Synchronization on small-world networks

Clustering in a network of non-identical and mutually interacting agents

Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behaviour. It is observed in fields ranging from the exact sciences to social and life sciences; consider, for example, swarm behaviour of animals or social insects, the dynamics of opinion formation or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modelling brain or heart cells. We consider a clustering model with a general network structure and saturating interaction functions. We derive both necessary and sufficient conditions for clustering behaviour of the model and we investigate the cluster structure for varying coupling strength. Generically, each cluster asymptotically reaches a (relative) equilibrium state. We discuss the relationship of the model to swarming, and we explain how the model equations naturally arise in a system of interconnected water basins. We also indicate how the model applies to opinion formation dynamics.

Strong effect of dispersal network structure on ecological dynamics

A central question in ecology with great importance for management, conservation and biological control is how changing connectivity affects the persistence and dynamics of interacting species. Researchers in many disciplines have used large systems of coupled oscillators to model the behaviour of a diverse array of fluctuating systems in nature. In the well-studied regime of weak coupling, synchronization is favoured by increases in coupling strength and large-scale network structures (for example 'small worlds') that produce short cuts and clustering. Here we show that, by contrast, randomizing the structure of dispersal networks in a model of predators and prey tends to favour asynchrony and prolonged transient dynamics, with resulting effects on the amplitudes of population fluctuations. Our results focus on synchronization and dynamics of clusters in models, and on timescales, more appropriate for ecology, namely smaller systems with strong interactions outside the weak-coupling regime, rather than the better-studied cases of large, weakly coupled systems. In these smaller systems, the dynamics of transients and the effects of changes in connectivity can be well understood using a set of methods including numerical reconstructions of phase dynamics, examinations of cluster formation and the consideration of important aspects of cyclic dynamics, such as amplitude.

Efficient rewirings for enhancing synchronizability of dynamical networks

In this paper, we present an algorithm for optimizing synchronizability of complex dynamical networks. Starting with an undirected and unweighted network, we end up with an undirected and unweighted network with the same number of nodes and edges having enhanced synchronizability. To this end, based on some network properties, rewirings, i.e., eliminating an edge and creating a new edge elsewhere, are performed iteratively avoiding always self-loops and multiple edges between the same nodes. We show that the method is able to enhance the synchronizability of networks of any size and topological properties in a small number of steps that scales with the network size. For numerical simulations, an optimization algorithm based on simulated annealing is used. Also, the evolution of different topological properties of the network such as distribution of node degree, node and edge betweenness centrality is tracked with the iteration steps. We use networks such as scale-free, Strogatz-Watts and random to start with and we show that regardless of the initial network, the final optimized network becomes homogeneous. In other words, in the network with high synchronizability, parameters, such as, degree, shortest distance, node, and edge betweenness centralities are almost homogeneously distributed. Also, parameters, such as, maximum node and edge betweenness centralities are small for the rewired network. Although we take the eigenratio of the Laplacian as the target function for optimization, we show that it is also possible to choose other appropriate target functions exhibiting almost the same performance. Furthermore, we show that even if the network is optimized taking into account another interpretation of synchronizability, i.e., synchronization cost, the optimal network has the same synchronization properties. Indeed, in networks with optimized synchronizability, different interpretations of synchronizability coincide. The optimized networks are Ramanujan graphs, and thus, this rewiring algorithm could be used to produce Ramanujan graphs of any size and average degree.

Evolving functional network properties and synchronizability during human epileptic seizures

We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with different anatomical onset locations by statistical and spectral properties of functionally defined networks. We observe a concave-like temporal evolution of characteristic path length and cluster coefficient indicative of a movement from a more random toward a more regular and then back toward a more random functional topology. Surprisingly, synchronizability was significantly decreased during the seizure state but increased already prior to seizure end. Our findings underline the high relevance of studying complex systems from the viewpoint of complex networks, which may help to gain deeper insights into the complicated dynamics underlying epileptic seizures.

