时滞耦合惯性项神经系统的多混沌路径共存

混沌及其共存是神经动力学的一个重要研究内容.该文基于非单调激活函数的惯性项神经元时滞耦合系统,在固定系统参数的情况下,以耦合时滞τ作为参变量,取不同的初始条件,利用Poincaré截面技术,展现了系统多个不同的倍周期分岔序列和概周期分岔序列,并给出了系统相应的相图.研究结果表明,时滞耦合神经系统具有多级倍周期分岔序列和概周期分岔序列的稳态共存,展现了系统更加丰富的多混沌和多周期解的多稳态共存.     

耦合混沌Hindmarsh-Rose神经元系统中相互作用诱导的周期解研究

神经元细胞作为构成神经系统结构和功能的基本单位,在神经信号传输过程中具有非常重要的作用.采用Hindmarsh-Rose神经元模型,探究与细胞中钙离子浓度有关的一个恢复变量参数在神经元信号传递中的影响.研究表明,当改变恢复变量参数值时,单个神经元会出现周期或混沌的放电行为,并且对该参数值变化比较敏感.此外,当单个神经元为混沌放电时,随着相互作用强度的变化,耦合神经元系统不仅会出现混沌放电行为,还会产生周期放电行为,周期解窗口和混沌解窗口交替出现.当恢复变量参数值不同时,周期解窗口的个数和周期解的性质明显不同.该结论表明,该恢复变量参数在调控神经元混沌放电和周期放电行为过程中扮演着非常重要的角色.     

复杂网络上耦合神经系统的非聚类相同步

考虑了不同复杂网络结构(小世界、无标度和随机网络)条件下的耦合神经元系统,针对其进入相同步的同步化路径进行了建模与仿真,发现系统呈现出非聚类相同步现象,并对其形成原因进行了定性分析.结果表明:复杂网络上的耦合神经元系统与其在规则网络下有相同的同步行为,系统均不出现通常耦合相振子中的聚类成群现象,而表现为随着耦合强度的增加所有神经元渐进趋于同步.另外,随着放电尖峰的插入与弥合,最终导致系统个体平均频率先增强后衰减的变化.这些结果将丰富对于网络动力学行为(尤其是相同步)的认识,对理解神经认知科学具有一定意义.     

非恒同耦合系统的同步

在过去的几十年,由于同步在通信、光学、神经生物网络等不同领域的广泛应用,使得耦合动力系统的同步行为吸引了很多的注意.除了关于周期信号的经典同步概念,还引入了许多新的同步的类型:如混沌同步,相同步,广义同步等等.利用不变流形理论讨论非恒同耦合系统的同步.     

一类具有扩散和时滞的离散复合种群模型的Hopf分岔

本文讨论了生物上一类有时滞和扩散(迁移)的离散复合种群模型．利用离散系统相关结果分析了该模型的正不动点的类型及稳定性,并用中心流形方法对原系统降维从而讨论了它的Hopf分岔问题以及扩散和时滞对种群生态学的意义．     

周期激励下的前包钦格呼吸神经元的簇振荡现象研究

研究周期激励作用下的非自治前包钦格呼吸神经元模型,结果表明当外界激励频率与系统固有频率存在着量级差距时,系统可以产生典型的簇放电模式.由于激励频率远小于系统的固有频率,因此将整个周期激励项视为慢变参数,从而可以利用稳定性分析理论研究慢变参数变化下的平衡点的分岔类型,进一步应用快慢动力学分析方法给出簇模式产生的动力学机理.本文的结果说明外界激励对神经元的动力学行为有着重要影响,为进一步揭示呼吸节律的产生机制提供了重要帮助.     

一类竞争系统的Hopf分岔及分岔周期解的稳定性

首先研究具有时滞的竞争三种群平衡点的存在性,接着应用特征方程,发现当τ穿过某些数时出现了Hopf分岔,并用规范型方法和中心流形定理得到Hopf分岔和分岔周期解的稳定性的计算公式.并举例当τ变化时该模型会出现混沌现象.     

互联网双时滞TCP-RED模型Hopf分岔和Fold分岔的数值分析

利用时滞动力学分析软件DDE-BIFTOOL研究了时滞互联网TCP-RED拥塞控制模型的动力学.传输时滞τ_1可以使得该系统发生Hopf分岔、Fold分岔,使得平均队列长度和到达速率出现近似恒速运动或周期波动,揭示了时滞对其动力学的重要影响.     

一类双面碰撞振子的对称性尖点分岔与混沌

讨论了一类单自由度双面碰撞振子的对称型周期n-2运动以及非对称型周期n-2运动．把映射不动点的分岔理论运用到该模型,并通过分析对称系统的Poincaré映射的对称性,证明了对称型周期运动只能发生音叉分岔．数值模拟表明:对称系统的对称型周期n-2运动,首先由一条对称周期轨道通过音叉分岔形成具有相同稳定性的两条反对称的周期轨道;随着参数的持续变化,两条反对称的周期轨道经历两个同步的周期倍化序列各自生成一个反对称的混沌吸引子．如果对称系统演变为非对称系统,非对称型周期n-2运动的分岔过程可用一个两参数开折的尖点分岔描述,音叉分岔将会演变为一支没有分岔的分支以及另外一个鞍结分岔的分支．     

