Burst synchronization transitions in a neuronal network of subnetworks

Analyzing bursting synchronization in structural connectivity matrix of a human brain under external pulsed currents

Cognitive tasks in the human brain are performed by various cortical areas located in the cerebral cortex. The cerebral cortex is separated into different areas in the right and left hemispheres. We consider one human cerebral cortex according to a network composed of coupled subnetworks with small-world properties. We study the burst synchronization and desynchronization in a human neuronal network under external periodic and random pulsed currents. With and without external perturbations, the emergence of bursting synchronization is observed. Synchronization can contribute to the processing of information, however, there are evidences that it can be related to some neurological disorders. Our results show that synchronous behavior can be suppressed by means of external pulsed currents.

Connecting curve: A new tool for locating hidden attractors

Attractors in nonlinear dynamical systems can be categorized as self-excited attractors and hidden attractors. In contrast to self-excited attractors, which can be located by the standard numerical computational method, hidden attractors are hard to detect due to the fact that its basin of attraction is away from the proximity to equilibrium. In multistable systems, many attractors, including self-excited and hidden ones, co-exist, which makes locating each different oscillation more difficult. Hidden attractors are frequently connected to rare or abnormal oscillations in the system and often lead to unpredicted behaviors in many engineering applications, and, thus, the research in locating such attractors is considerably significant. Previous work has proposed several methods for locating hidden attractors but these methods all have their limitations. For example, one of the methods suggests that perpetual points are useful in locating hidden and co-existing attractors, while an in-depth examination suggests that they are insufficient in finding hidden attractors. In this study, we propose that the method of connecting curves, which is a collection of points of analytical inflection including both perpetual points and fixed points, is more reliable to search for hidden attractors. We analyze several dynamical systems using the connecting curve, and the results demonstrate that it can be used to locate hidden and co-existing oscillations.

Towards proteinoid computers. Hypothesis paper

Proteinoids - thermal proteins - are produced by heating amino acids to their melting point and initiation of polymerisation to produce polymeric chains. Proteinoids swell in aqueous solution into hollow microspheres. The proteinoid microspheres produce endogenous burst of electrical potential spikes and change patterns of their electrical activity in response to illumination. The microspheres can interconnect by pores and tubes and form networks with a programmable growth. We speculate on how ensembles of the proteinoid microspheres can be developed into unconventional computing devices.

Effects of temperature and ion channel blocks on propagation of action potential in myelinated axons

Potassium ion and sodium ion channels play important roles in the propagation of action potentials along a myelinated axon. The random opening and closing of ion channels can cause the fluctuation of action potentials. In this paper, an improved Hodgkin-Huxley chain network model is proposed to study the effects of ion channel blocks, temperature, and ion channel noise on the propagation of action potentials along the myelinated axon. It is found that the chain network has minimum coupling intensity threshold and maximum tolerance temperature threshold that allow the action potentials to pass along the whole axon, and the blockage of ion channels can change these two thresholds. A striking result is that the simulated value of the optimum membrane size (inversely proportional to noise intensity) coincides with the area range of feline thalamocortical relay cells in biological experiments.

Cluster burst synchronization in a scale-free network of inhibitory bursting neurons

We consider a scale-free network of inhibitory Hindmarsh-Rose (HR) bursting neurons, and make a computational study on coupling-induced cluster burst synchronization by varying the average coupling strength J 0 . For sufficiently small J 0 , non-cluster desynchronized states exist. However, when passing a critical point J c ∗ ( ≃ 0.16 ) , the whole population is segregated into 3 clusters via a constructive role of synaptic inhibition to stimulate dynamical clustering between individual burstings, and thus 3-cluster desynchronized states appear. As J 0 is further increased and passes a lower threshold J l ∗ ( ≃ 0.78 ) , a transition to 3-cluster burst synchronization occurs due to another constructive role of synaptic inhibition to favor population synchronization. In this case, HR neurons in each cluster make burstings every 3rd cycle of the instantaneous burst rate R w ( t ) of the whole population, and exhibit burst synchronization. However, as J 0 passes an intermediate threshold J m ∗ ( ≃ 5.2 ) , HR neurons fire burstings intermittently at a 4th cycle of R w ( t ) via burst skipping rather than at its 3rd cycle, and hence they begin to make intermittent hoppings between the 3 clusters. Due to such intermittent intercluster hoppings via burst skippings, the 3 clusters become broken up (i.e., the 3 clusters are integrated into a single one). However, in spite of such break-up (i.e., disappearance) of the 3-cluster states, (non-cluster) burst synchronization persists in the whole population, which is well visualized in the raster plot of burst onset times where bursting stripes (composed of burst onset times and indicating burst synchronization) appear successively. With further increase in J 0 , intercluster hoppings are intensified, and bursting stripes also become dispersed more and more due to a destructive role of synaptic inhibition to spoil the burst synchronization. Eventually, when passing a higher threshold J h ∗ ( ≃ 17.8 ) a transition to desynchronization occurs via complete overlap between the bursting stripes. Finally, we also investigate the effects of stochastic noise on both 3-cluster burst synchronization and intercluster hoppings.

