Entrainment and synchronization in networks of Rayleigh-van der Pol oscillators with diffusive and Haken-Kelso-Bunz couplings

Third party stabilization of unstable coordination in systems of coupled oscillators

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.

Bridging the gap between emotion and joint action

Our daily human life is filled with a myriad of joint action moments, be it children playing, adults working together (i.e., team sports), or strangers navigating through a crowd. Joint action brings individuals (and embodiment of their emotions) together, in space and in time. Yet little is known about how individual emotions propagate through embodied presence in a group, and how joint action changes individual emotion. In fact, the multi-agent component is largely missing from neuroscience-based approaches to emotion, and reversely joint action research has not found a way yet to include emotion as one of the key parameters to model socio-motor interaction. In this review, we first identify the gap and then stockpile evidence showing strong entanglement between emotion and acting together from various branches of sciences. We propose an integrative approach to bridge the gap, highlight five research avenues to do so in behavioral neuroscience and digital sciences, and address some of the key challenges in the area faced by modern societies.

Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

The widely cited Haken-Kelso-Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

The Haken-Kelso-Bunz (HKB) model: from matter to movement to mind

This article presents a brief retrospective on the Haken-Kelso-Bunz (HKB) model of certain dynamical properties of human movement. Though unanticipated, HKB introduced, and demonstrated the power of, a new vocabulary for understanding behavior, cognition and the brain, revealed through a visually compelling mathematical picture that accommodated highly reproducible experimental facts and predicted new ones. HKB stands as a harbinger of paradigm change in several scientific fields, the effects of which are still being felt. In particular, HKB constitutes the foundation of a mechanistic science of coordination called Coordination Dynamics that extends from matter to movement to mind, and beyond.

Unifying Large- and Small-Scale Theories of Coordination

Coordination is a ubiquitous feature of all living things. It occurs by virtue of informational coupling among component parts and processes and can be quite specific (as when cells in the brain resonate to signals in the environment) or nonspecific (as when simple diffusion creates a source–sink dynamic for gene networks). Existing theoretical models of coordination—from bacteria to brains to social groups—typically focus on systems with very large numbers of elements (N→∞) or systems with only a few elements coupled together (typically N = 2). Though sharing a common inspiration in Nature’s propensity to generate dynamic patterns, both approaches have proceeded largely independent of each other. Ideally, one would like a theory that applies to phenomena observed on all scales. Recent experimental research by Mengsen Zhang and colleagues on intermediate-sized ensembles (in between the few and the many) proves to be the key to uniting large- and small-scale theories of coordination. Disorder–order transitions, multistability, order–order phase transitions, and especially metastability are shown to figure prominently on multiple levels of description, suggestive of a basic Coordination Dynamics that operates on all scales. This unified coordination dynamics turns out to be a marriage of two well-known models of large- and small-scale coordination: the former based on statistical mechanics (Kuramoto) and the latter based on the concepts of Synergetics and nonlinear dynamics (extended Haken–Kelso–Bunz or HKB). We show that models of the many and the few, previously quite unconnected, are thereby unified in a single formulation. The research has led to novel topological methods to handle the higher-dimensional dynamics of coordination in complex systems and has implications not only for understanding coordination but also for the design of (biorhythm inspired) computers.

Relaxation oscillations and frequency entrainment in quantum mechanics

Frequency entrainment of continuous-variable oscillators has to date been restrained to the weakly nonlinear regime. Here we overcome this bottleneck and extend frequency entrainment of quantum continuous-variable oscillators to arbitrary nonlinearities. The previously known steady state of such quantum oscillators in the weakly nonlinear regime (also known as a Stuart-Landau oscillator) is shown to emerge as a special case. Most importantly, the hallmark of strong nonlinearity-relaxation oscillations-is shown in quantum mechanics. Depending on the oscillator's nonlinearity, relaxation oscillations are found to occur via two distinct mechanisms in phase space.

Social Synchrony on Complex Networks

Social synchrony (SS) is an emergent phenomenon in human society. People often mimic others which, over time, can result in large groups behaving similarly. Drawing from prior empirical studies of SS in online communities, here we propose a discrete network model of SS based on four attributes: 1) depth of action; 2) breadth of impact, i.e., a large number of actions are performed with a large group of people involved; 3) heterogeneity of role, i.e., people of higher degree play more important roles; and 4) lastly, emergence of phenomenon, i.e., it is far from random. We analyze our model both analytically and with simulations, and find good agreement between the two. We find this model can well explain the four characters of SS, and thus hope it can help researchers better understand human collective behavior.

Critical diversity: Divided or united states of social coordination

Much of our knowledge of coordination comes from studies of simple, dyadic systems or systems containing large numbers of components. The huge gap 'in between' is seldom addressed, empirically or theoretically. We introduce a new paradigm to study the coordination dynamics of such intermediate-sized ensembles with the goal of identifying key mechanisms of interaction. Rhythmic coordination was studied in ensembles of eight people, with differences in movement frequency ('diversity') manipulated within the ensemble. Quantitative change in diversity led to qualitative changes in coordination, a critical value separating régimes of integration and segregation between groups. Metastable and multifrequency coordination between participants enabled communication across segregated groups within the ensemble, without destroying overall order. These novel findings reveal key factors underlying coordination in ensemble sizes previously considered too complicated or 'messy' for systematic study and supply future theoretical/computational models with new empirical checkpoints.

