Low-Dimensional Dynamics of Populations of Pulse-Coupled Oscillators

Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree—consisting of a large population of all-to-all pulse-coupled oscillators—is still missing. Here, we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and also allows for the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles that fail to synchronize with broad pulses, especially for certain phase-response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of “chimera” states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here, we derive three ordinary differential equations describing two coupled populations and uncover a variety of chimera states, including a new class with chaotic dynamics.

Popular Summary

Clapping audiences, pedestrians on the London’s millennium bridge, and flashing fireflies are spectacular examples of collective synchronization in living systems. This ubiquitous tendency towards synchrony also occurs at the microscopic level, where thousands of heart pacemaker cells self-organize their rhythmic activities to initiate the heartbeat. In 1967, Arthur Winfree, an American biologist, proposed a mathematical model that successfully replicated this natural phenomenon of self-organization and initiated a prolific research program on collective synchronization. Numerical simulations of the model revealed a striking transition to collective synchronization. Because of its mathematical complexity, however, the model has remained largely unexplored. In this paper, we present a powerful mathematical reduction of the Winfree model to a small number of equations, which should enable the model to become a widely used workhorse in modeling of synchronization phenomena.

The Winfree model involves a large number of nonlinear differential equations that govern the dynamics of individuals (or “oscillators”) in a population interacting via pulselike signals. We have been able to show that this high-dimensional system can be exactly reduced to two ordinary differential equations for two global variables. Such a dramatic reduction is highly enabling. As an example of its utility we have investigated the dynamics of populations of identical pulse-coupled oscillators and discovered the existence of a variety of “chimera states,” including a novel class with chaotic dynamics. In a chimera state, a population of identical oscillators counterintuitively breaks into a synchronous and an asynchronous part, and this phenomenon is currently attracting intense experimental and theoretical interest in a number of fields of physics.

Our methods can be readily extended to a number of problems of current interest. We expect that this will start new fundamental advances in our understanding of collective synchronization phenomena in physical, biological, and social settings.

Figure 1

(a) Phase diagram of model (1) obtained from the reduced Eq. (12), with $\beta =\sigma =0$ and $n=10$. Inset: Pulselike function [Eq. (3)]. Panels (b)–(g) show results obtained from the numerical integration of the Winfree model [Eq. (1)], with $N=2000$ oscillators and the natural frequencies selected deterministically to represent the Lorentzian distribution: ${\omega }_i=1+\mathrm{\Delta }\tan [\pi /2(2i-N-1)/(N+1)]$, for $i=1,\dots ,N$. Two different points ( open square, open circle) corresponding to the synchronous (b),(d),(f), and asynchronous (c),(e),(g) states were chosen. (b),(c) Coupling-modified frequencies: ${\mathrm{\Omega }}_i=\underset{t\to \infty }{lim}{t}^{-1}{\int }_0^t{\stackrel{.}{\theta }}_i(t)dt$ versus the oscillators’ index $i$. We observe that in the synchronization region, plateaus in $\mathrm{\Omega }$ appear at a basic frequency and its integer multiples; other plateaus at rational multiples of the basic frequency are absent due to the purely sinusoidal form of the PRC; see Ref. [33]. (d),(e) Raster plots (points depicted whenever ${\theta }_i=0$). (f),(g) Time series of the modulus of the Kuramoto order parameter $R(t)=|N^{-1}\sum _j{e}^{i{\theta }_j}|$ and the mean field $h(t)$.

Figure 2

Synchronization boundaries for PRCs with $\sigma =\sin \beta$, where (a) $\beta =0$ and (b) $\beta =1$, and for pulselike interactions (3) with $n=1$, 10, and $\infty$ (Dirac’s delta). Insets: PRCs $Q(\theta )$, see Eq. (2). The region of synchronization for $n=10$ appears shaded.

Figure 3

(a) Location of different chimera types in the $(\beta ,A)$ plane for $\sigma =0$, $n=1$, and $S=0.5$ (similar results are obtained in a wide range of $\sigma$, $S$, and $n$). Chimeras exist between the dashed line (the locus of a saddle-node bifurcation of limit cycles) and the solid line (corresponding to both the boundary crisis [46, 47] and the saddle-node bifurcation of cycles). Above the thick (blue) line, quasiperiodic and chaotic chimeras are found in the light (green) and dark (red) shaded regions, respectively. (b)–(e) Trajectories projected onto $R_2{e}^{i{\mathrm{\Psi }}_2}$ and $R_2(t)$ for the parameter values with distinct behaviors: $(\beta ,A)=(1.4,0.265)$, (1.42, 0.24), (1.5,0.35), and (1.45,0.19) from (b) to (e), corresponding to chaotic, quasiperiodic, and periodic chimera states above and below the thick (blue) line, respectively.