Synchronization of integrate and fire oscillators with global coupling

Clustering behavior is studied in a model of integrate-and-fire oscillators with excitatory pulse coupling. When considering a population of identical oscillators, the main result is a proof of global convergence to a phase-locked clustered behavior. The robustness of this clustering behavior is then investigated in a population of nonidentical oscillators by studying the transition from total clustering to the absence of clustering as the group coherence decreases. Clustering is a fragmentation of the collective behavior in locally synchronized but well separated subgroups. It is also observed in numerous contexts and has led to distinct contributions. 2,9,15 The robustness of these ensemble phenomena in relatively heterogeneous populations is a mathematical puzzle for the dynamical systems community 10,17 and a source of inspiration for the synthesis of collective behaviors. 11,13,16 We study the occurrence and robustness of clustering in one of the simplest models of spiking neurons: A network of integrate-and-fire oscillators with excitatory pulse coupling. A unique parameter in the model dictates the synchronization or the clustering behavior of the network. Because the synchronization behavior has been studied previously, we focus on the clustering behavior in this paper. The main mathematical result is a global convergence analysis of the clustering behavior in the ideal situation of identical oscillators. It is shown that, for all initial conditions, the oscillators aggregate into clusters that asymptotically converge to a phase-locked configuration. In the more realistic situation of oscillators with different natural frequencies, we study how the clustering behavior degrades with the heterogeneity of the population. This part of the analysis is mostly based on numerical experiments but reveals an interesting phenomenon of partial clustering characterized by few traveling oscillators between well separated clusters. These mathematical and numerical results suggest that the integrate-and-fire model is an interesting candidate for a better understanding of clustering behavior.

show abstract

“…Networks of nonidentical oscillators were analyzed in Ref. 3, with simplified models, and synchronization properties were studied. In Refs.…”

Section: Introductionmentioning

confidence: 99%

Clustering behaviors in networks of integrate-and-fire oscillators

Mauroy

Sepulchre

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…Although different processes can underlie synchronization (see Pikovsky, Rosemblum, & Kurths, 2001& Strogatz, 2003 for reviews), spontaneous phase synchrony has been observed among very different entities in a broad range of physical, biological and social systems ranging from Josephson junctions (Tsygankov & Wiesenfeld, 2002) to fireflies (Winfree, 1967), sinoatrial pacemakers (Michaels, Matyas, & Jalife, 1987), columns in the visual cortex (Gray, Konig, Engel, & Singer, 1989) and firing neurons (Nunez, Panetsos, & Avendano, 2000). Following on Huygens's analysis of two clocks synchronizing on a wall (Bennett, Schatz, Rockwood, & Weisenfield, 2002;Hugenii, 1673), many studies have since framed the problem of mutual synchronization in terms of a network of oscillators whose individual behavior is altered by nearest neighbor interactions (Bottani, 1996;Kuramoto, 1984;Pikovsky et al, 2001;Strogatz, 2003;Winfree, 1967Winfree, , 1980. The coordination dynamics of human brain and behavior has proven no exception to the principles of self-organized synchronization (Kelso, 1995;Fuchs, Kelso, & Haken, 1992;Kelso, Bressler, Buchanan, de Guzman, Ding, Fuchs, & Holroyd, 1992;Kelso, Fuchs, Lancaster, Holroyd, Cheyne, & Weinberg, 1998).…”

Section: Introductionmentioning

confidence: 99%

Social coordination dynamics: Measuring human bonding

Oullier¹,

Guzman²,

Jantzen³

et al. 2008

Social Neuroscience

325

282

View full text Add to dashboard Cite

Spontaneous social coordination has been extensively described in natural settings but so far no controlled methodological approaches have been employed that systematically advance investigations into the possible self-organized nature of bond formation and dissolution between humans. We hypothesized that, under certain contexts, spontaneous synchrony -a well-described phenomenon in biological and physical settings-could emerge spontaneously between humans as a result of information exchange. Here, a new way to quantify interpersonal interactions in real time is proposed. In a simple experimental paradigm, pairs of participants facing each other were required to actively produce actions, while provided (or not) with the vision of similar actions being performed by someone else. New indices of interpersonal coordination, inspired by the theoretical framework of coordination dynamics (based on relative phase and frequency overlap between movements of individuals forming a pair) were developed and used. Results revealed that spontaneous phase synchrony (i.e., unintentional in-phase coordinated behavior) between two people emerges as soon as they exchange visual information, even if they are not explicitly instructed to coordinate with each other. Using the same tools, we also quantified the degree to which the behavior of each individual remained influenced by the social encounter even after information exchange had been removed, apparently a kind of social memory.

show abstract

“…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.…”

mentioning

confidence: 89%

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

2015

View full text Add to dashboard Cite

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony.PACS numbers: 05.45. Xt, 05.70.Ln In one of the first experiments on firefly synchronization, the biologists John and Elizabeth Buck captured hundreds of male fireflies along a tidal river near Bangkok and then released them at night, fifty at a time, in their darkened hotel room [1]. As they looked on in wonder, they observed that "centers of synchrony began to build up slowly among the fireflies on the wall. In one area we would notice that a pair had begun to pulse in unison; in another part of the room a group of three would be flashing together, and so on." Synchronized groups continued to emerge and grow, until as many as a dozen fireflies were blinking on and off in concert. The Bucks realized that the fireflies were phase shifting each other with their flashes, driving themselves into sync.Here we study stylized models of oscillators akin to the fireflies, in which synchrony builds up stepwise, in expanding clusters. By borrowing techniques used to analyze aggregation phenomena in polymer physics, materials science, and related subjects [2, 3], we give the first analytical description of how these synchronized clusters emerge, coalesce, and grow. We hasten to add, however, that the models we discuss are not even remotely realistic descriptions of fireflies; they are merely intended as tractable first steps toward understanding how clusters evolve en route to synchrony.Our work is part of a broader interdisciplinary effort [4,5]. Oscillators coupled by sudden pulses have been used to model sensor networks [6][7][8][9][10], earthquakes [11,12], economic booms and busts [13], firing neurons [14,15], and cardiac pacemaker cells [16]. 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. 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. Aggregation theory offers a new set of tools to explore this prelude to synchrony.We introduce a toy model, which we call scrambler oscillators. It consists of N 1 identical integrate-andfire oscillators coupled all to all. Each oscillator has a ...

show abstract

Synchronization of integrate and fire oscillators with global coupling

Cited by 34 publications

References 61 publications

Clustering behaviors in networks of integrate-and-fire oscillators

Clustering behaviors in networks of integrate-and-fire oscillators

Social coordination dynamics: Measuring human bonding

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

Contact Info

Product

Resources

About