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 and spiking due to additional direct and stochastic currents in neuron models

Neurons at rest can exhibit diverse firing activities patterns in response to various external deterministic and random stimuli, especially additional currents. In this paper, neuronal firing patterns from bursting to spiking, induced by additional direct and stochastic currents, are explored in rest states corresponding to two values of the parameter $V_{\rm K}$ in the Chay neuron system. Three cases are considered by numerical simulation and fast/slow dynamic analysis, in which only the direct current or the stochastic current exists, or the direct and stochastic currents coexist. Meanwhile, several important bursting patterns in neuronal experiments, such as the period-1 ``circle/homoclinic" bursting and the integer multiple ``fold/homoclinic" bursting with one spike per burst, as well as the transition from integer multiple bursting to period-1 ``circle/homoclinic" bursting and that from stochastic ``Hopf/homoclinic" bursting to ``Hopf/homoclinic" bursting, are investigated in detail.     

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.     

Synchronization transition in gap-junction-coupled leech neurons

Real neurons can exhibit various types of firings including tonic spiking, bursting as well as silent state, which are frequently observed in neuronal electrophysiological experiments. More interestingly, it is found that neurons can demonstrate the co-existing mode of stable tonic spiking and bursting, which depends on initial conditions. In this paper, synchronization in gap-junction-coupled neurons with co-existing attractors of spiking and bursting firings is investigated as the coupling strength gets increased. Synchronization transitions can be identified by means of the bifurcation diagram and the correlation coefficient. It is illustrated that the coupled neurons can exhibit different types of synchronization transitions between spiking and bursting when the coupling strength increases. In the course of synchronization transitions, an intermittent synchronization can be observed. These results may be instructive to understand synchronization transitions in neuronal systems.     

神经元电活动不同节律模式的几种变化过程

利用神经元Chay模型,对实验中观察到的三种放电节律模式序列进行数值模拟,并应用余维1极限环分岔分析研究了其产生机理.首先考虑的是周期性放电模式的变化过程;其次,具有不同表象的一种超临界和一种亚临界倍周期簇放电序列产生并导致混沌现象的出现,然后以不同的方式转迁到逆超临界倍周期峰放电序列;最后研究无混沌的加周期簇放电序列,得出加周期分岔仅是一种与倍周期分岔密切相关的分岔现象.     

三个不同时间尺度的电耦合模型的组合簇放电

胰岛中间隙连接的胰腺β细胞的簇放电行为对胰岛素分泌起着重要的作用. 本文利用了最小的phantom 簇放电模型, 研究两个电耦合胰腺β细胞具有簇同步的组合簇放电, 其膜电位表现出一个长簇和几个短簇组成的放电簇集和振幅先减小后增大的小振幅阈下振荡的相互转迁. 在两个慢变量和快的膜电位的三维空间中, 分别考虑了中慢变量和慢慢变量作为分岔参数的多层次的快慢动力学分析, 研究这两个时间尺度不同的慢变量如何共同或单独地控制着这种组合簇放电的复杂动力学行为. 特别地, 探讨了耦合强度引起的组合簇放电的每个簇集中短簇个数变化的内在机理. 关键词： 电耦合 具有不同时间尺度的慢变量 组合簇放电 快慢动力学分析     

Phase synchronization and its transition in two coupled bursting neurons: theoretical and numerical analysis

It is crucially important to study different synchronous regimes in coupled neurons because different regimes may correspond to different cognitive and pathological states. In this paper, phase synchronization and its transitions are discussed by means of theoretical and numerical analyses. In two coupled modified Morris--Lecar neurons with a gap junction, we show that the occurrence of phase synchronization can be investigated from the dynamics of phase equation, and the analytical synchronization condition is derived. By defining the phase of spike and burst, the transitions from burst synchronization to spike synchronization and then toward nearly complete synchronization can be identified by bifurcation diagrams, the mean frequency difference and time series of neurons. The simulation results suggest that the synchronization of bursting activity is a multi-time-scale phenomenon and the phase synchronization deduced by the phase equation is actually spike synchronization.     

Transitions to synchrony in coupled bursting neurons

Certain cells in the brain, for example, thalamic neurons during sleep, show spike-burst activity. We study such spike-burst neural activity and the transitions to a synchronized state using a model of coupled bursting neurons. In an electrically coupled network, we show that the increase of coupling strength increases incoherence first and then induces two different transitions to synchronized states, one associated with bursts and the other with spikes. These sequential transitions to synchronized states are determined by the zero crossings of the maximum transverse Lyapunov exponents. These results suggest that synchronization of spike-burst activity is a multi-time-scale phenomenon and burst synchrony is a precursor to spike synchrony.     

Synchronization and propagation of bursts in networks of coupled map neurons

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.     

Interaction function of coupled bursting neurons

The interaction functions of electrically coupled Hindmarsh–Rose(HR) neurons for different firing patterns are investigated in this paper.By applying the phase reduction technique,the phase response curve(PRC) of the spiking neuron and burst phase response curve(BPRC) of the bursting neuron are derived.Then the interaction function of two coupled neurons can be calculated numerically according to the PRC(or BPRC) and the voltage time course of the neurons.Results show that the BPRC is more and more complicated with the increase of the spike number within a burst,and the curve of the interaction function oscillates more and more frequently with it.However,two certain things are unchanged:Φ = 0,which corresponds to the in-phase synchronization state,is always the stable equilibrium,while the anti-phase synchronization state with Φ = 0.5 is an unstable equilibrium.     

