Author response: Collective dynamics support group drumming, reduce variability, and stabilize tempo drift

Abstract

Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract Humans are social animals who engage in a variety of collective activities requiring coordinated action. Among these, music is a defining and ancient aspect of human sociality. Human social interaction has largely been addressed in dyadic paradigms, and it is yet to be determined whether the ensuing conclusions generalize to larger groups. Studied more extensively in non-human animal behavior, the presence of multiple agents engaged in the same task space creates different constraints and possibilities than in simpler dyadic interactions. We addressed whether collective dynamics play a role in human circle drumming. The task was to synchronize in a group with an initial reference pattern and then maintain synchronization after it was muted. We varied the number of drummers from solo to dyad, quartet, and octet. The observed lower variability, lack of speeding up, smoother individual dynamics, and leader-less inter-personal coordination indicated that stability increased as group size increased, a sort of temporal wisdom of crowds. We propose a hybrid continuous-discrete Kuramoto model for emergent group synchronization with a pulse-based coupling that exhibits a mean field positive feedback loop. This research suggests that collective phenomena are among the factors that play a role in social cognition. Editor's evaluation Taking joint drumming as a model of collective dynamics, and combining solid quantitative methods, the authors characterize how human behavior changes, at the individual- and group-level, as a function of group numerosity. A take-home message of this important work is that not everything we know from studies involving dyads should be necessarily generalized to larger groups. This study will be of great interest to scientists looking for new approaches to understanding group behavior, especially within the fields of human cognition, neurosciences, and musicology. https://doi.org/10.7554/eLife.74816.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Humans are social animals who engage in a variety of collective activities requiring coordinated joint action. Collective goals can be achieved through spontaneously distributed workloads among group members, such that the emerging collective dynamics confer benefits to performance not available to individuals—the ‘wisdom of crowds’ (Galton, 1907; Surowiecki, 2005). Principles of collective dynamics explain the adaptive value of collective behavior in some species (Couzin, 2018). For example, collective dynamics can overcome the limitations of individuals’ knowledge and ability to communicate (Goldstone and Roberts, 2006) by integrating information quickly during group decision making (Ispolatov, 2015; Miller et al., 2013; Rosenthal et al., 2015). Large coherent swarming and flocking in groups can arise from short-range interactions among proximal individuals in non-human animals including primates (Farine et al., 2017), insects (Jacobs et al., 2007), fish, and birds (Miller et al., 2013; Parrish et al., 2002). As such dynamics have been demonstrated in numerous species, theoretical models have been developed to provide additional quantitative explanatory support. In contrast, in spatial tasks such as navigation, aggregate group behavior, formalized as a mean field, can serve to stabilize feedback to individuals by virtue of being a group average (Berdahl et al., 2018; Sumpter, 2006; Torney et al., 2009). Here, we test whether mean field behavior accounts for stability of temporal group dynamics in humans during a drumming task. Collective dynamics are observable in animal as well as in human group behavior. Crowds of walking individuals achieve globally coherent states based on local inter-individual interactions (Rio et al., 2018; Warren, 2018). In audiences, individual-group interactions and social contagion govern the spontaneous onset and offset of applause (Mann et al., 2013). Collective dynamics can also play a constitutive role in sports by permitting advantages not available to individuals alone (Vilar et al., 2013). Evidence of collective dynamics is seen even in the minimal group, a dyad: spontaneous synchronization between two individuals emerges from constraints such as weak coupling (Oullier et al., 2008; Schmidt et al., 1990; Schmidt and O’Brien, 1997). Synchronized group action is an essential element of music making, a defining social behavior of human interaction (Honing et al., 2015; Patel and Iversen, 2014; Salimpoor et al., 2011; Savage et al., 2015; Trainor, 2015), and the value of precise synchronization may vary from culture to culture and across musical contexts (Benadon et al., 2018; Davies et al., 2013; Lucas et al., 2011).The archaeological evidence of musical instruments