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.     

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.     

Ordered bursting synchronization and complex wave propagation in a ring neuronal network

Ordered bursting synchronization and complex propagation are investigated for a ring neuronal network in which each neuron exhibits chaotic bursting behaviour. The neurons become more and more synchronous in chaotic bursting as the synaptic strength is increased. It is shown that excitatory chemical synapses can effectively tame the chaos, and ordered bursting synchronization can be observed as the synaptic strength is further increased. However, synchronization among neurons is weakened as the number of neurons is increased. More importantly, it is shown that ordered bursting synchronization can be turned into spiking synchronization at certain noise intensity. Complex spatio-temporal patterns propagating towards both sides of pacemaker are found in this network before the emergence of spiking synchronization.     

Pre-B?tzinger复合体的从簇到峰放电的同步转迁及分岔机制

Pre-Bötzinger复合体是兴奋性耦合的神经元网络,通过产生复杂的放电节律和节律模式的同步转迁参与调控呼吸节律.本文选用复杂簇和峰放电节律的单神经元数学模型构建复合体模型,仿真了与生物学实验相关的多类同步节律模式及其复杂转迁历程,并利用快慢变量分离揭示了相应的分岔机制.当初值相同时,随着兴奋性耦合强度的增加,复合体模型依次表现出完全同步的“fold/homoclinic”,“subHopf/subHopf”簇放电和周期1峰放电.当初值不同时,随耦合强度增加,表现为由“fold/homoclinic”,到“fold/fold limit cycle”、到“subHopf/subHopf”与“fold/fold limit cycle”的混合簇放电、再到“subHopf/subHopf”簇放电的相位同步转迁,最后到反相同步周期1峰放电.完全(同相)同步和反相同步的周期1节律表现出了不同分岔机制.反相峰同步行为给出了与强兴奋性耦合容易诱发同相同步这一传统观念不同的新示例.研究结果给出了preBötzinger复合体的从簇到峰放电节律的同步转迁规律及复杂分岔机制,反常同步行为丰富了非线性动力学的内涵.     

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.     

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.     

Cluster synchronization in networks of distinct groups of maps

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.     

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.     

Taming desynchronized bursting with delays in the Macaque cortical network

Inhibitory coupled bursting Hindmarsh-Rose neurons are considered as constitutive units of the Macaque cortical network. In the absence of information transmission delay the bursting activity is desynchronized,giving rise to spatiotemporally disordered dynamics. This paper shows that the introduction of finite delays can lead to the synchronization of bursting and thus to the emergence of coherent propagating fronts of excitation in the space-time domain. Moreover,it shows that the type of synchronous bursting is uniquely determined by the delay length,with the transitions from one type to the other occurring in a step-like manner depending on the delay. Interestingly,as the delay is tuned close to the transition points,the synchronization deteriorates,which implies the coexistence of different bursting attractors. These phenomena can be observed by different but fixed coupling strengths,thus indicating a new role for information transmission delays in realistic neuronal networks.     

Nonlocal Mechanism for Synchronization of Time Delay Networks

We present the interplay between synchronization of networks with heterogeneous delays and the greatest common divisor (GCD) of loops composing the network. We distinguish between two types of networks; (I) chaotic networks and (II) population dynamic networks with periodic activity driven by external stimuli. For type (I), in the weak chaos region, the units of a chaotic network characterized by GCD=1 are in a chaotic zero-lag synchronization, whereas for GCD>1, the network splits into GCD-clusters in which clustered units are in zero-lag synchronization. These results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps. Type (II) is exemplified by simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity, synaptic noise and distribution of delays within neurons belonging to a node and between connecting nodes. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the network splits into clusters equal to the greatest common divisor of loops composing the network (spatial) and the periodicity of the external stimuli (temporal). The results suggest that neural information processing may take place in the transient to synchronization and imply a much shorter time scale for the inference of a perceptual entity.     

“Hopf/homoclinic”簇放电和“SubHopf/homoclinic"簇放电之间的同步

以修正过的Morris-Lecar神经元模型为例,讨论了“Hopf/homoclinic”簇放电和“SubHopf/homoclinic"簇放电之间的同步行为.首先,分别考察了同一拓扑类型的两个耦合簇放电神经元的同步行为,发现“Hopf/homoclinic”簇放电比“SubHopf/homoclinic”簇放电达到膜电位完全同步所需要的耦合强度小,即前者比后者更容易达到膜电位完全同步.其次,对这两个不同拓扑类型的簇放电神经元的耦合同步行为进行了讨论.通过数值分析发现随着耦合强度的增加,两种不同类型的簇放电首先达到簇放电同步,然后当耦合强度足够大时甚至可以达到膜电位完全同步,并且同步后的放电类型更接近容易同步的簇放电类型,即“Hopf/homoclinic”簇放电.然而令人奇怪的是此时慢变量并没有达到完全同步,而是相位同步;慢变量之间呈现为一种线性关系.这一点和现有文献的结果截然不同.     

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.     

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.     

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.     

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.     

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.     

Self-organized and driven phase synchronization in coupled maps

We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation: (a) self-organized phase synchronization which leads to clusters with dominant intracluster couplings and (b) driven phase synchronization which leads to clusters with dominant intercluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.     

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.     

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.     

Synchronization in the network of chaotic microwave oscillators

Time scale synchronization in networks of chaotic microwave oscillators with the different topologies of the links between nodes has been studied. As a node element of the network the one-dimensional distributed model of the low-voltage vircator has been used. To characterize the degree of synchronization in the whole network the synchronization index has been introduced. The transition to the synchronous regime is shown to take place via cluster time scale synchronization. Meanwhile, the spectral structure of the output signals is complicated sufficiently which allows using such devices in a number of practical applications.