Globally coupled stochastic two-state oscillators: synchronization of infinite and finite arrays

Abstract

We consider arrays of the simplest two-state (on-off) stochastic units. The units are Markovian, that is, the transitions between the two states occur at a given rate. We construct arrays of N globally coupled binary units, and observe a remarkable richness of behavior as the control parameter that measures the coupling strength is increased. In the mean field limit as N  ¥ we consider the four simplest polynomial forms of coupling that lead to bifurcations, and characterize the associated phase transitions of the arrays. When N is finite there are fluctuations about the well-defined steady states of the infinite arrays. We study the nature of these fluctuations and their effects on the bifurcations in all cases by constructing the appropriate Langevin equations and the associated Fokker-Planck equations.

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

“…(10) due to the finite size effects, the minima of the two potentials coincide at F (n 1 ) = 0. We note that in contrast with our previous work [28,29], the choices of F (n 1 ) and G(n 1 ) in the Langevin equation (13) insure that the transition rates to/from the two states are symmetric, that is, the fluctuations do not add a bias to one or the other state.…”

“…Transitions to motionless ordered phases in which the fraction of oscillators in each state is not 1/2 even if the potential shows no asymmetry, on the other hand, are possible, and the study of the transition to the ordered phase in uniformly globally coupled arrays can be carried out analytically, as opposed to the numerical work needed for the threestate array. In fact, uniformly globally coupled networks of two-state Markovian stochastic oscillators can even be handled analytically for finite arrays [28,29]. We can describe the transition to an ordered phase taking into account the fluctuations induced by the finite size of the system via a Fokker-Planck equation, with an exact solution for the steady state distribution.…”

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

“…Discrete stochastic models for synchronization phenomena have been increasing in popularity as a simple paradigm of synchronization [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Of course, this simplicity is related precisely to the relative ease of dealing with only a few states.…”

“…If N is finite, we need to work with a set of coupled Langevin equations, as in Ref. [12,14]. However, except for our numerical simulations, finite size effects are beyond the scope of this paper; here we focus on the mean-field theory.…”

“…The transitions between the states of individual units are governed by a rate process. This rate process might be Markovian [11][12][13][14][15][16][17] or might involve distributed delays (such as, for instance, a refractory period) [20,21]. Interactions among units in our model appear as a dependence of the transition rates of a particular unit on the states of the other units to which it is coupled.…”