Winfree loop sustained oscillation in two-dimensional excitable lattices: Prediction and realization

The problem of self-sustained oscillations in excitable complex networks is the central issue under investigation, among which the prediction and the realization of self-sustained oscillations in different kinds of excitable networks are the challenging tasks. In this paper, we extensively investigate the prediction and the realization of a Winfree loop sustained oscillation (WLSO) in two-dimensional (2D) excitable lattices. By analyzing the network structure, the fundamental oscillation source structure (FOSS) of WLSO in a 2D excitable lattice is exposed explicitly. For the suitable combinations of system parameters, the Winfree loop can self-organize on the FOSS to form an oscillation source sustaining the oscillation, and these suitable parameter combinations are predicted by calculating the minimum Winfree loop length and have been further confirmed in numerical simulations. However, the FOSS cannot spontaneously offer the WLSO in 2D excitable lattices in usual cases due to the coupling bidirectionality and the symmetry properties of the lattice. A targeted protection scheme of the oscillation source is proposed by overcoming these two drawbacks. Finally, the WLSO is realized in the 2D excitable lattice successfully.

Fast regular firings induced by intra- and inter-time delays in two clustered neuronal networks

In this paper, we consider two clustered neuronal networks with dense intra-synaptic links within each cluster and sparse inter-synaptic links between them. We focus on the effects of intra- and inter-time delays on the spiking regularity and timing in both clusters. With the aid of simulation results, we show that intermediate intra- and inter-time delays are able to separately induce fast regular firing - spiking activity with a high firing rate as well as a high spiking regularity. Moreover, when both intra- and inter-time delays are present, we find that fast regular firings are induced much more frequently than if only a single type of delay is present in the system. Our results indicate that appropriately adjusted intra- and inter-time delays can significantly facilitate fast regular firing in neuronal networks. Based on a detailed analysis, we conjecture that this is most likely when the largest value of common divisors of the intra- and inter-time delays falls into a range where fast regular firings are induced by suitable intra- or inter-time delays alone.

Burst synchronization in a scale-free neuronal network with inhibitory spike-timing-dependent plasticity

We are concerned about burst synchronization (BS), related to neural information processes in health and disease, in the Barabási-Albert scale-free network (SFN) composed of inhibitory bursting Hindmarsh-Rose neurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without considering iSTDP, BS was found to appear in a range of noise intensities for fixed synaptic inhibition strengths. In contrast, in our present work, we take into consideration iSTDP and investigate its effect on BS by varying the noise intensity. Our new main result is to find occurrence of a Matthew effect in inhibitory synaptic plasticity: good BS gets better via LTD, while bad BS get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). We note that, due to inhibition, the roles of LTD and LTP in inhibitory synaptic plasticity are reversed in comparison with those in excitatory synaptic plasticity. Moreover, emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic burst onset times. Finally, in the presence of iSTDP we investigate the effects of network architecture on BS by varying the symmetric attachment degree l ∗ and the asymmetry parameter Δ l in the SFN.

Effect of spike-timing-dependent plasticity on stochastic burst synchronization in a scale-free neuronal network

We consider an excitatory population of subthreshold Izhikevich neurons which cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings. This neuronal population has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). However, STDP was not considered in previous works on stochastic burst synchronization (SBS) between noise-induced burstings of sub-threshold neurons. Here, we study the effect of additive STDP on SBS by varying the noise intensity D in the Barabási-Albert scale-free network (SFN). One of our main findings is a Matthew effect in synaptic plasticity which occurs due to a positive feedback process. Good burst synchronization (with higher bursting measure) gets better via long-term potentiation (LTP) of synaptic strengths, while bad burst synchronization (with lower bursting measure) gets worse via long-term depression (LTD). Consequently, a step-like rapid transition to SBS occurs by changing D, in contrast to a relatively smooth transition in the absence of STDP. We also investigate the effects of network architecture on SBS by varying the symmetric attachment degree [Formula: see text] and the asymmetry parameter [Formula: see text] in the SFN, and Matthew effects are also found to occur by varying [Formula: see text] and [Formula: see text]. Furthermore, emergences of LTP and LTD of synaptic strengths are investigated in details via our own microscopic methods based on both the distributions of time delays between the burst onset times of the pre- and the post-synaptic neurons and the pair-correlations between the pre- and the post-synaptic instantaneous individual burst rates (IIBRs). Finally, a multiplicative STDP case (depending on states) with soft bounds is also investigated in comparison with the additive STDP case (independent of states) with hard bounds. Due to the soft bounds, a Matthew effect with some quantitative differences is also found to occur for the case of multiplicative STDP.

