Chaotic phase synchronization in small-world networks of bursting neurons

Phase slips in coupled oscillator systems

Phase slips are a typical dynamical behavior in coupled oscillator systems: the route to phase synchrony is characterized by intervals of constant phase difference interrupted by abrupt changes in the phase difference. Qualitatively similar to stick-slip phenomena, analysis of phase slip has mainly relied on identifying remnants of saddle-nodes or "ghosts." We study sets of phase oscillators and by examining the dynamics in detail, offer a more precise, quantitative description of the phenomenon. Phase shifts and phase sticks, namely, the temporary locking of phases required for phase slips, occur at stationary points of phase velocities. In networks of coupled phase oscillators, we show that phase slips between pairs of individual oscillators do not occur simultaneously, in general. We consider additional systems that show phase synchrony: one where saddle-node ghosts are absent, one where the coupling is similarity dependent, and two cases of coupled chaotic oscillators.

Synchronous patterns and intermittency in a network induced by the rewiring of connections and coupling

The connection architecture plays an important role in the synchronization of networks, where the presence of local and nonlocal connection structures are found in many systems, such as the neural ones. Here, we consider a network composed of chaotic bursting oscillators coupled through a Watts-Strogatz-small-world topology. The influence of coupling strength and rewiring of connections is studied when the network topology is varied from regular to small-world to random. In this scenario, we show two distinct nonstationary transitions to phase synchronization: one induced by the increase in coupling strength and another resulting from the change from local connections to nonlocal ones. Besides this, there are regions in the parameter space where the network depicts a coexistence of different bursting frequencies where nonstationary zig-zag fronts are observed. Regarding the analyses, we consider two distinct methodological approaches: one based on the phase association to the bursting activity where the Kuramoto order parameter is used and another based on recurrence quantification analysis where just a time series of the network mean field is required.

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.

A Scalable FPGA Architecture for Randomly Connected Networks of Hodgkin-Huxley Neurons

Human intelligence relies on the vast number of neurons and their interconnections that form a parallel computing engine. If we tend to design a brain-like machine, we will have no choice but to employ many spiking neurons, each one has a large number of synapses. Such a neuronal network is not only compute-intensive but also memory-intensive. The performance and the configurability of the modern FPGAs make them suitable hardware solutions to deal with these challenges. This paper presents a scalable architecture to simulate a randomly connected network of Hodgkin-Huxley neurons. To demonstrate that our architecture eliminates the need to use a high-end device, we employ the XC7A200T, a member of the mid-range Xilinx Artix®-7 family, as our target device. A set of techniques are proposed to reduce the memory usage and computational requirements. Here we introduce a multi-core architecture in which each core can update the states of a group of neurons stored in its corresponding memory bank. The proposed system uses a novel method to generate the connectivity vectors on the fly instead of storing them in a huge memory. This technique is based on a cyclic permutation of a single prestored connectivity vector per core. Moreover, to reduce both the resource usage and the computational latency even more, a novel approximate two-level counter is introduced to count the number of the spikes at the synapse for the sparse network. The first level is a low cost saturated counter implemented on FPGA lookup tables that reduces the number of inputs to the second level exact adder tree. It, therefore, results in much lower hardware cost for the counter circuit. These techniques along with pipelining make it possible to have a high-performance, scalable architecture, which could be configured for either a real-time simulation of up to 5120 neurons or a large-scale simulation of up to 65536 neurons in an appropriate execution time on a cost-optimized FPGA.

Cortical Network Synchrony Under Applied Electrical Field in vitro

Synchronous network activity plays a crucial role in complex brain functions. Stimulating the nervous system with applied electric field (EF) is a common tool for probing network responses. We used a gold wire-embedded silk protein film-based interface culture to investigate the effects of applied EFs on random cortical networks of in vitro cultures. Two-week-old cultures were exposed to EF of 27 mV/mm for <1 h and monitored by time-lapse calcium imaging. Network activity was represented by calcium signal time series mapped to source neurons and analyzed by using a community detection algorithm. Cortical cultures exhibited large scale, synchronized oscillations under alternating EF of changing frequencies. Field polarity and frequency change were both found to be necessary for network synchrony, as monophasic pulses of similar frequency changes or EF of a constant frequency failed to induce correlated activities of neurons. Group-specific oscillatory patterns were entrained by network-level synchronous oscillations when the alternating EF frequency was increased from 0.2 Hz to 200 kHz. Binary responses of either activity increase or decrease contributed to the opposite phase patterns of different sub-populations. Conversely, when the EF frequency decreased over the same range span, more complex behavior emerged showing group-specific amplitude and phase patterns. These findings formed the basis of a hypothesized network control mechanism for temporal coordination of distributed neuronal activity, involving coordinated stimulation by alternating polarity, and time delay by change of frequency. These novel EF effects on random neural networks have important implications for brain functional studies and neuromodulation applications.

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.

Chaotic Resonance in Typical Routes to Chaos in the Izhikevich Neuron Model

Chaotic resonance (CR), in which a system responds to a weak signal through the effects of chaotic activities, is a known function of chaos in neural systems. The current belief suggests that chaotic states are induced by different routes to chaos in spiking neural systems. However, few studies have compared the efficiency of signal responses in CR across the different chaotic states in spiking neural systems. We focused herein on the Izhikevich neuron model, comparing the characteristics of CR in the chaotic states arising through the period-doubling or tangent bifurcation routes. We found that the signal response in CR had a unimodal maximum with respect to the stability of chaotic orbits in the tested chaotic states. Furthermore, the efficiency of signal responses at the edge of chaos became especially high as a result of synchronization between the input signal and the periodic component in chaotic spiking activity.