goes back 30,000 years, and singing and drumming are thought to be even older (Conard et al., 2009). The evolutionary origins of musical rhythmic actions may relate to social motor behaviors in non-human species, such as the synchronization and desynchronization of vocalizations between individuals in group chorusing, arising from pressures to either collaborate or compete (Gamba et al., 2016; Greenfield et al., 2017; Ravignani et al., 2014; Ravignani et al., 2019). Fundamental aspects of music are present in humans from early stages in development: infants show early musical preferences, social-emotional responses to music, and rate-sensitive motoric responses to musical rhythm (Cirelli et al., 2014; Cirelli et al., 2018; Trainor and Marsh-Rollo, 2019; Zentner and Eerola, 2010). Although musical behavior is found in individuals alone, it occurs mainly in groups ranging from duets to hundreds of participants. Yet, the role of collective dynamics, especially in groups larger than dyads, has largely remained untested. Here, we consider a musical task in which timing consistency and synchrony are crucial. We investigate in what ways the group average performs better than the individuals, a temporal version of the wisdom of the crowd phenomenon. To account for group interaction, one popular strategy is to first develop a theory for single-person tasks and then extrapolate into dyadic contexts. For example, the idea that the brain is a prediction machine that enables timely motor control in an uncertain environment can be extrapolated as two brains mutually predicting each other (Friston and Frith, 2015; Wolpert et al., 2003). This means that each individual instantiates two processes: the first for controlling one’s own rhythm and timing and the second for predicting the rhythm and timing of the partner in the dyad (Heggli et al., 2019; Heggli et al., 2021; Keller et al., 2014; van der Steen and Keller, 2013). It is not clear, however, if this can serve as an adequate foundation for behavior in larger groups. Every individual would have to predict every other individual and potentially factor-in higher-order predictions. Furthermore, at the perceptual level, group size changes the auditory information available to an individual. A listener is likely to hear an entire choir as one, or a small number of, coherent sounds rather than perceive every singer’s voice within that choir. Group interaction can also be addressed in terms of theoretical models explicitly developed for synchronization in systems made of many dynamic units (Alderisio et al., 2016; Oullier et al., 2008; Schmidt et al., 1990; Schmidt and O’Brien, 1997). The individual units in such models usually do not contain processes dedicated separately to self-timing and other-predicting. Instead, each unit has only a self-timing and a phase-correction coupling term. The propagation of phase adjustment across all units is sufficient for the collective to enter a group-synchronized state, despite that each unit is only trying to cancel a phase difference. An important benefit of this approach is that it is inherently collective, converging on the same formalisms used for small or large groups of animals (Kelso, 2021; Zhang et al., 2019). Here, we consider such a system of coupled oscillators to account for empirical data on individual and group variability in a drumming task. Group music making constitutes an ecologically valid and convenient paradigm for studying group action and collective experience in the laboratory (e.g. D’Ausilio et al., 2015). We used a timing task performed by groups of different sizes. Larger group dynamics are less studied because measuring highly precise timing while collecting group data is difficult for both logistical and methodological reasons (cf. Alderisio et al., 2016; Chang et al., 2017; Chang et al., 2019; Chauvigné et al., 2019; Shahal et al., 2020). Participants completed a group synchronization-continuation task (SCT; Figure 1A). It required them to drum in synchrony with an isochronous auditory stimulus and continue drumming at the same rate after the stimulus stopped, while we collected the onset times of each drum hit (Figure 1B). Stimulus tempo was varied across trials. We tested groups of two (dyads), four (quartets), and eight (octets) participants (Figure 1C). The task tested participants’ ability to synchronize to an external reference as well as to other participants, while also minimizing temporal variability and maintaining the initial stimulus rate. This ensemble drumming was also compared against the solo condition. Specifically, participants tested in duets or quartets completed both solo and ensemble conditions. The octet group did not complete the solo conditions for logistical reasons, but they instead completed a control condition in which the synchronization phase continued for the whole trial (i.e. there was no continuation