Relaxation of synchronization on complex networks

We study collective synchronization in a large number of coupled oscillators on various complex networks. In particular, we focus on the relaxation dynamics of the synchronization, which is important from the viewpoint of information transfer or the dynamics of system recovery from a perturbation. We measure the relaxation time tau that is required to establish global synchronization by varying the structural properties of the networks. It is found that the relaxation time in a strong-coupling regime (K>Kc) logarithmically increases with network size N , which is attributed to the initial random phase fluctuation given by O(N-1/2) . After elimination of the initial-phase fluctuation, the relaxation time is found to be independent of the system size; this implies that the local interaction that depends on the structural connectivity is irrelevant in the relaxation dynamics of the synchronization in the strong-coupling regime. The relaxation dynamics is analytically derived in a form independent of the system size, and it exhibits good consistency with numerical simulations. As an application, we also explore the recovery dynamics of the oscillators when perturbations enter the system.

Resonances and entrainment breakup in Kuramoto models with multimodal frequency densities

We characterize some intriguing aspects of the entrainment behavior of coupled oscillators by means of a perturbation analysis of the partially synchronized solution of the classical Kuramoto-Sakaguchi model. The analysis reveals that partial entrainment may disappear with increasing coupling strength. It also predicts the occurrence of resonances: partial entrainment is induced in oscillators with natural frequencies in specific intervals not corresponding to high oscillator densities. The results are illustrated by simulations.

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.

Transition to global synchronization in clustered networks

A clustered network is characterized by a number of distinct sparsely linked subnetworks (clusters), each with dense internal connections. Such networks are relevant to biological, social, and certain technological networked systems. For a clustered network the occurrence of global synchronization, in which nodes from different clusters are synchronized, is of interest. We consider Kuramoto-type dynamics and obtain an analytic formula relating the critical coupling strength required for global synchronization to the probabilities of intracluster and intercluster connections, and provide numerical verification. Our work also provides direct support for a previous spectral-analysis-based result concerning the role of random intercluster links in enhancing the synchronizability of a clustered network.

Population synchrony in small-world networks

Network topography ranges from regular graphs (linkage between nearest neighbours only) via small-world graphs (some random connections between nodes) to completely random graphs. Small-world linkage is seen as a revolutionary architecture for a wide range of social, physical and biological networks, and has been shown to increase synchrony between oscillating subunits. We study small-world topographies in a novel context: dispersal linkage between spatially structured populations across a range of population models. Regular dispersal between population patches interacting with density-dependent renewal provides one ecological explanation for the large-scale synchrony seen in the temporal fluctuations of many species, for example, lynx populations in North America, voles in Fennoscandia and grouse in the UK. Introducing a small-world dispersal kernel leads to a clear reduction in synchrony with both increasing dispersal rate and small-world dispersal probability across a variety of biological scenarios. Synchrony is also reduced when populations are affected by globally correlated noise. We discuss ecological implications of small-world dispersal in the frame of spatial synchrony in population fluctuations.

Synchronizability of network ensembles with prescribed statistical properties

It has been shown that synchronizability of a network is determined by the local structure rather than the global properties. With the same global properties, networks may have very different synchronizability. In this paper, we numerically studied, through the spectral properties, the synchronizability of ensembles of networks with prescribed statistical properties. Given a degree sequence, it is found that the eigenvalues and eigenratios characterizing network synchronizability have well-defined distributions, and statistically, the networks with extremely poor synchronizability are rare. Moreover, we compared the synchronizability of three network ensembles that have the same nodes and average degree. Our work reveals that the synchronizability of a network can be significantly affected by the local pattern of connections, and the homogeneity of degree can greatly enhance network synchronizability for networks of a random nature.

