Symmetry of Interpersonal Rhythmic Coordination: The Case of a Three-Person Drumming

Task

A dissertation submitted to the

Graduate School

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in the Department of Psychology

of the McMicken College of Arts and Sciences

by

Kris Ariyabuddhiphongs

M.S. Illinois State University, 2011

February 2017

Committee Chair: Rachel W. Kallen, Ph.D.

Committee: Michael J. Richardson, Ph.D.

Michael A. Riley, Ph.D.

ii

Abstract

The mathematical theory of symmetry provides a framework to understand higher order

structures of behavioral organization across various contexts; the same principle that explains the

organization of quadruped gaits can also be applied to behavioral coordination in interpersonal

contexts. The current studies examined how symmetries of perceptual coupling and social

information influenced interpersonal coordination during a three-person drumming task. In Study

1, triads of participants performed a drumming task without explicit instructions to coordinate;

each participant drummed to given metronome beats for 10 seconds and maintained his or her

rhythm for the rest of the trial. Half of the 24 triads drummed at 60 bpm, and the other drummed

at 45 bpm. Each triad performed the task under five auditory coupling conditions: the all-,

rotation-, partial-, clamped-, and no-coupling conditions. During the task, participants could hear

but not see each other’s drumming. The results showed that when coupling was present, the

spontaneous coordination mode that emerged tended to be inphase. Regardless of drumming

frequency, coordination in the all- and clamped-coupling conditions was more stable than in the

partial-coupling conditions, indicating the effect of asymmetric coupling functions. In addition,

period shifts were observed in the 45-bpm all-, rotation-, and clamped-coupling conditions. In

Study 2, the minimal group paradigm was used to manipulate the symmetry of social identity

among a triad. Fifteen triads were assigned to the heterogeneous condition, where two

participants were in the minimal ingroup—the red group—and one in the minimal outgroup—the

blue group. The other 14 triads were in the homogeneous condition (i.e., the control group) with

all of them assigned to the red group. Beside the minimal group manipulation, there was no

constraint on either visual or auditory information in Study 2. The participants first performed

the drumming task without explicit instructions to coordinate (i.e., spontaneous coordination

iii

task) and, then, with explicit instructions to coordinate in a partial-inphase pattern (two

participants inphase with each other and the third antiphase relate to the other two). The results

showed that asymmetric minimal group identity had no effect on the spontaneous or explicit

coordination. Plausible explanations for the null effects are discussed.

iv

v

Acknowledgements

First and foremost, I would like to thank my advisors and dissertation committees, Dr.

Rachel Kallen, Dr. Michael Richardson, and Dr. Michael Riley, for their invaluable advice,

insight, knowledge, and generous patience in the development and completion of this

dissertation. I would also like to thank my research assistants, Eduardo J. Rivera Pichardo and

Dalton Bettendorf, for their contribution during the data collection phase. I greatly appreciate my

friends and colleagues for their help on the participant recruitment and for their continuous

support and feedback on the project. To my partner and family, I am thankful for your endless

love and understanding during this journey. Lastly, I would like to express my gratitude to

Chulalongkorn University for their scholarship program and to all staffs at OEADC for their

fantastic assistance throughout these years.

This research was supported by Seeman Research Fund, Department of Psychology,

University of Cincinnati and the National Institutes of Health (R01MH094659).

vi

Table of Contents

List of Tables .............................................................................................................................. viii

List of Figures ............................................................................................................................... ix

Chapter 1: Introduction ............................................................................................................... 1

Interpersonal Rhythmic Coordination ................................................................................. 5

Synchrony in Social Contexts: Its Function and Constraints .............................................. 8

Symmetry of Nonlinear Coupled Oscillators .................................................................... 10

Interaction-Dominant Dynamical Systems ....................................................................... 17

The Current Studies .......................................................................................................... 20

Chapter 2: Study 1 ...................................................................................................................... 29

Overview ........................................................................................................................... 29

Participants ........................................................................................................................ 29

Instruments ........................................................................................................................ 29

Procedure .......................................................................................................................... 30

Data Preparation, Reduction, and Analysis ...................................................................... 32

Results ............................................................................................................................... 35

Discussion ......................................................................................................................... 48

Chapter 3: Study 2 ...................................................................................................................... 52

Overview ........................................................................................................................... 52

Participants ........................................................................................................................ 52

Instruments ........................................................................................................................ 52

Procedure .......................................................................................................................... 55

Data Preparation, Reduction, and Analysis ...................................................................... 56

vii

Results ............................................................................................................................... 58

Spontaneous Drumming Task ............................................................................... 58

Explicit Drumming Task: Partial-Inphase ............................................................ 59

Questionnaire ........................................................................................................ 62

Discussion ......................................................................................................................... 63

Chapter 4: General Discussion .................................................................................................. 65

Asymmetric Informational Coupling Constraints ............................................................. 65

Drumming Frequency ....................................................................................................... 68

Social Constraints ............................................................................................................. 69

Limitations & Future Research ......................................................................................... 72

References .................................................................................................................................... 74

viii

List of Tables

Table 1. Estimated marginal means of average periods and period instability for Study 1 ......... 37

Table 2. Number of time series pairs categorized as different phase modes for Study 1 ............ 44

Table 3. Estimated marginal means of coordination stability (r) for Study 1 ............................. 47

Table 4. Estimated marginal means of average periods and period instability for Study 2 ......... 60

Table 5. Estimated marginal means of coordination stability (r) for Study 2 ............................. 61

Table 6. Frequency of total chosen drumming roles during the explicit partial-inphase drumming

task for Study 2 ............................................................................................................................. 62

Table 7. Means of liking, similarity, and perceived coordination of ingroup and outgroup targets

for Study 2..................................................................................................................................... 63

ix

List of Figures

Figure 1. Schematic representation of the coupling configurations in Study 1 ........................... 24

Figure 2. Mean period for Study 1 ............................................................................................... 38

Figure 3. Mean period for the partial- and the clamped-coupling conditions for Study 1 ........... 40

Figure 4. Mean period stability (coefficient of variation) for Study 1 ......................................... 41

Figure 5. Mean frequency distribution of relative phase for Study 1 .......................................... 45

Figure 6. Mean coordination stability (r) for Study 1 ................................................................. 46

Figure 7. Mean coordination stability (r) in the partial-coupling condition for Study 1 ............ 48

Figure 8. Mean frequency distribution of relative phase for the spontaneous drumming task in

Study 2 .......................................................................................................................................... 59

Figure 9. Mean frequency distribution of relative phase for the explicit drumming task in Study

2..................................................................................................................................................... 61

1

Chapter 1

Introduction

Social behavior is rich with order and patterns. From unintentionally running in step with

a stranger to forming complex social networks, social scientists seek to identify and explain the

origins of patterns in social interactions. The question has been tackled from various approaches

and perspectives, ranging from neurocognitive investigations of joint actions (e.g., Hommel,

Müsseler, Aschersleben, & Prinz, 2001; Iacoboni et al., 2005), to computational models of social

cognition and personality systems (e.g., Mischel & Shoda, 2008; Read et al., 2010), to

dynamical complex systems explorations of social interactions (e.g., Eiler, Kallen, Harrison, &

Richardson, 2013; Schmidt, Fitzpatrick, Caron, & Mergeche, 2011). Despite any discrepancies

between these theoretical approaches, they all share the same essential overarching question,

“how do order and patterns in social interaction come to be?”

Within the broader domains of mathematics and science, the concepts of symmetry and

symmetry breaking were proposed as a unifying framework to analyze how order arises in nature

(Curie, 1894; Golubitsky & Stewart, 2003; Rosen, 1995). In a physical system, such as

Rayleigh–Bénard convection, the emergence of convection is a result of symmetry-breaking

bifurcation. At room temperature, molecules of oil in a pan move at random in a disorderly, yet

symmetrical state. The system is symmetrical because it is invariant to any transformation; no

matter how molecules are exchanged within the liquid, the system still looks the same. When the

liquid is heated from underneath and passes a critical temperature gradient (i.e., the difference in

temperature between the top and bottom of the preparation), random molecular movement no

longer efficiently dissipates energy. Instead, convection offers better heat dissipation, where hot

molecules rise up to cool down at the surface and cool molecules move down and retake energy.

