Connection Graph Stability Method Research Articles

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.

Nonlinear synchronization on connected undirected networks

In this paper, we give sufficient conditions to have a complete synchronization of oscillators in connected undirected networks. The oscillators we are considering are not necessarily identical and the synchronization terms can be nonlinear. Many results in the literature deal with sufficient conditions insuring complete synchronization. This is a difficult problem since such conditions require to take into account the individual dynamics of the oscillators and also the graph topology. In this paper, we extend one of these results, the connection graph stability method, to nonlinear coupling functions and we also give an existence condition of trajectories of the oscillators. The sufficient conditions for synchronization presented in this paper are based on the study of a Lyapunov function and on the use of pseudometrics which enable us to link network dynamics and graph theory. These results are applied to a network of Chua’s oscillators and lead to an explicit condition insuring the complete synchronization of the oscillators.

A graph approach to synchronization in complex networks of asymmetrically nonlinear coupled dynamical systems

In this paper, we investigate the global exponential synchronization in complex networks of nonlinearly coupled dynamical systems with an asymmetric outer-coupling matrix. Employing the Lyapunov function approach with some graph theory techniques, we improve the so-called connection graph stability method for the synchronization analysis, that was originally developed by Belykh et al. for symmetrically linear coupled dynamical systems, to fit the asymmetrically nonlinear coupled case. We derive some criteria that ensure the nonlinearly coupled as well as linearly coupled dynamical systems to be globally exponentially synchronized. An illustrative example of a regular network with a modular structure of nonlinearly coupled Hindmarsh–Rose neurons is provided. We further consider a small-world dynamical network of nonlinearly coupled Chua's circuits and demonstrate both theoretically and numerically that the small-world dynamical network is easier to synchronize than the original regular dynamical network. More interestingly, numerical results of a real-world network of the cat cortex modelled by the asymmetrically linear coupled FitzHugh–Nagumo equations are also presented.

DYNAMICS AND SYNCHRONIZATION OF NOISE PERTURBED ENSEMBLES OF PERIODICALLY ACTIVATED NEURON CELLS

The role of noise for a single neuron and for an ensemble of mutually coupled neurons is investigated. For a single element we show that an increase in noise intensity in the regime of irregular firing enhances the coherence of the neuronal response. For this regime of spiking a study of the collective behavior for a chain composed of n mutually coupled Hodgkin–Huxley elements that are subjected simultaneously to sinusoidal drive and to a common random forcing is performed. The conditions for complete synchronization for this case are examined both within the framework of a theoretical approach based on the connection graph stability method and through numerical simulation.

Cluster Synchronization in a Complex Dynamical Network with Two Nonidentical Clusters

This paper further investigates cluster synchronization in a complex dynamical network with two-cluster. Each cluster contains a number of identical dynamical systems, however, the sub-systems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. Complete synchronization within each cluster is possible only if each node from one cluster receives the same input from nodes in other cluster. In this case, the stability condition of one-cluster synchronization is known to contain two terms: the first accounts for the contribution of the inner-cluster coupling structure while the second is simply an extra linear term, which can be deduced by the “same-input” condition. Applying the connection graph stability method, the authors obtain an upper bound of input strength for one cluster if the first account is known, by which the synchronizability of cluster can be scaled. For different clusters, there are different upper bound of input strength by virtue of different dynamics and the corresponding cluster structure. Moreover, two illustrative examples are presented and the numerical simulations coincide with the theoretical analysis.

Enhancing synchronizability of dynamical networks using the connection graph stability method

AbstractDynamical networks with diffusive couplings are investigated from the point of view of synchronizability. Arbitrary connection graphs are admitted but the coupling is symmetric. Networks with equal interaction coefficients for all edges of the interaction graph are compared with networks where the interaction coefficients vary from edge to edge according to the bounds for global synchronization obtained by the connection graph stability method. Synchronizability is tested numerically by establishing the time to decrease the synchronization error from 1 to 10−5 in the case of networks of identical Lorenz or Rössler systems. Synchronizability from the point of view of phase synchronization is also tested for networks of non‐identical Lorenz or Rössler systems. In this case the phase‐order parameters are compared, as a function of the mean interaction strength. Throughout, as network structures, scale‐free and Watts–Strogatz small‐world networks are used. Copyright © 2007 John Wiley & Sons, Ltd.