Optimization of synchronization in complex clustered networks

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

The application of graph theoretical analysis to complex networks in the brain

Considering the brain as a complex network of interacting dynamical systems offers new insights into higher level brain processes such as memory, planning, and abstract reasoning as well as various types of brain pathophysiology. This viewpoint provides the opportunity to apply new insights in network sciences, such as the discovery of small world and scale free networks, to data on anatomical and functional connectivity in the brain. In this review we start with some background knowledge on the history and recent advances in network theories in general. We emphasize the correlation between the structural properties of networks and the dynamics of these networks. We subsequently demonstrate through evidence from computational studies, in vivo experiments, and functional MRI, EEG and MEG studies in humans, that both the functional and anatomical connectivity of the healthy brain have many features of a small world network, but only to a limited extent of a scale free network. The small world structure of neural networks is hypothesized to reflect an optimal configuration associated with rapid synchronization and information transfer, minimal wiring costs, resilience to certain types of damage, as well as a balance between local processing and global integration. Eventually, we review the current knowledge on the effects of focal and diffuse brain disease on neural network characteristics, and demonstrate increasing evidence that both cognitive and psychiatric disturbances, as well as risk of epileptic seizures, are correlated with (changes in) functional network architectural features.

Graph theoretical analysis of complex networks in the brain

Since the discovery of small-world and scale-free networks the study of complex systems from a network perspective has taken an enormous flight. In recent years many important properties of complex networks have been delineated. In particular, significant progress has been made in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks. For instance, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments in the theory of complex networks have inspired new applications in the field of neuroscience. Graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity based upon fMRI, EEG and MEG. These studies suggest that the human brain can be modelled as a complex network, and may have a small-world structure both at the level of anatomical as well as functional connectivity. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, as well as a balance between local processing and global integration. The topological structure of functional networks is probably restrained by genetic and anatomical factors, but can be modified during tasks. There is also increasing evidence that various types of brain disease such as Alzheimer's disease, schizophrenia, brain tumours and epilepsy may be associated with deviations of the functional network topology from the optimal small-world pattern.

Finite-size scaling in complex networks

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.

Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons

We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or "shortcuts", and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponentially distributed.

Graph theory and networks in Biology

A survey of the use of graph theoretical techniques in Biology is presented. In particular, recent work on identifying and modelling the structure of bio-molecular networks is discussed, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronisation and disease propagation.

Synchronization in weighted uncorrelated complex networks in a noisy environment: optimization and connections with transport efficiency

Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to (kikj)beta with ki and kj being the degrees of the nodes connected by the link. Subject to the constraint that the total edge cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at beta*= -1 . Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of beta*. Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.

Small-world brain networks

Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections. This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated/specialized and distributed/integrated information processing. Moreover, small-world networks are economical, tending to minimize wiring costs while supporting high dynamical complexity. The authors introduce some of the key mathematical concepts in graph theory required for small-world analysis and review how these methods have been applied to quantification of cortical connectivity matrices derived from anatomical tract-tracing studies in the macaque monkey and the cat. The evolution of small-world networks is discussed in terms of a selection pressure to deliver cost-effective information-processing systems. The authors illustrate how these techniques and concepts are increasingly being applied to the analysis of human brain functional networks derived from electroencephalography/magnetoencephalography and fMRI experiments. Finally, the authors consider the relevance of small-world models for understanding the emergence of complex behaviors and the resilience of brain systems to pathological attack by disease or aberrant development. They conclude that small-world models provide a powerful and versatile approach to understanding the structure and function of human brain systems.

Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methods

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronization at low coupling noted elsewhere is associated with high-eigenvalue modes.

Synchronization transition of heterogeneously coupled oscillators on scale-free networks

We investigate the synchronization transition of the modified Kuramoto model where the oscillators form a scale-free network with degree exponent lambda . An oscillator of degree k_{i} is coupled to its neighboring oscillators with asymmetric and degree-dependent coupling in the form of Jk_{i};{eta-1} . By invoking the mean-field approach, we find eight different synchronization transition behaviors depending on the values of eta and lambda , and derive the critical exponents associated with the order parameter and the finite-size scaling in each case. The synchronization transition point J_{c} is determined as being zero (finite) when eta>lambda-2 (eta<lambda-2) . The synchronization transition is also studied from the perspective of cluster formation of synchronized vertices. The cluster-size distribution and the largest cluster size as a function of the system size are derived for each case using the generating function technique. Our analytic results are confirmed by numerical simulations.