2

During the critical point when the system makes a transition from random movement to orderly

convection, small fluctuations in molecular movement may cause the system to adopt the first

convection in either a clockwise or counterclockwise direction. Once the symmetry is broken and

the convection is formed, the direction of the rest of the convections are now constrained and

determined.

In biology, bilateral symmetry (i.e., left-right symmetry) is commonly found in animals.

The symmetry on the lateral plane results in symmetrical limbs that enable legged locomotion.

On the other hand, asymmetry on the transverse plane results in the anterior and posterior ends of

the body (i.e., front and rear) that provide an organism the primary movement direction. In other

words, asymmetry in animal locomotion results from asymmetry in anatomy. At the behavioral

level, the principle of symmetry can also be used to explain complex motor coordination

patterns, such as animal gaits (Collins & Stewart, 1993; Turvey, 1990), rhythmic movement (Yu,

Russell, & Sternad, 2003), and interpersonal coordination (Richardson et al., 2015; Yokoyama &

Yamamoto, 2011). Similarly, at the societal level, slight asymmetry in individuals’ preferences

to live in the neighborhood with those who share the same race/ethnicity may result in racial

segregation according to Schelling’s model (1971). In sum, “when certain effects show a certain

asymmetry, this asymmetry must be found in the causes which gave rise to it” (Richardson &

Kallen, 2016, p. 229; translated from Curie, 1894).

While some human behaviors are complex and irregular, many behaviors are rhythmic

and symmetrical. Some of these rhythms involve our body and the environment, such as walking

or sleeping; some involve the use of tools, such as hammering a nail or weaving cloth; and some

involve other agents, such as taking turns in a conversation or playing Frisbee. In repetitive

behavior, each component of a system (e.g., a leg in walking or an individual in coordination)

3

can be modeled as a nonlinear oscillator (e.g., Collins & Stewart, 1993; Kelso, 1984; Schmidt,

Bienvenu, Fitzpatrick, & Amazeen, 1998). However, if the components are independent of each

other, there is no order apparent in the collective behavior level of the system; two legs moving

at different frequencies do not produce stable gaits and two individuals talking at the same time

do not carry a conversation. Instead, collective order emerges when the components interact with

each other and become coordinated. Hence, the system is more than a sum of its parts. The extent

to which each oscillator influences each of the other oscillators is a function of the degree of

coupling between each of them. Coupling constraints and the intrinsic dynamics of the oscillators

determine the whole system’s dynamics and its coordination patterns. If the oscillator’s intrinsic

frequencies are identical or similar, then the symmetry of the coordination is determined by the

symmetry of the coupling (Strogatz & Stewart, 1993). Golubitsky and Stewart (1985, 1986)

developed the group theory of symmetric Hopf bifurcation, which explains how different

symmetries of coupling functions in a ring of n nonlinear oscillators result in complex behavioral

coordination, such as patterns and transitions of quadrupedal gaits. Again, asymmetry found in

the collective, global behavior of the system can be traced to asymmetry of the coupling of the

oscillators, one of the primary constraints that shapes the coordinated behavior of the system.

How might the symmetry framework be applied to the study of social behavior?

Traditional social psychology usually posits a linear, component-dominant explanation (e.g.,

Adolphs, 2009; Ajzen, 1991; Gilbert & Malone, 1995; Kelley & Michela, 1980; but see also

Tesser, McMillen, & Collins, 1997). For example, social cognitive processes, realized by

specific neurocognitive component mechanisms, are an intermediary between the social input

and the behavioral output. Situational and dispositional factors are weighted and summed in the

course of these processes to determine an individual’s behavior. In a similar way, group behavior

4

is a sum of each member’s behavior. Because the linear model of behavior by itself does not

often produce complex behavior, the order of social coordination must be originated by some

pre-planned, organized processes—coordinated behavior is attributed to a component that

produces coordination.

On the other hand, the dynamical systems approach and the theory of symmetry suggest

that behavioral coordination can be self-organized, whereby in the absence of a central planner,

order emerges as an inevitable consequence of nonlinear interactions among components and the

stability of the system as a whole. Instead of positing social cognition as a direct causal

explanation of behavioral patterns, social cognition rather acts as constraints on the dynamics of

the agent-(agent-)environment interactions. Social cognition brings about new patterns by

breaking symmetries of the system. In other words, social behaviors are not always a product of

downward causation from cognitive processes, but an emergent phenomenon from nonlinear

interactions between components under physical, cognitive, and social constraints. Slower time

scale dynamics, such as social attitudes, constrain faster time scale group-level behavioral

dynamics that emerge as a result of interactions between even faster dynamics of individuals.

The emergent behavior exerts its influence upward to the slower timescale dynamics via the self-

organizing process by either sustaining or perturbing the existing constraints. By adopting this

interaction-dominant perspective (Eiler et al., 2013; Jensen, 1998; Van Orden & Holden, 2002),

one can appreciate the nature of circular causality and nonlinearity between social situations,

social cognition, and social behavior.

The current studies aim to demonstrate the usefulness of the symmetry framework in

explaining spontaneous, self-organized rhythmic coordination during a three-person drumming

task. A group of three oscillators presents an opportunity for a variety of behavioral patterns at a

5

manageable level for interpersonal experiments. The first study investigated whether different

patterns of coordination emerged as a result of asymmetry in informational coupling between

individuals. The second study was designed to investigate whether (minimal) asymmetries in

social group membership can impact the coupling between individuals and, consequently, the

patterns of coordination. Asymmetry in a slower timescale process of group membership was

predicted to modulate interpersonal coupling of rhythmic coordination and result in asymmetry

in behavioral patterns.

Interpersonal Rhythmic Coordination

Surprisingly, interpersonal coordination or joint actions did not receive much attention

from mainstream social psychological scientists until recently. Many pioneers in this area of

study came from the field of motor control, developmental psychology, and ecological

psychology. While many lines of research adopt the representational approach, including the

theory of common coding (Hommel et al., 2001), action simulation (Decety & Chaminade,

2003), shared attention (Tomasello & Carpenter, 2007), or shared intention (Knoblich & Sebanz,

2008), research from the ecological approach focuses on behavioral entrainment between

‘intentional-blind’ individuals. Studies in this line of research have suggested that individuals

can become spontaneously (i.e., unintentionally) synchronized without any explicit plans or

instructions (Richardson, Marsh, Isenhower, Goodman, & Schmidt, 2007; Richardson, Marsh, &

Schmidt, 2005; Shockley, Richardson, & Dale, 2009). In other words, spontaneous interpersonal

coordination is a self-organized process.

To say that spontaneous synchronization is a self-organized process is to say that it

occurs without a plan or central control. Biologists have documented spectacular spontaneous

synchronization in animals, such as fireflies flashing in perfect synchrony when in a line of

6

mangrove trees stretched over a river front in South East Asia (Buck & Buck, 1976) or migrating

birds flapping their wings in synchrony when they travel in a V-formation (Muijres & Dickinson,

2014). There is no firefly or bird ‘conductor’ that orchestrates such phenomena, nor is there a

blueprint or plan that they follow. Rather, each individual animal is itself an oscillator interacting

nonlinearly with other oscillators under a particular task constraint. Fireflies flash together

because their sensorimotor systems are sensitive to other firefly’s flashes as a part of their mating

behavior. The birds fly in a V-shape formation and flap their wings synchronously to reduce

energy expenditure during their long migration. A skeptic perhaps would argue that those

animals are hard wired for synchrony and that the self-organization principle is not applicable to

the behavior of complex animals like humans. However, for example, on the opening day of the

London Millennium Bridge in 2000, the crowded pedestrians walked side-to-side in lockstep as

they crossed the bridge. This perplexing phenomenon occurred because the bridge wobbled from

side to side. This lateral movement of the bridge destabilized pedestrians’ regular walking

pattern and forced them to balance themselves by walking side-to-side. As more and more

people walked in this fashion, they unintentionally amplified the lateral sway of the bridge,

which further destabilized the regular walking pattern and forced all pedestrians to walk side-to-

side in lockstep with the bridge’s sway (Macdonald, 2008, 2009; Strogatz, Abrams, McRobie,

Eckhardt, & Ott, 2005). This kind of positive feedback loop between an emergent collective

behavior and its constraints on the system’s components is the essential mechanism of any self-

organized phenomenon. Given the boundary conditions for the systems described above, the self-

organization of spontaneous synchrony emerged as a stable solution of the system’s behavioral

dynamics.

