Pulse-Coupled Relaxation Oscillators: From Biological Synchronization to Self-Organized Criticality

Abstract

It is shown that globally coupled oscillators with pulse interaction can synchronize under broader conditions than widely believed from a theorem of Mirollo and Strogatz. This behavior is stable against frozen disorder. Beside the relevance to biology, it is argued that synchronization in relaxation oscillator models is related to self-organized criticality in stick-slip-like models.PACS numbers: 05.45.+b, 05.40.+j Large assemblies of oscillator units can spontaneously evolve to a state of large scale organization. Collective synchronization is the best known phenomenon of this kind, where after some transient regime a coherent oscillatory activity of the set of oscillators emerges. This effect has attracted much interest in biology for the study of large scale rhythms in populations of interacting elements [1]. The southeastern fireflies, where thousands of individuals gathered on trees flash in unison, is the most cited example [2]. Other examples are the rhythmic activity of cells of the heart pacemaker, of cells of the pancreas, and of neural networks [2 -4]. Most of the works on synchronization have used models in which the interaction between the oscillators is smooth and continuous in time (see, e. g., [5]). Comparatively, few analytical results are known on models where the interactions are episodic and pulselike [1,3,6,7], although they are relevant to several biological situations as fireflies and neural networks. This Letter deals with the emergence of synchronization in a very simple model of globally coupled integrate and fire (IF) oscillators; see Eq. (1) for the definition and [2]. It is shown that synchronization occurs under broad conditions on the properties of the oscillators, thus generalizing a theorem of Mirollo and Strogatz [1], the usual interpretation of which restricted the synchronization conditions in a too drastic way. Furthermore, I show that the synchronized state is stable against a frozen disorder of the oscillator properties, a subject much studied in models with continuous interaction [5]. This result is different from that of a previous study [7] on a closely related model.Besides synchronization, another form of collective organized behavior is known to occur in large assemblies of elements with pulse interaction, that is, self-organized criticality (SOC). This concept has been proposed in [8] to describe out of equilibrium systems that are generically critical, i.e. , that organize into a scale invariant critical state spontaneously, without tuning of a control parameter. Systems displaying SOC are externally driven with a drive slower than any other characteristic time. These models are made critical by the choice of a threshold dynamics that forbids them to follow adiabatically the exter-nal drive. They can be modeled as coupled map lattices and can be divided into two subclasses based on the concept of oscillators. In subclass (a) the external drive acts globally and continuously on all the lattice sites of the coupled map, until one of them reaches the threshold, in wh...

Synchronization of pulse-coupled oscillators with a refractory period and frequency distribution for a wireless sensor network

“…In Fig. 2(a), the initial condition satisfies condition (8), so the oscillators synchronize from the first fire to the 10th fire. After the addition of a new oscillator, the synchronization breaks down.…”

“…The phases are within the shaded regions at t k± . The initial condition is given by condition (8). Since all the oscillators are in the active region at t = 0, an oscillator's fire induces the other oscillators' fire.…”

Spike Train Dynamics Underlying Pattern Formation in Integrate-and-Fire Oscillator Networks

“…+ e, 05.45. + b There has been much recent interest in studying the dynamics of pulse-coupled integrate-and-fire (IF) oscillators with applications to a wide range of systems including flashing fireflies [1], cardiac pacemaker cells [2], biological neural networks [3][4][5][6][7][8][9][10], digital phase-locked loops [11], and stick-slip models [12,13]. Most of the work has been concerned with the existence and stability of phaselocked solutions in which all oscillators have the same common frequency.…”

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

“…In contrast, systems with global coupling, where every oscillator interacts equally with every other, tend to fall into perfect synchrony. Rigorous convergence results have been proven for this case [20,[22][23][24][25]. But the techniques used previously have not revealed much about the transient dynamics leading up to synchrony-the opening and middle game, as opposed to the end game.…”

“…Diverse forms of collective behavior can occur in these pulse-coupled systems, depending on how the oscillators are connected in space. Systems with local coupling often display waves [17,18] or self-organized criticality [11,19,20], with possible relevance to neural computation [15] and epilepsy [21]. In contrast, systems with global coupling, where every oscillator interacts equally with every other, tend to fall into perfect synchrony.…”