Physics of the rhythmic applause

Influences on and Measures of Unintentional Group Synchrony

Many instances of large-scale coordination occur in real-life social situations without the explicit awareness of the individuals involved. While the majority of research to date has examined dyadic interactions – those between two individuals – during intentional or deliberate coordination, the present review surveys the handful of recent studies investigating behavioral and physiological synchrony across groups of more than two people when coordination was not an explicit goal. Both minimal (e.g., visual information, shared location) and naturalistic (e.g., choir voice section, family relationship) group interactions appear to promote unintentional group synchrony although they have so far only been studied separately. State differences in unintentional group synchrony, or the relative presence of coordination in various conditions, have tended to be assessed differently, such as using correlation-type relationships, compared to its temporal dynamics, or changes over time in the degree of coordination, which appear to be best captured using phase differences. Simultaneously evaluating behavioral, physiological, and social responses as well systematically comparing different synchrony measures could further our understanding of the influences on and measures of group synchrony, allowing us to move away from studying individual persons responding to static laboratory stimuli and toward investigating collective experiences in natural, dynamic social interactions.

One in the Dance: Musical Correlates of Group Synchrony in a Real-World Club Environment

Previous research on interpersonal synchrony has mainly investigated small groups in isolated laboratory settings, which may not fully reflect the complex and dynamic interactions of real-life social situations. The present study expands on this by examining group synchrony across a large number of individuals in a naturalistic environment. Smartphone acceleration measures were recorded from participants during a music set in a dance club and assessed to identify how group movement synchrony covaried with various features of the music. In an evaluation of different preprocessing and analysis methods, giving more weight to front-back movement provided the most sensitive and reliable measure of group synchrony. During the club music set, group synchrony of torso movement was most strongly associated with pulsations that approximate walking rhythm (100-150 beats per minute). Songs with higher real-world play counts were also correlated with greater group synchrony. Group synchrony thus appears to be constrained by familiarity of the movement (walking action and rhythm) and of the music (song popularity). These findings from a real-world, large-scale social and musical setting can guide the development of methods for capturing and examining collective experiences in the laboratory and for effectively linking them to synchrony across people in daily life.

Correlated disorder in the Kuramoto model: Effects on phase coherence, finite-size scaling, and dynamic fluctuations

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction p of oscillators are positively coupled, attracting all others, while the remaining fraction 1-p are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a critical width γ_c in the frequency distribution below which the system spontaneously synchronizes. Moreover, this γ_c is independent of p. Hence, our model and the traditional Kuramoto model (recovered when p = 1) have the same critical width γ_c. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of p < 1.

Describing synchronization and topological excitations in arrays of magnetic spin torque oscillators through the Kuramoto model

The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength between STO we show that this imposes an upper limit for the maximum number of oscillators that can be synchronized. Further, we show that the lack of long range order is associated with the formation of topological defects in the phase field similar to the two-dimensional XY model of ferromagnetism. Our results shed new light on the synchronization of STO, where controlling the mutual synchronization of several oscillators is considered crucial for applications.

Criterion for noise-induced synchronization: Application to colloidal alignment

Colloidal bodies of irregular shape rotate as they descend under gravity in solution. This rotational response provides a means of bringing a dispersion of identical bodies into a synchronized rotation with the same orientation using programed forcing. We use the notion of statistical entropy to derive bounds on the rate of synchronization. These bounds apply generally to dynamical systems with stable periodic motion with a phase ϕ(t), when subjected to an impulsive perturbation. The impulse causes a change of phase expressible as a phase map ψ(ϕ). We derive an upper limit on the average change of entropy 〈ΔH〉 in terms of this phase map; when this limit is negative, alignment must occur. For systems that have achieved a low entropy, the 〈ΔH〉 approaches this upper limit.

Collective dynamics of identical phase oscillators with high-order coupling

In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. Furthermore, a detailed linear stability analysis proves that the stationary symmetric distribution is only neutrally stable in the marginal regime which stems from the generalized time-reversal symmetry. Moreover, the critical parameters of the transition among various regimes are determined analytically by both the Ott-Antonsen method and linear stability analysis, the transient dynamics are further revealed in terms of the characteristic curves method. Finally, for the more general initial condition the symmetric dynamics could be reduced to a rigorous three-dimensional manifold which shows that the neutrally stable chaos could also occur in this model for particular parameters. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the dynamical properties in general systems with higher-order harmonics couplings.

Synchronization and plateau splitting of coupled oscillators with long-range power-law interactions

We investigate synchronization and plateau splitting of coupled oscillators on a one-dimensional lattice with long-range interactions that decay over distance as a power law. We show that in the thermodynamic limit the dynamics of systems of coupled oscillators with power-law exponent α≤1 is identical to that of the all-to-all coupling case. For α>1, oscillatory behavior of the phase coherence appears as a result of single plateau splitting into multiple plateaus. A coarse-graining method is used to investigate the onset of plateau splitting. We analyze a simple oscillatory state formed by two plateaus in detail and propose a systematic approach to predict the onset of plateau splitting. The prediction of breaking points of plateau splitting is in quantitatively good agreement with numerical simulations.

Switching synchronization in one-dimensional memristive networks

We report on a switching synchronization phenomenon in one-dimensional memristive networks, which occurs when several memristive systems with different switching constants are switched from the high- to low-resistance state. Our numerical simulations show that such a collective behavior is especially pronounced when the applied voltage slightly exceeds the combined threshold voltage of memristive systems. Moreover, a finite increase in the network switching time is found compared to the average switching time of individual systems. An analytical model is presented to explain our observations. Using this model, we have derived asymptotic expressions for memory resistances at short and long times, which are in excellent agreement with results of our numerical simulations.

Dynamics of globally coupled oscillators: Progress and perspectives

In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.

Intercommunity resonances in multifrequency ensembles of coupled oscillators

We generalize the Kuramoto model of globally coupled oscillators to multifrequency communities. A situation when mean frequencies of two subpopulations are close to the resonance 2:1 is considered in detail. We construct uniformly rotating solutions describing synchronization inside communities and between them. Remarkably, cross coupling across the frequencies can promote synchrony even when ensembles are separately asynchronous. We also show that the transition to synchrony due to the cross coupling is accompanied by a huge multiplicity of distinct synchronous solutions, which is directly related to a multibranch entrainment. On the other hand, for synchronous populations, the cross-frequency coupling can destroy phase locking and lead to chaos of mean fields.

Interactional synchrony in chimpanzees: Examination through a finger-tapping experiment

Humans often unconsciously coordinate behaviour with that of others in daily life. This interpersonal coordination, including mimicry and interactional synchrony, has been suggested to play a fundamental role in social interaction. If this coordinative behavior is socially adaptive, it may be shared with other highly social animal species. The current study targeted chimpanzees, which phylogenetically are the closest living relatives of humans and live in complex social groups, and examined whether interactional synchrony would emerge in pairs of chimpanzees when auditory information about a partner's movement was provided. A finger-tapping task was introduced via touch panels to elicit repetitive and rhythmic movement from each chimpanzee. We found that one of four chimpanzees produced significant changes in both tapping tempo and timing of the tapping relative to its partner's tap when auditory sounds were provided. Although the current results may have limitations in generalizing to chimpanzees as a species, we suggest that a finger-tapping task is one potential method to investigate interactional synchrony in chimpanzees under a laboratory setup.

The performative pleasure of imprecision: a diachronic study of entrainment in music performance

This study focuses in on a moment of live performance in which the entrainment amongst a musical quartet is threatened. Entrainment is asymmetric in so far as there is an ensemble leader who improvises and expands the structure of a last chorus of a piece of music beyond the limits tacitly negotiated during prior rehearsals and performances. Despite the risk of entrainment being disturbed and performance interrupted, the other three musicians in the quartet follow the leading performer and smoothly transition into unprecedented performance territory. We use this moment of live performance to work back through the fieldwork data, building a diachronic study of the development and bases of entrainment in live music performance. We introduce the concept of entrainment and profile previous theory and research relevant to entrainment in music performance. After outlining our methodology, we trace the evolution of the structure of the piece of music from first rehearsal to final performance. Using video clip analysis, interviews and field notes we consider how entrainment shaped and was shaped by the moment of performance in focus. The sense of trust between quartet musicians is established through entrainment processes, is consolidated via smooth adaptation to the threats of disruption. Non-verbal communicative exchanges, via eye contact, gesture, and spatial proximity, sustain entrainment through phase shifts occurring swiftly and on the fly in performance contexts. These exchanges permit smooth adaptation promoting trust. This frees the quartet members to play with the potential disturbance of equilibrium inherent in entrained relationships and to play with this tension in an improvisatory way that enhances audience engagement and the live quality of performance.

Self-organizing patterns in an evolutionary rock-paper-scissors game for stochastic synchronized strategy updates

We study a spatial evolutionary rock-paper-scissors game with synchronized strategy updating. Players gain their payoff from games with their four neighbors on a square lattice and can update their strategies simultaneously according to the logit rule, which is the noisy version of the best-response dynamics. For the synchronized strategy update two types of global oscillations (with an ordered strategy arrangement and periods of three and six generations) can occur in this system in the zero noise limit. At low noise values, all nine oscillating phases are present in the system by forming a self-organizing spatial pattern due to the comprising invasion and speciation processes along the interfaces separating the different domains.

Dynamics in the Sakaguchi-Kuramoto model with two subpopulations [corrected]

The dynamics in a variant of globally coupled Sakaguchi-Kuramoto [corrected]. phase oscillators is studied. The model consists of two subpopulations, each with a different phase lag and interaction strength. Using Ott-Antonson ansatz, we analyze the dynamics in the model and present the numerical results. There exist stationary synchronous states which are generalized π states and two types of traveling wave states. We find that the traveling wave states are the dominant dynamics in comparison with the stationary states. Particularly, we find that the stationary and traveling wave states can be smoothly connected through the properly chosen parameter paths.

Transition to synchronization in a Kuramoto model with the first- and second-order interaction terms

We investigate a Kuramoto model incorporated with the first-order and the second-order interaction terms. We show that the model displays the coexistence of multiattractors and different attractors may be characterized by the phase distributions of oscillators. By investigating the transition diagrams in both forward continuation and backward continuation, we find that the synchronous state with unimodal phase distribution is the most stable one while the state in cluster synchrony with evenly distributed bimodal phase distribution is the least stable one. We also present the phase diagram of the model in the parameter space.

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

Dynamics of the Kuramoto model in the presence of correlation between distributions of frequencies and coupling strengths

As a paradigmatic model, the Kuramoto model has provided a platform for investigating synchronization among nonidentical oscillators. In this work, we consider the Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength. We introduce the correlation between the distributions of natural frequencies and the coupling strengths of oscillators. Three different types of correlations are considered. We find rich dynamics result from the correlation such as different types of traveling wave states and, most interestingly, another type of nonstationary state: an oscillating π state.

Breathing synchronization in interconnected networks

Global synchronization in a complex network of oscillators emerges from the interplay between its topology and the dynamics of the pairwise interactions among its numerous components. When oscillators are spatially separated, however, a time delay appears in the interaction which might obstruct synchronization. Here we study the synchronization properties of interconnected networks of oscillators with a time delay between networks and analyze the dynamics as a function of the couplings and communication lag. We discover a new breathing synchronization regime, where two groups appear in each network synchronized at different frequencies. Each group has a counterpart in the opposite network, one group is in phase and the other in anti-phase with their counterpart. For strong couplings, instead, networks are internally synchronized but a phase shift between them might occur. The implications of our findings on several socio-technical and biological systems are discussed.

Multiplicity of singular synchronous states in the Kuramoto model of coupled oscillators

We study the Kuramoto model of globally coupled oscillators with a biharmonic coupling function. We develop an analytic self-consistency approach to find stationary synchronous states in the thermodynamic limit and demonstrate that there is a huge multiplicity of such states, which differ microscopically in the distributions of locked phases. These synchronous regimes already exist prior to the linear instability transition of the fully asynchronous state. In the presence of white Gaussian noise, the multiplicity is lifted, but the dependence of the order parameters on coupling constants remains nontrivial.

Dynamics in hybrid complex systems of switches and oscillators

While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on, they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.

Patterns of horse-rider coordination during endurance race: a dynamical system approach

In riding, most biomechanical studies have focused on the description of the horse locomotion in unridden condition. In this study, we draw the prospect of how the basic principles established in inter-personal coordination by the theory of Coordination Dynamics may provide a conceptual and methodological framework for understanding the horse-rider coupling. The recent development of mobile technologies allows combined horse and rider recordings during long lasting natural events such as endurance races. Six international horse-rider dyads were thus recorded during a 120 km race by using two tri-axial accelerometers placed on the horses and riders, respectively. The analysis concentrated on their combined vertical displacements. The obtained shapes and angles of Lissajous plots together with values of relative phase between horse and rider displacements at lower reversal point allowed us to characterize four coordination patterns, reflecting the use of two riding techniques per horse's gait (trot and canter). The present study shows that the concepts, methods and tools of self-organizing dynamic system approach offer new directions for understanding horse-rider coordination. The identification of the horse-rider coupling patterns constitutes a firm basis to further study the coalition of multiple constraints that determine their emergence and their dynamics in endurance race.

The dynamics of audience applause

The study of social identity and crowd psychology looks at how and why individual people change their behaviour in response to others. Within a group, a new behaviour can emerge first in a few individuals before it spreads rapidly to all other members. A number of mathematical models have been hypothesized to describe these social contagion phenomena, but these models remain largely untested against empirical data. We used Bayesian model selection to test between various hypotheses about the spread of a simple social behaviour, applause after an academic presentation. Individuals' probability of starting clapping increased in proportion to the number of other audience members already ‘infected’ by this social contagion, regardless of their spatial proximity. The cessation of applause is similarly socially mediated, but is to a lesser degree controlled by the reluctance of individuals to clap too many times. We also found consistent differences between individuals in their willingness to start and stop clapping. The social contagion model arising from our analysis predicts that the time the audience spends clapping can vary considerably, even in the absence of any differences in the quality of the presentations they have heard.

Cluster explosive synchronization in complex networks

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.

Dynamics of multifrequency oscillator communities

We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The simplest situation of three resonantly interacting groups is analyzed in detail. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations and present examples where the interaction between communities leads to their synchrony or to a partially asynchronous state or to a chaotic dynamics of order parameters.

A spatially explicit model of synchronization in fiddler crab waving displays

Fiddler crabs (Uca spp., Decapoda: Ocypodidae) are commonly found forming large aggregations in intertidal zones, where they perform rhythmic waving displays with their greatly enlarged claws. While performing these displays, fiddler crabs often synchronize their behavior with neighboring males, forming the only known synchronized visual courtship displays involving reflected light and moving body parts. Despite being one of the most conspicuous aspects of fiddler crab behavior, little is known about the mechanisms underlying synchronization of male displays. In this study we develop a spatially explicit model of fiddler crab waving displays using coupled logistic map equations. We explored two alternative models in which males either direct their attention at random angles or preferentially toward neighbors. Our results indicate that synchronization is possible over a fairly large region of parameter space. Moreover, our model was capable of generating local synchronization neighborhoods, as commonly observed in fiddler crabs under natural conditions.

Optimal synchronizability of bearings

Bearings are mechanical dissipative systems that, when perturbed, relax toward a synchronized (bearing) state. Here we find that bearings can be perceived as physical realizations of complex networks of oscillators with asymmetrically weighted couplings. Accordingly, these networks can exhibit optimal synchronization properties through fine-tuning of the local interaction strength as a function of node degree [Motter, Zhou, and Kurths, Phys. Rev. E 71, 016116 (2005)]. We show that, in analogy, the synchronizability of bearings can be maximized by counterbalancing the number of contacts and the inertia of their constituting rotor disks through the mass-radius relation, m~r(α), with an optimal exponent α=α(×) which converges to unity for a large number of rotors. Under this condition, and regardless of the presence of a long-tailed distribution of disk radii composing the mechanical system, the average participation per disk is maximized and the energy dissipation rate is homogeneously distributed among elementary rotors.

Synchronization in complex oscillator networks and smart grids

The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A widely adopted model of a coupled oscillator network is characterized by a population of heterogeneous phase oscillators, a graph describing the interaction among them, and diffusive and sinusoidal coupling. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here, we present a unique, concise, and closed-form condition for synchronization of the fully nonlinear, nonequilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters; they are statistically correct for almost all networks; and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks, such as electrical power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex network scenarios and in smart grid applications.

Measuring group synchrony: a cluster-phase method for analyzing multivariate movement time-series

A new method for assessing group synchrony is introduced as being potentially useful for objectively determining degree of group cohesiveness or entitativity. The cluster-phase method of Frank and Richardson (2010) was used to analyze movement data from the rocking chair movements of six-member groups who rocked their chairs while seated in a circle facing the center. In some trials group members had no information about others' movements (their eyes were shut) or they had their eyes open and gazed at a marker in the center of the group. As predicted, the group level synchrony measure was able to distinguish between situations where synchrony would have been possible and situations where it would be impossible. Moreover, other aspects of the analysis illustrated how the cluster phase measures can be used to determine the type of patterning of group synchrony, and, when integrated with multi-level modeling, can be used to examine individual-level differences in synchrony and dyadic level synchrony as well.

How to suppress undesired synchronization

Examples of synchronization can be found in a wide range of phenomena such as neurons firing, lasers cascades, chemical reactions, and opinion formation. However, in many situations the formation of a coherent state is not pleasant and should be mitigated. For example, the onset of synchronization can be the root of epileptic seizures, traffic congestion in networks, and the collapse of constructions. Here we propose the use of contrarians to suppress undesired synchronization. We perform a comparative study of different strategies, either requiring local or total knowledge, and show that the most efficient one solely requires local information. Our results also reveal that, even when the distribution of neighboring interactions is narrow, significant improvement is observed when contrarians sit at the highly connected elements. The same qualitative results are obtained for artificially generated networks and two real ones, namely, the Routers of the Internet and a neuronal network.

Mean-field behavior in coupled oscillators with attractive and repulsive interactions

We consider a variant of the Kuramoto model of coupled oscillators in which both attractive and repulsive pairwise interactions are allowed. The sign of the coupling is assumed to be a characteristic of a given oscillator. Specifically, some oscillators repel all the others, thus favoring an antiphase relationship with them. Other oscillators attract all the others, thus favoring an in-phase relationship. The Ott-Antonsen ansatz is used to derive the exact low-dimensional dynamics governing the system's long-term macroscopic behavior. The resulting analytical predictions agree with simulations of the full system. We explore the effects of changing various parameters, such as the width of the distribution of natural frequencies and the relative strengths and proportions of the positive and negative interactions. For the particular model studied here we find, unexpectedly, that the mixed interactions produce no new effects. The system exhibits conventional mean-field behavior and displays a second-order phase transition like that found in the original Kuramoto model. In contrast to our recent study of a different model with mixed interactions [Phys. Rev. Lett. 106, 054102 (2011)], the π state and traveling-wave state do not appear for the coupling type considered here.

Three people can synchronize as coupled oscillators during sports activities

We experimentally investigated the synchronized patterns of three people during sports activities and found that the activity corresponded to spatiotemporal patterns in rings of coupled biological oscillators derived from symmetric Hopf bifurcation theory, which is based on group theory. This theory can provide catalogs of possible generic spatiotemporal patterns irrespective of their internal models. Instead, they are simply based on the geometrical symmetries of the systems. We predicted the synchronization patterns of rings of three coupled oscillators as trajectories on the phase plane. The interactions among three people during a 3 vs. 1 ball possession task were plotted on the phase plane. We then demonstrated that two patterns conformed to two of the three patterns predicted by the theory. One of these patterns was a rotation pattern (R) in which phase differences between adjacent oscillators were almost 2π/3. The other was a partial anti-phase pattern (PA) in which the two oscillators were anti-phase and the third oscillator frequency was dead. These results suggested that symmetric Hopf bifurcation theory could be used to understand synchronization phenomena among three people who communicate via perceptual information, not just physically connected systems such as slime molds, chemical reactions, and animal gaits. In addition, the skill level in human synchronization may play the role of the bifurcation parameter.

Synchronization is very interesting as both a natural phenomenon and scientific topic in physical and biological systems. Examples include the Belousov-Zhabotinsky (BZ) reaction, the oscillation of metronomes, the flash of fireflies, and the calling behavior of Japanese tree frogs. The symmetric Hopf bifurcation theory, which is based on group theory, has been proposed as a useful approach for spatiotemporal pattern formation in coupled oscillator systems. This theory has been applied to various types of quadrupedal gaits in terms of symmetrically coupled oscillators, and to rings and chains of coupled oscillators of plasmodial slime molds. Here we report that the spatiotemporal pattern formation in three-person coupling during dynamic human movement, such as sports activity, conforms to symmetry-breaking theory. Our present study is salient because the spatiotemporal synchronization patterns among three people corresponded to the predicted patterns derived from symmetric Hopf bifurcation theory, as with pattern formation in slime molds, even though the actors were not connected physically but informationally. Moreover, although informational coupling between two people has been shown previously in well controlled experiments, we demonstrate three-person coupling using perceptual information in a real-life setting.

Conformists and contrarians in a Kuramoto model with identical natural frequencies

We consider a variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These contrarian oscillators tend to align in antiphase with the mean field, whereas, the positively coupled conformist oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.

On following behaviour as a mechanism for collective movement

During collective movement, animals display a wide variety of mechanisms to maintain cohesion. In some species, individuals rely mainly on following their direct predecessor, thereby forming spectacular processions of individuals in single file. Despite being the simplest case of following behaviour, it is largely absent from the theoretical literature on collective migrations. The objective of this study is to quantify the efficiency of following the predecessor, in terms of ensuring cohesion. The situation we consider is a sequence of individuals facing a bifurcation. The choice between left and right is influenced by the choice of the predecessor. First, we model this situation with a two-state Markov chain with a symmetric transition matrix. Cohesion is quantified as the expected number of individuals on either side, and the expected number of consecutive followers. Although cohesion increases with the probability of following, it remains surprisingly low unless the probability is almost equal to one. Furthermore, cohesion decreases with group size regardless of the probability of following. Then, we generalise our model to situations in which individuals have a preference for one of the two choices (asymmetric transition matrix). For some parameter sets, the tendency to follow each other leads a large fraction of the individuals to the non preferred side. Moreover, this fraction increases with the total population size. Finally, we include the possibility to follow N individuals. This provides the link between our model and other collective migration models. If enough individuals are perceived, the results shift from symmetrical (low cohesion) to asymmetrical (high cohesion) distribution of the individuals. All in all, our results suggest that following the direct predecessor must be complemented with other cohesive behaviours (involving the perception of more individuals or a navigation system) to guarantee its efficiency. We discuss our findings in the context of the different following behaviours covered in the literature.

Spontaneous synchronization of coupled oscillator systems with frequency adaptation

We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy frequency adaptation. In this paper, we develop a model for oscillators, which adapt both their phases and frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling constant k , separated by critical points k{1} and k{2}: (i) for k<k{1} only the stable incoherent state exists; (ii) for k>k{2}, the incoherent state becomes unstable and only the synchronized state exists; and (iii) for k{1}<k<k{2} both the incoherent and synchronized states are stable. In the bistable regime spontaneous transitions between the incoherent and synchronized states are observed for finite ensembles. These transitions are well described as a stochastic process on the order parameter r undergoing fluctuations due to the system's finite size, leading to the following conclusions: (a) in the bistable regime, the average waiting time of an incoherent-->coherent transition can be predicted by using Kramer's escape time formula and grows exponentially with the number of oscillators; (b) when the incoherent state is unstable (k>k{2}), the average waiting time grows logarithmically with the number of oscillators.

Nontrivial spontaneous synchronization

The collective behavior of an ensemble of multimode stochastic oscillators is investigated. The oscillators are pulse coupled; they are able to emit pulses and to detect the pulses emitted by the others. As a function of the output intensity in the system they can operate in different modes having different pulsing periods. The system is designed to optimize the output intensity around a fixed f* output threshold. In order to do so a simple dynamics is considered. Whenever the total output intensity in the system is lower than f*, a mode with a higher interpulse period is chosen. If the light intensity in the system is higher than f*, a mode with a lower interpulse period is selected. As a side effect of this simple optimization rule, for a given f* interval a nontrivial synchronization of the oscillators is observed. The synchronization level is studied by computer simulations, investigating the influence of model parameters (number of modes, stochasticity of the oscillators, the f* threshold value, and interaction topology). An experimental realization of this system is also considered; an ensemble of electronic oscillators communicating with light pulses was constructed and studied. The experimental system behaves in many ways similar to the theoretically considered multimode stochastic oscillator ensemble.

A quantitative dynamical systems approach to differential learning: self-organization principle and order parameter equations

Differential learning is a learning concept that assists subjects to find individual optimal performance patterns for given complex motor skills. To this end, training is provided in terms of noisy training sessions that feature a large variety of between-exercises differences. In several previous experimental studies it has been shown that performance improvement due to differential learning is higher than due to traditional learning and performance improvement due to differential learning occurs even during post-training periods. In this study we develop a quantitative dynamical systems approach to differential learning. Accordingly, differential learning is regarded as a self-organized process that results in the emergence of subject- and context-dependent attractors. These attractors emerge due to noise-induced bifurcations involving order parameters in terms of learning rates. In contrast, traditional learning is regarded as an externally driven process that results in the emergence of environmentally specified attractors. Performance improvement during post-training periods is explained as an hysteresis effect. An order parameter equation for differential learning involving a fourth-order polynomial potential is discussed explicitly. New predictions concerning the relationship between traditional and differential learning are derived.

Social coordination dynamics: measuring human bonding

Spontaneous social coordination has been extensively described in natural settings but so far no controlled methodological approaches have been employed that systematically advance investigations into the possible self-organized nature of bond formation and dissolution between humans. We hypothesized that, under certain contexts, spontaneous synchrony-a well-described phenomenon in biological and physical settings-could emerge spontaneously between humans as a result of information exchange. Here, a new way to quantify interpersonal interactions in real time is proposed. In a simple experimental paradigm, pairs of participants facing each other were required to actively produce actions, while provided (or not) with the vision of similar actions being performed by someone else. New indices of interpersonal coordination, inspired by the theoretical framework of coordination dynamics (based on relative phase and frequency overlap between movements of individuals forming a pair) were developed and used. Results revealed that spontaneous phase synchrony (i.e., unintentional in-phase coordinated behavior) between two people emerges as soon as they exchange visual information, even if they are not explicitly instructed to coordinate with each other. Using the same tools, we also quantified the degree to which the behavior of each individual remained influenced by the social encounter even after information exchange had been removed, apparently a kind of social memory.

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.

The principles of collective animal behaviour

In recent years, the concept of self-organization has been used to understand collective behaviour of animals. The central tenet of self-organization is that simple repeated interactions between individuals can produce complex adaptive patterns at the level of the group. Inspiration comes from patterns seen in physical systems, such as spiralling chemical waves, which arise without complexity at the level of the individual units of which the system is composed. The suggestion is that biological structures such as termite mounds, ant trail networks and even human crowds can be explained in terms of repeated interactions between the animals and their environment, without invoking individual complexity. Here, I review cases in which the self-organization approach has been successful in explaining collective behaviour of animal groups and societies. Ant pheromone trail networks, aggregation of cockroaches, the applause of opera audiences and the migration of fish schools have all been accurately described in terms of individuals following simple sets of rules. Unlike the simple units composing physical systems, however, animals are themselves complex entities, and other examples of collective behaviour, such as honey bee foraging with its myriad of dance signals and behavioural cues, cannot be fully understood in terms of simple individuals alone. I argue that the key to understanding collective behaviour lies in identifying the principles of the behavioural algorithms followed by individual animals and of how information flows between the animals. These principles, such as positive feedback, response thresholds and individual integrity, are repeatedly observed in very different animal societies. The future of collective behaviour research lies in classifying these principles, establishing the properties they produce at a group level and asking why they have evolved in so many different and distinct natural systems. Ultimately, this research could inform not only our understanding of animal societies, but also the principles by which we organize our own society.

Enhanced synchronizability by structural perturbations

In this Brief Report, we investigate the collective synchronization of a system of coupled oscillators on a Barabási-Albert scale-free network. We propose an approach of structural perturbations aiming at those nodes with maximal betweenness. This method can markedly enhance the network synchronizability, and is easy to realize. The simulation results show that the eigenratio will sharply decrease by one-half when only 0.6% of those hub nodes occur under three-division processes when the network size . In addition, the present study also provides numerical evidence that the maximal betweenness plays a major role in network synchronization.

On partial contraction analysis for coupled nonlinear oscillators

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in "flocks" of oscillators or dynamic elements.

Synchronization of oscillators with long range interaction: Phase transition and anomalous finite size effects

Synchronization in a lattice of a finite population of phase oscillators with algebraically decaying, non-normalized coupling is studied by numerical simulations. A critical level of decay is found, below which full locking takes place if the population contains a sufficiently large number of elements. For large number of oscillators and small coupling constant, numerical simulations and analytical arguments indicate that a phase transition separating synchronization from incoherence appears at a decay exponent value equal to the number of dimensions of the lattice. In contrast with earlier results on similar systems with normalized coupling, we have indications that for the decay exponent less than the dimensions of the lattice and for large populations, synchronization is possible even if the coupling is arbitarily weak. This finding suggests that in organisms interacting through slowly decaying signals such as light or sound, collective oscillations can always be established if the population is sufficiently large.

Synchronization of Kauffman networks

We analyze the synchronization transition for a pair of coupled identical Kauffman networks in the chaotic phase. The annealed model for Kauffman networks shows that synchronization appears through a transcritical bifurcation and provides an approximate description for the whole dynamics of the coupled networks. We show that these analytical predictions are in good agreement with numerical results for sufficiently large networks and study finite-size effects in detail. Preliminary analytical and numerical results for partially disordered networks are also presented.