7

In human coordination, the dynamical, self-organization approach has successfully

explained both intra- and interpersonal spontaneous synchrony. For example, the HKB model

predicts stable patterns and phase transition of coordination between two limbs as a function of

oscillation frequency (Haken, Kelso, & Bunz, 1985; Kelso, 1984). For instance, as the frequency

increases, the oscillatory coordination between two index fingers changes from an antiphase

pattern (i.e., one oscillator is moving a half a period out-of-sync with the other) to an inphase

pattern (i.e., both oscillators are moving in synchrony). The coordination is possible because our

limbs are coupled via the haptic and neuromuscular systems, which allows the limbs to influence

each other. At the abstract level, the system consists of (a) nonlinear oscillators interacting via

(b) some coupling mechanisms under (c) a task constraint. A generalization of intrapersonal limb

coordination to the interpersonal case can be made by replacing the haptic connections between

limbs with visual coupling. In fact, Schmidt, Carello, and Turvey (1990) found that interpersonal

limb coordination of two individuals swinging one of their legs side by side also conformed to

the same dynamical principles that govern intrapersonal limb coordination (also see Schmidt &

O'Brien, 1997; Schmidt & Richardson, 2008; Schmidt & Turvey, 1994, for a review) As the

frequency of leg swings increased, the antiphase coordination between the two individuals

became unstable, and inphase coordination emerged as a stable solution instead. However, the

perceptual information required for interpersonal coupling depends on the task. For instance,

Richardson et al. (2007) demonstrated that emergence of synchrony between participants on a

rocking chair was determined by participants’ attention and degree to which they detected visual

information. The participants who were asked to fix their gaze on their co-actor’s chair showed a

greater degree of synchrony and stability. On the other hand, in an auditory-based task like

rhythmic drumming, visual information was less crucial for interpersonal coupling

8

(Ariyabuddhiphongs & Richardson, 2015, April)—individuals were able to drum and coordinate

with others on the instructed patterns whether they could see other person drumming or not. In

sum, although the nature of interaction and coupling vary with the system in question, the same

self-organization principle that explains the emergence of order in physical and biological

systems can also be applied to interpersonal systems.

Synchrony in Social Contexts: Its Function and Constraints

The ubiquity of synchrony in nature has led many theorists to speculate about its

functional value. On one hand, entrainment can be a solution to accomplish a task at hand, such

as stabilizing oneself while crossing the swaying Millennium Bridge. On the other hand, our

ability to imitate and synchronize with others is thought to be a fundamental building block of

social coordination (Chartrand, Maddux, & Lakin, 2005; Gallese & Goldman, 1998; Iacoboni,

2009; Schmidt & Richardson, 2008). Although synchrony itself does not require a biological

basis (e.g., inanimate objects such as metronomes placed on a movable platform can become

synchronized), biological systems successfully build upon this entrainment tendency to open new

possibilities for behavior. For instance, rhythmic contraction of the body at different regions can

allow an organism to move around. Once locomotion is possible, the organism can find food and

escape predators. In other words, the emergent behavior of an autocatalytic system, such as an

organism, allows the system to be ‘about’ the context from which the behavior emerges (Jordan

& Ghin, 2006). Entrainment opens new possibilities for affordance beyond a mere covariation in

movement.

In humans, the tendency for entrainment enables us to imitate and synchronize our

behavior with others. Sharing bodily states with others may enable us to feel what others feel and

develop empathy (Gallese, 2003; Meltzoff, 2007). In addition, by being in the same behavioral

9

dynamics with others, one can experience what the behavior is about and gain insight to others’

intentions (Goldman, 2006). More than other species, humans are concerned with others’

intentions (Adolphs, 2009; Buckner & Carroll, 2007; Herrmann, Call, Hernàndez-Lloreda, Hare,

& Tomasello, 2007; D. Premack & Woodruff, 1978; cf. Premack, 2010). Horner and Whiten

(2004) found that 3- to 4-year-old children imitated both the relevant and irrelevant behavior of a

demonstrator solving a puzzle. On the other hand, young chimpanzees only imitated the relevant

behavior in order to achieve the goal. Horner and Whiten argued that even though children at this

age were capable of identifying relevant causal information, the prevalence of their imitation

suggested that their predominant focus was on the actions and intentions of the demonstrator.

Many researchers also argued that our ability to entrain to or match others’ behavior facilitates

affiliation, cooperation, and social exchanges (Lumsden, Miles, Richardson, Smith, & Macrae,

2012; van Baaren, Janssen, Chartrand, & Dijksterhuis, 2009). Studies have shown that

behavioral synchrony was associated with cooperation (Valdesolo, Ouyang, & DeSteno, 2010;

Wiltermuth & Heath, 2009) and rapport (Miles, Nind, & Macrae, 2009). In sum, certain aspects

of social cognition are rooted in our ability to coordinate with others.

On one hand, behavioral coordination, such as synchrony, is thought to be a foundation

of our complex social cognition. On the other hand, synchrony is an emergent behavior that can

be constrained by social cognitive processes. For example, individuals with prosocial orientation

were more likely to coordinate with others (Lumsden et al., 2012). When being assigned into

different minimal groups, dyads were more likely to move in synchrony (Miles, Lumsden,

Richardson, & Macrae, 2011). A strong social distress like ostracism prompted individuals to

seek affiliation by mimicking others (Lakin, Chartrand, & Arkin, 2008). Also, when facing a

disliked other (Stel et al., 2010) or one with inappropriate behavior (e.g., being late, Miles,

10

Griffiths, Richardson, & Macrae, 2010), participants were less inclined to mimic or coordinate

with the unlikable other than with a likable one. Taken together, the evidence suggests that

interpersonal coordination is not only constrained by physical properties of the task but also by

the social environment and social cognitive contents.

How does social cognition influence behavioral dynamics, such as synchrony, that seem

to be fully defined by dynamical rules? The symmetry approach to this question suggests that

social influences create asymmetry in the system of nonlinear coupled oscillators by either

modifying their coupling functions or intrinsic dynamics. The group theory of symmetry

provides a general framework for analyzing such coordination.

Symmetry of Nonlinear Coupled Oscillators

Formally defined, the symmetry of an object refers to properties of the object that are

invariant with respect to a given transformation. For example, a geometric shape like a square

can be rotated in any multiple of 90°, and the square will still look the same. Under this

particular transformation, the object is invariant; all rotated squares in these particular angles are

equivalent. However, a square is not invariant with respect to, say, a 45° rotation or to color

transformation. In the case of oscillators, the state of the system can be described with two types

of symmetry: spatial and temporal. If the state of oscillator i at time t is represented by a vector

variable oi t , the state of a system of n oscillators is

o t =(o1 t ,o2 t ,o3 t ,...,on t ). (1)

If the oscillators are synchronized inphase, their states are identical. In that case, the system is

spatially symmetrical as you can swap any oscillators with each other in any permutation, and

the system will still look the same, that is,

o1 t = o2 t =o3 t =...on t ), (2)

11

and, for instance,

(o1 t ,o2 t ,o3 t ,...,on t ) = (o3 t ,o2 t ,o1 t ,...,on t ). (3)

In other words, inphase synchrony is invariant to spatial transformation or the change of an

oscillator’s position in the system.

Temporal symmetries deal with transformation regarding time, such as a phase shift. For

example, if two oscillators are moving antiphase at a period T, the second oscillator will be half a

period (T/2) away from the first oscillator. Thus,

o1 t = o2 t+T/2 . (4)

In order to achieve the spatial symmetry again, one oscillator must be shifted by half a period.

