Bursting synchronization in scale-free networks

Neuronal networks in some areas of the brain cortex present the scale-free property, i.e., the neuron connectivity is distributed according to a power-law, such that neurons are more likely to couple with other already well-connected ones. Neuron activity presents two timescales, a fast one related to action-potential spiking, and a slow timescale in which bursting takes place. Some pathological conditions are related with the synchronization of the bursting activity in a weak sense, meaning the adjustment of the bursting phase due to coupling. Hence it has been proposed that an externally applied time-periodic signal be applied in order to control undesirable synchronized bursting rhythms. We investigated this kind of intervention using a two-dimensional map to describe neurons with spiking–bursting activity in a scale-free network.     

Bursting mechanism in a time-delayed oscillator with slowly varying external forcing

This paper investigates the generation of complex bursting patterns in the Duffing oscillator with time-delayed feedback. We present the bursting patterns, including symmetric fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.     

Environmentally induced amplitude death and firing provocation in large-scale networks of neuronal systems

We study the dynamics of multielement neuronal systems taking into account both the direct interaction between the cells via linear coupling and nondiffusive cell-to-cell communication via common environment. For the cells exhibiting individual bursting behavior, we have revealed the dependence of the network activity on its scale. Particularly, we show that small-scale networks demonstrate the inability to maintain complicated oscillations: for a small number of elements in an ensemble, the phenomenon of amplitude death is observed. The existence of threshold network scales and mechanisms causing firing in artificial and real multielement neural networks, as well as their significance for biological applications, are discussed.     

Synchronization analysis through coupling mechanism in realistic neural models

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

Community structure in real-world networks from a non-parametrical synchronization-based dynamical approach

This work analyzes the problem of community structure in real-world networks based on the synchronization of nonidentical coupled chaotic Rössler oscillators each one characterized by a defined natural frequency, and coupled according to a predefined network topology. The interaction scheme contemplates an uniformly increasing coupling force to simulate a society in which the association between the agents grows in time. To enhance the stability of the correlated states that could emerge from the synchronization process, we propose a parameterless mechanism that adapts the characteristic frequencies of coupled oscillators according to a dynamic connectivity matrix deduced from correlated data. We show that the characteristic frequency vector that results from the adaptation mechanism reveals the underlying community structure present in the network.     

Delay-induced diversity of firing behavior and ordered chaotic firing in adaptive neuronal networks

In this paper, we study the effect of time delay on the firing behavior and temporal coherence and synchronization in Newman–Watts thermosensitive neuron networks with adaptive coupling. At beginning, the firing exhibit disordered spiking in absence of time delay. As time delay is increased, the neurons exhibit diversity of firing behaviors including bursting with multiple spikes in a burst, spiking, bursting with four, three and two spikes, firing death, and bursting with increasing amplitude. The spiking is the most ordered, exhibiting coherence resonance (CR)-like behavior, and the firing synchronization becomes enhanced with the increase of time delay. As growth rate of coupling strength or network randomness increases, CR-like behavior shifts to smaller time delay and the synchronization of firing increases. These results show that time delay can induce diversity of firing behaviors in adaptive neuronal networks, and can order the chaotic firing by enhancing and optimizing the temporal coherence and enhancing the synchronization of firing. However, the phenomenon of firing death shows that time delay may inhibit the firing of adaptive neuronal networks. These findings provide new insight into the role of time delay in the firing activity of adaptive neuronal networks, and can help to better understand the complex firing phenomena in neural networks.     

两个非线性耦合网络间的自适应同步

本文研究了带非线性信号连接的两个复杂网络间的同步问题,引入非线性耦合参数α来调节两个复杂网络的同步．若耦合参数不能保证网络达到外部同步,这里我们提出了一种自适应同步方式,通过此方式可以使两个复杂网络达到同步,最后通过简单的数值算例来阐述得到的理论结果,包括网络具有相同和不相同的拓扑结构两种情形．     

Synchronization in complex networks with distinct chaotic nodes

We present an approach to the chaos synchronization of complex networks with distinct nodes. The chaotic synchronization is achieved by adding a derivative coupling term in the network equation. We assume that node in networks are different and are given by the Lorenz, Rössler, Chen and Sprott chaotic systems. The derivative term is capable to induce the synchronous behavior in the network. Moreover such a coupling leads the global behavior to a chaotic attractor. We found that without derivative coupling the network is leaded only to an equilibrium point or a limit cycle. Numerical simulations are provided to illustrate the result. Complementary the network synchrony can be chaotic in presence of the derivative coupling.     

Ordering chaos and synchronization transitions by chemical delay and coupling on scale-free neuronal networks

Chemical synaptic connections are more common than electric ones in neurons, and information transmission delay is especially significant for the synapses of chemical type. In this paper, we report a phenomenon of ordering spatiotemporal chaos and synchronization transitions by the delays and coupling through chemical synapses of modified Hodgkin–Huxley (MHH) neurons on scale-free networks. As the delay τ is increased, the neurons exhibit transitions from bursting synchronization (BS) to intermittent multiple spiking synchronizations (SS). As the coupling g_syn is increased, the neurons exhibit different types of firing transitions, depending on the values of τ. For a smaller τ, there are transitions from spatiotemporal chaotic bursting (SCB) to BS or SS; while for a larger τ, there are transitions from SCB to intermittent multiple SS. These findings show that the delays and coupling through chemical synapses can tame the chaotic firings and repeatedly enhance the firing synchronization of neurons, and hence could play important roles in the firing activity of the neurons on scale-free networks.     

