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.     

Small-World Connections to Induce Firing Activity and Phase Synchronization in Neural Networks

We investigate how the firing activity and the subsequent phase synchronization of neural networks with smallworld topological connections depend on the probability p of adding-links. Network elements are described by two-dimensional map neurons （2DMNs） in a quiescent original state. Neurons burst for a given coupling strength when the topological randomness p increases, which is absent in a regular-lattice neural network. The bursting activity becomes frequent and synchronization of neurons emerges as topological randomness further increases. The maximal firing frequency and phase synchronization appear at a particular value of p. However, if the randomness p further increases, the firing frequency decreases and synchronization is apparently destroyed.     

Evolutionary snowdrift game on heterogeneous Newman--Watts small-world network

We study the evolutionary snowdrift game in a heterogeneous Newman-Watts small-world network. The heterogeneity of the network is controlled by the number of hubs. It is found that the moderate heterogeneity of the network can promote the cooperation best. Besides, we study how the hubs affect the evolution of cooperative behaviours of the heterogeneous Newman-Watts small-world network. Simulation results show that both the initial states of hubs and the connections between hubs can play an important role. Our work gives a further insight into the effect of hubs on the heterogeneous networks.     

When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons

We show that weak common inhibition applied to a network of bursting neurons with strong desynchronizing connections can induce burst and complete synchronization. We demonstrate that the weak synchronizing inhibition from the same pacemaker neuron can win out over much stronger desynchronizing connections within the network, provided that the neuron's duty cycle is sufficiently long. We also gain insight into how the changes in burst duty cycles can trigger unexpected clusters of synchrony in bursting networks.     

Bifurcation in neuronal networks with hub structure

We investigate bifurcations in neuronal networks with a hub structure. It is known that hubs play a leading role in characterizing the network dynamical behavior. However, the dynamics of hubs or star-coupled systems is not well understood. Here, we study rather subnetworks with a star-like configuration. This coupled system is an important motif in complex networks. Thus, our study is a basic step for understanding structure formation in large networks. We use the Morris-Lecar neuron with class I and class II excitabilities as a node. Homogeneous (coupling the same class neurons) and heterogeneous (coupling different class neurons) cases are considered for both excitatory and inhibitory coupling. For the homogeneous system class II neurons are suitable for achieving both complete and cluster synchronization in excitatory and inhibitory coupling, respectively. For the heterogeneous system with inhibitory coupling, the class I hub neuron has a wider parameter region of synchronous firings than the class II hub. Moreover, the class I hub neuron with the excitatory synapse gives rise to bifurcations of synchronized states and multi-stability (coexistence of a few different states) is observed.     

Generalized synchronization of coupled chaotic systems

In this paper we briefly report some recent developments on generalized synchronization. We discuss different methods of detecting generalized synchronization. We first consider two unidirectionally coupled systems and then two mutually coupled systems. We then extend the study to a network of coupled systems. In the study of generalized synchronization of coupled nonidentical systems we discuss the Master Stability Function (MSF) formalism for coupled nearly identical systems. Later we use this MSF to construct synchronized optimized networks. In the optimized networks the nodes which have parameter value at one extreme are chosen as hubs and the pair of nodes with larger difference in parameter are chosen to create links.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Role of connectivity in congestion and decongestion in a communication network

We study network traffic dynamics in a two dimensional communication network with regular nodes and hubs. If the network experiences heavy message traffic, congestion occurs due to finite capacity of the nodes. We discuss strategies to manipulate hub capacity and hub connections to relieve congestion and define a coefficient of betweenness centrality (CBC), a direct measure of network traffic, which is useful for identifying hubs which are most likely to cause congestion. The addition of assortative connections to hubs of high CBC relieves congestion very efficiently. An erratum to this article is available at .     

Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling

We study synchronization transitions and pattern formation on small-world networks consisting of Morris-Lecar excitable neurons in dependence on the information transmission delay and the rewiring probability. In addition, networks formed via gap junctional connections and coupling via chemical synapses are considered separately. For gap-junctionally coupled networks we show that short delays can induce zigzag fronts of excitations, whereas long delays can further detriment synchronization due to a dynamic clustering anti-phase synchronization transition. For the synaptically coupled networks, on the other hand, we find that the clustering anti-phase synchronization can appear as a direct consequence of the prolongation of information transmission delay, without being accompanied by zigzag excitatory fronts. Irrespective of the coupling type, however, we show that an appropriate small-world topology can always restore synchronized activity if only the information transmission delays are short or moderate at most. Long information transmission delays always evoke anti-phase synchronization and clustering, in which case the fine-tuning of the network topology fails to restore the synchronization of neuronal activity.     

