Interaction patterns and individual dynamics shape the way we move in synchrony

An important open problem in Human Behaviour is to understand how coordination emerges in human ensembles. This problem has been seldom studied quantitatively in the existing literature, in contrast to situations involving dual interaction. Here we study motor coordination (or synchronisation) in a group of individuals where participants are asked to visually coordinate an oscillatory hand motion. We separately tested two groups of seven participants. We observed that the coordination level of the ensemble depends on group homogeneity, as well as on the pattern of visual couplings (who looked at whom). Despite the complexity of social interactions, we show that networks of coupled heterogeneous oscillators with different structures capture well the group dynamics. Our findings are relevant to any activity requiring the coordination of several people, as in music, sport or at work, and can be extended to account for other perceptual forms of interaction such as sound or feel.

graph configuration.   Fig. 1d). Despite each of them achieving full frame synchronisation up to 350Hz, data was recorded 31 with a sampling frequency of 100Hz, with an estimated error of 0.01mm for each coordinate. 32 In order for the cameras to detect the position of each player's hand, circular markers were attached 33 on top of their index fingers; such positions were provided as triplets of (x, y, z) coordinates in a Cartesian 34 frame of reference ( Supplementary Fig. 2, black axes). In rare occasions (0.54% of the total number of 35 data points for Group 1, never for Group 2) it was necessary to deprive these trajectories of possible 36 undesired spikes caused by the cameras not being able to appropriately detect the position of the markers 37 for the whole duration of the trial. As for Group 1, spikes were found in: 38 • 1 trajectory of Player 2 in the Ring graph topology; 39 • 2 trajectories of Player 2 in the Path graph topology; 40 • 4 trajectories of Player 2 and 8 trajectories of Player 7 in the Star graph topology. 41 After removing possible spikes, classical interpolation techniques were used to fill the gap previously 42 occupied by the spikes themselves. Besides, since the players' positions were provided as triplets of 43 (x, y, z) coordinates but essentially the motion of each player could be described as a one-dimensional 44 movement, it was necessary to perform principal component analysis (PCA) on the collected trajectories 45 to find such direction, which turns out to correspond to the x P CA axis ( Supplementary Fig. 2, in red). Cartesian frame of reference (x, y, z) and principal components (x P CA , y P CA , z P CA ). The axes x and y lie on a plane which is parallel to the ground, while the z axis is orthogonal to it. The axes x P CA , y P CA and z P CA individuate the principal components: x P CA is the direction where most of the movement takes place.
PCA was applied to the collected players' trajectories defined by x and y, since the hand motion 47 took place mostly in the (x, y) plane so that the z-coordinate could be neglected, obtaining the principal 48 components x P CA and y P CA . Since the motion along the component y P CA turns out to be negligible 49 compared to that along x P CA , it is possible to further assume that the motion of each player is one-50 dimensional ( Supplementary Fig. 3  Supplementary Figure 4: Probability distribution function of natural oscillation frequencies ω k . The probability distribution functions evaluated from the M values of ω k obtained experimentally in the eyes-closed trials are represented as black solid lines, whereas the fitted normal distributions used in the numerical simulations are represented as red dashed lines. The null hypothesis that the experimental data comes from a normal distribution was tested. Such hypothesis could never be rejected, as specified by a p-value always greater than 5%. The top row refers to players of Group 1, while the bottom row to those of Group 2, whereas each column refers to a different player in the group. The mean value of the frequencies is indicated with µ (ω k ), while their standard deviation is indicated with σ (ω k ), ∀k ∈ [1, N ].

Supplementary
Furthermore, as confirmation of the fact that players in both groups exhibited time-varying natural 68 oscillation frequencies ω k , we also computed the Hilbert transform of each position trajectory x k (t) 69 collected in the M eyes-closed trials, then we evaluated its first time derivative, and finally observed that By definingω := [µ (ω 1 ) µ (ω 2 ) ... µ (ω 7 )] T ∈ R 7 , it is possible to obtain the coefficient of variation: which is equal to c v1 0.13 for Group 1 and c v2 0.21 for Group 2. Such coefficient of variations 75 quantify the overall dispersions of the natural oscillation frequencies of the players, respectively for the 76 two groups. Analogously, it is possible to define the individual coefficient of variation: as a measure of the individual variability of the natural oscillation frequency of each kth player.

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As for the coupling strength c, we found that setting the same constant value for all the topologies 79 under investigation captures well the experimental observations (see Supplementary Section 4 below).

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As for the initial values of the phases, since before starting any trial all the human players were asked to 81 completely extend their arm so that the first movement would be pulling their arm back towards their 82 torso from the same initial conditions, we set θ k (0) = π 2 for all the nodes, trials and topologies.   Table 3 Similarly, the standard deviation is given by  Supplementary Fig. 7), 99 meaning that players managed to maximise synchronisation with those they were visually coupled with.

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In particular:

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• Complete graph: all means ρ µ,hk are higher than 0.82 for Group 1 and 0.86 for Group 2.

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• Ring graph: for each player of Group 1 the highest values of ρ µ,hk are obtained with respect to the two agents that player was asked to be topologically connected with, whereas for each player 104 of Group 2 at least either of the two values of ρ µ,hk related to her/his partners turns out to be the 105 highest, and that related to the other partner is either the second highest (nodes 2, 4 and 7), the 106 third highest (nodes 1, 3 and 6) or the fourth highest (node 5).

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• Path graph: remarks analogous to those of the Ring graph configuration can be made. The only     • for Group 1 (Supplementary Fig. 8), ρ g (t) never achieves a constant value at steady state but 130 exhibits persistent oscillations (less noticeable in the Complete graph);

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• for Group 2 ( Supplementary Fig. 9), ρ g (t) exhibits higher oscillations in Ring and Path graphs,   For each topology, the ten panels on the left show the trend of ρ g (t) observed experimentally in the ten eyes-open trials, respectively, whereas the panel on the right shows a typical trend of ρ g (t) obtained numerically for that topology. a Complete graph     Fig. 10b shows the values of the individual synchronisation indices ρ k as a function of 166 the intra-individual variability σ(ω k ) of the natural oscillation frequencies. A One-way ANOVA revealed 167 no statistically significant effect of σ(ω k ) (F (3, 24) = 0.170, p = 0.916, η 2 = 0.019).

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Overall, these results suggest that, unlike the intra-individual variability of oscillation frequencies 169 σ(ω k ), the overall dispersion c v has a significant effect on the coordination levels of the group members.