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Sparsely-synchronized cortical rhythms, associated with diverse cognitive functions, have been observed in electric recordings of brain activity. At the population level, cortical rhythms exhibit small-amplitude fast oscillations while at the cellular level, individual neurons show stochastic firings sparsely at a much lower rate than the population rate. We study the effect of network architecture on sparse synchronization in an inhibitory population of subthreshold Morris-Lecar neurons (which cannot fire spontaneously without noise). Previously, sparse synchronization was found to occur for cases of both global coupling (i.e., regular all-to-all coupling) and random coupling. However, a real neural network is known to be non-regular and non-random. Here, we consider sparse Watts-Strogatz small-world networks which interpolate between a regular lattice and a random graph via rewiring. We start from a regular lattice with only short-range connections and then investigate the emergence of sparse synchronization by increasing the rewiring probability p for the short-range connections. For p = 0, the average synaptic path length between pairs of neurons becomes long; hence, only an unsynchronized population state exists because the global efficiency of information transfer is low. However, as p is increased, long-range connections begin to appear, and global effective communication between distant neurons may be available via shorter synaptic paths. Consequently, as p passes a threshold pth (}~ 0.044), sparsely-synchronized population rhythms emerge. However, with increasing p, longer axon wirings become expensive because of their material and energy costs. At an optimal value p*DE (}~ 0.24) of the rewiring probability, the ratio of the synchrony degree to the wiring cost is found to become maximal. In this way, an optimal sparse synchronization is found to occur at a minimal wiring cost in an economic small-world network through trade-off between synchrony and wiring cost.

We consider a population of subthreshold Izhikevich neurons that cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings (i.e., discrete groups or bursts of noise-induced spikes). We investigate stochastic bursting synchronization by varying the noise intensity. Through competition between the constructive and the destructive roles of noise, collective coherence between noise-induced burstings is found to occur over a large range of intermediate noise intensities. This kind of stochastic bursting synchronization is well characterized by using the techniques of statistical mechanics and nonlinear dynamics, such as the order parameter, the raster plot of neural spikes, the time series of the ensemble-averaged global potential, and the phase portraits of limit cycles. In contrast to spiking neurons showing only spike synchronization (characterizing a temporal relationship between spikes), bursting neurons are found to exhibit both spike synchronization and burst synchronization (characterizing a temporal relationship between the onset times of the active phases of repetitive spikings). The degree of stochastic bursting synchronization is also measured in terms of a synchronization measure that reflects the resemblance of the global potential to the individual potential.

We study the transition from a silent state to a bursting state by varying the dc stimulus in the Hindmarsh-Rose neuron under quasiperiodic stimulation. For this quasiperiodically-forced case, a new type of strange nonchaotic (SN) bursting state is found to occur between the silent state and the chaotic bursting state. This is in contrast to the periodically-forced case where the silent state transforms directly to a chaotic bursting state. Using a rational approximation to the quasiperiodic forcing, we investigate the mechanism for the appearance of such an SN bursting state. Thus, a smooth torus (corresponding to a silent state) is found to transform to an SN bursting attractor through a phase-dependent subcritical period-doubling bifurcation. These SN bursting states, together with chaotic bursting states, are characterized in terms of the interburst interval, the bursting length, and the number of spikes in each burst. Both bursting states are found to be aperiodic complex ones. Consequently, aperiodic complex burstings may result from two dynamically different states with strange geometry (one is chaotic and the other is nonchaotic). Thus, in addition to chaotic burstings, SN burstings may become a dynamical origin for the complex physiological rhythms that are ubiquitous in organisms.

Recently, much attention has been paid to brain rhythms [1,2]. Synchronous neural oscillations may be used for efficient sensory processing (e.g., visual binding) [3]. In addition, neural synchronization is also correlated with pathological rhythms associated with neural diseases (e.g., epileptic seizures and tremors in Parkinson’s disease) [4]. Here, we are interested in the characterization of these synchronized neural rhythms. Collective coherence in a neural population may be well described by the (population-averaged) collective potential VC . For a coherent case, an oscillating (macroscopic) collective potential VC emerges via cooperation of (microscopic) potentials of individual neurons. This collective neural coherence has been characterized mostly in terms of two coherence measures [5]. The first is a “thermodynamic” fluctuation-based coherence measure Mf . Neural coherence is directly related to fluctuations of the collective potential VC . In the thermodynamic limit (N → ∞), a collective state becomes coherent if an oscillating collective potential VC appears. Otherwise (i.e., when the collective potential VC is stationary), it becomes incoherent. Thus, the mean square deviation of the collective potential VC (i.e., time-averaged fluctuations of VC) plays the role of an order parameter to de

We consider an ensemble of globally coupled subthreshold Morris-Lecar neurons. As the coupling strength passes a lower threshold, the coupling stimulates coherence between noise-induced spikings. This coherent transition is well described in terms of an order parameter. However, for su ciently large J , \stochastic oscillator death" (i.e., quenching of noise-induced spikings), leading to breakup of collective spiking coherence, is found to occur. Using the techniques of nonlinear dynamics, we investigate the dynamical origin of stochastic oscillator death. Thus, we show that stochastic oscillator death occurs because each local neuron is attracted to a noisy equilibrium state via an in nite-period bifurcation. Furthermore, we introduce a new \statistical-mechanical" parameter, called the average ring probability Pf and quantitatively characterize a transition from ring to nonring states which results from stochastic oscillator death. For a ring (nonring) state, Pf tends to be non-zero (zero) in the thermodynamic limit. We note that the role of Pf for the ringnon ring transition is similar to that of the order parameter used for the coherence-incoherence transition.

We consider a large population of globally coupled subthreshold Morris-Lecar neurons. By varying the noise intensity D, we investigate numerically stochastic spiking coherence (i.e., noise-induced coherence between neural spikings). As D passes a threshold, a transition from an incoherent to a coherent state occurs. This coherent transition is described in terms of the “thermodynamic” order parameter O, which concerns a macroscopic time-averaged fluctuation of the global potential. We note that such stochastic spiking coherence may be well visualized in terms of the raster plot of neural spikings (i.e., spatiotemporal plot of neural spikings), which is directly obtained in experiments. To quantitatively measure the degree of stochastic spiking coherence (seen in the raster plot), we introduce a new type of “spiking coherence measure,” Ms, by taking into consideration the average contribution of (microscopic) local neural spikings to the (macroscopic) global membrane potential. Hence, the spiking coherence measure may be regarded as a “statistical-mechanical” measure. Through competition between the constructive and the destructive roles of noise, stochastic spiking coherence is found to occur over a large range of intermediate noise intensities and to be well characterized in terms of the mutually complementary quantities of O and Ms. Particularly, Ms reflects the degree of stochastic spiking coherence seen in the raster plot very well.