SYNCHRONIZATION AND CONTROL IN ENSEMBLES OF PERIODIC AND CHAOTIC NEURONAL ELEMENTS WITH TIME DEPENDENT COUPLING

The study of complete synchronization in networks of periodic and chaotic neurodynamical elements with different coupling configurations is performed. Using the connection graph stability method we obtain the sufficient conditions for achievement of synchronous behavior of all elements involved in these ensembles. The theoretical predictions we compare with the numerical results obtained for the networks composed of the classical Hodgkin-Huxley neuronal elements. The problem how to control the synchronization of networks growing in time is discussed. Copyright © 2007 IFAC

Synchronization Stability in Oscillator Networks: Solution for Asymmetrical Configurations

We extend the connection graph stability method, originally developed for symmetrically coupled networks, to the general asymmetrical case. The principal new component of the method is the transformation of the directed connection graph into an undirected graph. The extension of the method to asymmetrical coupling consists in symmetrizing the graph and associating a weight to each path. This weight involves the node unbalance of the two nodes. The synchronization condition for this symmetrized-and-weighted network then also guarantees synchronization in the original non-symmetrical network.

Chaotic synchronization in ensembles of coupled neurons modeled by the FitzHugh-Rinzel system

We consider various networks of diffusively coupled identical neurons modeled by a system of FitzHugh-Rinzel coupled differential equations. The mathematical model of a solitary nerve cell is analyzed theoretically and numerically. Regions corresponding to the qualitatively different behavior of a neuron are separated in the parameter space of the system. We present synchronization conditions in a network in which the central element modeling the pacemaking neuron is linked to the group of uncoupled neurons (star configuration). Within the framework of the connection graph stability method [1], which allows us to determine the character of variation in the threshold values of the coupling strengths sufficient for establishing the complete-synchronization regime, we study the influence of the network structure on the synchronization thresholds in different ensembles. In particular, we consider a configuration in the form of diffusively coupled stars structures.

Generalized connection graph method for synchronization in asymmetrical networks

We present a general framework for studying global complete synchronization in networks of dynamical systems with asymmetrical connections. We extend the connection graph stability method, originally developed for symmetrically coupled networks, to the general asymmetrical case. The principal new component of the method is the transformation of the directed connection graph into an undirected graph. In our method for symmetrically coupled networks we have to choose a path between each pair of nodes. The extension of the method to asymmetrical coupling consists in symmetrizing the graph and associating a weight to each path. This weight involves the “node unbalance” of the two nodes. This quantity is defined to be the difference between the sum of connection coefficients of the outgoing edges and the sum of the connection coefficients of the incoming edges to the node. The synchronization condition for this symmetrized-and-weighted network then also guarantees synchronization in the original asymmetrical network.

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.

SYNCHRONIZATION AND GRAPH TOPOLOGY

This paper clarifies the relation between synchronization and graph topology. Applying the Connection Graph Stability method developed by Belykh et al. [2004a] to the study of synchronization in networks of coupled oscillators, we show which graph properties matter for synchronization. In particular, while we explicitly link the stability of synchronization with the average path length for a wide class of coupling graphs, we prove by a simple argument that the average path length is not always the crucial quantity for synchronization. We also show that synchronization in scale-free networks can be described by means of regular networks with a star-like coupling structure. Finally, by considering an example of coupled Hindmarsh–Rose neuron models, we demonstrate how global stability of synchronization depends on the parameters of the individual oscillator.

Blinking model and synchronization in small-world networks with a time-varying coupling

The paper proposes a new type of small-world networks of cells with chaotic behavior. This network consists of a regular lattice of cells with constant 2 K-nearest neighbor couplings and time-dependent on–off couplings between any other pair of cells. In each time interval of duration τ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment, the coupling structure corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. It is to be distinguished from networks that have randomly chosen shortcuts, fixed in time. Here, we apply the Connection Graph Stability method, developed in the companion paper (“Connection graph stability method for synchronized coupled chaotic systems”, see this issue), to the study of global synchronization in this network with the time-varying coupling structure, in the case where the on–off switching is fast with respect to the characteristic synchronization time of the network. The synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.

Connection graph stability method for synchronized coupled chaotic systems

This paper elucidates the relation between network dynamics and graph theory. A new general method to determine global stability of total synchronization in networks with different topologies is proposed. This method combines the Lyapunov function approach with graph theoretical reasoning. In this context, the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph. In particular, the method is applied to the study of synchronization in rings of 2 K-nearest neighbor coupled oscillators. A rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators. This bound is explicitly linked with the average path length of the coupling graph. Contrary to the master stability function approach developed by Pecora and Carroll, the connection graph stability method leads to global stability of synchronization, and it permits not only constant, but also time-dependent interaction coefficients. In a companion paper (“Blinking model and synchronization in small-world networks with a time-varying coupling,” see this issue), this method is extended to the blinking model of small-world networks where, in addition to the fixed 2 K-nearest neighbor interactions, all the remaining links are rapidly switched on and off independently of each other.