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.     

振幅耦合动态网络中相邻结点间的相同步

研究用适当的量化指标来刻画动态网络的相同步,为此定义了新的量化指标:相邻结点的网络平均锁相值和网络平均相频差.动态网络结点选择的是多旋转中心的Lorenz混沌振子,对Lorenz系统进行柱面坐标变换,用振幅耦合方法构造动态网络.分别对星形网络和小世界网络进行了仿真计算,结果表明随着耦合强度的增大,网络中相邻结点的两个系统之间存在相同步现象,而且相同步行为与定义的量化指标之间存在较准确的对应关系.     

The adaptive coupling scheme and the heterogeneity in intrinsic frequency and degree distributions of the complex networks

In the paper, we applied an adaptive principle to three kinds of complex networks as well as a random network within the context of the Kuramoto model. We found that the adaptive scheme could suppress the negative effect of the heterogeneity in the networks and the phase synchronization is enhanced obviously. The paper mainly investigates the adaptive coupling scheme in the small-world network, the scale-free network, and the modular network. Comparing with other weighted or unweighted static coupling schemes, the adaptive coupling scheme has a better performance in synchronization and communication efficiency, and provides a more realistic picture of synchronization in complex networks.     

Delay-induced synchronization transitions in small-world neuronal networks with hybrid electrical and chemical synapses

We study the dependence of synchronization transitions in small-world networks of bursting neurons with hybrid electrical–chemical synapses on the information transmission delay, the probability of electrical synapses, and the rewiring probability. It is shown that, irrespective of the probability of electrical synapses, the information transmission delay can always induce synchronization transitions in small-world neuronal networks, i.e., regions of synchronization and nonsynchronization appear intermittently as the delay increases. In particular, all these transitions to burst synchronization occur approximately at integer multiples of the bursting period of individual neurons. In addition, for larger probability of electrical synapses, the intermittent synchronization transition is more profound, due to the stronger synchronization ability of electrical synapses compared with chemical ones. More importantly, chemical and electrical synapses can perform complementary roles in the synchronization of hybrid small-world neuronal networks: the larger the electrical synapse strength is, the smaller the chemical synapse strength needed to achieve burst synchronization. Furthermore, the small-world topology has a significant effect on the synchronization transition in hybrid neuronal networks. It is found that increasing the rewiring probability can always enhance the synchronization of neuronal activity. The results obtained are instructive for understanding the synchronous behavior of neural systems.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Generalized synchronization in complex dynamical networks via adaptive couplings

This paper investigates generalized synchronization of three typical classes of complex dynamical networks: scale-free networks, small-world networks, and interpolating networks. The proposed synchronization strategy is to adjust adaptively a node’s coupling strength based on the node’s local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. It is demonstrated that the coupling strengths are affected by topologies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we discuss in detail how a network’s degree of heterogeneity affects its generalization synchronization behavior.     

Enhancing the network synchronizability

The structural and dynamical properties, particularly the small-world effect and scale-free feature, of complex networks have attracted tremendous interest and attention in recent years. This article offers a brief review of one focal issue concerning the structural and dynamical behaviors of complex network synchronization. In the presentation, the notions of synchronization of dynamical systems on networks, stability of dynamical networks, and relationships between network structure and synchronizability, will be first introduced. Then, various technical methods for enhancing the network synchronizability will be discussed, which are roughly divided into two classes: Structural Modification and Coupling-Pattern Regulation, where the former includes three typical methods—dividing hub nodes, shortening average distances, and deleting overload edges, while the latter mainly is a method of strengthening the hub-nodes’ influence on the network.      

Stochastic resonance and synchronization behaviors of excitatory-inhibitory small-world network subjected to electromagnetic induction