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

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

Broad-scale small-world network topology induces optimal synchronization of flexible oscillators

The discovery of small-world and scale-free properties of many man-made and natural complex networks has attracted increasing attention. Of particular interest is how the structural properties of a network facilitate and constrain its dynamical behavior. In this paper we study the synchronization of weakly coupled limit-cycle oscillators in dependence on the network topology as well as the dynamical features of individual oscillators. We show that flexible oscillators, characterized by near zero values of divergence, express maximal correlation in broad-scale small-world networks, whereas the non-flexible (rigid) oscillators are best correlated in more heterogeneous scale-free networks. We found that the synchronization behavior is governed by the interplay between the networks global efficiency and the mutual frequency adaptation. The latter differs for flexible and rigid oscillators. The results are discussed in terms of evolutionary advantages of broad-scale small-world networks in biological systems.     

Frequency transitions in synchronized neural networks

Temporal organization of events can emerge in complex systems, like neural networks. Here, random graph and cellular automaton are used to represent coupled neural structures, in order to investigate the occurrence of synchronization. The connectivity pattern of this toy model of neural system is of Newman–Watts type, formed from a regular lattice with additional random connections. Two networks with this coupling topology are connected by extra random links and an impulse stimulus is either constantly or periodically applied to a unique neuron. Numerical simulations reveal that this model can exhibit a variety of dynamic behaviors. Usually, the whole system achieves synchronization; however, the oscillation frequencies of the stimulus and of each network can be different. The dynamics is evaluated in function of the network size, the amount of the randomly added edges and the number of time steps in which a neuron can remain firing. The biological relevance of these results is discussed.     

Adaptive synchronization in an array of chaotic neural networks with mixed delays and jumping stochastically hybrid coupling

In this paper, a general model of an array of N linearly coupled delayed neural networks with Markovian jumping hybrid coupling is introduced. The hybrid coupling consists of constant coupling, discrete and distributed time-varying delay coupling. The complex dynamical network jumps from one mode to another according to a Markovian chain, where all the coupling configurations are also dependent on mode switching. Meanwhile, all the coupling terms are subjected to stochastic disturbances which are described in terms of a Brownian motion. By adaptive approach, some sufficient criteria have been derived to ensure the synchronization in an array of jump neural networks with mixed delays and hybrid coupling in mean square. Surprisingly, it is found that complex networks with two different structure can also be synchronized according to known probability matrix. Finally, an example illustrated by switching between small-world networks and nearest-neighbor networks is given to show the effectiveness of the proposed criteria.     

Natural synchronization in power-grids with anti-correlated units

Synchronization between spatially distributed nodes of a power-grid is a crucial requirement for its proper operation. Using a Kuramoto-like network as a realistic physical model for the distribution of electrical power in a power-grid, we obtain coupling strengths and topological characteristics that favor the synchronous state of those technological networks. Power-grids are highly heterogeneous. They are composed of different classes of nodes and each node behaves differently. We show in this work that if a power-grid is extensive and nodes are highly connected, the coupling strength that leads to synchronization depends mainly on the eigenvalues of the Laplacian matrix, as it happens in homogeneous networks composed of equal nodes. On the other hand, if a power-grid is sparsely connected, the coupling strength that leads to synchronization is also strongly related to the correlation coefficient of the network, which means that a high number of connections between similar nodes (two power plants or two consumer centers) disfavor the synchronizability of the power-grid. We apply our results to the Brazilian power-grid system.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Ordered chaotic bursting and multiple coherence resonance by time-periodic coupling strength in Newman–Watts neuronal networks

In this paper, we study the effect of time-periodic coupling strength (TPCS) on the temporal coherence of the chaotic bursting of Newman–Watts thermosensitive neuron networks. It is found that the chaotic bursting can exhibit coherence resonance and multiple coherence resonance behavior when TPCS amplitude and frequency is varied, respectively. It is also found that TPCS can also enhance the temporal coherence and spatial synchronization of the optimal spatio-temporal bursting in the case of fixed coupling strength. These results show that TPCS can tame the chaotic bursting and can repeatedly enhance the temporal coherence of the chaotic bursting neuronal networks. This implies that TPCS may play a more efficient role for improving the time precision of the information processing in chaotic bursting neurons.     

Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes

In this paper, complex networks with community structure and nonidentical nodes are investigated. The cluster mixed synchronization of these networks is studied by using some linear pinning control schemes. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Adaptive coupling strength method is adopted to achieve the synchronization as well. According to Lyapunov stability theory, several sufficient conditions for the network to achieve cluster mixed synchronization are derived. Numerical simulations are provided to verify the correctness and the effectiveness of the theoretical results.     

Generalized outer synchronization between non-dissipatively coupled complex networks with different-dimensional nodes

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.     

Chaotic synchronization of nearest-neighbor diffusive coupling Hindmarsh–Rose neural networks in noisy environments

The chaotic synchronization of Hindmarsh–Rose neural networks linked by a nonlinear coupling function is discussed. The HR neural networks with nearest-neighbor diffusive coupling form are treated as numerical examples. By the construction of a special nonlinear-coupled term, the chaotic system is coupled symmetrically. For three and four neurons network, a certain region of coupling strength corresponding to full synchronization is given, and the effect of network structure and noise position are analyzed. For five and more neurons network, the full synchronization is very difficult to realize. All the results have been proved by the calculation of the maximum conditional Lyapunov exponent.     

Local and global synchronization in general complex dynamical networks with delay coupling

Local and global synchronization of complex dynamical networks are studied in this paper. Some simple yet generic criteria ensuring delay-independent and delay-dependent synchronization are derived in terms of linear matrix inequalities (LMIs), which can be verified easily via interior-point algorithm. The assumption that the coupling configuration matrix is symmetric and irreducible, which is frequently used in other literatures, is removed. A network with a fixed delay and a special coupling scheme is given as an example to illustrate the theoretical results and the effectiveness of the proposed synchronization scheme.