Minimum Winfree loop determines self-sustained oscillations in excitable Erdös-Rényi random networks

The investigation of self-sustained oscillations in excitable complex networks is very important in understanding various activities in brain systems, among which the exploration of the key determinants of oscillations is a challenging task. In this paper, by investigating the influence of system parameters on self-sustained oscillations in excitable Erdös-Rényi random networks (EERRNs), the minimum Winfree loop (MWL) is revealed to be the key factor in determining the emergence of collective oscillations. Specifically, the one-to-one correspondence between the optimal connection probability (OCP) and the MWL length is exposed. Moreover, many important quantities such as the lower critical connection probability (LCCP), the OCP, and the upper critical connection probability (UCCP) are determined by the MWL. Most importantly, they can be approximately predicted by the network structure analysis, which have been verified in numerical simulations. Our results will be of great importance to help us in understanding the key factors in determining persistent activities in biological systems.

Partial coupling delay induced multiple spatiotemporal orders in a modular neuronal network

The influence of partial coupling delay on the spatiotemporal spiking dynamics is explored in a modular neuronal network. The modular neuronal network is composed of two subnetworks which present the small-world property and scale-free property, respectively. Numerical results show that spatiotemporal order that the modular network is most coherent in time and nearly synchronized in space can emerge intermittently when the coupling delays among neurons are appropriately tuned. The appropriately tuned delays are further detected to be integer multiples of the intrinsic spiking period of the modular neuronal network, which implies that the phenomenon of multiple spatiotemporal orders could be the result of a locking between the length of coupling delay and the intrinsic spiking period of the modular neuronal network. Moreover, the multiple spatiotemporal orders are verified to be robust against variations of the fraction of delayed connection as well as the key parameters of network architecture such as the rewiring probability, the average degree of small-world subnetwork, the initial nodes of scale-free subnetwork and the total size of the modular network.

Pinning Impulsive Synchronization of Reaction-Diffusion Neural Networks With Time-Varying Delays

This paper investigates the exponential synchronization of reaction-diffusion neural networks with time-varying delays subject to Dirichlet boundary conditions. A novel type of pinning impulsive controllers is proposed to synchronize the reaction-diffusion neural networks with time-varying delays. By applying the Lyapunov functional method, sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction-diffusion neural networks with large and small delay sizes. It is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network; the technique used in this paper is also applicable to reaction-diffusion networks with Neumann boundary conditions. Numerical examples are presented to demonstrate the effectiveness of the theoretical results.

Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks

In this paper, we study effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks. Our focus is on the impact of two parameters, namely the time delay τ and the probability of partial time delay p_delay, whereby the latter determines the probability with which a connection between two neurons is delayed. Our research reveals that partial time delays significantly affect phase synchronization in this system. In particular, partial time delays can either enhance or decrease phase synchronization and induce synchronization transitions with changes in the mean firing rate of neurons, as well as induce switching between synchronized neurons with period-1 firing to synchronized neurons with period-2 firing. Moreover, in comparison to a neuronal network where all connections are delayed, we show that small partial time delay probabilities have especially different influences on phase synchronization of neuronal networks.

Cooperative effect of random and time-periodic coupling strength on synchronization transitions in one-way coupled neural system: mean field approach

The cooperative effect of random coupling strength and time-periodic coupling strengh on synchronization transitions in one-way coupled neural system has been investigated by mean field approach. Results show that cooperative coupling strength (CCS) plays an active role for the enhancement of synchronization transitions. There exist an optimal frequency of CCS which makes the system display the best CCS-induced synchronization transitions, a critical frequency of CCS which can not further affect the CCS-induced synchronization transitions, and a critical amplitude of CCS which can not occur the CCS-induced synchronization transitions. Meanwhile, noise intensity plays a negative role for the CCS-induced synchronization transitions. Furthermore, it is found that the novel CCS amplitude-induced synchronization transitions and CCS frequency-induced synchronization transitions are found.

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

Activity of neurons in the pre-Bötzinger complex within the mammalian brain stem has an important role in the generation of respiratory rhythms. Previous experimental results have shown that the dynamics of sodium and calcium within each cell may be responsible for various bursting mechanisms. In this paper, we study the bursting dynamics of the two-coupled pre-Bötzinger complex neurons. Using a combination of fast-slow decomposition and two-parameter bifurcation analysis, we explore the possible forms of dynamics that the model network can produce as well the transitions of in-phase and anti-phase bursting respectively.

