Ordering chaos and synchronization transitions by chemical delay and coupling on scale-free neuronal networks

Chemical synaptic connections are more common than electric ones in neurons, and information transmission delay is especially significant for the synapses of chemical type. In this paper, we report a phenomenon of ordering spatiotemporal chaos and synchronization transitions by the delays and coupling through chemical synapses of modified Hodgkin–Huxley (MHH) neurons on scale-free networks. As the delay τ is increased, the neurons exhibit transitions from bursting synchronization (BS) to intermittent multiple spiking synchronizations (SS). As the coupling g_syn is increased, the neurons exhibit different types of firing transitions, depending on the values of τ. For a smaller τ, there are transitions from spatiotemporal chaotic bursting (SCB) to BS or SS; while for a larger τ, there are transitions from SCB to intermittent multiple SS. These findings show that the delays and coupling through chemical synapses can tame the chaotic firings and repeatedly enhance the firing synchronization of neurons, and hence could play important roles in the firing activity of the neurons on scale-free networks.     

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.     

Delay-induced synchronization transitions in small-world neuronal networks with hybrid synapses

We study delay-induced synchronization transitions in small-world networks of bursting neurons with hybrid excitatory-inhibitory synapses. Numerical results show that transitions of the spatiotemporal synchrony of neurons can be induced not only by the variations of the information transmission delay but also by changing the probability of inhibitory synapses and the rewiring probability. The delay can either promote or destroy synchronization of neuronal activity in the hybrid small-world neuronal network. In particular, regions of synchronization and nonsynchronization appear intermittently as the delay increases. In addition, for smaller and higher probability of inhibitory synapses, the intermittent synchronization transition is relative profound, while for the moderate probability of inhibitory synapses, synchronization transition seems less profound. More importantly, it is found that a suitable rewired network topology can always enhance the synchronized neuronal activity if only the delay is appropriate.     

Optimal autaptic and synaptic delays enhanced synchronization transitions induced by each other in Newman–Watts neuronal networks

In this paper, we numerically study the effect of electrical autaptic and synaptic delays on synchronization transitions induced by each other in Newman–Watts Hodgkin–Huxley neuronal networks. It is found that the synchronization transitions induced by synaptic delay vary with varying autaptic delay and become strongest when autaptic delay is optimal. Similarly, the synchronization transitions induced by autaptic delay vary with varying synaptic delay and become strongest at optimal synaptic delay. Also, there is optimal coupling strength by which the synchronization transitions induced by either synaptic or autaptic delay become strongest. These results show that electrical autaptic and synaptic delays can enhance synchronization transitions induced by each other in the neuronal networks. This implies that electrical autaptic and synaptic delays can cooperate with each other and more efficiently regulate the synchrony state of the neuronal networks. These findings could find potential implications for the information transmission in neural systems.     

The effects of time delay on the synchronization transitions in a modular neuronal network with hybrid synapses

Delay-induced synchronization transitions are studied in a modular neuronal network of small-world subnetworks with hybrid synapses in this paper. Numerical results show that the spatiotemporal synchronization transitions in a modular neuronal network not only depend on the information transmission delay, but also can be induced by the variations of the probability of inhibitory synapses and the number of subnetworks in the modular networks. In the hybrid modular network, the information transmission delay is shown to be significant, which can either promote or destroy synchronization of neuronal activity. In particular, the increasing delays can induce the intermittent appearance of regions of synchronization and non-synchronization. Interestingly, it is found that intermittent synchronization transition is relatively profound for smaller and larger probability of inhibitory synapses, while synchronization transition seems less profound for the moderate probability of inhibitory synapses. In addition, if only the delay is appropriate, there exists a suitable modular network topology structure enhancing the synchronized neuronal activity.     

Multiple firing coherence resonances in excitatory and inhibitory coupled neurons

The impact of inhibitory and excitatory synapses in delay-coupled Hodgkin–Huxley neurons that are driven by noise is studied. If both synaptic types are used for coupling, appropriately tuned delays in the inhibition feedback induce multiple firing coherence resonances at sufficiently strong coupling strengths, thus giving rise to tongues of coherency in the corresponding delay-strength parameter plane. If only inhibitory synapses are used, however, appropriately tuned delays also give rise to multiresonant responses, yet the successive delays warranting an optimal coherence of excitations obey different relations with regards to the inherent time scales of neuronal dynamics. This leads to denser coherence resonance patterns in the delay-strength parameter plane. The robustness of these findings to the introduction of delay in the excitatory feedback, to noise, and to the number of coupled neurons is examined. Mechanisms underlying our observations are revealed, and it is suggested that the regularity of spiking across neuronal networks can be optimized in an unexpectedly rich variety of ways, depending on the type of coupling and the duration of delays.     