The phenomenon of stochastic resonance and synchronization on some complex neuronal networks have been investigated extensively.These studies are of great significance for us to understand the weak signal detection and information transmission in neural systems.Moreover,the complex electrical activities of a cell can induce time-varying electromagnetic fields,of which the internal fluctuation can change collective electrical activities of neuronal networks.However,in the past there have been a few corresponding research papers on the influence of the electromagnetic induction among neurons on the collective dynamics of the complex system.Therefore,modeling each node by imposing electromagnetic radiation on the networks and investigating stochastic resonance in a hybrid network can extend the interest of the work to the understanding of these network dynamics.In this paper,we construct a small-world network consisting of excitatory neurons and inhibitory neurons,in which the effect of electromagnetic induction that is considered by using magnetic flow and the modulation of magnetic flow on membrane potential is described by using memristor coupling.According to our proposed network model,we investigate the effect of induced electric field generated by magnetic stimulation on the transition of bursting phase synchronization of neuronal system under electromagnetic radiation.It is shown that the intensity and frequency of the electric field can induce the transition of the network bursting phase synchronization.Moreover,we also analyze the effect of magnetic flow on the detection of weak signals and stochastic resonance by introducing a subthreshold pacemaker into a single cell of the network and we find that there is an optimal electromagnetic radiation intensity,where the phenomenon of stochastic resonance occurs and the degree of response to the weak signal is maximized.Simulation results show that the extension of the subthreshold pacemaker in the network also depends greatly on coupling strength.The presented results may have important implications for the theoretical study of magnetic stimulation technology,thus promoting further development of transcranial magnetic stimulation(TMS) as an effective means of treating certain neurological diseases.     

Chaotic phase synchronization in small-world networks of bursting neurons

We investigate the chaotic phase synchronization in a system of coupled bursting neurons in small-world networks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that phase synchronization is largely facilitated by a large fraction of shortcuts, but saturates when it exceeds a critical value. We also study the external chaotic phase synchronization of bursting oscillators in the small-world network by a periodic driving signal applied to a single neuron. It is demonstrated that there exists an optimal small-world topology, resulting in the largest peak value of frequency locking interval in the parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this interval increases with the driving amplitude, but decrease rapidly with the network size. We infer that the externally applied driving parameters outside the frequency locking region can effectively suppress pathologically synchronized rhythms of bursting neurons in the brain.     

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.     

Vibrational resonance in excitable neuronal systems

In this paper, we investigate the effect of a high-frequency driving on the dynamical response of excitable neuronal systems to a subthreshold low-frequency signal by numerical simulation. We demonstrate the occurrence of vibrational resonance in spatially extended neuronal networks. Different network topologies from single small-world networks to modular networks of small-world subnetworks are considered. It is shown that an optimal amplitude of high-frequency driving enhances the response of neuron populations to a low-frequency signal. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, such as the coupling strength between neurons, network size, and rewiring probability of single small-world networks, as well as the number of links between different subnetworks and the number of subnetworks in the modular networks. All these parameters play a key role in determining the ability of the network to enhance the outreach of the localized subthreshold low-frequency signal. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for the weak signal detection and information propagation across neuronal systems.     

Synchronization of Phase Oscillators in Networks with Certain Frequency Sequence

Synchronization of Kuramoto phase oscillators arranged in real complex neural networks is investigated. It is shown that the synchronization greatly depends on the sets of natural frequencies of the involved oscillators. The influence of network connectivity heterogeneity on synchronization depends particularly on the correlation between natural frequencies and node degrees. This finding implies a potential application that inhibiting the effects caused by the changes of network structure can be bManced out nicely by choosing the correlation parameter appropriately.     

Synchronization in complex clustered networks

Synchronization in complex networks has been an active area of research in recent years. While much effort has been devoted to networks with the small-world and scale-free topology, structurally they are often assumed to have a single, densely connected component. Recently it has also become apparent that many networks in social, biological, and technological systems are clustered, as characterized by a number (or a hierarchy) of sparsely linked clusters, each with dense and complex internal connections. Synchronization is fundamental to the dynamics and functions of complex clustered networks, but this problem has just begun to be addressed. This paper reviews some progress in this direction by focusing on the interplay between the clustered topology and network synchronizability. In particular, there are two parameters characterizing a clustered network: the intra-cluster and the inter-cluster link density. Our goal is to clarify the roles of these parameters in shaping network synchronizability. By using theoretical analysis and direct numerical simulations of oscillator networks, it is demonstrated that clustered networks with random inter-cluster links are more synchronizable, and synchronization can be optimized when inter-cluster and intra-cluster links match. The latter result has one counterintuitive implication: more links, if placed improperly, can actually lead to destruction of synchronization, even though such links tend to decrease the average network distance. It is hoped that this review will help attract attention to the fundamental problem of clustered structures/synchronization in network science.      