一类位于加周期分岔中的貌似混沌的随机神经放电节律的识别

识别非周期神经放电节律是混沌还是随机一直是一个重要的科学问题. 在神经起步点实验中发现了一类介于周期k和周期k+1(k=1,2)节律之间非周期自发放电节律, 其行为是长串的周期k簇和周期k+1簇的交替. 确定性理论模型Chay模型展示出了周期k和周期k+1节律的共存行为. 噪声在共存区诱发出了与实验结果类似的非周期节律, 说明该类节律是噪声引起的两类簇的跃迁. 非线性预报及其回归映射揭示该节律具有确定性机理; 将两类簇分别转换为0和1得到一个二进制序列, 对该序列进行概率分析获得了两类簇跃迁的随机机理. 这不仅说明该节律是具有确定性结构的随机节律而不是混沌, 还为深入识别现实神经系统的混沌和随机节律提供了典型示例和有效方法.     

Burst synchronization of electrically and chemically coupled map-based neurons

Burst synchronization and burst dynamics of a system consisting of two map-based neurons coupled through electrical or chemical synapses are discussed. Some basic characteristic quantities are introduced to describe burst synchronization and burst dynamics of neurons. It is observed that excitatory coupling leads to in-phase burst synchronization but inhibitory coupling results in anti-phase one. By using the basic characteristics of burst dynamics, the effects of the intrinsic bursting properties and the coupling schemes on complex bursting behaviors are also presented for both inhibitory and excitatory couplings. The results are instructive to identify bursting behaviors through experimental data.     

Synchronization of coupled metronomes on two layers

Coupled metronomes serve as a paradigmatic model for exploring the collective behaviors of complex dynamical systems, as well as a classical setup for classroom demonstrations of synchronization phenomena. Whereas previous studies of metronome synchronization have been concentrating on symmetric coupling schemes, here we consider the asymmetric case by adopting the scheme of layered metronomes. Specifically, we place two metronomes on each layer, and couple two layers by placing one on top of the other. By varying the initial conditions of the metronomes and adjusting the friction between the two layers, a variety of synchronous patterns are observed in experiment, including the splay synchronization (SS) state, the generalized splay synchronization (GSS) state, the anti-phase synchronization (APS) state, the in-phase delay synchronization (IPDS) state, and the in-phase synchronization (IPS) state. In particular, the IPDS state, in which the metronomes on each layer are synchronized in phase but are of a constant phase delay to metronomes on the other layer, is observed for the first time. In addition, a new technique based on audio signals is proposed for pattern detection, which is more convenient and easier to apply than the existing acquisition techniques. Furthermore, a theoretical model is developed to explain the experimental observations, and is employed to explore the dynamical properties of the patterns, including the basin distributions and the pattern transitions. Our study sheds new lights on the collective behaviors of coupled metronomes, and the developed setup can be used in the classroom for demonstration purposes.     

间接延迟耦合可激发介质中螺旋波的演化

采用Bär 模型研究了通过被动介质间接延迟耦合的两层可激发介质中螺旋波的相互作用. 数值模拟结果表明: 延迟耦合可以促进两个螺旋波的同步, 也可导致从螺旋波到集体振荡、各种靶波、时空混沌态或静息态的转变; 在这个耦合系统中还观察到周期 2和周期3螺旋波以及螺旋波漫游和漂移现象; 对产生这些现象的物理机制做了讨论. 关键词： 螺旋波 被动介质 时间延迟耦合 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.     

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.     

Delayed excitatory self-feedback-induced negative responses of complex neuronal bursting patterns

In traditional viewpoint, excitatory modulation always promotes neural firing activities. On contrary, the negative responses of complex bursting behaviors to excitatory self-feedback mediated by autapse with time delay are acquired in the present paper. Two representative bursting patterns which are identified respectively to be "Fold/Big Homoclinic"bursting and "Circle/Fold cycle" bursting with bifurcations are studied. For both burstings, excitatory modulation can induce less spikes per burst for suitable time delay and strength of the self-feedback/autapse, because the modulation can change the initial or termination phases of the burst. For the former bursting composed of quiescent state and burst, the mean firing frequency exhibits increase, due to that the quiescent state becomes much shorter than the burst. However, for the latter bursting pattern with more complex behavior which is depolarization block lying between burst and quiescent state, the firing frequency manifests decrease in a wide range of time delay and strength, because the duration of both depolarization block and quiescent state becomes long. Therefore, the decrease degree of spike number per burst is larger than that of the bursting period, which is the cause for the decrease of firing frequency. Such reduced bursting activity is explained with the relations between the bifurcation points of the fast subsystem and the bursting trajectory. The present paper provides novel examples of paradoxical phenomenon that the excitatory effect induces negative responses, which presents possible novel modulation measures and potential functions of excitatory self-feedback/autapse to reduce bursting activities.     

Parameter-dependent synchronization transition of coupled neurons with co-existing spiking and bursting

A firing pattern transition is simulated in the Leech neuron model, firstly from bursting to co-existence of spiking and bursting and then to spiking. The attraction domain of spiking and bursting for three different parameter values are calculated. Synchronization transition processes of two coupled Leech neurons, one is bursting and the other the co-existing spiking, are simulated for the three parameters. The three synchronization processes appear similar as the coupling strength increases, beginning from non-synchronization to complete synchronization through a complex dynamical procedure, but their detailed processes are different depending on the parameter values. The transition procedure is complex and the complete synchronization is in bursting for larger parameter values, while the process is simple with complete synchronization of spiking for smaller values. The potential relationship between complete synchronization and the attraction domain is also discussed. The results are instructive to understanding the synchronization behaviors of the coupled neuronal system with co-existing attractors.     

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.     

“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.