Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh–Rose and FitzHugh–Nagumo neurons with two time delays

A memristor-coupled heterogenous neural network consisting of two-dimensional(2D)FitzHugh–Nagumo(FHN)and Hindmarsh–Rose(HR)neurons with two time delays is established.Taking the time delays as the control parameters,the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically,and the validity of the Hopf bifurcation condition is verified by numerical analysis.The results show that the two time delays can make the stable equilibrium point unstable,thus leading to periodic oscillations induced by Hopf bifurcation.Furthermore,the time delays in FHN and HR neurons have different effects on the firing activity of neural network.Complex firing patterns,such as quiescent state,chaotic spiking,and periodic spiking can be induced by the time delay in FHN neuron,while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron.Especially,phase synchronization between the heterogeneous neurons is explored,and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron.Finally,the theoretical analysis is verified by circuit simulations.

Modeling and dynamics of double Hindmarsh-Rose neuron with memristor-based magnetic coupling and time delay

The firing of a neuron model is mainly affected by the following factors:the magnetic field,external forcing current,time delay,etc.In this paper,a new time-delayed electromagnetic field coupled dual Hindmarsh-Rose neuron network model is constructed.A magnetically controlled threshold memristor is improved to represent the self-connected and the coupled magnetic fields triggered by the dynamic change of neuronal membrane potential for the adjacent neurons.Numerical simulation confirms that the coupled magnetic field can activate resting neurons to generate rich firing patterns,such as spiking firings,bursting firings,and chaotic firings,and enable neurons to generate larger firing amplitudes.The study also found that the strength of magnetic coupling in the neural network also affects the number of peaks in the discharge of bursting firing.Based on the existing medical treatment background of mental illness,the effects of time lag in the coupling process against neuron firing are studied.The results confirm that the neurons can respond well to external stimuli and coupled magnetic field with appropriate time delay,and keep periodic firing under a wide range of external forcing current.

Adaptive coupling-enhanced spiking synchronization in Newman-Watts neuronal networks with time delays

In this paper, we study how adaptive coupling with time-periodic growth speed (TPGS) affects the spiking synchronization of weighted adaptive Newman-Watts Hodgkin-Huxley neuron networks with time delays. It is found that the neuronal spiking intermittently exhibits synchronization transitions between desynchronization and in-phase synchronization or anti-phase synchronization as TPGS amplitude or frequency is varied, showing multiple synchronization transitions. These transitions depend on the values of time delay and can occur only when time delay is close to those values that can induce synchronization transitions when the growth speed is fixed. These results show that the adaptive coupling with TPGS has great influence on the spiking synchronization of the neuronal networks and thus plays a crucial role in the information processing and transmission in neural systems.

Novel criteria for exponential synchronization of inner time-varying complex networks with coupling delay

This paper mainly investigates the exponential synchronization of an inner time-varying complex network with coupling delay. Firstly, the synchronization of complex networks is decoupled into the stability of the corresponding dynamical systems. Based on the Lyapunov function theory, some sufficient conditions to guarantee its stability with any given convergence rate are derived, thus the synchronization of the networks is achieved. Finally, the results are illustrated by a simple time-varying network model with a coupling delay. All involved numerical simulations verify the correctness of the theoretical analysis.

Chemical synaptic coupling-induced delay-dependent synchronization transitions in scale-free neuronal networks

Chemical synaptic couplings are more common than electric(gap junction) connections in neurons.In this paper,the firing synchronizations induced by chemical synaptic coupling in chemically delayed scale-free networks of modified Hodgkin-Huxley neurons have been studied.It was found that the chemical coupling-induced synchronization transitions are delay-dependent and much different for various delay lengths.In the absence of delay,the neurons exhibit a transition from chaotic bursting(CB) to bursting synchronization(BS) with desynchronized spikes in each burst;for smaller delay lengths,the firing evolves from CB to spiking synchronization(SS),but for larger delay lengths,there are transitions from CB to intermittently multiple SS behaviors.These findings show that the chemical coupling-induced firing synchronization transitions strongly depend on the chemical delay lengths,and intermittently multiple SS can only occur for larger delay lengths.This result would be helpful for better understanding the joint roles of the chemical coupling and chemical delay in the firing activity of the neurons.

Chaos synchronization with dual-channel time-delayed couplings

While the significance of oscillator dynamics and coupling structure to synchronization behaviors has been well addressed in the literature, little attention has been paid to the possible influence of coupling functions. In the present paper, adopting the scheme of dual-channel time-delayed couplings, we investigate how the synchronization behaviors of networked chaotic oscillators are influenced by parameters in the coupling functions. It is found that, with the introduction of the second coupling channel, the synchronization region, as calculated according to the method of master stability function(MSF), can be largely modified. In particular, by a slight change of the time delay, it is found that the synchronization region can be significantly adjusted, or even switched from non-existing to existing. We demonstrate this interesting phenomenon for both situations of processing and propagation induced time delays, as well as for different coupling functions. Our studies shed new light on the mechanism of chaos synchronization, and may potentially be used for the control of complex network dynamics.