Chaotic phase synchronization in a modular neuronal network of small-world subnetworks

We investigate the onset of chaotic phase synchronization of bursting oscillators in a modular neuronal network of small-world subnetworks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that this bursting synchronization transition can be induced not only by the variations of inter- and intra-coupling strengths but also by changing the probability of random links between different subnetworks. We also analyze the effect of external chaotic phase synchronization of bursting behavior in this clustered network by an external time-periodic signal applied to a single neuron. Simulation results demonstrate a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this synchronization region increases with the signal amplitude and the number of driven neurons but decreases rapidly with the network size. Considering that the synchronization of bursting neurons is thought to play a key role in some pathological conditions, the presented results could have important implications for the role of externally applied driving signal in controlling bursting activity in neuronal ensembles.     

Modeling and adaptive pinning synchronization control for a chaotic-motion motor in complex network

We introduce a chaos model for a permanent-magnet synchronous motor and construct a coupled chaotic motor in a complex dynamic network using the Newman–Watts small-world network algorithm. We apply adaptive pinning control theory for complex networks to obtain suitable adaptive feedback gain and the number of nodes to be pinned. Nodes of low degree are pinned to realize global asymptotic synchronization in the complex network. The proposed adaptive pinning controller is added to the complex motor network for simulation and verification.     

Explosive synchronization: From synthetic to real-world networks

Synchronization is a widespread phenomenon in both synthetic and real-world networks. This collective behavior of simple and complex systems has been attracting much research during the last decades. Two different routes to synchrony are defined in networks; first-order, characterized as explosive, and second-order, characterized as continuous transition. Although pioneer researches explained that the transition type is a generic feature in the networks, recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization. The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions. Despite different theoretical analyses about the appearance of the first-order transition, studies are limited to the mean-field theory, which cannot be generalized to all networks. There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization, e.g., the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks. In this review article, explosive synchronization is discussed from two main aspects. First, pioneer articles are categorized from the dynamical-structural framework point of view. Then, articles that considered different oscillators in the explosive synchronization frameworks are studied. In this article, the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators. Also, efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.     

Global synchronization in general complex delayed dynamical networks and its applications

The main objective of the present paper is further to investigate global synchronization of a general model of complex delayed dynamical networks. Based on stability theory on delayed dynamical systems, some simple yet less conservative criteria for both delay-independent and delay-dependent global synchronization of the networks are derived analytically. It is shown that under some conditions, if the uncoupled dynamical node is stable itself, then the network can be globally synchronized for any coupling delays as long as the coupling strength is small enough. On the other hand, if each dynamical node of the network is chaotic, then global synchronization of the networks is heavily dependent on the effects of coupling delays in addition to the connection configuration. Furthermore, the results are applied to some typical small-world (SW) and scale-free (SF) complex networks composing of coupled dynamical nodes such as the cellular neural networks (CNNs) and the chaotic FHN neuron oscillators, and numerical simulations are given to verify and also visualize the theoretical results.     

不同耦合下混沌神经元网络的同步

采用Hindmarsh-Rose神经元动力学模型, 对二维点阵上的神经元网络的同步进行了研究. 为了解不同耦合对网络同步的影响, 提出了一般反馈耦合、分层反馈耦合和分层局域平均场反馈耦合三种方案.研究表明:在耦合强度较小的近邻耦合下, 一般反馈耦合不能使网络达到完全同步, 而分层反馈耦合和分层局域平均场反馈耦合可以使网络出现局部同步和全局同步. 不同形式的耦合会导致网络出现不同的斑图, 随着耦合强度的增大, 网络从不同步到同步的过程也不相同, 一般反馈耦合和分层反馈耦合网络是突然出现全局同步, 同步之前网络出现非周期性的相干斑图; 对于分层局域平均场反馈耦合网络, 同层神经元之间先出现从簇放电同步到同步的转变, 形成靶波, 然后同步区由中心向外逐渐扩大, 最终达到网络的全局同步. 这些结果表明, 只有适当的耦合才能实现信号的无损耗的传递. 此外我们发现分层局域平均场反馈耦合可以促进网络的同步.     

Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network

The spatiotemporal patterns and chaotic burst synchronization of a small-world neuronal network are studied in this paper. The synchronization parameter, similarity parameter and order parameter are introduced to investigate the dynamics behaviour of the neurons. Chaotic burst synchronization and nearly complete synchronization can be observed if the link probability and the coupling strength are large enough. It is found that with increasing link probability and the coupling strength chaotic bursts become appreciably synchronous in space and coherent in time, and the maximal spatiotemporal order appears at some particular values of the probability and the coupling strength, respectively. The larger the size of the network, the smaller the probability and the coupling strength are needed for the network to achieve burst synchronization. Moreover, the bursting activity and the spatiotemporal patterns are robust to small noise.     

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.