Dynamics of Time Delay-Induced Multiple Synchronous Behaviors in Inhibitory Coupled Neurons

The inhibitory synapse can induce synchronous behaviors different from the anti-phase synchronous behaviors, which have been reported in recent studies. In the present paper, synchronous behaviors are investigated in the motif model composed of reciprocal inhibitory coupled neurons with endogenous bursting and time delay. When coupling strength is weak, synchronous behavior appears at a single interval of time delay within a bursting period. When coupling strength is strong, multiple synchronous behaviors appear at different intervals of time delay within a bursting period. The different bursting patterns of synchronous behaviors, and time delays and coupling strengths that can induce the synchronous bursting patterns can be well interpreted by the dynamics of the endogenous bursting pattern of isolated neuron, which is acquired by the fast-slow dissection method, combined with the inhibitory coupling current. For an isolated neuron, when a negative impulsive current with suitable strength is applied at different phases of the bursting, multiple different bursting patterns can be induced. For a neuron in the motif, the inhibitory coupling current, of which the application time and strength is modulated by time delay and coupling strength, can cause single or multiple synchronous firing patterns like the negative impulsive current when time delay and coupling strength is suitable. The difference compared to the previously reported multiple synchronous behaviors that appear at time delays wider than a period of the endogenous firing is discussed. The results present novel examples of synchronous behaviors in the neuronal network with inhibitory synapses and provide a reasonable explanation.     

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling

This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.     

Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.     

时滞援助的两抑制性突触耦合的Chay 神经元的同步

一对抑制性突触耦合的混沌Chay神经元的同步模式被研究。结果表明当耦合强度超过临界值时,两抑制耦合的混沌Chay神经元能达到反相的同步。与此同时,两混沌的神经元变为周期而不是原来的混沌运动。然而,如果考虑耦合神经元信息的传导时滞,在有效的时滞下,两个耦合神经元的在相簇同步能增加。在相簇同步窗口的大小随着耦合强度的增加而增加。此结果对于我们理解神经元集群的运动是一个指导。     

A method for the estimation of functional brain connectivity from time-series data

A central issue in cognitive neuroscience is which cortical areas are involved in managing information processing in a cognitive task and to understand their temporal interactions. Since the transfer of information in the form of electrical activity from one cortical region will in turn evoke electrical activity in other regions, the analysis of temporal synchronization provides a tool to understand neuronal information processing between cortical regions. We adopt a method for revealing time-dependent functional connectivity. We apply statistical analyses of phases to recover the information flow and the functional connectivity between cortical regions for high temporal resolution data. We further develop an evaluation method for these techniques based on two kinds of model networks. These networks consist of coupled Rössler attractors or of coupled stochastic Ornstein–Uhlenbeck systems. The implemented time-dependent coupling includes uni- and bi-directional connectivities as well as time delayed feedback. The synchronization dynamics of these networks are analyzed using the mean phase coherence, based on averaging over phase-differences, and the general synchronization index. The latter is based on the Shannon entropy. The combination of these with a parametric time delay forms the basis of a connectivity pattern, which includes the temporal and time lagged dynamics of the synchronization between two sources. We model and discuss potential artifacts. We find that the general phase measures are remarkably stable. They produce highly comparable results for stochastic and periodic systems. Moreover, the methods proves useful for identifying brief periods of phase coupling and delays. Therefore, we propose that the method is useful as a basis for generating potential functional connective models.     

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.     

Phase-Coherence Transitions and Communication in the Gamma Range between Delay-Coupled Neuronal Populations

Synchronization between neuronal populations plays an important role in information transmission between brain areas. In particular, collective oscillations emerging from the synchronized activity of thousands of neurons can increase the functional connectivity between neural assemblies by coherently coordinating their phases. This synchrony of neuronal activity can take place within a cortical patch or between different cortical regions. While short-range interactions between neurons involve just a few milliseconds, communication through long-range projections between different regions could take up to tens of milliseconds. How these heterogeneous transmission delays affect communication between neuronal populations is not well known. To address this question, we have studied the dynamics of two bidirectionally delayed-coupled neuronal populations using conductance-based spiking models, examining how different synaptic delays give rise to in-phase/anti-phase transitions at particular frequencies within the gamma range, and how this behavior is related to the phase coherence between the two populations at different frequencies. We have used spectral analysis and information theory to quantify the information exchanged between the two networks. For different transmission delays between the two coupled populations, we analyze how the local field potential and multi-unit activity calculated from one population convey information in response to a set of external inputs applied to the other population. The results confirm that zero-lag synchronization maximizes information transmission, although out-of-phase synchronization allows for efficient communication provided the coupling delay, the phase lag between the populations, and the frequency of the oscillations are properly matched.     

Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction

The properties of equilibria and phase synchronization involving burst synchronization and spike synchronization of two electrically coupled HR neurons are studied in this paper. The findings reveal that in the non-delayed system the existence of equilibria can be turned into intersection of two odd functions, and two types of equilibria with symmetry and non-symmetry can be found. With the stability and bifurcation analysis, the bifurcations of equilibria are investigated. For the delayed system, the equilibria remain unchanged. However, the Hopf bifurcation point is drastically affected by time delay. For the phase synchronization, we focus on the synchronization transition from burst synchronization to spike synchronization in the non-delayed system and the effect of coupling strength and time delay on spike synchronization in delayed system. In addition, corresponding firing rhythms and spike synchronized regions are obtained in the two parameters plane. The results allow us to better understand the properties of equilibria, multi-time-scale properties of synchronization and temporal encoding scheme in neuronal systems.     

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

Weak electric fields detectability in a noisy neural network

We investigate the detectability of weak electric field in a noisy neural network based on Izhikevich neuron model systematically. The neural network is composed of excitatory and inhibitory neurons with similar ratio as that in the mammalian neocortex, and the axonal conduction delays between neurons are also considered. It is found that the noise intensity can modulate the detectability of weak electric field. Stochastic resonance (SR) phenomenon induced by white noise is observed when the weak electric field is added to the network. It is interesting that SR almost disappeared when the connections between neurons are cancelled, suggesting the amplification effects of the neural coupling on the synchronization of neuronal spiking. Furthermore, the network parameters, such as the connection probability, the synaptic coupling strength, the scale of neuron population and the neuron heterogeneity, can also affect the detectability of the weak electric field. Finally, the model sensitivity is studied in detail, and results show that the neural network model has an optimal region for the detectability of weak electric field signal.     

Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.     

Synchronization of an Excitatory Integrate-and-Fire Neural Network

In this paper, we study the influence of the coupling strength on the synchronization behavior of a population of leaky integrate-and-fire neurons that is self-excitatory with a population density approach. Each neuron of the population is assumed to be stochastically driven by an independent Poisson spike train and the synaptic interaction between neurons is modeled by a potential jump at the reception of an action potential. Neglecting the synaptic delay, we will establish that for a strong enough connectivity between neurons, the solution of the partial differential equation which describes the population density function must blow up in finite time. Furthermore, we will give a mathematical estimate on the average connection per neuron to ensure the occurrence of a burst. Interpreting the blow up of the solution as the presence of a Dirac mass in the firing rate of the population, we will relate the blow up of the solution to the occurrence of the synchronization of neurons. Fully stochastic simulations of a finite size network of leaky integrate-and-fire neurons are performed to illustrate our theoretical results.     

A cross-interval spike train analysis: the correlation between spike generation and temporal integration of doublets

A stochastic spike train analysis technique is introduced to reveal the correlation between the firing of the next spike and the temporal integration period of two consecutive spikes (i.e., a doublet). Statistics of spike firing times between neurons are established to obtain the conditional probability of spike firing in relation to the integration period. The existence of a temporal integration period is deduced from the time interval between two consecutive spikes fired in a reference neuron as a precondition to the generation of the next spike in a compared neuron. This analysis can show whether the coupled spike firing in the compared neuron is correlated with the last or the second-to-last spike in the reference neuron. Analysis of simulated and experimentally recorded biological spike trains shows that the effects of excitatory and inhibitory temporal integration are extracted by this method without relying on any subthreshold potential recordings. The analysis also shows that, with temporal integration, a neuron driven by random firing patterns can produce fairly regular firing patterns under appropriate conditions. This regularity in firing can be enhanced by temporal integration of spikes in a chain of polysynaptically connected neurons. The bandpass filtering of spike firings by temporal integration is discussed. The results also reveal that signal transmission delays may be attributed not just to conduction and synaptic delays, but also to the delay time needed for temporal integration. Received: 3 March 1997 / Accepted in revised form: 6 November 1997     

Local and global synchronization transitions induced by time delays in small-world neuronal networks with chemical synapses

Effects of time delay on the local and global synchronization in small-world neuronal networks with chemical synapses are investigated in this paper. Numerical results show that, for both excitatory and inhibitory coupling types, the information transmission delay can always induce synchronization transitions of spiking neurons in small-world networks. In particular, regions of in-phase and out-of-phase synchronization of connected neurons emerge intermittently as the synaptic delay increases. For excitatory coupling, all transitions to spiking synchronization occur approximately at integer multiples of the firing period of individual neurons; while for inhibitory coupling, these transitions appear at the odd multiples of the half of the firing period of neurons. More importantly, the local synchronization transition is more profound than the global synchronization transition, depending on the type of coupling synapse. For excitatory synapses, the local in-phase synchronization observed for some values of the delay also occur at a global scale; while for inhibitory ones, this synchronization, observed at the local scale, disappears at a global scale. Furthermore, the small-world structure can also affect the phase synchronization of neuronal networks. It is demonstrated that increasing the rewiring probability can always improve the global synchronization of neuronal activity, but has little effect on the local synchronization of neighboring neurons.     