The temporal symmetry only occurs when the system exhibits some kinds of phase-locked

behavior.

In dynamical systems theory, the Hopf bifurcation theorem describes how limit cycles or

periodic states arise from a stable steady state as the system’s control parameters vary. However,

the standard Hopf bifurcation cannot be applied to dynamical systems with symmetry because

simple imaginary eigenvalues may not occur (Golubitsky, Stewart, & Schaeffer, 1988).

Golubitsky and Stewart (1985) developed a general theory of spatiotemporal symmetry in a

symmetric network of nonlinear oscillators undergoing Hopf bifurcation. The symmetric Hopf

bifurcation theorem suggests that oscillation patterns of a ring of n coupled nonlinear oscillators

can be predicted in terms of its symmetry, without investigating the detailed dynamical

equations. Hence, the general theory can be applied to different dynamical systems with different

makeup.

For a ring of n coupled oscillators, the three common symmetry groups are as follows

(Golubitsky et al., 1988). First, the symmetric group Sn involves all permutation of n objects. All

12

oscillators are identically coupled to each of the other oscillators. In a larger network, this

formation creates a star-like configuration. Second, the dihedral group Dn describes the

symmetry of n-sided polygons. The oscillators are coupled to the two nearest neighbors. Hence,

any oscillators are bidirectionally coupled to their adjacent neighbors. For a system of three

coupled oscillators, S3 = D3. Third, the cyclic group Zn is the symmetry group for a directed n-

sided polygon. The coupling between each oscillator is unidirectional.

In the case of two coupled oscillators, which are typically investigated in interpersonal

synchrony research, there are two behavioral solutions to the system: inphase and antiphase

patterns (under some circumstances, a four-phase pattern is also possible, Katsuta & Kawakami,

2006). As the number of oscillators in the system increases, more and more symmetries become

possible and the system may exhibit even more complex behavior. For the purpose of the current

studies, only the case of a three-oscillator ring will be considered. The system has two types of

symmetry: the dihedral D3 and the cyclic Z3. The cyclic symmetry has three isotropic subgroups.

As a result, four oscillation patterns are predicted by the symmetric Hopf bifurcation.

1. The all-inphase pattern preserves D3 symmetry. All three oscillators, o, have

identical waveforms and are inphase, that is,

o1 t = o2 t =o3 t . (5)

2. The rotation pattern is described by an isotropic subgroup Z3. All oscillators have

identical waveforms and are phase-shifted by 1/3 of a cycle (note that a cycle is 2π).

If o1 t =A, then (6)

o2 t =A+ 2π3

, and (7)

o3 t =A+ 4π3

. (8)

13

3. The partial-inphase pattern occurs in an isotropic subgroup Z2 K . Two oscillators

are identical and move inphase, and the third oscillator has a different waveform and

moves out of phase with the others. That is,

o1 t = o2 t =A, and (9)

o3 t =B, where B is out of phase with A. (10)

One special case of this pattern is when two oscillators move inphase and the other

moves antiphase.

4. The half-period pattern shows a 2:1 phase-locking symmetry of an isotropic

subgroup Z2 K, π . In this pattern, two oscillators have identical waveforms but are

phase-shifted by half a period (i.e., antiphase), while the third one oscillates at twice

the frequency of the first two, which can be described as

o1 t = A, (11)

o2 t =A+ π, and (12)

o3 t =B, where B=B+π. (13)

In addition to the behavioral patterns, the symmetry groups imply different coupling

configurations among the three oscillators. The D3 symmetry implies that all oscillators have

bidirectional coupling between each of them. The Z3 symmetry implies the cyclical

unidirectional coupling among the oscillators. The Z2 K symmetry suggests that two oscillators

are coupled and the third one is uncoupled from those two. The 2:1 phase-locking of Z2 K, π

suggests that the higher-frequency oscillator is driven by the antiphase coordination of the other

two. By manipulating the coupling among the oscillators, one should be able to induce symmetry

breaking in the system and observe an emergence of the predicted patterns.

14

Experiments with real biological systems support the emergence of the patterns predicted

by the symmetry groups. A ring of three plasmodial slime mold colonies exhibited protoplasmic

streaming oscillation in three modes: rotation, partial-inphase, and half-period oscillation

(Takamatsu et al., 2001). In humans, Yokoyama and Yamamoto (2011) studied a system of three

soccer players in a 3 versus 1 ball-possession task and found that two prominent patterns

emerged during this interaction: The rotation and the half-period patterns (or as they called it the

partial-antiphase pattern). Moreover, in a three-person drumming task, participants could

intentionally generate all of the four patterns described in the symmetric Hopf bifurcation

(Ariyabuddhiphongs & Richardson, 2015, April). Because symmetry breaking of coupled

nonlinear oscillators is a result of changes in their coupling function (Strogatz & Stewart, 1993),

it is possible that participants achieved coordination by perceptually or attentionally modulating

coupling strength. By doing so, participants in the drumming task could exploit dynamical

stability resulting from symmetry constraints to achieve stable coordination.

Social Influences as Symmetry Breaker

How does the symmetry of coupled oscillators relate to social phenomena? As Curie’s

(1894) principle suggests, symmetry of symmetry-breaking effects must be found in the

symmetry of causes that give rise to the effect. In the three-person drumming task described

above (Ariyabuddhiphongs & Richardson, 2015, April), the symmetry breaking occurred as a

result of explicit instructions to the participants. The task constraint was an apparent cause of

symmetry breaking in that case. However, it is less obvious how other social influences would

cause symmetry breaking. One plausible explanation is that social factors break the symmetry of

the system by changing how its components are coupled. Instead of directly modeling social

factors as extra parameters of task dynamics, one may adopt a model in which social factors

15

operate at a higher level by modulating parameter dynamics (e.g., influencing how coupling

functions change), which in turn control the lower-level task (state) dynamics. In terms of

hierarchy, social influences can be the slower timescale dynamics that constrain the cognition-

perception-action system, which, in turn, constrains the task behavior. Because the relation

between each hierarchy of the system is nonlinear (Van Orden & Holden, 2002), one can expect

that social influences do not always translate one-to-one to behavior. For example, when an

individual is in the same room with another unlikable person, reduced affiliative motives might

lead the first individual to pay less attention to the second person, resulting in weak coupling

between them, and, hence, less spontaneous coordination. However, if the task constraints are

strong (e.g., the participants were explicitly asked to coordinate), social influences might not be

able to resist the attraction of synchronicity. In such a case, the attractor landscape of the

behavioral dynamics would be dominated by the task parameters rather than social influences. In

other words, social factors may not always directly cause behavior. Rather, they should be

understood as cascades of influences among nested system hierarchies.

Contemporary approaches in social psychology posit that perception of social stimuli

activates associated behavioral tendencies, mostly in the form of imitation or behavioral

resonance (Bargh & Chartrand, 1999; Dijksterhuis & Bargh, 2001). The automaticity of

perception-behavior links has been supported by evidence from behavioral priming research. For

example, after taking an ostensible language test with words related to stereotypes of the elderly,

participants walked slower down the hallway after the experiment (Bargh, Chen, & Burrows,

1996). In social interaction, Chartrand and Bargh (1999) found that individuals unconsciously

mimicked the behavior of a confederate who was in the same room. Also, participants who

received high-power primes tended to write a letter E on their own forehead from their own

16

perspective (i.e., a mirrored E from an observer’s viewpoint), but those who received low-power

primes wrote the letter on their forehead from others’ perspective (i.e., a normal E from an

observer’s viewpoint). From the symmetry theory standpoint, the primes might work by creating

asymmetry in social cognitive contents, so that they constrained the behavior as it unfolded

within the task context. As the activation of power-related concepts increased to the symmetry-

breaking point, the writing behavior switched from one mode to another. However, social

cognitive contents, such as primed thoughts, did not directly dictate the behavior. Specifically,

the power prime by itself did not cause people to become poor perspective takers. Rather, such

effects emerged in the nested task context that demanded perspective tasking. On the contrary,

power, in a different context like an interview, could lead individuals to appear more confident

and enthusiastic (Cuddy, Wilmuth, Yap, & Carney, 2015). Therefore, it is important to stress the

significance of context dependency in the social cognition-behavior system. The manifestation of

our social cognitive contents is codetermined by social and task contexts. Social influences instill

asymmetry into the system, but how the asymmetry affects behavioral dynamics will depend on

the particular task context.