phase) as well as trials using a more complex and musically realistic rhythm. Analyses of the more complex rhythm will be reported separately. Figure 1 Download asset Open asset Experiment setup. (A) In ensemble condition, drummers faced each other in a circle. (B) The main task was synchronization-continuation where participants were paced initially by an auditory stimulus and then had to maintain the rhythm, tempo, and synchronization among each other. The inter-onset intervals between drum hits, shown schematically as vertical lines, were used to obtain individual-level measures of variability and speeding up. Cross-correlation and transfer entropy were used as pair-level measures of synchronization and interaction. (C) Transfer entropies, color, and width-coded, from three sample trials from different groups. Network analysis was applied to these graphs to obtain group-level characterization. Inspired by theoretical models from the animal literature (Sumpter, 2006), we assumed that individual interactions average to overall group behavior, a mean field, which provides stabilizing feedback to individual group members (Figure 2). A key prediction was that the relative influence of the stabilizing feedback would increase with increasing ensemble size. To this end, we measured separately the variability of individual drummers and the group. We constructed a theoretical group-aggregate onset time as the center of clusters of individual onset times. We expected that larger groups would exhibit lower variability as measured using the coefficient of variation of inter-onset intervals. We also tested this idea formally by adapting a Kuramoto model of group synchronization. Figure 2 with 1 supplement see all Download asset Open asset Group synchronization of eight oscillators with random initial conditions (A) and coherent phases later in the trial (B). The middle inset shows individual trajectories (red lines) and the mean field (blue line) in time (t). Averaging the phase oscillators θ (red lines) gives a so-called mean field (blue lines) with phase Ψ and amplitude r. As the individual oscillators become more coherent from (A) to (B), r increases which leads to stronger influence from Ψ to diverging θ’s (weight of the downward vectors). The (Kuramoto, 1975) dynamic system of coupled phase oscillators, Equation 1, was conceptualized as a large population of oscillators with different natural frequencies capable of spontaneously locking to a common frequency. (1) θ˙i=ωi+KN∑j=1Nsin⁡(θj−θi) Here, θi is the phase of oscillator i, ωi is its preferred frequency, i.e., how fast around the unit circle it likes to go, K is coupling strength, and N is the number of oscillators. The model gives a mathematical account of group synchronization as dependent on a mean field, referred to here as group aggregate. A central feature of the model is that the feedback is positive: the amplitude of the mean field grows as a function of inter-individual synchronization, and reciprocally, the individual oscillators are affected more by the mean field if its amplitude is larger, see Figure 2. This is shown by using the definition of the mean of phases, Equation 2, to express Equation 1 equivalently (Strogatz, 2000) in terms of the coupling between individual oscillators and the mean field, Equation 3. Ψ is the mean field phase, and r is the mean field coherence, also called order parameter (Kuramoto, 1975). (2) reiΨ=1N∑j=1Neiθj (3) θ˙i= ωi+rKsin⁡(Ψ−θi) Such individual-collective positive feedback loops enable ant trails and other phenomena in swarming animals (Sumpter, 2006) as well as acoustic herding in chorusing animals (Ravignani et al., 2014). The system also serves as a model for neuronal collective synchronization (Breakspear et al., 2010; Frank et al., 2000; Noori et al., 2020). Theoretically, the principles embodied by this system should apply to group action in humans too (Zhang et al., 2019). It has found application in understanding inter-personal synchronization of dyads (Dotov et al., 2019; Heggli et al., 2019; Roman et al., 2019), individual rate preferences in the dyad (Bégel et al., 2022), and effects of coupling topology in larger groups (Alderisio et al., 2016). In the present context, the model predicts that ensembles will be more stable than individuals because of the feedback loop between the timing of individuals and the group aggregate of individuals. The original Kuramoto model involves continuous dynamics and continuous coupling with constant gain and no delay. In a drumming task, participants are coupled by way of discrete acoustic onset times. Here, we propose a hybrid continuous-discrete system of oscillators with event-based feedback updated once per cycle, Equation 4. It utilizes a pulse function shaped like the acoustic envelope of a drum hit. To this end, Equation 5 specifies an asymmetric gamma distribution that rises sharply