Ensembles of excitable two-state units with delayed feedback

Synchronization and fluctuations: Coupling a finite number of stochastic units

It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions

We study the role of the tail and the range of interaction in a spatially structured population of two-state on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long-range interactions) to a disordered one (short-range interactions). Depending on the interaction kernel, the transition may be smooth (second order) or abrupt (first order). We analyze the transient, which may present different routes to the steady state with vastly different time scales.

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators.

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.

Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation

We consider an array of units each of which can be in one of three states. Unidirectional transitions between these states are governed by Markovian rate processes. The interactions between units occur through a dependence of the transition rates of a unit on the states of the units with which it interacts. This coupling is nonlocal, that is, it is neither an all-to-all interaction (referred to as global coupling), nor is it a nearest neighbor interaction (referred to as local coupling). The coupling is chosen so as to disfavor the crowding of interacting units in the same state. As a result, there is no global synchronization. Instead, the resultant spatiotemporal configuration is one of clusters that move at a constant speed and that can be interpreted as traveling waves. We develop a mean field theory to describe the cluster formation and analyze this model analytically. The predictions of the model are compared favorably with the results obtained by direct numerical simulations.

Globally coupled stochastic two-state oscillators: fluctuations due to finite numbers

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a 1/N expansion and the Langevin equation via standard methods of Itô calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits N → ∞ and t → ∞ (t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

Synchronization of globally coupled two-state stochastic oscillators with a state-dependent refractory period

We present a model of identical coupled two-state stochastic units, each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition.

Species mobility induces synchronization in chaotic population dynamics

A prototype population dynamics model with cyclic domination of four species and empty sites is proposed for studying transition to synchronization. At the mean-field level the dynamics shows quasiperiodicity and chaos depending on the parameter values. The realization of the model on a square lattice shows that spatial restrictions and intrinsic stochasticity change the whole picture. The mean-field dynamics qualitatively remains only under global reactions, while local reactions drive the lattice to poisoning, where only some of the species survive. Nontrivial oscillatory steady states are developed if long-distance exchange is introduced due to gradual mixing with a certain probability. The mixing probability is shown to control the transition to synchronization which emerges abruptly following a phase slip scenario. Near the transition a typical intermittency crisis takes place, with phase slips becoming more infrequent as the transition is approached.