In spontaneous rhythmic coordination, Miles and colleagues (2010) demonstrated that

social constraints could create asymmetry in coordination between individuals. Participants

synchronized less with a target person when the person committed undesirable behavior by

arriving late for the experiment. The negative characteristics of others could repel us from

synchronizing with them, resulting in less coordination. Moreover, asymmetry in social identities

among individuals moderated their entrainment. When individuals in a dyad were assigned to

different minimal groups, they might be motivated to resolve in the disparate identities by

interacting more with each other, which led to a higher level of spontaneous synchrony (Miles et

17

al., 2011). Also, individuals who were socially excluded became more motivated to affiliate with

others. Lakin, Chartrand, and Arkin (2008) found that participants who previously experienced

social exclusion were more likely to mimic their partner in the subsequent part of the experiment.

In sum, the degree to which individuals let themselves be influenced by and coupled with others

seems to be affected by social factors and social cognitive processes.

Interaction-Dominant Dynamical Systems

In classical conceptions of systems, a system could be nearly decomposed into a

hierarchy of nested timescales (Simon, 1973). For example, an organism can be analyzed at the

molecular level, cellular level, tissue level, organ level, organ system level, body level,

behavioral level, etc. Each level changes at its own timescale, and the shorter timescales are

nested within the longer ones. For example, cultures change on a longer timescale than social

attitudes, which change slower than daily social interactions. Within the classical approach, the

concept of timescale decomposition or vertical separation is central to the analysis of the system.

Behaviors at a faster timescale change against a backdrop of a relatively static longer timescale.

Hence, causal properties at different timescales can be isolated. In addition to vertical separation,

within each timescale, the system is assumed to exhibit loose horizontal coupling, which means

that the behavior is a result of linear interactions among independent components within the

same level. In other words, the system’s behavior can be parsed into the summation of the work

by its components at different timescales. Because causal properties of the system’s behavior can

be reduced to the system’s components, this kind of system is called a component-dominant

system. To study such systems, researchers employ a subtractive method, where one could

hypothesize the mediating components by subtracting the sum of behavior (e.g., reaction time)

under different conditions (Donders, 1868/1969; Gottsdanker & Shragg, 1985; Sternberg, 1969).

18

For example, differences in response time to act aggressively toward African American versus

Caucasian American targets could be attributed to a facilitation of social cognitive shortcuts such

as stereotypes. In addition, vertical separation suggests that fluctuations or noises within each

timescale are isolated from other timescales, and loose horizontal coupling implies that the

components’ noises are added to the behavior. Therefore, the noises are uncorrelated; they are

random factors that should be treated as measurement error.

The perspective of interaction-dominant dynamical systems (IDDS), on the other hand,

posits that nested hierarchies are both horizontally and vertically coupled (Eiler et al., 2013; Van

Orden & Holden, 2002). Components acting at the same timescale interact nonlinearly, and the

relationship between different timescale activities are coupled via feedback loops. The

component and timescale interactions are the causal driver behind the system’s behavior, hence

the term ‘interaction-dominant dynamics.’ The nonlinear interaction and feedback processes

enable complex phenomena, such as self-organization, to occur in IDDS. The collective behavior

of a self-organized system is a longer timescale behavior that emerges from faster timescale local

interactions. In turn, the collective behavior enslaves the local components’ behavior via

feedback loops. The system is, thus, vertically coupled, and causality cannot be solely attributed

to only one timescale. In other words, an IDDS exhibits circular causality. In addition, because

the interactions are multiplicative rather than additive, the noises of each component and each

timescale propagate throughout the system, resulting in a long-range correlation of behavior at

different timescales. Evidence supports the nature of long-range correlations or fractal noises in

human behavior such as postural sway (Blaszczyk & Klonowski, 2001), gait (Hausdorff, 2007;

Hausdorff et al., 1997), rhythmic behavior (Coey, Hassebrock, Kloos, & Richardson, 2015),

cognition (Holden & Rajaraman, 2012; Van Orden, Holden, & Turvey, 2003; Van Orden,

19

Holden, & Turvey, 2005) and social cognition (Correll, 2008; Wong, Vallacher, & Nowak,

2014), supporting the notion that human behavioral and cognitive dynamics are interaction-

dominant.

The assumption of vertical coupling provides a framework for studying how order in

social behavior emerges and sustains itself (Eiler et al., 2013). As mentioned earlier, the self-

organization process requires the emergence-constraints relationship, where global, collective

order emerges from the system’s components and, in turn, constrains how the local components

behave. In other words, the global behavioral patterns now become a context that enslaves the

local components. The nature of reciprocal causality allows influences from multiple timescales

to cascade through the nested hierarchies. For example, behavioral outcomes in our daily life

may give rise to our attitudes and world view. Furthermore, the pattern of interaction between

individuals who share the same belief system defines social norms and the culture of the society.

Simultaneously, societal norms dictate what is right and wrong and shape how we form attitudes

about people and things around us. Those attitudes, in turn, constrain our actions. Such positive

feedback loops strengthen the emergent patterns and sustain the system dynamics. For example,

previous research has shown that by seeding identity asymmetry via the minimal group

manipulation, people became more attached to their group and escalated hostility toward the

outgroups over time, which became a positive feedback loop that sustains the intergroup

dynamics (Sherif, Harvey, White, Hood, & Sherif, 1961). Nonetheless, the cascade of faster

timescales or the enslavement of slower timescales is not always linear. A system may be able to

maintain its stable dynamics in the face of influences from faster or slower timescale events, and,

then, the system may suddenly shift to a new pattern when the influences move beyond the

critical transition point (Tesser, 1980). Therefore, the IDDS framework inherently

20

accommodates nonlinearity typically found in social psychology, such as attitude-behavior

inconsistency.

The IDDS perspective provides a framework to apply the symmetry theory to rhythmic

coordination. From the IDDS perspective, spontaneous coordination emerges from local

nonlinear interaction of coupled oscillators. If a certain asymmetry is observed in an effect, this

asymmetry should also be found in the process that gives rise to the outcome. On one hand,

asymmetries in global behavioral patterns can be found in asymmetries of local interaction. The

interaction between oscillators is determined by their coupling functions. Hence, global

behavioral patterns of the whole system are determined by asymmetries of the coupling

constraints. In the case of interpersonal coordination, symmetries of coupling functions can be

determined by the availability of perceptual information about the oscillatory behavior (Schmidt

et al., 1990). On the other hand, the IDDS perspective suggests that slower timescale processes,

such as social cognition, can also constrain the dynamics of faster timescale processes, such as

interpersonal coordination. Asymmetries in social factors, such as group memberships, may

constrain the local interaction between individuals and, thus, lead to asymmetries in their

coordination. In sum, IDDS is a framework to integrate influences from multiple timescales into

the symmetry theory.

The Current Studies

Following the symmetric group theory of coupled nonlinear oscillators, the current

research focused on coordination among a group of three individuals during a drumming task, in

which each individual could be conceived as a nonlinear oscillator. While of the task varied in

each study, the three-person drumming task can generally be described as follows. Each of the

three individuals listened and drummed to simple metronome beats that were slightly offset from

21

one another, which prevented the group from being trapped in inphase coordination right away.

After 10 seconds, the metronomes stopped and each individual maintained the drum beats for the

rest of the trial. Coupling manipulations were achieved by limiting access to auditory and/or

visual information between the individuals. Because the individuals were not explicitly told to

coordinate with other group members, the emergence of coordination patterns during the task

could be attributed to the self-organized dynamics of coupled nonlinear oscillators. The goal of

the current research was to demonstrate how asymmetry in perceptual (Study 1) and social

information (Study 2) constrained and gave rise to asymmetry in behavioral coordination

patterns.