Design and Validation of a Virtual Player for Studying Interpersonal Coordination in the Mirror Game

The mirror game has been recently proposed as a simple, yet powerful paradigm for studying interpersonal interactions. It has been suggested that a virtual partner able to play the game with human subjects can be an effective tool to affect the underlying neural processes needed to establish the necessary connections between the players, and also to provide new clinical interventions for rehabilitation of patients suffering from social disorders. Inspired by the motor processes of the central nervous system (CNS) and the musculoskeletal system in the human body, in this paper we develop a novel interactive cognitive architecture based on nonlinear control theory to drive a virtual player (VP) to play the mirror game with a human player (HP) in different configurations. Specifically, we consider two cases: 1) the VP acts as leader and 2) the VP acts as follower. The crucial problem is to design a feedback control architecture capable of imitating and following or leading an HP in a joint action task. The movement of the end-effector of the VP is modeled by means of a feedback controlled Haken-Kelso-Bunz (HKB) oscillator, which is coupled with the observed motion of the HP measured in real time. To this aim, two types of control algorithms (adaptive control and optimal control) are used and implemented on the HKB model so that the VP can generate a human-like motion while satisfying certain kinematic constraints. A proof of convergence of the control algorithms is presented together with an extensive numerical and experimental validation of their effectiveness. A comparison with other existing designs is also discussed, showing the flexibility and the advantages of our control-based approach.

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.

A Novel Computer-Based Set-Up to Study Movement Coordination in Human Ensembles

Existing experimental works on movement coordination in human ensembles mostly investigate situations where each subject is connected to all the others through direct visual and auditory coupling, so that unavoidable social interaction affects their coordination level. Here, we present a novel computer-based set-up to study movement coordination in human groups so as to minimize the influence of social interaction among participants and implement different visual pairings between them. In so doing, players can only take into consideration the motion of a designated subset of the others. This allows the evaluation of the exclusive effects on coordination of the structure of interconnections among the players in the group and their own dynamics. In addition, our set-up enables the deployment of virtual computer players to investigate dyadic interaction between a human and a virtual agent, as well as group synchronization in mixed teams of human and virtual agents. We show how this novel set-up can be employed to study coordination both in dyads and in groups over different structures of interconnections, in the presence as well as in the absence of virtual agents acting as followers or leaders. Finally, in order to illustrate the capabilities of the architecture, we describe some preliminary results. The platform is available to any researcher who wishes to unfold the mechanisms underlying group synchronization in human ensembles and shed light on its socio-psychological aspects.

Reconstructing the structure of directed and weighted networks of nonlinear oscillators

The formalism of complex networks is extensively employed to describe the dynamics of interacting agents in several applications. The features of the connections among the nodes in a network are not always provided beforehand, hence the problem of appropriately inferring them often arises. Here, we present a method to reconstruct directed and weighted topologies of networks of heterogeneous nonlinear oscillators. We illustrate the theory on a set of representative examples.

A Novel Computer-Based Set-Up to Study Movement Coordination in Human Ensembles

Existing experimental works on movement coordination in human ensembles mostly investigate situations where each subject is connected to all the others through direct visual and auditory coupling, so that unavoidable social interaction affects their coordination level. Here, we present a novel computer-based set-up to study movement coordination in human groups so as to minimize the influence of social interaction among participants and implement different visual pairings between them. In so doing, players can only take into consideration the motion of a designated subset of the others. This allows the evaluation of the exclusive effects on coordination of the structure of interconnections among the players in the group and their own dynamics. In addition, our set-up enables the deployment of virtual computer players to investigate dyadic interaction between a human and a virtual agent, as well as group synchronization in mixed teams of human and virtual agents. We show how this novel set-up can be employed to study coordination both in dyads and in groups over different structures of interconnections, in the presence as well as in the absence of virtual agents acting as followers or leaders. Finally, in order to illustrate the capabilities of the architecture, we describe some preliminary results. The platform is available to any researcher who wishes to unfold the mechanisms underlying group synchronization in human ensembles and shed light on its socio-psychological aspects.

Design of a Virtual Player for Joint Improvisation with Humans in the Mirror Game

Joint improvisation is often observed among humans performing joint action tasks. Exploring the underlying cognitive and neural mechanisms behind the emergence of joint improvisation is an open research challenge. This paper investigates jointly improvised movements between two participants in the mirror game, a paradigmatic joint task example. First, experiments involving movement coordination of different dyads of human players are performed in order to build a human benchmark. No designation of leader and follower is given beforehand. We find that joint improvisation is characterized by the lack of a leader and high levels of movement synchronization. Then, a theoretical model is proposed to capture some features of their interaction, and a set of experiments is carried out to test and validate the model ability to reproduce the experimental observations. Furthermore, the model is used to drive a computer avatar able to successfully improvise joint motion with a human participant in real time. Finally, a convergence analysis of the proposed model is carried out to confirm its ability to reproduce joint movements between the participants.