Spiking sychronization regulated by noise in three types of Hodgkin—Huxley neuronal networks

In this paper,we study spiking synchronization in three different types of Hodgkin-Huxley neuronal networks,which are the small-world,regular,and random neuronal networks.All the neurons are subjected to subthreshold stimulus and external noise.It is found that in each of all the neuronal networks there is an optimal strength of noise to induce the maximal spiking synchronization.We further demonstrate that in each of the neuronal networks there is a range of synaptic conductance to induce the effect that an optimal strength of noise maximizes the spiking synchronization.Only when the magnitude of the synaptic conductance is moderate,will the effect be considerable.However,if the synaptic conductance is small or large,the effect vanishes.As the connections between neurons increase,the synaptic conductance to maximize the effect decreases.Therefore,we show quantitatively that the noise-induced maximal synchronization in the Hodgkin-Huxley neuronal network is a general effect,regardless of the specific type of neuronal network.     

The role of neural architecture and the speed of signal propagation in the process of synchronization of bursting neurons

Synchronized neuronal activity has been observed at all levels of human and any other nervous systems and was suggested as particularly relevant in information processing and coding. In the present paper we investigate the synchronization of bursting neuronal activity. Motivated by the fact that in neural systems the interplay between the network structure and the dynamics taking place on it is closely interrelated, we develop a spatial network representation of neural architecture in which we can tune the network organization between a scale-free network with dominating long-range connections and a homogeneous network with mostly adjacent neurons connected. Our results reveal that the most synchronized response is obtained for the intermediate regime where long- as well as short-range connections constitute the neural architecture. Moreover, the optimal response is additionally enhanced when the speed of signal propagation is optimized.     

Multi-time-scale synchronization and information processing in bursting neuron networks

We analyze the effect of synchronization in networks of chemically coupled multi-time-scale (spiking-bursting) neurons on the process of information transmission within the network. Although, synchronization occurs first in the slow time-scale (burst) and then in the fast time-scale (spike), we show that information can be transmitted with low probability of errors in both time scales when the bursts become synchronized. Furthermore, we show that for networks of non-identical multi-time-scales neurons, complete synchronization is no longer possible, but instead burst phase synchronization. Our analysis shows that clusters of burst phase synchronized neurons are very likely to appear in a network for parameters far smaller than the ones for which the onset of burst phase synchronization in the whole network takes place.     

Explosive phenomena in complex networks

The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in an attempt to delay percolation. This observation of ‘explosive percolation’ was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena in networked systems and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.     

Synchronization in complex networks with a modular structure

Networks with a community (or modular) structure arise in social and biological sciences. In such a network individuals tend to form local communities, each having dense internal connections. The linkage among the communities is, however, much more sparse. The dynamics on modular networks, for instance synchronization, may be of great social or biological interest. (Here by synchronization we mean some synchronous behavior among the nodes in the network, not, for example, partially synchronous behavior in the network or the synchronizability of the network with some external dynamics.) By using a recent theoretical framework, the master-stability approach originally introduced by Pecora and Carroll in the context of synchronization in coupled nonlinear oscillators, we address synchronization in complex modular networks. We use a prototype model and develop scaling relations for the network synchronizability with respect to variations of some key network structural parameters. Our results indicate that random, long-range links among distant modules is the key to synchronization. As an application we suggest a viable strategy to achieve synchronous behavior in social networks.     

Synchronization scenarios of chimeras in multiplex networks

Complex networks consisting of several interacting layers allow for remote synchronization of distant layers via an intermediate relay layer. We extend the notion of relay synchronization to chimera states, and study the scenarios of relay synchronization in a three-layer network of FitzHugh–Nagumo (FHN) oscillators, where each layer has a nonlocal coupling topology. Varying the coupling strength and time delay in the inter-layer connections, we observe relay synchronization between chimera states, i.e., complex spatio-temporal patterns of coexisting coherent and incoherent domains, in the outer network layers. Special regimes where only the coherent domains of chimeras are synchronized, and the incoherent domains remain desynchronized, as well as transitions between different synchronization regimes are analyzed.     

