Abstract
Coordination in living systems—from cells to people—must be understood at multiple levels of description. Analyses and modelling of empirically observed patterns of biological coordination often focus either on ensemble-level statistics in large-scale systems with many components, or on detailed dynamics in small-scale systems with few components. The two approaches have proceeded largely independent of each other. To bridge this gap between levels and scales, we have recently conducted a human experiment of mid-scale social coordination specifically designed to reveal coordination at multiple levels (ensemble, subgroups and dyads) simultaneously. Based on this experiment, the present work shows that, surprisingly, a single system of equations captures key observations at all relevant levels. It also connects empirically validated models of large- and small-scale biological coordination—the Kuramoto and extended Haken–Kelso–Bunz (HKB) models—and the hallmark phenomena that each is known to capture. For example, it exhibits both multistability and metastability observed in small-scale empirical research (via the second-order coupling and symmetry breaking in extended HKB) and the growth of biological complexity as a function of scale (via the scalability of the Kuramoto model). Only by incorporating both of these features simultaneously can we reproduce the essential coordination behaviour observed in our experiment.
Highlights
Coordination is central to living systems and biological complexity at large, where the whole can be more than and different from the sum of its parts
Similar to the HKB model [29], secondorder coupling is demanded by the experimental observation of antiphase but in eightperson coordination; and similar to the extended HKB [37], the model captures how increasing frequency difference δf weakens inphase and antiphase patterns, leading to segregation but between two groups instead of two persons
This cross-scale consistency of experimental observations may be explained by the scale-invariant nature of the critical coupling ratio κc = 1, the transition point between monostability and multistability
Summary
Coordination is central to living systems and biological complexity at large, where the whole can be more than and different from the sum of its parts. Existing studies of phase coordination often focus on systems of either very few (small-scale, mostly N = 2) [13,19,20], or very many oscillators (large-scale, N → ∞) [21,22,23]. The present work answers this question by modelling empirically observed coordinative behaviour in midscale systems (N = 8), based on data collected in a specially designed human experiment [24]. The resultant model that captures all key experimental observations happens to connect previous theories of small- and large-scale biological coordination in a single mathematical formulation
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