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.     

Effects of time delay and coupling strength on synchronization transitions in excitable homogeneous random network

In this paper we numerically investigate the effects of time delay and coupling strength on synchronization transitions in excitable homogeneous random network. Different roles of time delay and coupling strength have been discovered by synchronization parameter and space–time plots. Specifically, we have found three distinct parameter regions, i.e., asynchronous region (domain I for small time delay), transition region (domain II for moderate time delay) and synchronous region (domain III for large time delay) as time delay is increased. The phenomenon of multi-stability is observed in the transition region. While coupling strength can enhance synchronization in the transition region and can reduce synchronization time in the synchronous region. All these results are independence on the system size.     

时滞对磁通耦合及化学耦合神经元分岔及同步的影响

以化学突触耦合神经元模型为基础,讨论了抑制性及兴奋性条件下达到同步的区别及同步的类型。并根据磁通耦合对神经元放电的影响,讨论了具有时滞、磁通耦合和化学耦合Morris-Lecar (ML)神经元模型的放电状态、分岔类型及其同步情况。发现具有磁通耦合和化学耦合ML神经元系统在不同参数下会产生丰富的逆倍周期分岔或加周期分岔行为。而时滞的引入,虽然可以增加系统的周期性,但同时也会破环系统同步。相反,适当的耦合强度能够增加同步。     

Quantification of synchronization phenomena in two reciprocally gap-junction coupled bursting pancreatic β-cells

This paper aims to discuss our research into synchronized transitions in two reciprocally gap-junction coupled bursting pancreatic β-cells. Numerical results revealed that propagations of synchronous states could be induced not only by changing the coupling strength, but also by varying the slow time constant. Firstly, these asynchronous and synchronous states such as out-of-phase, almost in-phase and in-phase synchronization were specifically demonstrated by phase portraits and time evolutions. By comparing interspike intervals (ISI) bifurcation diagrams of two coupled neurons with an individual neuron, we found that coupling strength played a critical role in tonic-to-bursting transitions. In particular, with the phase difference and ISI-distance being introduced, regions of various synchronous and asynchronous states were plotted in a two-dimensional parameter space. More interestingly, it was found that the coupled neurons could always realize complete synchronization as long as the coupling strength was appropriate.     

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.     

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.     

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.     

Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.     

Chaotic synchronization of a fractional-order system based on washout filter control

In this paper, chaos in a fractional-order neural network system with varying time delays is presented, and chaotic synchronization system with varying time delays is constructed. The stability of constructed synchronization system is analyzed by Laplace transformation theory. In addition, the bifurcation graph of the chaotic system is illustrated. The study results show that the chaos in such fractional-order neural networks with varying time delay can be synchronized, and Washout filter control can be used to reduce the range of coupled parameter.     

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.     

Synchronization of neural networks based on parameter identification and via output or state coupling

For neural networks with all the parameters unknown, we focus on the global robust synchronization between two coupled neural networks with time-varying delay that are linearly and unidirectionally coupled. First, we use Lyapunov functionals to establish general theoretical conditions for designing the coupling matrix. Neither symmetry nor negative (positive) definiteness of the coupling matrix are required; under less restrictive conditions, the two coupled chaotic neural networks can achieve global robust synchronization regardless of their initial states. Second, by employing the invariance principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the robust synchronization of almost all kinds of coupled neural networks with time-varying delay based on the parameter identification of uncertain delayed neural networks. Finally, numerical simulations validate the effectiveness and feasibility of the proposed technique.     

Bursting and synchronization transition in the coupled modified ML neurons

Bursting is an important electrical behavior in neuron’s firing. In this paper, based on the fast/slow dynamical bifurcation analysis and the phase plane analysis, two types of bursting are presented in the modified Morris–Lecar neuronal model, and the associated bifurcation mechanisms of switching between the active phase and the silent phase are analyzed. For two coupled bursters, it is found that the same type of coupled bursters may have different synchronization transition path from that of two different types of coupled bursters. The analysis of bursting types and the transition to synchronization may provide us with better insight into neuronal encoding and information transmission.     

