals, Inc. Complexity 21: 38–46, 2015     

Synchronization of hyperchaotic oscillators via single unidirectional chaotic-coupling

In this paper, synchronization of two hyperchaotic oscillators via a single variable’s unidirectional coupling is studied. First, the synchronizability of the coupled hyperchaotic oscillators is proved mathematically. Then, the convergence speed of this synchronization scheme is analyzed. In order to speed up the response with a relatively large coupling strength, two kinds of chaotic coupling synchronization schemes are proposed. In terms of numerical simulations and the numerical calculation of the largest conditional Lyapunov exponent, it is shown that in a given range of coupling strengths, chaotic-coupling synchronization is quicker than the typical continuous-coupling synchronization. Furthermore, A circuit realization based on the chaotic synchronization scheme is designed and Pspice circuit simulation validates the simulated hyperchaos synchronization mechanism.     

Mutual synchronization behavior for chaotic systems via limited capacity communication channels

Several important properties of chaos synchronization of bidirectional coupled systems remain still unexplored. This article investigates synchronization behavior for chaotic systems subject to states quantization. Based on the invariance principle of differential equations, an adaptive feedback scheme is proposed to strictly synchronize chaotic systems via limited capacity communication channels. Furthermore, it is important to point out that the mutual synchronization behavior for bidirectional coupled systems is determined by the amount of transmitting information and the initial states of coupled systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 335–342, 2016     

Synchronization in nonlinear oscillators with conjugate coupling

In this work, we investigate the synchronization in oscillators with conjugate coupling in which oscillators interact via dissimilar variables. The synchronous dynamics and its stability are investigated theoretically and numerically. We find that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We also find that the synchronization may be independent of the number of oscillators. Numerical demonstrations with Lorenz oscillators are provided.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Broad-scale small-world network topology induces optimal synchronization of flexible oscillators

The discovery of small-world and scale-free properties of many man-made and natural complex networks has attracted increasing attention. Of particular interest is how the structural properties of a network facilitate and constrain its dynamical behavior. In this paper we study the synchronization of weakly coupled limit-cycle oscillators in dependence on the network topology as well as the dynamical features of individual oscillators. We show that flexible oscillators, characterized by near zero values of divergence, express maximal correlation in broad-scale small-world networks, whereas the non-flexible (rigid) oscillators are best correlated in more heterogeneous scale-free networks. We found that the synchronization behavior is governed by the interplay between the networks global efficiency and the mutual frequency adaptation. The latter differs for flexible and rigid oscillators. The results are discussed in terms of evolutionary advantages of broad-scale small-world networks in biological systems.     

On–off intermittency and coherent bursting in stochastically-driven coupled maps

On–off intermittency is a phase space mechanism for bursting in dynamical systems. Here we recall how the simple example of a logistic map with a time-dependent control parameter, considered as a dynamical variable of the system, gives rise to bursting or on–off behavior. We show that, for a given realization of the driver, a stochastically driven logistic map in the on–off intermittent regime always converges to the same temporal dynamics, independently of initial conditions. In that sense, the map is not chaotic. We then explore the behavior of two coupled on–off logistic maps, each driven by a separate random process, and show that, for a wide range of coupling strengths, bursting becomes at least partially coherent. The bursting coherence has a smooth dependence on the coupling parameter and no sharp transition from coherence to incoherence is detected. In the system of two coupled on–off maps studied here, coherent bursting is rooted in the behavior during off phases when the mapped coordinates take on extremely small values.     

Adaptive coupled synchronization among multi-Lorenz systems family

In this paper, the adaptive synchronization method of coupled system is proposed for multi-Lorenz systems family. This method can avoid estimating the value of coupling coefficient. Strict theoretical proofs are given. And we derived a sufficient condition of synchronization for a general unidirectional coupling ring network with N identical Lorenz systems. The network is coupled through the first state variable of each equation. In fact, the whole unidirectional coupling network will synchronize by adding only one adaptive feedback gain equation. Numerical simulations show the effectiveness of the methods.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

Synchronization of complex chaotic systems in series expansion form

This paper studies the synchronization of complex chaotic systems in series expansion form by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of an error dynamics, and is applied to guiding the design of the secure communication. Finally, numerical results are studied for the Quantum-CNN oscillators synchronizing with unidirectional/bidirectional linear coupling to show the effectiveness of the proposed synchronization strategy.     

Noise and coupling induced synchronization in a network of chaotic neurons

The synchronization in four forced FitzHugh–Nagumo (FHN) systems is studied, both experimentally and by numerical simulations of a model. We show that synchronization may be achieved either by coupling of systems through bidirectional diffusive interactions, by introducing a common noise to all systems or by combining both ingredients, noise and coupling together. Here we consider white and colored noises, showing that the colored noise is more efficient in synchronizing the systems respect to white noise. Moreover, a small addition of common noise allows the synchronization to occur at smaller values of the coupling strength. When the diffusive coupling in the absence of noise is considered, the system undergoes the transition to subthreshold oscillations, giving a spike suppression regime. We show that noise destroys the appearance of this dynamical regime induced by coupling.     

Various synchronization phenomena in bidirectionally coupled double scroll circuits

In this paper, the various cases of synchronization phenomena investigated in a system of two bidirectionally coupled double scroll circuits, were studied. Complete synchronization, inverse lag synchronization, and inverse π-lag synchronization are the observed synchronization phenomena, as the coupling factor is varied. The inverse lag synchronization phenomenon in mutually coupled identical oscillators is presented for the first time. As the coupling factor is increased, the system undergoes a transition from chaotic desynchronization to chaotic complete synchronization, while inverse lag synchronization and inverse π-lag synchronization are observed for greater values of the coupling factor, depending on the initial conditions of the state variables of the system. Inverse π-lag synchronization in coupled nonlinear oscillators is a special case of lag synchronization, which is also presented for the first time.     

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.     

How single node dynamics enhances synchronization in neural networks with electrical coupling

The stability of the completely synchronous state in neural networks with electrical coupling is analytically investigated applying both the Master Stability Function approach (MSF), developed by Pecora and Carroll (1998), and the Connection Graph Stability method (CGS) proposed by Belykh et al. (2004). The local dynamics is described by Morris–Lecar model for spiking neurons and by Hindmarsh–Rose model in spike, burst, irregular spike and irregular burst regimes. The combined application of both CGS and MSF methods provides an efficient estimate of the synchronization thresholds, namely bounds for the coupling strength ranges in which the synchronous state is stable. In all the considered cases, we observe that high values of coupling strength tend to synchronize the system. Furthermore, we observe a correlation between the single node attractor and the local stability properties given by MSF. The analytical results are compared with numerical simulations on a sample network, with excellent agreement.     

Burst-duration mechanism of in-phase bursting in inhibitory networks

We study the emergence of in-phase and anti-phase synchronized rhythms in bursting networks of Hodgkin-Huxley-type neurons connected by inhibitory synapses. We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burst-duration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in- and anti-phase synchronized bursting.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.