Third party stabilization of unstable coordination in systems of coupled oscillators

Abstract

The Haken-Kelso-Bunz (HKB) system of equations is a well-developed model for dyadic rhythmic coordination in biological systems. It captures ubiquitous empirical observations of bistability - the coexistence of in-phase and antiphase motion - in neural, behavioral, and social coordination. Recent work by Zhang and colleagues has generalized HKB to many oscillators to account for new empirical phenomena observed in multiagent interaction. Utilising this generalization, the present work examines how the coordination dynamics of a pair of oscillators can be augmented by virtue of their coupling to a third oscillator. We show that stable antiphase coordination emerges in pairs of oscillators even when their coupling parameters would have prohibited such coordination in their dyadic relation. We envision two lines of application for this theoretical work. In the social sciences, our model points toward the development of intervention strategies to support coordination behavior in heterogeneous groups (for instance in gerontology, when younger and older individuals interact). In neuroscience, our model will advance our understanding of how the direct functional connection of mesoscale or microscale neural ensembles might be switched by their changing coupling to other neural ensembles. Our findings illuminate a crucial property of complex systems: how the whole is different than the system's parts.

Keywords: HKB; attractor; complex systems; coordination dynamics; multistability; synergetics.

Figure 1.

A plot of the relative phase potential function landscape for A_ij = 2B_ij = 1 for each i, j. Note the many valleys (marked with red asterisks) in which an oscillator moving around in this landscape will become trapped. These valleys are the local minima corresponding to the coordination states. There are two types of valleys in this landscape: in-phase valleys, which have relatively very deep and wide basins of attraction, and antiphase valleys, which are narrower and shallower, reflecting the fact that the in-phase state is more stable than the antiphase state. Each of these valleys is separated by a distance of π, and repeats infinitely on the potential surface in a 2π-periodic pattern.

Figure 2.

Top-down views of the potential function used to visualize stable fixed points (marked with asterisks) for different values their pairwise coupling parameters: in (a) setting all pairs in bistable regimes (when considered in isolation); or with one (b), two (c), or all three sets of coupling parameters (d) set in monostable regimes. In (a), the potential function landscape possesses four stable fixed points at in-phase and antiphase (white asterisks). In (b) with one dyadic relation (ϕ₁) in the monostable regime, four stable fixed points were also observed, two of them (grey asterisks) would not exist in their isolated dyad but are rescued by the coupling to the third oscillator. In (c) with two dyads in the monostable regime, three stable fixed points were found, including two (grey asterisks) that would not exist for their pairs in isolation. In (d), with all dyads in the monostable regime, all the antiphase regimes vanished, leaving a single stable fixed point with all oscillators coordinated in-phase.