at time 0 and then decays slowly (Figure 2—figure supplement 1). In the model, an oscillator emits a pulse once per cycle when its phase angle crosses zero. Furthermore, we posit additive phase variability with a Gaussian distribution N(0,σ). (4) θ˙i=ωi+KN∑j=1NPj≠i(t)sinθi+N(0,σ),⋅⋅⋅⋅⋅⋅i=1,...,N (5) P(t)=f(t|a,b)=1baΓ(a)ta−1e−tb Like the original system, this model involves intrinsic dynamics (ωi) and interaction dynamics given by the whole coupling term. For comparison, we tested two additional models: the classic Kuramoto system with constant full coupling, Equation 1, as well as the hybrid continuous-discrete model, Equations 4; 5, with a sparse coupling matrix in a ring topology. Specifically, the individual units were only coupled to their two immediate neighbors, j=i±1, with a periodic boundary condition, θN+1=θ1 and θ0=θN . Results In Performance, Dynamics and coordination in ensemble, SCT conditions, and Group dynamics, we present the analyses of the participants’ drumming behavior. In Variability in the mean-field model with discretely updated feedback, we present the results from our adapted hybrid model consisting of a continuous-discrete system of oscillators with pulse-based feedback updated once per cycle. Performance Variability Does individual variability increase in ensemble compared to solo conditions? To answer this question, we fitted a model including only individuals’ data in dyads and quartets (octets did not complete solo conditions). As Figure 3A shows, individuals’ variability increased when playing in an ensemble relative to playing solo (β=0.009, SE=0.001, t=6.65, and ω2=0.10). Figure 3 with 2 supplements see all Download asset Open asset Task performance measured in terms of mean (SE) coefficient of variation of inter-onset intervals (IOIs). (A) Synchronization-continuation task (SCT) drumming trials (no data collected in N8 solo condition). (B) Pulse-coupled Kuramoto model. N2=dyad; N4=quartet; N8=octet; Inds = individual participants; Aggr = group-aggregate; Solo = solo condition; Ensemble = ensemble condition. Error bars are standard errors. With missing trials, the number of observations in the respective conditions was n=(88, 88, 44, 111, 110, 32, ∅, 953, 133) in (A), and n=(200, 198, 99, 400, 400, 100, 800, 688, 86) in (B). Figure 3—source data 1 https://cdn.elifesciences.org/articles/74816/elife-74816-fig3-data1-v2.xlsx Download elife-74816-fig3-data1-v2.xlsx In the context of true ensemble playing, how does variability of the individuals and group aggregate change across dyads, quartets, and octets? To specifically test the main hypothesis, we fitted a model to continuation (SCT) trials from all three group sizes in the ensemble playing condition. We found that individual variability increased with group size (β=0.001, SE=0.0005, t=2.20, and ω2=0.09), group-aggregate variability was lower than individual variability (β=−0.010, SE=0.003, t=−3.29, and ω2=0.01), and the difference between individual and group-aggregate variability increased for larger group sizes (β=−0.0025, SE=0.0005, t=−5.19, and ω2=0.03), see Figure 3A. As a sanity check, we verified that the procedure for group-aggregate onset times did not lead to spuriously periodic data. We observed very high variability for the pseudo-group-aggregate in solo condition where individuals did not hear each other, see Figure 3—figure supplement 1A. How does variability compare between continuation and synchronization? The octet group completed trials in conditions of both SCT (i.e. after the offset of the reference metronome) and synchronization-only (with a constant reference metronome). Surprisingly, there were no differences between these conditions either in individual (t<1) or in group-aggregate variability (t<1). Speeding up Do individuals speed up more when playing solo than in dyad or quartet ensembles? Speeding up was defined as the linear increase in tempo (i.e. decrease of IOIs) over the course of a trial. As expected, individuals sped up even when playing solo (β=0.018, SE=0.007, t=2.67, and ω2=0.10), and greater speeding up was observed in ensemble than in solo conditions (β=0.022, SE=0.007, t=3.29, and ω2=0.16). The effect of group size was not significant (t<1). Do dyad, quartet, and octet ensembles speed up to different extents? To test the effect of group size across the duets, quartets, and octets, a second model was fitted with group size as a continuous predictor, but only including trials in SCT ensemble playing. As suggested by Figure 4, there was a trend for speeding up to decrease as group size increased. The best linear model included an effect of tempo (β=−0.001, SE=0.00005, t=−1.98, and ω2=0.005) and an interaction between group size and tempo (β=−.00004, SE=0.000007, t=−6.20, and ω2=0.11), reflecting that at higher tempos larger ensembles sped up less than smaller ensembles, see Figure 4—figure supplement 1. Figure 