Enhanced vibrational resonance in a single neuron with chemical autapse for signal detection

Many animals can detect the multi-frequency signals from their external surroundings.The understanding for underlying mechanism of signal detection can apply the theory of vibrational resonance,in which the moderate high frequency driving can maximize the nonlinear system's response to the low frequency subthreshold signal.In this work,we study the roles of chemical autapse on the vibrational resonance in a single neuron for signal detection.We reveal that the vibrational resonance is strengthened significantly by the inhibitory autapse in the neuron,while it is weakened typically by the excitatory autapse.It is generally believed that the inhibitory synapse has a suppressive effect in neuronal dynamics.However,we find that the detection of the neuron to the low frequency subthreshold signal can be improved greatly by the inhibitory autapse.Our finding indicates that the inhibitory synapse may act constructively on the detection of weak signal in the brain and neuronal system.

Synchronization Transition of Time Delayed Complex Dynamical Networks with Discontinuous Coupling

This paper investigates the synchronization of time delayed complex dynamical networks with periodical on-off coupling. Both the theoretical and numerical results show that, in spite of time delays and on-off coupling, two networks may synchronize if the coupling strength and the on-off rate are large enough. It is shown that, for undirected and strongly connected networks, the upper bound of time delays for synchronization is a decreasing function of the absolute value of the minimum eigenvalue of the adjacency matrix. The theoretical analysis confirms the numerical results and provides a better understanding of the influence of time delays and on-off coupling on the synchronization transition. The influence of random delays on the synchronization is also discussed.

Synchronization of complex switched delay dynamical networks with simultaneously diagonalizable coupling matrices

This paper studies local exponential synchronization of complex delayed networks with switching topology via switched system stability theory. First, by a common unitary matrix, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when all subnetworks are synchronizable, a delay-dependent sufficient condition is given in terms of linear matrix inequalities （LMIs） which guarantees the solvability of the synchronization problem under an average dwell time scheme. We extend this result to the case that not all subnetworks are synchronizable. It is shown that in addition to average dwell time, if the ratio of the total activation time of synchronizable and non-synchronizable subnetworks satisfy an extra condition, then the problem is also solvable. Two numerical examples of delayed dynamical networks with switching topology are given, which demonstrate the effectiveness of obtained results.

Delay-induced firing behavior and transitions in adaptive neuronal networks with two types of synapses

In this paper, we study delay-induced firing behavior and transitions in adaptive Newman-Watts networks of thermosensitive neurons with electrical or chemical synapses. It is found that electrical and chemical synapse time delay-induced firing behavior and transitions differ significantly. In the case of electrical synapses, the bursts for a fixed delay involve equal number of spikes in each burst, and for certain time delays the firing can be inhibited. However, in the case of chemical synapses the bursts for a fixed delay involve different numbers of spikes in each burst, and no firing inhibition is observed. It is also shown that larger growth rates of adaptive coupling strength or larger network randomness can enhance the synchronization of bursting in the case of electrical synapses but reduce it in the case of chemical synapses. These results show that electrical and chemical synapses have different effects on delay-induced firing behavior and dynamical evolution. Compared to electrical synapses, chemical synapses might be more beneficial to the generation of firing and abundant firing transitions in adaptive and delayed neuronal networks. These findings can help to better understand different firing behaviors in neuronal networks with electrical and chemical synapses.

Synchronization of Transient Delay-Coupled Network

A transient delay-coupled network was proposed by modifying the standard delay-coupled network with a transient coupling technique to enlarge the synchronization domain of the network, where the synchronization domain is the interval of the coupling strength for which the network gets synchronized. The coupling of the transient delay-coupled network is activated when the systems are in a particular region (coupling region) of the phase space and inactivated otherwise, which is different from the standard coupling. The specific synchronization performance of the transient delay-coupled network was investigated through case studies. The relationships between the synchronization domain and the coupling region were obtained by gauging the synchronization index. It is understood that the synchronization domain changes in a non-smooth manner with the variation of the coupling region. In particular, the synchronization domain of a transient delay-coupled network is much larger than that of the standard delay-coupled network when the coupling region is appropriately determined.