Impulsive Effects and Stability Analysis on Memristive Neural Networks With Variable Delays

In this brief, hybrid impulsive and adaptive feedback controllers are simultaneously exerted on a general delayed memristive neural network (MNN) model to formulate a novel impulsive controlled MNN (IMNN) model with variable delays. By means of Lyapunov-Razumikhin technique and other analytical ways, several new stability criteria of the proposed IMNN model are obtained. In addition, by choosing appropriate impulses and external inputs, the convergence speed of IMNN can be increased, which implies that its dynamic behaviors will be optimized. Finally, the effectiveness of the obtained results is illustrated by one numerical example.

Analysis and application of neuronal network controllability and observability

Controllability and observability analyses are important prerequisite for designing suitable neural control strategy, which can help lower the efforts required to control and observe the system dynamics. First, 3-neuron motifs including the excitatory motif, the inhibitory motif, and the mixed motif are constructed to investigate the effects of single neuron and synaptic dynamics on network controllability (observability). Simulation results demonstrate that for networks with the same topological structure, the controllability (observability) of the node always changes if the properties of neurons and synaptic coupling strengths vary. Besides, the inhibitory networks are more controllable (observable) than the excitatory networks when the coupling strengths are the same. Then, the numerically determined controllability results of 3-neuron excitatory motifs are generalized to the desynchronization control of the modular motif network. The control energy and neuronal synchrony measure indexes are used to quantify the controllability of each node in the modular network. The best driver node obtained in this way is the same as the deduced one from motif analysis.

Spike-timing-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman-Watts neuronal networks

In this paper, we numerically study the effect of spike-timing-dependent plasticity (STDP) on synchronization transitions induced by autaptic activity in adaptive Newman-Watts Hodgkin-Huxley neuron networks. It is found that synchronization transitions induced by autaptic delay vary with the adjusting rate A_p of STDP and become strongest at a certain A_p value, and the A_p value increases when network randomness or network size increases. It is also found that the synchronization transitions induced by autaptic delay become strongest at a certain network randomness and network size, and the values increase and related synchronization transitions are enhanced when A_p increases. These results show that there is optimal STDP that can enhance the synchronization transitions induced by autaptic delay in the adaptive neuronal networks. These findings provide a new insight into the roles of STDP and autapses for the information transmission in neural systems.

Chaotic, informational and synchronous behaviour of multiplex networks

The understanding of the relationship between topology and behaviour in interconnected networks would allow to charac- terise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex net- works with constant Jacobian. We also study this relationship numerically in multiplex networks of Hindmarsh-Rose neurons. Whereas behaviour in the analytically tractable network is a direct but not trivial consequence of the spectra of eigenvalues of the Laplacian matrix, where behaviour may strongly depend on the break of symmetry in the topology of interconnections, in Hindmarsh-Rose neural networks the nonlinear nature of the chemical synapses breaks the elegant mathematical connec- tion between the spectra of eigenvalues of the Laplacian matrix and the behaviour of the network, creating networks whose behaviour strongly depends on the nature (chemical or electrical) of the inter synapses.

Pinning Synchronization of Directed Networks With Switching Topologies: A Multiple Lyapunov Functions Approach

This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from M -matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results.

Effect of intermodular connection on fast sparse synchronization in clustered small-world neural networks

We consider a clustered network with small-world subnetworks of inhibitory fast spiking interneurons and investigate the effect of intermodular connection on the emergence of fast sparsely synchronized rhythms by varying both the intermodular coupling strength J(inter) and the average number of intermodular links per interneuron M(syn)(inter). In contrast to the case of nonclustered networks, two kinds of sparsely synchronized states such as modular and global synchronization are found. For the case of modular sparse synchronization, the population behavior reveals the modular structure, because the intramodular dynamics of subnetworks make some mismatching. On the other hand, in the case of global sparse synchronization, the population behavior is globally identical, independently of the cluster structure, because the intramodular dynamics of subnetworks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching degree between the instantaneous subpopulation spike rates of the subnetworks, and examine whether the sparse synchronization is global or modular. Depending on its magnitude, the intermodular coupling strength J(inter) seems to play "dual" roles for the pacing between spikes in each subnetwork. For large J(inter), due to strong inhibition it plays a destructive role to "spoil" the pacing between spikes, while for small J(inter) it plays a constructive role to "favor" the pacing between spikes. Through competition between the constructive and the destructive roles of J(inter), there exists an intermediate optimal J(inter) at which the pacing degree between spikes becomes maximal. In contrast, the average number of intermodular links per interneuron M(syn)(inter) seems to play a role just to favor the pacing between spikes. With increasing M(syn)(inter), the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes. Furthermore, we employ the realistic sub- and whole-population order parameters, based on the instantaneous sub- and whole-population spike rates, to determine the threshold values for the synchronization-unsynchronization transition in the sub- and whole populations, and the degrees of global and modular sparse synchronization are also measured in terms of the realistic sub- and whole-population statistical-mechanical spiking measures defined by considering both the occupation and the pacing degrees of spikes. It is expected that our results could have implications for the role of the brain plasticity in some functional behaviors associated with population synchronization.

