Abstract
Synchrony is inevitable in many oscillating systems -- from the canonical alignment of two ticking grandfather clocks, to the mutual entrainment of beating flagella or spiking neurons. Yet both biological and manmade systems provide striking examples of spontaneous desynchronization, such as failure cascades in alternating current power grids or neuronal avalanches in the mammalian brain. Here, we generalize classical models of synchronization among heterogenous oscillators to include short-range phase repulsion among individuals, a property that abets the emergence of a stable desynchronized state. Surprisingly, we find that our model exhibits self-organized avalanches at intermediate values of the repulsion strength, and that these avalanches have similar statistical properties to cascades seen in real-world systems such as neuronal avalanches. We find that these avalanches arise due to a critical mechanism based on competition between mean field recruitment and local displacement, a property that we replicate in a classical cellular automaton model of traffic jams. We exactly solve our system in the many-oscillator limit, and obtain analytical results relating the onset of avalanches or partial synchrony to the relative heterogeneity of the oscillators, and their degree of mutual repulsion. Our results provide a minimal analytically-tractable example of complex dynamics in a driven critical system.
Highlights
Spontaneous synchronization occurs in diverse systems such as robotic swarms, power grids, neuronal ensembles, and even social networks [1,2,3,4,5]
While the theory of phase oscillators is well established, in recent years a variety of novel physical phenomena have been discovered in variants of phase oscillator models [13,14,15,16], including chimera states [17,18], glasslike relaxation mediated by a “volcano” transition [19], and oscillation death via broken rotational symmetry [20]
These fluctuations occur independently of the choice of repulsive kernel V (·); we show results for Gaussian, Cauchy, and triangular potentials in Fig. 2, we otherwise focus on Gaussian repulsion for simplicity
Summary
Spontaneous synchronization occurs in diverse systems such as robotic swarms, power grids, neuronal ensembles, and even social networks [1,2,3,4,5]. Many real-world oscillator populations, such as neuronal ensembles, exhibit a maximally desynchronous state, in which the phases of individual oscillators become evenly spaced apart on the unit circle [21,22] Such dynamics are achievable, in principle, by introducing negative couplings into standard phase oscillator models [23,24]; alternative approaches include introducing phase offsets or time delays in the interactions among oscillators, or introducing specific pairwise couplings among oscillators that embed them on a complex graph [3,25,26]. We solve our model exactly in the many-oscillator limit and show that avalanches in the system originate in the amplification of noiselike excitations provided by local rearrangements
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