4 with 1 supplement see all Download asset Open asset Mean (SE) speeding up defined as the linear slope of tempo over time, computed from the IOIs. N2=dyad; N4=quartet; N8=octet (no data collected in solo N8 condition). Error bars are standard errors. With missing trials, the number of observations in the respective conditions was n=(88, 111, ∅, 88, 110, 482). Figure 4—source data 1 https://cdn.elifesciences.org/articles/74816/elife-74816-fig4-data1-v2.xlsx Download elife-74816-fig4-data1-v2.xlsx Dynamics and coordination in ensemble, SCT conditions Individual and group-aggregate dynamics (autocorrelations) Individuals Figure 5A shows that individuals’ autocorrelations at lag 1 were negative across all group sizes and tempos. This is typical for the alternating long-short IOIs seen in synchronization tasks, reflecting alternating fast-slow synchronization errors when adapting to the previous interval. There was also a trend across all group sizes and tempo conditions for positive peaks at even lags (2, 4, 6, and 8) and negative peaks at odd lags (3, 5, and 7), see Figure 5—figure supplement 1. However, this general pattern became smoother with increasing group size. To test this statistically, in each trial, we took the average absolute difference between successive lags, thus measuring the average range of the autocorrelation function up to lag 8 and applied the same linear modeling approach as in Performance. The effect of group size was significant (β=−0.0054, SE=0.0023, t=−2.35, and ω2=0.13). There was also a decrease of range with increasing tempo (β=−0.00039, SE=0.0001, t=−4.92, and ω2=0.04). Additionally, we verified the signs, relative magnitudes, and the effects of group size and tempo on the autocorrelations by fitting separate linear models for each lag, see Supplementary file 1a. Figure 5 with 1 supplement see all Download asset Open asset Autocorrelations in the synchronization-continuation task (SCT) in ensemble conditions. (A) Autocorrelations of individual IOIs, averaged (SE) across participants’ trials and tempos, separately per group size (color-coded). (B) Same for group-aggregate IOIs. Error bars are bootstrap 95% confidence intervals. The number of observations in the respective group sizes was n=(88, 110, 474) in (A) and n=(44, 32, 66) in (B). Figure 5—source data 1 https://cdn.elifesciences.org/articles/74816/elife-74816-fig5-data1-v2.xlsx Download elife-74816-fig5-data1-v2.xlsx Group-aggregate Interestingly, group-aggregate IOIs revealed the same overall pattern of results even though by definition the group-aggregate timing was smoother and less variable (see Group-aggregate (mean field) measure for ensembles and pseudo-ensembles), see Figure 5B and Supplementary file 1b. There was a trend across all group sizes for relative positive peaks at even lags 2, 4, 6, and 8 and relative negative peaks at odd numbered lags 1, 3, 5, and 7. The similarity of the dynamics of individuals and the group aggregate is not trivial because it implies that even groups of up to eight participants spontaneously acquire the alternating long-short interval dynamics characteristic of when individual participants synchronize with a stimulus. Inter-personal coordination (cross-correlations) Cross-correlation applied to the pre-processed IOIs assessed inter-personal coordination between pairs of participants drumming together in the continuation phase of ensemble performances. On rare occasions, individuals produced diverging beat times by, for example, missing the drum or hitting it two times. The resulting outlier IOIs were removed, and the remaining were re-aligned relative to the other participants by temporal adjacency. The last pre-processing step consisted of whitening each time-series of IOIs (see From beat onset times to IOIs, outlier removal, and clustering–Pre-whitening). We analyzed lags in the range 0–4 because cross-correlation coefficients tended to be symmetric between positive and negative lags, see Figure 6. As expected, cross-correlations at lag 0 were negative in duets (β=−0.039, SE=0.011, t=−3.60, and ω2=0.04), consistent with the pattern of results for autocorrelations, and inter-personal coordination dynamics were smoother in larger group sizes. Furthermore, lag 0 cross-correlations became less negative with increasing group size (β=0.012, SE=0.002, t=5.93, and ω2=0.13). Lag 1 cross-correlations were positive (β=0.14, SE=0.012, t=11.80, and ω2=0.32), but their magnitude decreased with group size (β=−0.016, SE=0.002, t=−7.14, and ω2=0.20). There were also effects of tempo, see Figure 6—figure supplement 1. Figure 6 with 1 supplement see all Download asset Open asset Cross-correlations in the synchronization-continuation task (SCT) in ensemble conditions. Averages (SE) across participant pairs and tempos are shown separately per group size (color-coded lines). IOIs were aligned across participants and pre-whitened