Synchronization transitions induced by the fluctuation of adaptive coupling strength in delayed Newman-Watts neuronal networks

Introducing adaptive coupling in delayed neuronal networks and regulating the dissipative parameter (DP) of adaptive coupling by noise, we study the effect of fluctuations of the changing rate of adaptive coupling on the synchronization of the neuronal networks. It is found that time delay can induce synchronization transitions for intermediate DP values, and the synchronization transitions become strongest when DP is optimal. As the intensity of DP noise is varied, the neurons can also exhibit synchronization transitions, and the phenomenon is delay-dependent and is enhanced for certain time delays. Moreover, the synchronization transitions change with the change of DP and become strongest when DP is optimal. These results show that randomly changing adaptive coupling can considerably change the synchronization of the neuronal networks, and hence could play a crucial role in the information processing and transmission in neural systems.

Impacts of clustering on noise-induced spiking regularity in the excitatory neuronal networks of subnetworks

In this paper, we investigate how clustering factors influent spiking regularity of the neuronal network of subnetworks. In order to do so, we fix the averaged coupling probability and the averaged coupling strength, and take the cluster number M, the ratio of intra-connection probability and inter-connection probability R, the ratio of intra-coupling strength and inter-coupling strength S as controlled parameters. With the obtained simulation results, we find that spiking regularity of the neuronal networks has little variations with changing of R and S when M is fixed. However, cluster number M could reduce the spiking regularity to low level when the uniform neuronal network's spiking regularity is at high level. Combined the obtained results, we can see that clustering factors have little influences on the spiking regularity when the entire energy is fixed, which could be controlled by the averaged coupling strength and the averaged connection probability.

Frequency-domain order parameters for the burst and spike synchronization transitions of bursting neurons

We are interested in characterization of synchronization transitions of bursting neurons in the frequency domain. Instantaneous population firing rate (IPFR) [Formula: see text], which is directly obtained from the raster plot of neural spikes, is often used as a realistic collective quantity describing population activities in both the computational and the experimental neuroscience. For the case of spiking neurons, a realistic time-domain order parameter, based on [Formula: see text], was introduced in our recent work to characterize the spike synchronization transition. Unlike the case of spiking neurons, the IPFR [Formula: see text] of bursting neurons exhibits population behaviors with both the slow bursting and the fast spiking timescales. For our aim, we decompose the IPFR [Formula: see text] into the instantaneous population bursting rate [Formula: see text] (describing the bursting behavior) and the instantaneous population spike rate [Formula: see text] (describing the spiking behavior) via frequency filtering, and extend the realistic order parameter to the case of bursting neurons. Thus, we develop the frequency-domain bursting and spiking order parameters which are just the bursting and spiking "coherence factors" [Formula: see text] and [Formula: see text] of the bursting and spiking peaks in the power spectral densities of [Formula: see text] and [Formula: see text] (i.e., "signal to noise" ratio of the spectral peak height and its relative width). Through calculation of [Formula: see text] and [Formula: see text], we obtain the bursting and spiking thresholds beyond which the burst and spike synchronizations break up, respectively. Consequently, it is shown in explicit examples that the frequency-domain bursting and spiking order parameters may be usefully used for characterization of the bursting and the spiking transitions, respectively.

Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons

We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh-Rose neurons. For modeling complex synaptic connectivity, we consider the Watts-Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the [Formula: see text]-[Formula: see text] plane ([Formula: see text]: synaptic inhibition strength and [Formula: see text]: noise intensity). As the rewiring probability [Formula: see text] is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the [Formula: see text]-[Formula: see text] plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.

Autaptic activity-induced synchronization transitions in Newman-Watts network of Hodgkin-Huxley neurons

In this paper, we numerically study the effect of autapse on the synchronization of Newman-Watts small-world Hodgkin-Huxley neuron network. It is found that the neurons exhibit synchronization transitions as autaptic self-feedback delay is varied, and the phenomenon becomes strongest when autaptic self-feedback strength is optimal. This phenomenon also changes with the change of coupling strength and network randomness and become strongest when they are optimal. There are similar synchronization transitions for electrical and chemical autapse, but the synchronization transitions for chemical autapse occur more frequently and are stronger than those for electrical synapse. The underlying mechanisms are briefly discussed in quality. These results show that autaptic activity plays a subtle role in the synchronization of the neuronal network. These findings may find potential implications of autapse for the information processing and transmission in neural systems.