Study 1. This study aimed to demonstrate the first goal: To show how asymmetry in

perceptual constraints led to symmetry breaking of spontaneous coordination patterns in a ring of

three coupled nonlinear oscillators. Although there are four solutions to a ring of three oscillators

(i.e., D3, Z3, Z2 K , and Z2 K, π ), the Z2 K, π symmetry is not suitable for the spontaneous

three-person drumming task because one oscillator must move at twice the period of the other

two. Such symmetry is unlikely to occur when the task requires each individual to maintain a

similar drumming frequency. Since the focus of Study 1 was in asymmetry of the coupling

function, rather than asymmetry of period dynamics, the frequency was set to be similar across

individuals. Hence, the design of Study 1 ruled out the emergence of the pattern from Z2 K, π

symmetry. In addition, to achieve better control of the coupling manipulation, the participants

could not see each other and were only coupled via auditory information. Although visual

information about another’s movement might serve as a cue for coordination, the drum sound

was arguably the salient information that drives behavioral coordination in this task. In support

of this claim, previous work showed that participants were able to intentionally coordinate in the

22

drumming patterns described by the group symmetry whether visual information was available

or not (Ariyabuddhiphongs & Richardson, 2015, April). Following the Curie principle, the

symmetry of coupling functions should lead to a corresponding symmetry in the behavioral

organization. Hence,

Hypothesis 1.1: Symmetry of spontaneous coordination in the three-person rhythmic

drumming task can be found in the configuration of auditory coupling among individuals

performing the task.

Depending on the coupling condition, each of the three participants heard one, two, or

none of the other participants’ drumming. Emergent coordination patterns and their stability

were examined as a function of coupling configurations within a ring of three coupled oscillators

(see Figure 1). Four specific predictions and conditions follow Hypothesis 1.1.

1) The all-inphase pattern tends to emerge when all oscillators are identically and

bidirectionally coupled to each other (see Figure 1a). In this all-coupling condition,

all three participants could hear all other participant’s drumming.

2) The partial-inphase pattern tends to emerge when two oscillators are bidirectionally

coupled to each other and unidirectionally coupled to the third oscillator, while the

third oscillator is uncoupled from the previous two (see Figure 1b). In this partial-

coupling condition, the first and second participants could hear each other’s

drumming. They also could hear the third participant. Nonetheless, the third

participant could not hear the other two participants. Hence, the third person was

unaffected by others. At the same time, the other two participants were more likely

to adopt inphase synchrony.

23

3) The inphase pattern tends to emerge when two oscillators are bidirectionally

coupled, while the third oscillator is unidirectionally coupled to them (see Figure

1c). This clamped-coupling condition was an inverse of the previous one; the

coupling direction of the third oscillator was flipped. The first and second

participants could hear each other’s drumming and, at the same time, sent their

drumming to the third participant. However, the third participant’s drumming could

not be heard by any others. The inphase coordination from the first and second

participants was expected to drive the third participant to adopt the inphase pattern

as well.

4) The rotation pattern tends to emerge when each oscillator is unidirectionally coupled

(see Figure 1d). In this rotation-coupling condition, the first participant’s drum beats

could be heard by the second participant; the second participant’s beats could be

heard by the third participant; and the third participant’s beats could be heard by the

first. This setup created a ring of unidirectional flow of auditory information among

the three participants.

24

Figure 1. Schematic representation of the coupling configurations in Study 1. On denotes an oscillator n. The arrow direction represents the direction of auditory informational flows among the oscillators. Configuration (a), the all-coupling condition, allows all bidirectional coupling among the three oscillators. Configuration (b), the partial-coupling condition, allows bidirectional coupling between the first and second oscillators. Those oscillators are also affected by the third oscillator, but not vice versa. In Configuration (c), the clamped-coupling condition, the first and second oscillators are bidirectionally coupled, and the third one is unidirectionally affected by them (i.e., receiving signals from them). Configuration (d), the rotation-coupling condition, represents a ring with three unidirectional coupling. Configuration (e), the no-coupling condition, shows an uncoupled system.

In addition to the coupling constraints, the task frequency was limited to moderate and

slow frequencies as a high frequency might moderate stability of the emergent coordination

pattern. That is, a higher frequency tends to increase the attractor strength of inphase

coordination and destabilized other modes of coordination (Schmidt et al., 1998). Hence,

Hypothesis 1.2: The stability of the non-inphase coordination increases as the drumming

frequency decreases. Specifically, the rotation pattern was expected to be more stable in

the slow frequency drumming than in the moderate one.

Following Hypothesis 1.2, antiphase coordination was predicted to occur more often in

the lower frequency than in the moderate frequency condition in all of the four coupling

configurations. However, a slower drumming frequency should particularly facilitate

O1

O2 O3

a.

O1

O2 O3

b.

O1

O2 O3

c.

O1

O2 O3

d.

O1

O2 O3

e.

25

coordination at the 1/3 phase-shift (Z3 symmetry) in the unidirectional coupling condition

(Figure 1d).

Due to the unintentional nature and the fluctuation of initial conditions in the drumming

task, spontaneous coordination under this task might not always be perfect. Rather, intermittent

occurrences of the predicted patterns under each coupling configuration were expected. In sum,

Study 1 aimed to demonstrate that asymmetry in spontaneous behavioral coordination (i.e.,

drumming patterns) emerged as a result of asymmetry in the coupling constraints (i.e., auditory

information) among the nonlinear oscillators.

Study 2. To achieve the second goal, Study 2 attempted to demonstrate that asymmetry

in the social environment could constrain rhythmic behavioral coordination in a group of three

individuals. The dynamics of coupled nonlinear oscillators are contingent upon (a) coupling

functions among the oscillators and (b) the intrinsic dynamics of each oscillator. While Study 1

investigated the effect of asymmetry in coupling functions by a direct manipulation of the

informational flow, Study 2 was based on the notion that an individual’s social cognition could

modulate how individuals assemble their cognition-perception-action system under task

constraints. In other words, social cognition influences the behavioral dynamics by varying a

system’s parameters, such as the coupling strength. For example, a drummer who takes pride in

his or her ability to keep beats may put forth tremendous effort to uncouple him- or herself from

other group members (e.g., by closing his or her eyes). However, social cognitive processes are

not a sole determinant of behavior. Although the drummer tries hard to become uncoupled,

synchrony may still occur because other individuals in the group may still be susceptible to

entrain with the drummer. Moreover, the physical constraints on the task, such as the frequency

of oscillation, may help stabilize or destabilize the mode of coordination (Haken et al., 1985;

26

Schmidt et al., 1998). The notion that social cognition functions as a parameter constraint on the

system dynamics illustrates two major reasons why social cognition does not directly cause

behavior. First, at an individual level, social cognition influences only a subset of each

component’s intrinsic (e.g., frequency) or relational (e.g., coupling) parameters, leaving a

possibility for other influences, like a rigid physical constraint, to shape behavioral dynamics.

Thus, the relationship between social cognitive events and behavior is rarely one-to-one. Second,

at a system level, the collective behavior of a self-organized system (e.g., group coordination) is

an irreducible outcome of nonlinear interactions among the system’s components (e.g., each

person’s social cognition and neuromuscular system); no single component or hierarchy is solely

responsible for the collective behavior. Hence, the IDDS framework suggests that the dynamics

of social behavior, where social cognition is one of many constraints on the system, unfolds as a

result of nonlinear multiple-timescale interactions. A better understanding of the effect of social

cognition on behavior would require a good model on the level of state-, parameter-, and graph-

dynamics (Riley, Kuznetsov, & Bonnette, 2011; Saltzman & Munhall, 1992).