by filtering with an autoregressive model. Error bars are bootstrap 95% confidence intervals. The number of observations in the respective group sizes was n=(44, 140, 1479). Figure 6—source data 1 https://cdn.elifesciences.org/articles/74816/elife-74816-fig6-data1-v2.xlsx Download elife-74816-fig6-data1-v2.xlsx Group dynamics Network analysis We used network analysis in the continuation phase of trials of ensemble playing to describe group coordination at an even higher level of organization than interpersonal coordination. First, we obtained the directed graphs of functional connectivity among participant drummers. In each trial, the graph consisted of the real-valued directed links between pairs of drummers estimated by way of the transfer entropy (TE), see Figure 1C. Then, network properties of these graphs were computed, see Network analysis. There was a significant effect of group size on causal density (β=−0.0006, SE=0.0001, t=−5.88, and ω2=0.47), see Figure 7A. Causal density increased with tempo (β=0.008, SE=0.0019, t=4.01, and ω2=0.12). As Figure 7B shows, mean node strength increased with group size (β=0.0032, SE=0.00032, t=10.13, and ω2=0.73). Node strength was affected by tempo (β=0.016, SE=0.005, t=2.96, and ω2=0.07). For better model conditioning, tempos values were reduced and compressed by transforming them with a logistic function. Comparing between SCT trials and synchronization-only trials in the octet group found no difference for causal density (t<1) or mean node strength (t<1), suggesting group dynamics were similar regardless of whether a pacing stimulus was present or not. Figure 7 with 2 supplements see all Download asset Open asset Network dynamics. (A) Mean node strength increases with group size in the continuation phase of synchronization-continuation task (SCT) ensemble drumming trials. (B) Causal density decreases with group size (Abscissa jittered for visibility). Figure 7—source data 1 https://cdn.elifesciences.org/articles/74816/elife-74816-fig7-data1-v2.xlsx Download elife-74816-fig7-data1-v2.xlsx Network properties and task performance The relation between group dynamics and group performance was evaluated by regressing the network properties separately with respect to speeding up and IOI variability in ensemble playing. Speeding up decreased with increasing causal density (β=−0.005, SE=0.002, t=−2.22, and ω2=0.02) and was not associated with node strength (β=−0.01, SE=0.007, and t=−1.66), see Figure 7—figure supplement 1A–B. Variability was not associated with causal density (β=−.003, SE=0.0017, and t=−1.97) but decreased with increasing node strength (β=−0.007, SE=0.0019, t=−4.01, and ω2=0.09), see Figure 7—figure supplement 1C–D. Interestingly, we also observed that TE tended to be higher in trials that deviated less from the stimulus tempo, see Figure 7—figure supplement 2. Variability in the mean-field model with discretely updated feedback The results of the modeling showed that the cycle duration variabilities of the mean-field hybrid Kuramoto model, adapted to have discretely updated feedback, reproduced the main experimental findings. The model results are summarized in Figure 3B and can be compared to the experimental results in Figure 3A. The adapted model reproduced the pattern of behavioral results at both the level of individual oscillators (relating to individuals in our drumming data) and the mean field (relating to our group-aggregate analyses). Specifically, for all group sizes, variability was higher for individual oscillators when playing in the group than when playing solo, but variability was always lowest for the mean field (group aggregate). In addition, with increasing group size, the difference between individual variability and that of the mean field became greater, indicating increasing collective benefits with increasing group size. See Figure 3—figure supplement 1B for a comparison with the null (pseudo-ensemble) condition. A separate report will address the model’s individual unit dynamics and co-variation in more detail (Delasanta et al., in preparation). The importance of the mean-field hybrid model for understanding the present results can be seen in that other versions of the Kuramoto model did not successfully replicate the pattern of the data. Specifically, a pulse-coupled network in a neighbors-only ring-topology and a constant coupling network in full connectivity did not exhibit the same pattern of variability as the empirical data, see Figure 3—figure supplement 2A–B. Discussion The present study examined how group performance in a synchronization timing task depends on group size and interactions among group members. Our overall hypothesis was that the mean field would influence individuals in a group, thereby stabilizing group performance. We expected that larger groups would exhibit more stable performance due to a more stable mean field. This hypothesis