Intrinsic excitability state of local neuronal population modulates signal propagation in feed-forward neural networks

Reliable signal propagation across distributed brain areas is an essential requirement for cognitive function, and it has been investigated extensively in computational studies where feed-forward network (FFN) is taken as a generic model. But it is still unclear how distinct local network states, which are intrinsically generated by synaptic interactions within each layer, would affect the ability of FFN to transmit information. Here we investigate the impact of such network states on propagating transient synchrony (synfire) and firing rate by a combination of numerical simulations and analytical approach. Specifically, local network dynamics is attributed to the competition between excitatory and inhibitory neurons within each layer. Our results show that concomitant with different local network states, the performance of signal propagation differs dramatically. For both synfire propagation and firing rate propagation, there exists an optimal local excitability state, respectively, that optimizes the performance of signal propagation. Furthermore, we find that long-range connections strongly change the dependence of spiking activity propagation on local network state and propose that these two factors work jointly to determine information transmission across distributed networks. Finally, a simple mean field approach that bridges response properties of long-range connectivity and local subnetworks is utilized to reveal the underlying mechanism.

Diversity and time delays induce resonance in a modular neuronal network

This paper focuses on the resonance dynamics of a modular neuronal network consisting of several small-world subnetworks. The considered network is composed of delay-coupled FitzHugh-Nagumo (FHN) neurons, whose characteristic parameters present diversity in the form of quenched noise. Our numerical results indicate that when such a network is subjected to an external subthreshold periodic signal, its collective response is optimized for an intermediate level of diversity, namely, a resonant behavior can be induced by an appropriate level of diversity. How the probabilities of intramodule and intermodule connections, as well as the number of subnetworks influence the diversity-induced resonance are also discussed. Further, conclusive evidences demonstrate the nontrivial role of time-delayed coupling on the diversity-induced resonance properties. Especially, multiple resonance is obviously detected when time delays are located at integer multiples of the oscillation period of the signal. Moreover, the phenomenon of fine-tuned delays in inducing multiple resonance remains when diversity is within an intermediate range. Our findings have implications that neural systems may profit from their generic diversity and delayed coupling to optimize the response to external stimulus.

ADAPTATION-DEPENDENT SYNCHRONIZATION TRANSITIONS AND BURST GENERATIONS IN ELECTRICALLY COUPLED NEURAL NETWORKS

A typical feature of neurons is their ability to encode neural information dynamically through spike frequency adaptation (SFA). Previous studies of SFA on neuronal synchronization were mainly concentrated on the correlated firing between neuron pairs, while the synchronization of neuron populations in the presence of SFA is still unclear. In this study, the influence of SFA on the population synchronization of neurons was numerically explored in electrically coupled networks, with regular, small-world, and random connectivity, respectively. The simulation results indicate that cross-correlation indices decrease significantly when the neurons have adaptation compared with those of nonadapting neurons, similar to previous experimental observations. However, the synchronous activity of population neurons exhibits a rather complex adaptation-dependent manner. Specifically, synchronization strength of neuron populations changes nonmonotonically, depending on the degree of adaptation. In addition, single neurons in the networks can switch from regular spiking to bursting with the increase of adaptation degree. Furthermore, the connection probability among neurons exhibits significant influence on the population synchronous activity, but has little effect on the burst generation of single neurons. Accordingly, the results may suggest that synchronous activity and burst firing of population neurons are both adaptation-dependent.

Synchronization on complex networks of networks

In this paper, pinning synchronization on complex networks of networks is investigated, where there are many subnetworks with the interactions among them. The subnetworks and their connections can be regarded as the nodes and interactions of the networks, respectively, which form the networks of networks. In this new setting, the aim is to design pinning controllers on the chosen nodes of each subnetwork so as to reach synchronization behavior. Some synchronization criteria are established for reaching pinning control on networks of networks. Furthermore, the pinning scheme is designed, which shows that the nodes with very low degrees and large degrees are good candidates for applying pinning controllers. Then, the attack and robustness of the pinning scheme are discussed. Finally, a simulation example is presented to verify the theoretical analysis in this paper.

Synchronization of stochastic dynamical networks under impulsive control with time delays

In this paper, the stochastic synchronization problem is studied for a class of delayed dynamical networks under delayed impulsive control. Different from the existing results on the synchronization of dynamical networks under impulsive control, impulsive input delays are considered in our model. By assuming that the impulsive intervals belong to a certain interval and using the mathematical induction method, several conditions are derived to guarantee that complex networks are exponentially synchronized in mean square. The derived conditions reveal that the frequency of impulsive occurrence, impulsive input delays, and stochastic perturbations can heavily affect the synchronization performance. A control algorithm is then presented for synchronizing stochastic dynamical networks with delayed synchronizing impulses. Finally, two examples are given to demonstrate the effectiveness of the proposed approach.