Strong effects of network architecture in the entrainment of coupled oscillator systems

Random networks of coupled phase oscillators, representing an approximation for systems of coupled limit-cycle oscillators, are considered. Entrainment of such networks by periodic external forcing applied to a subset of their elements is numerically and analytically investigated. For a large class of interaction functions, we find that the entrainment window with a tongue shape becomes exponentially narrow for networks with higher hierarchical organization. However, the entrainment is significantly facilitated if the networks are directionally biased--i.e., closer to the feedforward networks. Furthermore, we show that the networks with high entrainment ability can be constructed by evolutionary optimization processes. The neural network structure of the master clock of the circadian rhythm in mammals is discussed from the viewpoint of our results.

Frequency-selective response of FitzHugh-Nagumo neuron networks via changing random edges

We consider a network of FitzHugh-Nagumo neurons; each neuron is subjected to a subthreshold periodic signal and independent Gaussian white noise. The firing pattern of the mean field changes from an internal-scale dominant pattern to an external-scale dominant one when more and more edges are added into the network. We find numerically that (a) this transition is more sensitive to random edges than to regular edges, and (b) there is a saturation length for random edges beyond which the transition is no longer sharpened. The influence of network size is also investigated.

Synchronizabilities of networks: a new index

The random matrix theory is used to bridge the network structures and the dynamical processes defined on them. We propose a possible dynamical mechanism for the enhancement effect of network structures on synchronization processes, based upon which a dynamic-based index of the synchronizability is introduced in the present paper.

Abnormal synchronization in complex clustered networks

Recent research has revealed that complex networks with a smaller average distance and more homogeneous degree distribution are more synchronizable. We find, however, that synchronization in complex, clustered networks tends to obey a different set of rules. In particular, the synchronizability of such a network is determined by the interplay between intercluster and intracluster links. The network is most synchronizable when the numbers of the two types of links are approximately equal. In the presence of a mismatch, increasing the number of intracluster links, while making the network distance smaller, can counterintuitively suppress or even destroy the synchronization. We provide theory and numerical evidence to establish this phenomenon.

Self-organized criticality and scale-free properties in emergent functional neural networks

Recent studies on complex systems have shown that the synchronization of oscillators, including neuronal ones, is faster, stronger, and more efficient in small-world networks than in regular or random networks. We show that the functional structures in the brain can be self-organized to both small-world and scale-free networks by synaptic reorganization via spike timing dependent synaptic plasticity instead of conventional Hebbian learning rules. We show that the balance between the excitatory and the inhibitory synaptic inputs is critical in the formation of the functional structure, which is found to lie in a self-organized critical state.

Dynamic behaviors in directed networks

Motivated by the abundance of directed synaptic couplings in a real biological neuronal network, we investigate the synchronization behavior of the Hodgkin-Huxley model in a directed network. We start from the standard model of the Watts-Strogatz undirected network and then change undirected edges to directed arcs with a given probability, still preserving the connectivity of the network. A generalized clustering coefficient for directed networks is defined and used to investigate the interplay between the synchronization behavior and underlying structural properties of directed networks. We observe that the directedness of complex networks plays an important role in emerging dynamical behaviors, which is also confirmed by a numerical study of the sociological game theoretic voter model on directed networks.

Periodic neural activity induced by network complexity

We study a model for neural activity on the small-world topology of Watts and Strogatz and on the scale-free topology of Barabási and Albert. We find that the topology of the network connections may spontaneously induce periodic neural activity, contrasting with nonperiodic neural activities exhibited by regular topologies. Periodic activity exists only for relatively small networks and occurs with higher probability when the rewiring probability is larger. The average length of the periods increases with the square root of the network size.

Network reorganization driven by temporal interdependence of its elements

We employ an adaptive parameter control technique based on detection of phase/lag synchrony between the elements of the system to dynamically modify the structure of a network of nonidentical, coupled Rossler oscillators. Two processes are simulated: adaptation, under which the initially different properties of the units converge, and rewiring, in which clusters of interconnected elements are formed based on the temporal correlations. We show how those processes lead to different network structures and investigate their optimal characteristics from the point of view of resulting network properties.