Firing responses of bursting neurons with delayed feedback

Thalamic neurons, which play important roles in the genesis of rhythmic activities of the brain, show various bursting behaviors, particularly modulated by complex thalamocortical feedback via cortical neurons. As a first step to explore this complex neural system and focus on the effects of the feedback on the bursting behavior, a simple loop structure delayed in time and scaled by a coupling strength is added to a recent mean-field model of bursting neurons. Depending on the coupling strength and delay time, the modeled neurons show two distinct response patterns: one entrained to the unperturbed bursting frequency of the neurons and one entrained to the resonant frequency of the loop structure. Transitions between these two patterns are explored in the model’s parameter space via extensive numerical simulations. It is found that at a fixed loop delay, there is a critical coupling strength at which the dominant response frequency switches from the unperturbed bursting frequency to the loop-induced one. Furthermore, alternating occurrence of these two response frequencies is observed when the delay varies at fixed coupling strength. The results demonstrate that bursting is coupled with feedback to yield new dynamics, which will provide insights into such effects in more complex neural systems.     

Noise effects on robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design

In this paper, we report on the synchronization of a pacemaker neuronal ensemble constituted of an AB neuron electrically coupled to two PD neurons. By the virtue of this electrical coupling, they can fire synchronous bursts of action potential. An external master neuron is used to induce to the whole system the desired dynamics, via a nonlinear controller. Such controller is obtained by a combination of sliding mode and feedback control. The proposed controller is able to offset uncertainties in the synchronized systems. We show how noise affects the synchronization of the pacemaker neuronal ensemble, and briefly discuss its potential benefits in our synchronization scheme. An extended Hindmarsh–Rose neuronal model is used to represent a single cell dynamic of the network. Numerical simulations and Pspice implementation of the synchronization scheme are presented. We found that, the proposed controller reduces the stochastic resonance of the network when its gain increases.     

Hindmarsh-Rose 神经网络的混沌同步

研究了通过特殊构造的非线性函数耦合连接的神经网络的混沌同步问题。在发展基于稳定性准则的混沌同步方法的基础上,给出了计算同步稳定性的误差发展方程,当耦合强度取参考值时,可实现稳定的混沌同步而不需要计算最大条件Lyapunov指数去判定是否稳定。通过对按照完全连接形式构成的Hindmarsh-Rose神经网络的数值模拟,显示可仅从两个耦合神经的耦合强度的稳定性范围预期到许多耦合神经实现同步的稳定性范围。该方法在噪声影响下,对实现神经元的混沌同步仍具有较强的鲁棒性。此外发现随着耦合神经数的增加,满足同步稳定性的耦合强度减小,与耦合神经的数量成反比。     

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.     

Spike propagation synchronized by temporally asymmetric Hebbian learning

 Synchronously spiking neurons have been observed in the cerebral cortex and the hippocampus. In computer models, synchronous spike volleys may be propagated across appropriately connected neuron populations. However, it is unclear how the appropriate synaptic connectivity is set up during development and maintained during adult learning. We performed computer simulations to investigate the influence of temporally asymmetric Hebbian synaptic plasticity on the propagation of spike volleys. In addition to feedforward connections, recurrent connections were included between and within neuron populations and spike transmission delays varied due to axonal, synaptic and dendritic transmission. We found that repeated presentations of input volleys decreased the synaptic conductances of intragroup and feedback connections while synaptic conductances of feedforward connections with short delays became stronger than those of connections with longer delays. These adaptations led to the synchronization of spike volleys as they propagated across neuron populations. The findings suggests that temporally asymmetric Hebbian learning may enhance synchronized spiking within small populations of neurons in cortical and hippocampal areas and familiar stimuli may produce synchronized spike volleys that are rapidly propagated across neural tissue. Received: 28 May 2002 / Accepted: 3 June 2002 RID="*" ID="*" Correspondence to: R. E. Suri Intelligent Optical Systems (IOS), 2520 W 237th St, Torrance, CA 90505-5217, USA (e-mail: rsuri@intopsys.com, Tel.: +1-310-5307130 ext. 108, Fax: +1-210-5307417)     

Coupling-induced synchronization in multicellular circadian oscillators of mammals

In mammals, circadian rhythms are controlled by the neurons located in the suprachiasmatic nucleus (SCN) of the hypothalamus. Each neuron in the SCN contains an autonomous molecular clock. The fundamental question is how the individual cellular oscillators, expressing a wide range of periods, interact and assemble to achieve phase synchronization. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light-dark cycle in neuronal synchronization. However, in natural conditions, the interaction between the SCN neurons is non-negligible and coupling between cells in the SCN is achieved partly by neurotransmitters. In this paper, we use a model of nonidentical, globally coupled cellular clocks considered as Goodwin oscillators. We mainly study the synchronization induced by coupling from an analytical way. Our results show that the role of the coupling is to enhance the synchronization to the external forcing. The conclusion of this paper can help us better understand the mechanism of circadian rhythm.