Synchronization and array-enhanced resonances in delayed coupled neuronal network with channel noise

This paper studies the combined effect of transmission delay and channel fluctuations on population behaviors of an excitatory Erdös-Rényi neuronal network. First, it is found that the network reaches a perfect spatial temporal coherence at a suitable membrane size. Such a coherence resonance is stimulus-free and is array-enhanced. Second, the presence of transmission delay can induce intermittent changes of the population dynamics. Besides, two resonant peaks of the population firing rate are observed as delay changes: one is at τd≈7ms for all membrane areas, which reflects the resonance between the delayed interaction and the intrinsic period of channel kinetics; the other occurs when the transmission delay equals to the mean inter-spike intervals of the population firings in the absence of delay, which reflects the resonance between the delayed interaction and the firing period of the non-delayed system. Third, concerning the impact of network topology and population size, it is found that decreasing the connection probability does not change the range of transmission delay but broadens the range of synaptic coupling that supports population neurons to generate action potentials synchronously and temporally coherently. Furthermore, there exists a critical connection probability that distinguishes the population dynamics into an asynchronous and synchronous state. All the results we obtained are based on networks of size N = 500, which are shown to be robust to further increasing the population size.

Decreased coherence and functional connectivity of electroencephalograph in Alzheimer's disease

In this paper, we investigate the abnormalities of electroencephalograph (EEG) signals in the Alzheimer's disease (AD) by analyzing 16-scalp electrodes EEG signals and make a comparison with the normal controls. Coherence is introduced to measure the pair-wise normalized linear synchrony and functional correlations between two EEG signals in different frequency domains, and graph analysis is further used to investigate the influence of AD on the functional connectivity of human brain. Data analysis results show that, compared with the control group, the pair-wise coherence of AD group is significantly decreased, especially for the theta and alpha frequency bands in the frontal and parieto-occipital regions. Furthermore, functional connectivity among different brain regions is reconstructed based on EEG, which exhibit obvious small-world properties. Graph analysis demonstrates that the local functional connections between regions for AD decrease. In addition, it is found that small-world properties of AD networks are largely weakened, by calculating its average path lengths, clustering coefficients, global efficiency, local efficiency, and small-worldness. The obtained results show that both pair-wise coherence and functional network can be taken as effective measures to distinguish AD patients from the normal, which may benefit our understanding of the disease.

Degree mixing in multilayer networks impedes the evolution of cooperation

Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play different roles in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider two-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as "fold/limit cycle" type of bursting, then anti-phase continuous spiking, followed by the "fold/torus" type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate "fold/Hopf" bursting via "fold/fold" hysteresis loop; whereas, the chemically coupled cells generate "fold/subHopf" bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast-slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.

Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model

Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.

Impact of delays on the synchronization transitions of modular neuronal networks with hybrid synapses

The combined effects of the information transmission delay and the ratio of the electrical and chemical synapses on the synchronization transitions in the hybrid modular neuronal network are investigated in this paper. Numerical results show that the synchronization of neuron activities can be either promoted or destroyed as the information transmission delay increases, irrespective of the probability of electrical synapses in the hybrid-synaptic network. Interestingly, when the number of the electrical synapses exceeds a certain level, further increasing its proportion can obviously enhance the spatiotemporal synchronization transitions. Moreover, the coupling strength has a significant effect on the synchronization transition. The dominated type of the synapse always has a more profound effect on the emergency of the synchronous behaviors. Furthermore, the results of the modular neuronal network structures demonstrate that excessive partitioning of the modular network may result in the dramatic detriment of neuronal synchronization. Considering that information transmission delays are inevitable in intra- and inter-neuronal networks communication, the obtained results may have important implications for the exploration of the synchronization mechanism underlying several neural system diseases such as Parkinson's Disease.

Adaptive coupling optimized spiking coherence and synchronization in Newman-Watts neuronal networks

In this paper, we have numerically studied the effect of adaptive coupling on the temporal coherence and synchronization of spiking activity in Newman-Watts Hodgkin-Huxley neuronal networks. It is found that random shortcuts can enhance the spiking synchronization more rapidly when the increment speed of adaptive coupling is increased and can optimize the temporal coherence of spikes only when the increment speed of adaptive coupling is appropriate. It is also found that adaptive coupling strength can enhance the synchronization of spikes and can optimize the temporal coherence of spikes when random shortcuts are appropriate. These results show that adaptive coupling has a big influence on random shortcuts related spiking activity and can enhance and optimize the temporal coherence and synchronization of spiking activity of the network. These findings can help better understand the roles of adaptive coupling for improving the information processing and transmission in neural systems.

Spontaneous scale-free structure in adaptive networks with synchronously dynamical linking

Inspired by the anti-Hebbian learning rule in neural systems, we study how the feedback from dynamical synchronization shapes network structure by adding new links. Through extensive numerical simulations, we find that an adaptive network spontaneously forms scale-free structure, as confirmed in many real systems. Moreover, the adaptive process produces two nontrivial power-law behaviors of deviation strength from mean activity of the network and negative degree correlation, which exists widely in technological and biological networks. Importantly, these scalings are robust to variation of the adaptive network parameters, which may have meaningful implications in the scale-free formation and manipulation of dynamical networks. Our study thus suggests an alternative adaptive mechanism for the formation of scale-free structure with negative degree correlation, which means that nodes of high degree tend to connect, on average, with others of low degree and vice versa. The relevance of the results to structure formation and dynamical property in neural networks is briefly discussed as well.