Bursting synchronization in clustered neuronal networks

Neuronal networks in the brain exhibit the modular (clustered) property, i.e., they are composed of certain subnetworks with differential internal and external connectivity. We investigate bursting synchronization in a clustered neuronal network. A transition to mutual-phase synchronization takes place on the bursting time scale of coupled neurons, while on the spiking time scale, they behave asynchronously. This synchronization transition can be induced by the variations of inter- and intra- coupling strengths, as well as the probability of random links between different subnetworks. Considering that some pathological conditions are related with the synchronization of bursting neurons in the brain, we analyze the control of bursting synchronization by using a time-periodic external signal in the clustered neuronal network. Simulation results show a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even in the presence of external driving. Hence, effective synchronization suppression can be realized with the driving parameters outside the frequency locking region.     

Synaptic plasticity modulates autonomous transitions between waking and sleep states: insights from a Morris-Lecar model

The transitions between waking and sleep states are characterized by considerable changes in neuronal firing. During waking, neurons fire tonically at irregular intervals and a desynchronized activity is observed at the electroencephalogram. This activity becomes synchronized with slow wave sleep onset when neurons start to oscillate between periods of firing (up-states) and periods of silence (down-states). Recently, it has been proposed that the connections between neurons undergo potentiation during waking, whereas they weaken during slow wave sleep. Here, we propose a dynamical model to describe basic features of the autonomous transitions between such states. We consider a network of coupled neurons in which the strength of the interactions is modulated by synaptic long term potentiation and depression, according to the spike time-dependent plasticity rule (STDP). The model shows that the enhancement of synaptic strength between neurons occurring in waking increases the propensity of the network to synchronize and, conversely, desynchronization appears when the strength of the connections become weaker. Both transitions appear spontaneously, but the transition from sleep to waking required a slight modification of the STDP rule with the introduction of a mechanism which becomes active during sleep and changes the proportion between potentiation and depression in accordance with biological data. At the neuron level, transitions from desynchronization to synchronization and vice versa can be described as a bifurcation between two different states, whose dynamical regime is modulated by synaptic strengths, thus suggesting that transition from a state to an another can be determined by quantitative differences between potentiation and depression.     

Synchronization structure of evolving epileptic networks using cross-entropy

In this paper we present connectivity patterns of evolving large scale epileptic networks. We employed a cross-entropy measure in the frequency domain on EEG signals to infer the networks, before and during episodes of epileptic seizures. This measure allowed us to make a richer portrait about the node interactions on the graph and to identify emergent structures associated with the synchronization of brain activity. Our results points to a more complex scenario of network organization than the synchronized/unsynchronized dichotomy, with two main results: first, showing regions with unsynchronized (or independent) behavior, even during absence seizures, contradicting the concept of hypersynchrony. Furthermore, we explore the cross-entropy fluctuations along the seizure: a group of nodes became more similar over time while another group became more different, showing a complementary behaviour and different local brain activities. These results bring new questions about the spreading and the sustenance of the epileptic seizures and others synchronization phenomena in living systems.     

Synchronization of bursting neurons: what matters in the network topology

We study the influence of coupling strength and network topology on synchronization behavior in pulse-coupled networks of bursting Hindmarsh-Rose neurons. Surprisingly, we find that the stability of the completely synchronous state in such networks only depends on the number of signals each neuron receives, independent of all other details of the network topology. This is in contrast with linearly coupled bursting neurons where complete synchrony strongly depends on the network structure and number of cells. Through analysis and numerics, we show that the onset of synchrony in a network with any coupling topology admitting complete synchronization is ensured by one single condition.     

Burst synchronization transitions in a neuronal network of subnetworks

In this paper, the transitions of burst synchronization are explored in a neuronal network consisting of subnetworks. The studied network is composed of electrically coupled bursting Hindmarsh-Rose neurons. Numerical results show that two types of burst synchronization transitions can be induced not only by the variations of intra- and intercoupling strengths but also by changing the probability of random links between different subnetworks and the number of subnetworks. Furthermore, we find that the underlying mechanisms for these two bursting synchronization transitions are different: one is due to the change of spike numbers per burst, while the other is caused by the change of the bursting type. Considering that changes in the coupling strengths and neuronal connections are closely interlaced with brain plasticity, the presented results could have important implications for the role of the brain plasticity in some functional behavior that are associated with synchronization.