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.

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.

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.

Phase synchronization of coupled bursting neurons and the generalized Kuramoto model

Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model.

Phase multistability in a dynamical small world network

The effect of phase multistability is explored in a small world network of periodic oscillators with diffusive couplings. The structure of the network represents a ring with additional non-local links, which spontaneously arise and vanish between arbitrary nodes. The dynamics of random couplings is modeled by "birth" and "death" stochastic processes by means of the cellular automate approach. The evolution of the network under gradual increasing of the number of random couplings goes through stages of phases fluctuations and spatial cluster formation. Finally, in the presence of non-local couplings the phase multistability "dies" and only the in-phase regime survives.

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN's parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.

Time delay and long-range connection induced synchronization transitions in Newman-Watts small-world neuronal networks

The synchronization transitions in Newman-Watts small-world neuronal networks (SWNNs) induced by time delay τ and long-range connection (LRC) probability P have been investigated by synchronization parameter and space-time plots. Four distinct parameter regions, that is, asynchronous region, transition region, synchronous region, and oscillatory region have been discovered at certain LRC probability P = 1.0 as time delay is increased. Interestingly, desynchronization is observed in oscillatory region. More importantly, we consider the spatiotemporal patterns obtained in delayed Newman-Watts SWNNs are the competition results between long-range drivings (LRDs) and neighboring interactions. In addition, for moderate time delay, the synchronization of neuronal network can be enhanced remarkably by increasing LRC probability. Furthermore, lag synchronization has been found between weak synchronization and complete synchronization as LRC probability P is a little less than 1.0. Finally, the two necessary conditions, moderate time delay and large numbers of LRCs, are exposed explicitly for synchronization in delayed Newman-Watts SWNNs.

Control of bursting synchronization in networks of Hodgkin-Huxley-type neurons with chemical synapses

Thermally sensitive neurons present bursting activity for certain temperature ranges, characterized by fast repetitive spiking of action potential followed by a short quiescent period. Synchronization of bursting activity is possible in networks of coupled neurons, and it is sometimes an undesirable feature. Control procedures can suppress totally or partially this collective behavior, with potential applications in deep-brain stimulation techniques. We investigate the control of bursting synchronization in small-world networks of Hodgkin-Huxley-type thermally sensitive neurons with chemical synapses through two different strategies. One is the application of an external time-periodic electrical signal and another consists of a time-delayed feedback signal. We consider the effectiveness of both strategies in terms of protocols of applications suitable to be applied by pacemakers.

Vibrational resonance in excitable neuronal systems

In this paper, we investigate the effect of a high-frequency driving on the dynamical response of excitable neuronal systems to a subthreshold low-frequency signal by numerical simulation. We demonstrate the occurrence of vibrational resonance in spatially extended neuronal networks. Different network topologies from single small-world networks to modular networks of small-world subnetworks are considered. It is shown that an optimal amplitude of high-frequency driving enhances the response of neuron populations to a low-frequency signal. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, such as the coupling strength between neurons, network size, and rewiring probability of single small-world networks, as well as the number of links between different subnetworks and the number of subnetworks in the modular networks. All these parameters play a key role in determining the ability of the network to enhance the outreach of the localized subthreshold low-frequency signal. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for the weak signal detection and information propagation across neuronal systems.

Chaotic phase synchronization in a modular neuronal network of small-world subnetworks

We investigate the onset of chaotic phase synchronization of bursting oscillators in a modular neuronal network of small-world subnetworks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that this bursting synchronization transition can be induced not only by the variations of inter- and intra-coupling strengths but also by changing the probability of random links between different subnetworks. We also analyze the effect of external chaotic phase synchronization of bursting behavior in this clustered network by an external time-periodic signal applied to a single neuron. Simulation results demonstrate a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this synchronization region increases with the signal amplitude and the number of driven neurons but decreases rapidly with the network size. Considering that the synchronization of bursting neurons is thought to play a key role in some pathological conditions, the presented results could have important implications for the role of externally applied driving signal in controlling bursting activity in neuronal ensembles.

Stochastic resonance on a modular neuronal network of small-world subnetworks with a subthreshold pacemaker

We study the phenomenon of stochastic resonance on a modular neuronal network consisting of several small-world subnetworks with a subthreshold periodic pacemaker. Numerical results show that the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the intensity of additive spatiotemporal noise. This effect of pacemaker-driven stochastic resonance of the system depends extensively on the local and the global network structure, such as the intra- and inter-coupling strengths, rewiring probability of individual small-world subnetwork, the number of links between different subnetworks, and the number of subnetworks. All these parameters play a key role in determining the ability of the network to enhance the noise-induced outreach of the localized subthreshold pacemaker, and only they bounded to a rather sharp interval of values warrant the emergence of the pronounced stochastic resonance phenomenon. Considering the rather important role of pacemakers in real-life, the presented results could have important implications for many biological processes that rely on an effective pacemaker for their proper functioning.