Spike phase synchronization in delayed-coupled neural networks: uniform vs. non-uniform transmission delay

In this paper, we investigated phase synchronization in delayed dynamical networks. Non-identical spiking Hindmarsh-Rose neurons were considered as individual dynamical systems and coupled through a number of network structures such as scale-free, Erdős-Rényi, and modular. The individual neurons were coupled through excitatory chemical synapses with uniform or distributed time delays. The profile of spike phase synchrony was different when the delay was uniform across the edges as compared to the case when it was distributed, i.e., different delays for the edges. When an identical transmission delay was considered, a quasi-periodic pattern was observed in the spike phase synchrony. There were specific values of delay where the phase synchronization reached to its peaks. The behavior of the phase synchronization in the networks with non-uniform delays was different with the former case, where the phase synchrony decreased as distributed delays introduced to the networks.

Spreading of cooperative behaviour across interdependent groups

Recent empirical research has shown that links between groups reinforce individuals within groups to adopt cooperative behaviour. Moreover, links between networks may induce cascading failures, competitive percolation, or contribute to efficient transportation. Here we show that there in fact exists an intermediate fraction of links between groups that is optimal for the evolution of cooperation in the prisoner's dilemma game. We consider individual groups with regular, random, and scale-free topology, and study their different combinations to reveal that an intermediate interdependence optimally facilitates the spreading of cooperative behaviour between groups. Excessive between-group links simply unify the two groups and make them act as one, while too rare between-group links preclude a useful information flow between the two groups. Interestingly, we find that between-group links are more likely to connect two cooperators than in-group links, thus supporting the conclusion that they are of paramount importance.

Impact of connection delays on noise-induced spatiotemporal patterns in neuronal networks

In the present work, we investigate the nontrivial roles of independent Gaussian noise and time-delayed coupling on the synchronous dynamics and coherence property of Fitz Hugh-Nagumo neurons on small-world networks by numerical simulations. First, it is shown that an intermediate level of noise in the neuronal networks can optimally induce a temporal coherence state when the delay in the coupling is absent. We find that this phenomenon is robust to changes of the coupling strength and the rewiring probability of small-world networks. Then, when appropriately tuned delays with moderate values are included in the coupling, the neurons on the networks can reach higher ordered spatiotemporal patterns which are the most coherent in time and almost synchronized in space. Moreover, the tuned delays are within a range, and the period of the firing activity is delay-dependent which equals nearly to the length of the coupling delay. This result implies that the higher ordered spatiotemporal dynamics induced by intermediate delays could be the result of a locking between the period-1 neuronal spiking activity and the delay. The performance of moderate delays in enhancing the ordered spatiotemporal patterns is also examined to be robust against variations of the network randomness.

Optimal spike-based communication in excitable networks with strong-sparse and weak-dense links

The connectivity of complex networks and functional implications has been attracting much interest in many physical, biological and social systems. However, the significance of the weight distributions of network links remains largely unknown except for uniformly- or Gaussian-weighted links. Here, we show analytically and numerically, that recurrent neural networks can robustly generate internal noise optimal for spike transmission between neurons with the help of a long-tailed distribution in the weights of recurrent connections. The structure of spontaneous activity in such networks involves weak-dense connections that redistribute excitatory activity over the network as noise sources to optimally enhance the responses of individual neurons to input at sparse-strong connections, thus opening multiple signal transmission pathways. Electrophysiological experiments confirm the importance of a highly broad connectivity spectrum supported by the model. Our results identify a simple network mechanism for internal noise generation by highly inhomogeneous connection strengths supporting both stability and optimal communication.

Phase synchronization of bursting neurons in clustered small-world networks

We investigate the collective dynamics of bursting neurons on clustered networks. The clustered network model is composed of subnetworks, each of them presenting the so-called small-world property. This model can also be regarded as a network of networks. In each subnetwork a neuron is connected to other ones with regular as well as random connections, the latter with a given intracluster probability. Moreover, in a given subnetwork each neuron has an intercluster probability to be connected to the other subnetworks. The local neuron dynamics has two time scales (fast and slow) and is modeled by a two-dimensional map. In such small-world network the neuron parameters are chosen to be slightly different such that, if the coupling strength is large enough, there may be synchronization of the bursting (slow) activity. We give bounds for the critical coupling strength to obtain global burst synchronization in terms of the network structure, that is, the probabilities of intracluster and intercluster connections. We find that, as the heterogeneity in the network is reduced, the network global synchronizability is improved. We show that the transitions to global synchrony may be abrupt or smooth depending on the intercluster probability.