时滞速度反馈对强迫自持系统动力学行为的影响

研究强迫自持振动系统因时滞反馈产生的主共振解及其分岔．通过对强迫非自治系统的时滞反馈控制,得到所要研究的数学模型．讨论对应的线性化系统使平凡平衡态失稳出现周期解的稳定性临界条件．特别关注主共振及分岔．结果表明,稳定的主共振解随着时滞的变化周期性地出现在系统中．同时,也给出了不稳定的主共振关于时滞变化的区域,在理论方面给出了系统出现概周期运动的时滞区域．数据模拟证实了理论结果．     

Single or multiple synchronization transitions in scale-free neuronal networks with electrical or chemical coupling

In this paper, we have studied time delay- and coupling strength-induced synchronization transitions in scale-free modified Hodgkin–Huxley (MHH) neuron networks with gap-junctions and chemical synaptic coupling. It is shown that the synchronization transitions are much different for these two coupling types. For gap-junctions, the neurons exhibit a single synchronization transition with time delay and coupling strength, while for chemical synapses, there are multiple synchronization transitions with time delay, and the synchronization transition with coupling strength is dependent on the time delay lengths. For short delays we observe a single synchronization transition, whereas for long delays the neurons exhibit multiple synchronization transitions as the coupling strength is varied. These results show that gap junctions and chemical synapses have different impacts on the pattern formation and synchronization transitions of the scale-free MHH neuronal networks, and chemical synapses, compared to gap junctions, may play a dominant and more active function in the firing activity of the networks. These findings would be helpful for further understanding the roles of gap junctions and chemical synapses in the firing dynamics of neuronal networks.     

Bifurcation and synchronization of synaptically coupled FHN models with time delay

This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

Bursting oscillations, bifurcation and synchronization in neuronal systems

This paper investigates bursting oscillations and related bifurcation in the modified Morris-Lecar neuron. It is shown that for some appropriate parameters, the modified Morris-Lecar neuron can exhibit two types of fast-slow bursters, that is “circle/fold cycle” bursting and “subHopf/homoclinic” bursting with class 1 and class 2 neural excitability, which have different neuro-computational properties. By means of the analysis of fast-slow dynamics and phase plane, we explore bifurcation mechanisms associated with the two types of bursters. Furthermore, the properties of some crucial bifurcation points, which can determine the type of the burster, are studied by the stability and bifurcation theory. In addition, we investigate the influence of the coupling strength on synchronization transition and the neural excitability in two electrically coupled bursters with the same bursting type. More interestingly, the multi-time-scale synchronization transition phenomenon is found as the coupling strength varies.     

Effect of an autapse on the firing pattern transition in a bursting neuron

On the basis of the Hindmarsh–Rose (HR) neuron model, the dynamics of electrical activity and the transition of firing patterns induced by three types of autapses have been investigated in detail. The dynamic effect of an autapse is detected by imposing a feedback term with a specific time-delay and autaptic intensity. We found that the delayed autaptic feedback connection switches the electrical activities of the HR neuron among quiescent, periodic and chaotic firing patterns. In the case of an electrical autapse, the transition from a periodic to a chaotic state occurs depending on the specific autaptic intensity and the time-delay. The excitatory chemical autapse plays a positive role in generating and enhancing the chaotic state. A time delay could decrease and suppress the chaotic state in the case of inhibitory chemical self-connections with a proper autaptic intensity. The bifurcation diagram vs. time-delay and autaptic intensity has been extensively studied, and the time series of membrane potentials and the distribution of information entropy have also been calculated to confirm the bifurcation analysis.     

The effects of time delay on the stochastic resonance in feed-forward-loop neuronal network motifs

The dependence of stochastic resonance in the feed-forward-loop neuronal network motifs on the noise and time delay are studied in this paper. By computational modeling, Izhikevich neuron model with the chemical coupling is used to build the triple-neuron feed-forward-loop motifs with all possible motif types. Numerical results show that the correlation between the periodic subthreshold signal’s frequency and the dynamical response of the network motifs is resonantly dependent on the intensity of additive spatiotemporal noise. Interestingly, the excitatory intermediate neuron could induce intermittent stochastic resonance, whereas the inhibitory one weakens its influence on the intermittent mode. More importantly, it is found that the increasing delays can induce the intermittent appearance of regions of stochastic resonance. Based on the effects of the time delay on the stochastic resonance, the reasons and conditions of such intermittent resonance phenomenon are analyzed.     

Spatio-temporal patterns of Hopf bifurcating periodic oscillations in a pair of identical tri-neuron network loops

Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations.     

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.     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

Stability,coherent spiking and synchronization in noisy excitable systems with coupling and internal delays

We study the onset and the adjustment of different oscillatory modes in a system of excitable units subjected to two forms of noise and delays cast as external or internal according to whether they are associated with inter- or intra-unit activity. Conditions for stability of a single unit are derived in case of the linearized perturbed system, whereas the interplay of noise and internal delay in shaping the oscillatory motion is analyzed by the method of statistical linearization. It is demonstrated that the internal delay, as well as its coaction with external noise, drive the unit away from the bifurcation controlled by the excitability parameter. For the pair of interacting units, it is shown that the external/internal character of noise primarily influences frequency synchronization and the competition between the noise-induced and delay-driven oscillatory modes, while coherence of firing and phase synchronization substantially depend on internal delay. Some of the important effects include: (i) loss of frequency synchronization under external noise; (ii) existence of characteristic regimes of entrainment, where under variation of coupling delay, the optimized unit (noise intensity fixed at resonant value) may be controlled by the adjustable unit (variable noise) and vice versa, or both units may become adjusted to coupling delay; (iii) phase synchronization achieved both for noise-induced and delay-driven modes.