Ordering spatiotemporal chaos in complex thermosensitive neuron networks

We have studied the effect of random long-range connections in chaotic thermosensitive neuron networks with each neuron being capable of exhibiting diverse bursting behaviors, and found stochastic synchronization and optimal spatiotemporal patterns. For a given coupling strength, the chaotic burst-firings of the neurons become more and more synchronized as the number of random connections (or randomness) is increased and, rather, the most pronounced spatiotemporal pattern appears for an optimal randomness. As the coupling strength is increased, the optimal randomness shifts towards a smaller strength. This result shows that random long-range connections can tame the chaos in the neural networks and make the neurons more effectively reach synchronization. Since the model studied can be used to account for hypothalamic neurons of dogfish, catfish, etc., this result may reflect the significant role of random connections in transferring biological information.

The size of the sync basin

We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the "sync basin" (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ring of n >> 1 identical phase oscillators with sinusoidal coupling. Each oscillator interacts equally with its k nearest neighbors on either side. For k/n greater than a critical value (approximately 0.34, obtained analytically), we show that the sync basin is the whole phase space, except for a set of measure zero. As k/n passes below this critical value, coexisting attractors are born in a well-defined sequence. These take the form of uniformly twisted waves, each characterized by an integer winding number q, the number of complete phase twists in one circuit around the ring. The maximum stable twist is proportional to n/k; the constant of proportionality is also obtained analytically. For large values of n/k, corresponding to large rings or short-range coupling, many different twisted states compete for their share of phase space. Our simulations reveal that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q. Furthermore, as n/k increases, the standard deviation of this distribution grows linearly with square root of n/k. We have been unable to explain either of these last two results by anything beyond a hand-waving argument.

Synchronization in uncertain complex networks

We consider the problem of synchronization in uncertain generic complex networks. For generic complex networks with unknown dynamics of nodes and unknown coupling functions including uniform and nonuniform inner couplings, some simple linear feedback controllers with updated strengths are designed using the well-known LaSalle invariance principle. The state of an uncertain generic complex network can synchronize an arbitrary assigned state of an isolated node of the network. The famous Lorenz system is stimulated as the nodes of the complex networks with different topologies. We found that the star coupled and scale-free networks with nonuniform inner couplings can be in the state of synchronization if only a fraction of nodes are controlled.

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.

Spatiotemporal dynamics of networks of excitable nodes

A network of excitable nodes based on the photosensitive Belousov-Zhabotinsky reaction is studied in experiments and simulations. The addressable medium allows both local and nonlocal links between the nodes. The initial spread of excitation across the network as well as the asymptotic oscillatory behavior are described. Synchronization of the spatiotemporal dynamics occurs by entrainment to high-frequency network pacemakers formed by excitation loops. Analysis of the asymptotic behavior reveals that the dynamics of the network is governed by a subnetwork selected during the initial transient period.

Nonlocal coupling can prevent the collapse of spatiotemporal chaos

Spatiotemporal chaos on a regular ring network of excitable Gray-Scott dynamical elements is transient. We find that the addition of very few nonlocal network connections drastically changes the average lifetime of spatiotemporal chaos. In the presence of a single shortcut local interface formation delays the collapse of spatiotemporal chaos. This competes with a reduced average characteristic path length that advances the collapse process. Two added shortcuts can prevent the collapse of spatiotemporal chaos by causing an asymptotic local collapse.

Synchronizations in small-world networks of spiking neurons: diffusive versus sigmoid couplings

By using a semianalytical dynamical mean-field approximation previously proposed by the author [H. Hasegawa, Phys. Rev. E 70, 066107 (2004)], we have studied the synchronization of stochastic, small-world (SW) networks of FitzHugh-Nagumo neurons with diffusive couplings. The difference and similarity between results for diffusive and sigmoid couplings have been discussed. It has been shown that with introducing the weak heterogeneity to regular networks, the synchronization may be slightly increased for diffusive couplings, while it is decreased for sigmoid couplings. This increase in the synchronization for diffusive couplings is shown to be due to their local, negative feedback contributions, but not due to the short average distance in SW networks. Synchronization of SW networks depends not only on their structure but also on the type of couplings.

