Dynamic Third-party Entrainment and Learning in Systems of Multistable Oscillators*

The generalized Haken-Kelso-Bunz system of coupled oscillators has been shown to model bistable rhythmic coordination in complex systems, including the coordination dynamics of an individual’s fingers and other limbs, the dynamic organization of multiple individuals in social systems, and the synchronization of the brain waves of various neural ensembles.  Additionally, experiments have shown that some systems described by the HKB model, including pairs of individuals coordinating the movement of their index fingers, can learn from their interactions with one another, inducing lasting changes in their intrinsic dynamics after the interaction has ceased.  This learning phenomenon, reminiscent of dynamic Hebbian learning from neuroscience, has been incorporated into the extended HKB model for pairs of oscillators.  However, recent work has demonstrated how the emergence of new coordination dynamics in complex systems depends on the scaling of system size: specifically, adding an appropriately tuned third oscillator to a pair of oscillators can allow that pair to coordinate in new ways, enriching the system’s coordinative repertoire.  Here, we propose a generalization of the learning rule for pairs of oscillators to model learning among an arbitrary number of multistable HKB oscillators and demonstrate how multi-oscillator coupling influences the learning dynamics of oscillator pairs.  Applications for this theoretical work include modelling learning in social systems at the triadic level and above, and neuroplastic learning in networks of neural oscillators.