In the three-person drumming task, social contexts should influence behavioral

coordination by changing relational parameters such as the degree of coupling between the

individuals. In Study 2, participants’ coupling tendency was expected to be affected by an

asymmetry of social identity created by the minimal group paradigm. The paradigm was

invented to study social cognitive biases without being confounded by history or stereotypes of

the group (Tajfel, 1970). Typical minimal group studies assign participants into different made-

up, meaningless groups (e.g., different color). Miles et al. (2011) have shown that minimal group

memberships influence behavioral synchrony between individuals; participants coordinated more

with a target from a different group. Dissimilarity in group memberships might induce the need

27

to re-establish social connection (Lakin et al., 2008; Maner, DeWall, Baumeister, & Schaller,

2007; Williams, Cheung, & Choi, 2000), which led to more synchrony. In Study 2, each

individual in a group of three was assigned into either the blue or the red group. Because

minimal groups were inherently hollow, there was no difference in which group held majority or

minority members. Therefore, the four permutations of the group assignment, which are 1) all

three reds, 2) two reds and one blue, 3) one reds and two blue, and 4) all three blues, are

symmetrical and can be collapsed into two conditions: homogenous and heterogeneous groups.

To simplify the experimental design, Permutation 3 and 4 were excluded from the study. After

the minimal group assignment, participants performed the spontaneous three-person drumming

task without any constraints on visual or auditory information. Because perceptual information

was not physically constrained in this study, asymmetry in behavioral coordination should be

attributed to asymmetry in participant’s social identity. Hence,

Hypothesis 2.1: Asymmetry in minimal group membership among individuals induces

asymmetry in the degree of coupling among them, which eventually leads to asymmetry in

spontaneous behavioral coordination during the spontaneous three-person drumming

task.

Per Hypothesis 2.1, specific predictions follow. First, the homogeneous group (i.e., all

three individuals in the same minimal group) embodies the D3 symmetry. Individuals exhibit

similar rhythmic behavior and are more likely to show the all-inphase coordination pattern.

Second, the heterogeneous group (i.e., two individuals in one minimal group and one individual

in the other group) resembles Z2 K symmetry, where two oscillators move inphase and the other

moves out of phase. Considering the dissimilarity in his or her social identity, the minority

participant is more likely to have different intrinsic dynamics (e.g., frequency) from the majority

28

participants. Nonetheless, the minority participant is also driven by the need to establish social

connection with the others. Therefore, the individual is more likely to show inphase coordination

with the majority participants.

In addition to spontaneous behavior, disparity of group membership within a triad might

also affect explicit coordination as well as role differentiation. In the explicit drumming task, the

participants had to differentiate their roles and control their rhythmic behavior to achieve the

partial-inphase drumming pattern. Because the heterogeneous group had similar configuration to

the Z2 K symmetry, the participants might be primed with their group identity to choose a

congruent drumming role in the partial-inphase pattern (i.e., the majority participants drummed

inphase, while the minority one drummed antiphase). Furthermore, the congruency between the

role and identity might also influence their coordination dynamics. Hence,

Hypothesis 2.2: In the explicit three-person drumming task, an individual is more likely

to adopt a drumming role that is congruent with his or her minimal group identity.

Hypothesis 2.3: When all individuals in a triad adopt drumming roles that are congruent

with their minimal group identity, their coordination is more stable than when they adopt

incongruent roles.

In sum, the effect of minimal group membership on both spontaneous and explicit

coordination were investigated in Study 2. Asymmetry in social cognitive processes, such as

intergroup biases or the need for social connections, was expected to constrain how a multi-agent

system organized its collective behavior.

29

Chapter 2

Study 1

Overview

The goal of Study 1 was to demonstrate how different spontaneous interpersonal

coordination in the three-person drumming task emerged as a function of asymmetry in auditory

coupling constraints. In addition, to determine whether oscillation frequency modulates

coordination stability, the triads of participants either drummed at a frequency of 45 bpm or 60

bpm

Participants

Seventy-five undergraduate and graduate students (Mage = 21.08 yrs, SDage = 3.70; 49

women and 26 men; 82.67% Caucasian, 9.33% Asian, 4% African American, 2.67% Other)

participated in the study in groups of three, resulting in 25 triads. A participant in one of the

triads did not drum at the instructed pace, and that triad was removed from the analysis1. The

data analyses were performed on the remaining 24 triads. Half of triads were assigned to drum in

the slower, 45-bpm condition, while the other 12 triads were assigned into the moderate, 60-bpm

drumming condition.

Instruments

Electronic drum set. Each participant in a three-person group stood in front of a

Yahama DTXPRESS drum pad (Yamaha Corporation, Buena Park, CA) with a drumstick in his

or her dominant hand. The drum pads were positioned to face up at each participant’s

approximate waist height. The MIDI signals from the drum set were fed to a Dell Optiplex 760

desktop computer (Dell Inc., Round Rock, TX) for recording and audio generation.

1 The participant drummed at twice the frequency of the rest of the group, which resembled the Z2 K, π symmetry. However, this behavior deviated from the task instruction.

30

Headphones and audio interface. Three pairs of headphones were connected to an

Focusrite Scarlett 18i8 audio interface (Focusrite Audio Engineering, Buckinghamshire, UK) for

an audio input/output routing. Audio generated from the electronic drum set was selectively

routed to each participant depending on their coupling conditions.

Metronome and background noise. For the 45-bpm condition, three 10-second

metronome audio clips with different tempos (i.e., 41.4, 45, and 48.6 bpm) were used to set up

the rhythm for the drumming task. The lower and upper bpm were ±8% of the middle 45 bpm

(0.75 Hz) and were under the range of ±10-15% period differences that encompassed the basin of

entrainment (Lopresti-Goodman, Richardson, Silva, & Schmidt, 2008). Unintentional

coordination is less likely to occur if period differences are greater than that range. The three

tempos were offset to prevent the participants from starting a trial in inphase coordination. In the

60-bpm condition, the three metronomes were set to 55.2, 60, and 64.8 bpm, which were in the

range of ±8% of 60 bpm (1 Hz).

In addition, white noise was played in the background during the experimental session to

minimize any ambient noises as well as a sound from physical contact between the drumsticks

and the pads. The generated drum sound was set to be louder than the white noise. None of the

participants had trouble hearing the generated audio.

Motion capture system. A Polhemus Fastrak magnetic motion tracking sensor

(Polhemus Corporation, Colchester, VT) was attached to each drumstick. The relative movement

position in X, Y, and Z planes of the drumsticks were recorded at 40 Hz.

Procedure

Each triad was randomly assigned to either the 45-bpm or 60-bpm condition. In each

session, each participant stood on each side of a triangle drum rack. The curtains were raised to

31

block any visual access among the participants. Hence, the only source of their informational

coupling was the auditory information generated by the electronic drums. An experimenter

explained that he was studying how people kept rhythm. The participant’s task was to drum to a

metronome played via his or her headphones for 10 s. Once the metronome stopped, the

participant should do his or her best to maintain the drumming rhythm for another 50 s. Hence,

the total length of each trial was 60 s. The participants were informed that their metronomes

might or might not be different from other participants and that they should not be worried if

they were not in sync with others. It is important to note that the participants were not given any

explicit instruction to coordinate with other participants. If a participant asked for clarification,

the experimenter would only repeat the instruction that the participants should try their best to

maintain the rhythm that they heard.

Five within-subject conditions were designed to investigate the effect of asymmetric

coupling on spontaneous coordination. First, the all-coupling condition (D3 symmetry; Figure

1a) allowed all participants to hear the drumming from all other participants. Second, the partial-

coupling condition (Z2 K symmetry; Figure 1b) allowed two participants to hear each other’s

drumming and the third participant’s drumming, but the third participant, the independent

oscillator, did not hear any drumming from the first two participants. Third, the clamped-

coupling condition (modified Z2 K symmetry; Figure 1c) allowed the first and second

participants to hear each other, and their drumming was sent to the third participant, the clamped

oscillator, but the first and second participants could not hear the third participant’s drumming.

Fourth, the rotation-coupling condition (Z3 symmetry; Figure 1d) allowed each participant to

hear only the drumming from the person to their right, forming a ring of unidirectional coupling.

Fifth, the no-coupling condition was a baseline control condition, where none of the participants

32

could hear any other’s drumming. Any intermittent coordination in this condition should occur

by chance.