Rate of afferent stimulus dependent synchronization and coding in coupled neurons system

Sequences of intervals between firing times (interspike interval (ISI)) from a pair of locus ceruleus (LC) neurons coupled by axon–dendrite synapse with stimulus of constant and chaos are investigated in this paper. We analyze how the dynamical properties of chaotic input determine those of the output ISI sequences, and assess how various strength of stimulus and coupling affects the input–output relationship. The attractors constructed from delay embeddings of ISIs and of chaotic input are compared from the points of view of geometry and nonlinear dynamics characteristics, i.e., Lyapunov exponent spectrum (LES), Kaplan–York fractal dimension (KY_D) and unstable periodic orbit (UPO). For the coupled LC neurons system investigated, with the moderate strength of stimulus and coupling, the synchronous oscillation of the two neurons is well preserved even if the external stimulus is chaotic; the similarity between these attractors is high only when the afferent stimulus strength is smaller and rate is lower. When these conditions are satisfied, the output two ISI sequences are reciprocally related to input signals, and their oscillation wave shape in time course can be derived from that of the input signals variation, furthermore, the similar input sequence of order of UPOs, distribution of LES and value of KY_D remain in attractors reconstructed from ISI sequences. But these phenomena will disappear in higher rate of stimulus activity or in changing of the strength of stimulus and coupling, for this situation, the ISIs shows bifurcate behavior. These results may be of vital importance for any kind of information processing based on the neurons and temporal coding.     

Adaptive cluster synchronization in networks with time-varying and distributed coupling delays

This paper studies the adaptive cluster synchronization of a generalized linearly coupled network with time-varying delay and distributed delays. This network includes nonidentical nodes displaying different local dynamical behaviors, while for each cluster of that network the internal dynamics is uniform (such as chaotic, periodic, or stable behavior). In particular, the generalized coupling matrix of this network can be asymmetric and weighted. Two different adaptive laws of time-varying coupling strength and a linear feedback control are designed to achieve the cluster synchronization of this network. Some sufficient conditions to ensure the cluster synchronization are obtained by using the invariant principle of functional differential equations and linear matrix inequality (LMI). Numerical simulations verify the efficiency of our proposed adaptive control method.     

Synchronization of multiple bursting neurons ring coupled via impulsive variables

Synchronization behavior of bursting neurons is investigated in a neuronal network ring impulsively coupled, in which each neuron exhibits chaotic bursting behavior. Based on the Lyapunov stability theory and impulsive control theory, sufficient conditions for synchronization of the multiple systems coupled with impulsive variables can be obtained. The neurons become synchronous via suitable impulsive strength and resetting period. Furthermore, the result is obtained that synchronization among neurons is weakened with the increasing of the reset period and the number of neurons. Finally, numerical simulations are provided to show the effectiveness of the theoretical results.© 2014 Wiley Periodicals, Inc. Complexity 21: 29–37, 2015     

Synchronization of quaternion-valued coupled systems with time-varying coupling via event-triggered impulsive control

In this paper, the problem of exponential synchronization of quaternion-valued coupled systems based on event-triggered impulsive control is investigated for the first time. It should be pointed out that the coupling strength is quaternion-valued and time-varying, which makes our model more in line with practical models. First, we prove that event-triggered impulsive control can exclude Zeno behavior. Then, based on the Lyapunov method and the graph theory, some sufficient conditions are derived to ensure that quaternion-valued coupled systems reach synchronization. Furthermore, as an application of our theoretical results, exponential synchronization of quaternion-valued Kuramoto oscillators is studied in detail and a synchronization criterion is presented. Finally, some numerical simulations are given to show the effectiveness of our theoretical results.     

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.     

Chaotic synchronization with gap junction of multi-neurons in external electrical stimulation

The synchronization of n(n  3) neurons coupled with gap junction in external electrical stimulation is investigated. In this paper, the coupled model is established on the basis of nonlinear cable model, and then the relation between coupling strength of the gap junction and the synchronization is discussed in detail. The sufficient condition of complete synchronization is attained from rigorous mathematical derivation. The synchronizations of periodic neurons and chaotic neurons are studied respectively.     

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.     

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.