Impacts of hybrid synapses on the noise-delayed decay in scale-free neural networks

We study the phenomenon of noise-delayed decay in a scale-free neural network consisting of excitable FitzHugh–Nagumo neurons. In contrast to earlier works, where only electrical synapses are considered among neurons, we primarily examine the effects of hybrid synapses on the noise-delayed decay in this study. We show that the electrical synaptic coupling is more impressive than the chemical coupling in determining the appearance time of the first-spike and more efficient on the mitigation of the delay time in the detection of a suprathreshold input signal. We obtain that hybrid networks including inhibitory chemical synapses have higher signal detection capabilities than those of including excitatory ones. We also find that average degree exhibits two different effects, which are strengthening and weakening the noise-delayed decay effect depending on the noise intensity.     

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.     

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.     

Ordered chaotic bursting and multiple coherence resonance by time-periodic coupling strength in Newman–Watts neuronal networks

In this paper, we study the effect of time-periodic coupling strength (TPCS) on the temporal coherence of the chaotic bursting of Newman–Watts thermosensitive neuron networks. It is found that the chaotic bursting can exhibit coherence resonance and multiple coherence resonance behavior when TPCS amplitude and frequency is varied, respectively. It is also found that TPCS can also enhance the temporal coherence and spatial synchronization of the optimal spatio-temporal bursting in the case of fixed coupling strength. These results show that TPCS can tame the chaotic bursting and can repeatedly enhance the temporal coherence of the chaotic bursting neuronal networks. This implies that TPCS may play a more efficient role for improving the time precision of the information processing in chaotic bursting neurons.     

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。     

Robust adaptive synchronization of uncertain complex networks with multiple time‐varying coupled delays

In this article, the synchronization problem of uncertain complex networks with multiple coupled time‐varying delays is studied. The synchronization criterion is deduced for complex dynamical networks with multiple different time‐varying coupling delays and uncertainties, based on Lyapunov stability theory and robust adaptive principle. By designing suitable robust adaptive synchronization controllers that have strong robustness against the uncertainties in coupling matrices, the all nodes states of complex networks globally asymptotically synchronize to a desired synchronization state. The numerical simulations are given to show the feasibility and effectiveness of theoretical results. © 2014 Wiley Periodicals, Inc. Complexity 20: 62–73, 2015     

Synchronization analysis of networks with both delayed and non-delayed couplings via adaptive pinning control method

In this paper, simple controllers are designed to realize the synchronization of complex networks with time delays, in which the coupling configuration matrix and inner coupling matrix are not restricted to be symmetric matrix. Several adaptive synchronization criteria are obtained based on Lyapunov stability theory. These criteria relay on the coupling strength and the number of nodes pinning to the networks. For a given complex dynamical network with both delayed and non-delayed couplings, we give the minimum number of controllers under which synchronization can be achieved. One example shows the effectiveness of the proposed pinning adaptive controller.     

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.     

Stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses

The dependence of stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses on the probability of chemical synapse and the rewiring probability is investigated. A subthreshold periodic signal is imposed on one single neuron within the neuronal network as a pacemaker. It is shown that, irrespective of the probability of chemical synapse, there exists a moderate intensity of external noise optimizing the response of neuronal networks to the pacemaker. Moreover, the effect of pacemaker driven stochastic resonance of the system depends largely on the probability of chemical synapse. A high probability of chemical synapse will need lower noise intensity to evoke the phenomenon of stochastic resonance in the networked neuronal systems. In addition, for fixed noise intensity, there is an optimal chemical synapse probability, which can promote the propagation of the localized subthreshold pacemaker across neural networks. And the optimal chemical synapses probability turns even larger as the coupling strength decreases. Furthermore, the small-world topology has a significant impact on the stochastic resonance in hybrid neuronal networks. It is found that increasing the rewiring probability can always enhance the stochastic resonance until it approaches the random network limit.     