Collective synchronization in spatially extended systems of coupled oscillators with random frequencies

We study collective behavior of locally coupled limit-cycle oscillators with random intrinsic frequencies, spatially extended over d -dimensional hypercubic lattices. Phase synchronization as well as frequency entrainment are explored analytically in the linear (strong-coupling) regime and numerically in the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator phases are always desynchronized up to d=4 , which implies the lower critical dimension dP(l) =4 for phase synchronization. On the other hand, the oscillators behave collectively in frequency (phase velocity) even in three dimensions (d=3) , indicating that the lower critical dimension for frequency entrainment is dF(l)=2 . Nonlinear effects due to the periodic nature of limit-cycle oscillators are found to become significant in the weak-coupling regime: So-called runaway oscillators destroy the synchronized (ordered) phase and there emerges a fully random (disordered) phase. Critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation. Collective behavior of globally coupled oscillators is also examined and compared with that of locally coupled oscillators.

Transition from local to global phase synchrony in small world neural network and its possible implications for epilepsy

Temporal correlations in the brain are thought to have very dichotomous roles. On one hand they are ubiquitously present in the healthy brain and are thought to underlie feature binding during information processing. On the other hand, large-scale synchronization is an underlying mechanism of epileptic seizures. In this paper we show a potential mechanism for the transition to pathological coherence underlying seizure generation. We show that properties of phase synchronization in a two-dimensional lattice of nonidentical coupled Hindmarsh-Rose neurons change radically depending on the connectivity structure of the network. We modify the connectivity using the small world network paradigm and measure properties of phase synchronization using a previously developed measure based on assessment of the distributions of relative interspike intervals. We show that the temporal ordering undergoes a dramatic change as a function of topology of the network from local coherence strongly dependent on the distance between two neurons, to global coherence exhibiting a larger degree of ordering and spanning the whole network.

Synchronization on Erdös-Rényi networks

In this Brief Report, by analyzing the spectral properties of the Laplacian matrix of Erdös-Rényi networks, we obtained the critical coupling strength of the complete synchronization analytically. In particular, for any size of the networks, when the average degree is greater than a threshold and the coupling strength is large enough, the networks can synchronize. Here, the threshold is determined by the value of the maximal Lyapunov exponent of each dynamical unit.

Synchronization transition in scale-free networks: clusters of synchrony

We study the synchronization transition in scale-free networks that display power-law asymptotic behaviors in their degree distributions. The critical coupling strength and the order-parameter critical exponent derived by the mean-field approach depend on the degree exponent lambda, which implies a close connection between structural organization and the emergence of dynamical order in complex systems. We also derive the finite-size scaling behavior of the order parameter, finding that the giant cluster of synchronized nodes is formed in different ways between scale-free networks with 2 < lambda < 3 and those with lambda > 3.

Spatiotemporal networks in addressable excitable media

Spatiotemporal networks are studied in a photosensitive Belousov-Zhabotinsky medium that allows both local and nonlocal transmission of excitation. Local transmission occurs via propagating excitation waves, while nonlocal transmission takes place by nondiffusive jumps to destination sites linked to excited sites in the medium. Static, dynamic, and domain link networks are experimentally and computationally characterized. Transitions to synchronized behavior are exhibited with increasing link density, and power-law relations are observed for first-coverage time as a function of link probability.

Synchronized clusters in coupled map networks. I. Numerical studies

We study the synchronization of coupled maps on a variety of networks including regular one- and two-dimensional networks, scale-free networks, small world networks, tree networks, and random networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. When nodes show synchronized behavior, we observe two interesting phenomena. First, there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently. Second, we identify two different ways of cluster formation, namely self-organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings. The synchronized clusters may be of dominant self-organized type, dominant driven type, or mixed type depending on the type of network and the parameters of the dynamics. We define different states of the coupled dynamics by considering the number and type of synchronized clusters. For the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant (epsilon) and the logistic map parameter (mu). For large coupling strengths and nonlinear coupling we find that the scale-free networks and the Caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity. For most of our study we use the number of connections of the order of the number of nodes. As the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase.