Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.     

具有不同耦合强度且带有时滞的振子网络上的同步

本文研究了具有不同耦合强度且带有时滞的振子网络上的同步问题．我们给出了该网络同步状态的稳定性准则，证实了其同步状态的稳定性与网络的拓扑性无关．最后，通过数值模拟验证了我们的理论结果．     

Exponential Synchronization of the Linearly Coupled Dynamical Networks with Delays

In this paper,the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling.Here the coupling matrix can be asymmetric and reducible.Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively.It is shown that if the coupling delay is less than a positive threshold,then the coupled network will be synchronized.On the other hand,with the increase of coupling delay,the synchronization stability of the network will be restrained,even eventually de-synchronized.     

Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling

This paper proposes an adaptive control method to achieve the lag synchronization between uncertain complex dynamical network having delayed coupling and a non-identical reference node. Unknown parameters of both the network and reference node are estimated by adaptive laws obtained by Lyapunov stability theory. With the estimated parameters, the proposed method guarantees the globally asymptotical synchronization of the network in spite of unknown bounded disturbances. The effectiveness of our work is verified through a numerical example and simulation.     

One- and two-cluster synchronized dynamics of non-diffusively coupled Tchebycheff map networks

We use the master stability formalism to discuss one- and two-cluster synchronization of coupled Tchebycheff map networks. For diffusively coupled map systems, the one-cluster synchronized dynamics is given by the behaviour of the individual maps, and the coupling only determines the stability of the coherent state. For the case of non-diffusive coupling and for two-cluster synchronization, the synchronized dynamics on networks is different from the behaviour of the single individual map. Depending on the coupling, we study numerically the characteristics of various forms of the resulting synchronized dynamics. The stability properties of the respective one-cluster synchronized states are discussed for arbitrary network structures. For the case of two-cluster synchronization on bipartite networks we also present analytical expressions for fixed points and zig-zag patterns, and explicitly determine the linear stability of these orbits for the special case of ring-networks.     

Dynamical behaviors in time-delay systems with delayed feedback and digitized coupling

We consider a network of delay dynamical systems connected in a ring via unidirectional positive feedback with constant delay in coupling. For the specific case of Mackey–Glass systems on the ring topology, we capture the phenomena of amplitude death, isochronous synchronization and phase-flip bifurcation as the relevant parameters are tuned. Using linear stability analysis and Master Stability Function approach, we predict the region of amplitude death and synchronized states respectively in the parameter space and study the nature of transitions between the different states. For a large number of systems in the same dynamical configuration, we observe splay states, mixed splay states and phase locked clusters. We extend the study to the case of digitized coupling and observe that these emergent states still persist. However, the sampling and quantization reduce the regions of amplitude death and induce phase-flip bifurcation.     

Stability of synchronized dynamics and pattern formation in coupled systems: Review of some recent results

In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. These two problems are often treated separately in the literature. Here, we present a unified framework in which we show that the eigenvalues of the coupling matrix determine the stability of the synchronized state, while the eigenvectors correspond to patterns emerging from desynchronization. Based on this simple framework three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable spatial patterns can be selectively realized by varying the coupling strengths. Distinct temporal evolution of the spatial pattern can be obtained depending on the bifurcating synchronized state. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves. Systems with specific coupling schemes are used as examples to illustrate the general methods.     

Adaptive synchronization of a hyperchaotic system with uncertain parameter

This paper addresses the synchronization problem of two Lü hyperchaotic dynamical systems in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is derived to make the states of two identical Lü hyperchaotic systems with unknown system parameters asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization schemes.     

三个耦合的非扩散Lorenz系统的全局混沌同步

以Lyapunov稳定性理论和矩阵论为基础,针对非扩散Lorenz系统,提出了一种三个耦合的恒等系统的全局混沌同步方案.该方案的关键是耦合参数的选取.通过选择适当的耦合参数,使得三个系统的状态变量达到同步,并利用Mathematic软件进行数值仿真.理论分析和仿真结果都表明了该方法的有效性.     

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.     

Synchronized states in a ring of four mutually coupled self-sustained electromechanical devices

In this paper, we study the dynamics of a ring of four mutually coupled identical self-sustained electromechanical devices both in their autonomous and nonautonomous chaotic states. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived. Numerical simulations are then performed to support the accuracy of the analytical approach.     

Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons

The use of spikes to carry information between brain areas implies complete or partial synchronization of the neurons involved. The degree of synchronization reached by two coupled systems and the energy cost of maintaining their synchronized behavior is highly dependent on the nature of the systems. For non-identical systems the maintenance of a synchronized regime is energetically a costly process. In this work, we study conditions under which two non-identical electrically coupled neurons can reach an efficient regime of synchronization at low energy cost. We show that the energy consumption required to keep the synchronized regime can be spontaneously reduced if the receiving neuron has adaptive mechanisms able to bring its biological parameters closer in value to the corresponding ones in the sending neuron.     

Optimal synchronization of Rössler system with complete uncertain parameters

The paper discusses the optimal control for the chaos synchronization of Rössler systems with complete uncertain parameters during finite and infinite time intervals. Based on the Liapunov–Bellman technique, optimal control laws are derived from the conditions that ensure asymptotic stability of the error dynamical system and minimizes the cost transfer of this system from arbitrary state to its equilibrium state. The derived control laws make the states of two identical Rössler systems asymptotically synchronized. Some special cases are introduced. Important numerical simulation is included to show the effectiveness of the optimal synchronization technique.     

Adaptive control and synchronization of a coupled dynamo system with uncertain parameters

This paper treats the adaptive control and synchronization of the coupled dynamo system with unknown parameters. Based on the Lyapunov stability technique, an adaptive control laws are derived such that the coupled dynamo system is asymptotically stable and the two identical dynamo systems are asymptotically synchronized. Also the update rules of the unknown parameters are derived. Finally, numerical simulation of the controlled and synchronized systems are presented.     

Cluster synchronization,switching and spatiotemporal coding in a phase oscillator network

A network of five globally-coupled identical phase oscillators is considered. Cluster states consisting of two synchronized pairs of oscillators and one singleton are investigated. Forcing the system with non-uniform constant inputs results in regular switches between cluster states. The resultant cyclic sequences of switches (spatiotemporal codes) are studied for different initial conditions and input configurations. Implications on information coding in neural systems are briefly discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Emergence of a multilayer structure in adaptive networks of phase oscillators

We report on self-organization of adaptive networks, where topology and dynamics evolve in accordance to a competition between homophilic and homeostatic mechanisms, and where links are associated to a vector of weights. Under an appropriate balance between the intra- and inter- layer coupling strengths, we show that a multilayer structure emerges due to the adaptive evolution, resulting in different link weights at each layer, i.e. different components of the weights’ vector. In parallel, synchronized clusters at each layer are formed, which may overlap or not, depending on the values of the coupling strengths. Only when intra- and inter- layer coupling strengths are high enough, all layers reach identical final topologies, collapsing the system into, in fact, a monolayer network. The relationships between such steady state topologies and a set of dynamical network’s properties are discussed.     

Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control

This paper presents a general model of singular complex switched networks, in which the nodes can be singular dynamic systems and switching behaviors act on both nodes and edges. The parametric uncertainties and unknown coupling topologies are also considered in this model. Two robust synchronization schemes are discussed respectively. In one scheme, the network is synchronized to a homogeneous orbit and in the other one the network is synchronized to a weighted average of all the nodes. Based on the Lyapunov stability theory, different robust synchronization conditions for the two schemes are obtained for this singular complex switched network model via impulsive control. The similarities and differences between these synchronization conditions for the two schemes are discussed. In addition, three useful robust results for the special cases of the singular complex switched networks are presented. Two systematic-design procedures are presented for the two schemes, and three numerical examples are provided for illustrations.     

Pinning adaptive synchronization of a class of uncertain complex dynamical networks with multi-link against network deterioration

For the reason that the uncertain complex dynamic network with multi-link is quite close to various practical networks, there is superiority in the fields of research and application. In this paper, we focus upon pinning adaptive synchronization for uncertain complex dynamic networks with multi-link against network deterioration. The pinning approach can be applied to adapt uncertain coupling factors of deteriorated networks which can compensate effects of uncertainty. Several new synchronization criterions for networks with multi-link are derived, which ensure the synchronized states to be local or global stable with uncertainty and deterioration. Results of simulation are shown to demonstrate the feasibility and usefulness of our method.     

Spatio-temporal patterns of Hopf bifurcating periodic oscillations in a pair of identical tri-neuron network loops

Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations.     

Synchronization analysis through coupling mechanism in realistic neural models

Synchronization which relates to the system’s stability is important to many engineering and neural applications. In this paper, an attempt has been made to implement response synchronization using coupling mechanism for a class of nonlinear neural systems. We propose an OPCL (open-plus-closed-loop) coupling method to investigate the synchronization state of driver-response neural systems, and to understand how the behavior of these coupled systems depend on their inner dynamics. We have investigated a general method of coupling for generalized synchronization (GS) in 3D modified spiking and bursting Morris–Lecar (M-L) neural models. We have also presented the synchronized behavior of a network of four bursting Hindmarsh–Rose (H-R) neural oscillators using a bidirectional coupling mechanism. We can extend the coupling scheme to a network of N neural oscillators to reach the desired synchronous state. To make the investigations more promising, we consider another coupling method to a network of H-R oscillators using bidirectional ring type connections and present the effectiveness of the coupling scheme.