单向耦合FitzHugh-Nagumo神经元的滞后同步研究

本文提出一种新的研究思路,将两个单向耦合的FitzHugh-Nagumo神经元之间的滞后同步视为一种特殊的广义同步,通过辅助系统方法来获得滞后同步发生的条件.首先建立原响应系统的辅助系统,将原系统中驱动系统和响应系统的滞后同步问题转换为响应系统与辅助系统之间的误差系统原点稳定问题.通过拉普拉斯变换方法,将原本用微分方程表示的误差系统等效为Volterra积分方程表示,而后利用积分方程理论中的逐次逼近方法解析地得到原误差系统原点稳定性条件.本文给出的滞后同步判据与信号传送中的时间滞后量大小无关,数值模拟验证了本文提出判据的有效性.

时滞影响下的环式耦合混沌神经元同步

由于突触连接,神经元之间的信息传递普遍存在着时滞效应.本文讨论时滞对四个环式耦合混沌HR(Hindmarsh-Rose)神经元的同步动力学行为影响.通过计算同步差,发现适当的时滞会诱发或增强混沌神经元间的完全同步,即时滞的作用使耦合系统在较小的耦合强度下存在稳定的同步状态.进一步还发现时滞会诱发神经元间出现相位同步,增大在相同步窗口.此外,在完全同步出现可以观察到由时滞诱发的一些复杂现象,包括近似同步与相位同步之间的转迁等.

时滞作用下的化学突触耦合的Hindmarsh-Rose神经元同步研究

本文通过数值模拟的方法研究了化学突触耦合的Hindmarsh-Rose神经元的同步问题,以及时滞对耦合的Hindmarsh-Rose神经元同步的影响.研究发现:耦合强度能够影响化学突触的耦合神经元系统的放电活动以及耦合神经元之间的同步状态.通过调整化学突触耦合的两个Hindmarsh-Rose神经元之间的耦合强度,可以使得神经元系统从非同步状态变为同步状态.同时也发现,适当的时滞会使得两个Hindmarsh-Rose化学突触耦合神经元系统从同步状态变为非同步状态,从而破坏系统的同步活动.

混合突触作用下耦合时滞对模块神经元网络簇同步的影响

针对电突触耦合和化学突触耦合混合作用下含有耦合时滞的模块神经元网络,利用非线性动力学理论和数值仿真方法,探讨了耦合强度及耦合时滞对模块神经元网络簇同步特性的影响.结果发现,模块神经元网络中子网络内、子网络间的耦合强度都能促使簇放电神经元取得簇同步,但是时滞却对耦合诱导的簇同步具有显著的抑制作用.进一步的研究证实了本文所得的研究结果不依赖于子网络的数目与子网络的节点个数.需要指出的是,耦合时滞对神经元网络簇同步的抑制作用对治疗簇同步引发的一些神经性疾病(如帕金森病、癫痫等)具有一定的理论指导意义.

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.

时滞对链式耦合混沌神经元同步的影响

针对神经元信息传递过程中存在的时滞,研究了3个链式耦合混沌HR(Hind-marsh-Rose)神经元的同步行为受时滞的影响。通过C语言编程和Matlab数值模拟,证明加入恰当的时滞可以加强混沌神经元之间的完全同步,即加入时滞之后,耦合系统较没有时滞情况下更容易达到同步状态。

SYNCHRONIZATION STABILITY OF COMPLEX CONNECTED NETWORKS WITH DELAYS

Recently, much work has been devoted to the study of a large-scale complex system described by a network or a graph with complex topology, whose nodes are the elements of the system and whose edges represent the interactions among them. On the other hand, realistic modelling of many large networks with nonlocal interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units. This paper gives the sufficient conditions guaranteeing the local and global synchronization stability of the complex connected networks by using Lyapunov functional.

多电机同步控制建模与仿真

采用偏差耦合控制结构控制4台永磁同步电机同步运动时,由于控制器结构复杂,计算量大,而且传统的PID控制算法不能在线实时调整参数,动态响应慢。因此,对4电机同步运动控制结构和控制算法进行改进,采用环形耦合控制结构和免疫单神经元PID控制器控制整个同步运动系统。仿真结果表明:该控制结构复杂度低、跟踪性能和稳态性能良好,抗干扰性强。

异质神经元的排列对环形耦合神经元网络频率同步的影响

以一维环形耦合的非全同FitzHugh-Nagumo神经元网络为研究对象,讨论这种异质神经元在环上的不同排列对其频率同步的影响.研究结果显示,异质神经元的排列不同,对应的神经元网络达到频率同步所需的临界耦合强度也不完全相同.在平均意义下,异质性较小的神经元在环上的距离越近,神经元网络达到频率同步所需的临界耦合强度越大;相反,异质性较大的神经元在环上的距离越近,神经元网络达到同步所需的临界耦合强度越小.通过对频率同步过程的分析,进一步给出了产生这一现象的动力学机理.