Cooperative dynamics in neuronal networks

There exist rich cooperative behaviors and their transitions in biological neuronal systems as some key biological factors are changed. Among all of cooperative behaviors of neuronal systems, the existing experiments have shown that the spatiotemporal pattern and synchronization dynamics are very crucial, which are closely related to normal function and dysfunction of neuronal systems. Based on different neuron models, the recent works have been made to explore the mechanisms of pattern formation and synchronization transition. This paper mainly overviews the recent studies of the cooperative dynamics including the pattern formation and synchronization transition in biological neuronal networks. Firstly, we review complicated spatiotemporal pattern dynamics of neuronal networks. Secondly, the interesting synchronization transition is reviewed. Finally, conclusion is given and we put forward some outlooks of research on the cooperative behaviors in real neuronal networks.     

Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties

The problem of robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties is investigated in this article. It is only assumed that the upper normal bound of uncertain inner and outer coupling matrices is positive but its concrete structure is not also required to be known. The time‐varying coupling delay is a any nonnegative continuous and bounded function and not require its derivative to be less than one, that is, general time‐varying coupling delays and uncertainties. For such a class of uncertain complex networks, a new synchronization scheme is presented by a class of continuous memoryless robust decentralized adaptive synchronization controllers. It is also shown that the synchronization error dynamics of uncertain complex networks can be guaranteed as uniformly exponentially convergent toward a ball that can be as small as desired. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of proposed complex networks synchronization schemes. © 2013 Wiley Periodicals, Inc. Complexity 19: 10–26, 2014     

Spatiotemporal dynamics on small-world neuronal networks: The roles of two types of time-delayed coupling

We investigate temporal coherence and spatial synchronization on small-world networks consisting of noisy Terman–Wang (TW) excitable neurons in dependence on two types of time-delayed coupling: {x_j(t − τ) − x_i(t)} and {x_j(t − τ) − x_i(t − τ)}. For the former case, we show that time delay in the coupling can dramatically enhance temporal coherence and spatial synchrony of the noise-induced spike trains. In addition, if the delay time τ is tuned to nearly match the intrinsic spike period of the neuronal network, the system dynamics reaches a most ordered state, which is both periodic in time and nearly synchronized in space, demonstrating an interesting resonance phenomenon with delay. For the latter case, however, we cannot achieve a similar spatiotemporal ordered state, but the neuronal dynamics exhibits interesting synchronization transitions with time delay from zigzag fronts of excitations to dynamic clustering anti-phase synchronization (APS), and further to clustered chimera states which have spatially distributed anti-phase coherence separated by incoherence. Furthermore, we also show how these findings are influenced by the change of the noise intensity and the rewiring probability of the small-world networks. Finally, qualitative analysis is given to illustrate the numerical results.     

The effects of synaptic time delay on motifs of chemically coupled Rulkov model neurons

Given the importance of the network motifs, we consider a pair of Rulkov chaotic map neurons, reciprocally coupled via symmetrical chemical synapses with the time delay τ. For the inhibitory and excitatory synapses, the system dynamics is determined by the synaptic weight g_c, synaptic gain parameter k, time delay τ and the external excitation σ. Due to chaotic nature of the map and synaptic model complexity, the appropriately averaged cross-correlation of membrane potentials represents a suitable numerical diagnostics to quantify mutual synchronization. Along with the expected phase and anti-phase synchronization regimes, we find the emergent phenomena that significantly influence the synchronization behavior.     

Dependence of inter-neuronal effective connectivity on synchrony dynamics in neuronal network motifs

Effective connectivity, characterized as directional causal influences among neural units, is functionally significant to be reconstructed. Various dynamic regimes have been considered to underlie reshaping of the effective connections. In this work, the impact of zero-lag synchronization on the reconstruction of effective connectivity in neuronal network motifs is investigated. The synchronization analysis and effective connectivity estimation by using Granger causality (GC) method are performed. It is shown that the synchronization of the neurons at zero lag contributes to the reconstruction of reciprocal effective connections without synaptic connections. In addition, delay-induced zero-lag synchronous transition facilitates dynamic transformation of the causal interactions. With the increase of synaptic coupling strength, the causal interplay undergoes the transition to be statistically significant at a critical value. Furthermore, it can be found that multiple effective motifs are extracted from different synchronization states of the underlying structural motifs. GC measures of effective connectivity are proved to be reliable compared with the Information Flow for causal analysis. The obtained results may be helpful to future research about information processes.