Each of the five conditions was a block of three trials, yielding in a total of 15 trials. The

order of the conditions was randomized. The coupling configurations and metronome

frequencies were rotated among participants from trial to trial. Therefore, each participant

experienced each role in each coupling configuration at least once. For example, in the partial-

coupling condition, which had two bidirectionally coupled oscillators and the independent

oscillator, the participants in a group took turns being assigned into the independent role during

the three trials of this condition. Participants were unaware of the conditions and roles to which

they were assigned.

Data Preparation, Reduction, and Analysis

Drumsticks position and MIDI audio time series for each trial were truncated from the

15th to the 55th second. The first 15 s were removed to eliminate any unstable movements that

might occur at the beginning of the trial when the metronomes were played. The remaining data

was then trimmed to an equal length at the 55th second, resulting in the remaining 40 s of time

series data.

Drumstick position and velocity. Instead of oscillating their drumstick in a stationary

fashion with respect to the target period, some participants kept their rhythm by tapping in

between the beats in mid-air. This behavior seems to be a solution to keeping the beats outside an

average preferred frequency of 120 bpm (Moelants, 2002). Although those participants still

produced the drumming sound with respect to the metronome’s period, the oscillation of their

movement was twice the target frequency. This behavior posed challenges in the identification of

drumming events and the calculation of relative phase. In relative phase analysis, the signals are

33

assumed to have the same or similar frequency components. Therefore, an analysis of two

movement position time series with different frequency components will result in spurious

relative phase angles, which do not reflect the coordination of the actual drumming events. For

the sake of accuracy and consistency in the relative phase calculation, the movement position

and its derivative (i.e., velocity) were forsaken. Instead, the MIDI audio data, which captured the

exact drumming events, was used.

MIDI data. The electronic drum pads generated digital MIDI signals that contained

information regarding how the instruments were played. Timestamps of the drumming events

were extracted from the MIDI data and used for the discrete relative phase analysis. To prepare

the MIDI time series, first, any drumming events that occurs within 0.33 s (~3 Hz) of the prior

event were considered spurious and removed from the analysis. Second, within each person’s

MIDI time series, abnormally large gaps between drumming signals were removed. The exact

cutoff varied in some trials but, on average, any gaps that were larger than 1.5 times the

metronome frequency were removed. Third, any time series with too few or too many beats were

removed. A time series with too few beats indicated that the participant could not continuously

maintain their drumming, while a time series with too many beats indicated that the participant

was drumming faster than the target frequency. Tukey’s hinges of the total number of beats for

each frequency condition were used as a cutoff. The lower and upper limits for the 60-bpm

condition were 31 and 47 periods, respectively. For the 45-bpm condition, the lower and upper

limits were 23.5 and 35.5 periods, respectively. Any trials that had one or more time series with

the total beats outside the range (i.e., outliers) were removed from the analysis. A total number of

removed trials was 37, which equals 111 removed time series from a total of 1080 (24 triads ´ 3

persons ´ 15 trials) or 4.1%. The removed trials accounted for 11.7% and 8.9% of the data in the

34

45-bpm and 60-bpm conditions, respectively. They also accounted for 18.1% in the all-coupling

condition, 6.9% in the partial-coupling condition, 11.1% in the clamped-coupling condition,

2.8% in the rotation-coupling condition, and 12.5% in the no-coupling condition.

Discrete relative phase analysis. The MIDI time series from the three participants were

subjected to a pair-wise discrete relative phase analysis (Varlet & Richardson, 2011; Wheat &

Glazier, 2006). The calculation of discrete relative phase is

ϕ= t1(j)-t2(j)

t1(j+1)-t1(j)×360°. (14)

In this equation, t1and t2 are time of the maximum or peak rotation (e.g., a drumming event) of

Oscillator 1 and 2, respectively. In this case, Oscillator 1 is a reference signal, and its period is

calculated from the peak of the cycle j to the next cycle, j+1. The latency of t2 to t1 is scaled to

the period of Oscillator 1 and, then, converted into a 360° scale to yield a relative phase angle, f.

Because f is a circular variable, the scale can be transformed from 0°–360° to -180°–180°. Then,

a positive value of f indicates how much t2(j) lags the reference signal, t1(j). On the other hand, a

negative f indicates how much t2(j) leads the next cycle of the reference signal, t1(j+1).

The discrete relative phase (f) for drumming events for each possible pair in a triad were

calculated in a -180° to 180° range. A frequency distribution of relative phase was constructed

with a bin size of 20°, and a percentage of occurrence (% occurrence) within each bin was

calculated by dividing the bin’s frequency by a total number of f. The distribution of ϕ around 0°

indicates inphase relationship, while the distribution around -180° or 180° indicates antiphase

relationship. In the rotation pattern, each oscillator is offset by one-third of a period, which

equals to -120° or 120° relative phase angle.

35

In addition, a degree of synchrony (r), which indicates the stability of the coordination

(Pikovsky, Rosenblum, & Kurths, 2001; Richardson, Garcia, Frank, Gregor, & Marsh, 2012) was

calculated for each pair in a triad. The r value is an inverse of circular variance of the relative

phase and ranges from zero (no coordination) to one (perfect synchrony). It is important to note

that a high r value can be achieved from a strong synchrony at any phase angle. For example,

strong antiphase coordination can also result with high r value.

Statistical analysis. The effect of between- (i.e., drumming frequencies) and within-

subject (i.e., coupling conditions) factors was modeled with the linear mixed model (LMM)

approach. Because random assignment was utilized throughout the experimental design, the

random effect of subjects was not included in the LMM, making it a marginal model. An

unstructured covariance matrix was used in the LMM. All means shown in the results sections

were estimated marginal means unless stated otherwise.

Results

Average drumming periods. An average drumming period for each participant in each

trial was calculated from the MIDI audio data. A larger period indicated slower frequency

drumming. The metronome target period was 1.33 s for 45 bpm and 1 s for 60 bpm. As expected,

the participants’ average drumming period in the 45-bpm condition (M = 1.289 s, SE = 0.007)

was longer than the period in the 60-bpm condition (M = 0.993 s, SE = 0.007), F(1, 63.89) =

914.48, p < .001. The participants were, on average, drumming close to the target frequency. In

addition, the main effect of coupling condition was significant, F(4, 63.40) = 14.01, p < .001).

The main focus of this analysis was a significant interaction between the frequency and coupling

conditions, F(4, 63.40) = 12.13, p < .001. In the 60-bpm condition, the participant’s average

periods were similar across all coupling conditions (Ms = 0.989–0.999 s, SEs = 0.007–0.011, ps

36

> .05 with Bonferroni adjustment). On the other hand, in the 45-bpm condition, the average

periods in the no-coupling (M = 1.326 s, SE = 0.009) and partial-coupling conditions (M =

1.329 s, SE = 0.010) were greater than those in the all-coupling condition (M = 1.285 s, SE =

0.010), which in turn were greater than those in the clamped-coupling condition (M = 1.248 s, SE

= 0.008) and the rotation coupling condition (M = 1.255 s, SE = .007). The drumming periods in

the 45-bpm no-coupling and partial-coupling condition were also closer the target frequency

(1.33 s). Table 1 and Figure 2 show the estimated marginal mean periods by frequency and

coupling conditions.

37

Table 1

Estimated marginal means of average period and period instability for each coupling condition

(in bold), asymmetric coupling function (in regular*), and drumming frequency. Standard errors

are in parentheses.

Coupling Conditions

No

All

Partial

Clamped

Rotation

Frequency Coupled

Independent

Total Coupled

Clamped

Total

Average Periods

45 bpm 1.326

(0.010)

1.285

(0.010)

1.311

(0.012)

1.361

(0.014)

1.329

(0.010)

1.245

(0.010)

1.268

(0.013)

1.248

(0.008)

1.255

(0.007)

60 bpm 0.999

(0.009)

0.991

(0.011)

0.975

(0.011)

1.027

(0.013)

0.991

(0.010)

0.981

(0.010)

1.005

(0.012)

0.989

(0.008)

0.993

(0.007)

Total 1.162

(0.007)

1.138

(0.007)

1.143

(0.008)

1.194

(0.009)

1.160

(0.007)

1.113

(0.007)

1.137

(0.009)

1.118

(0.006)

1.124

(0.005)

Period Instability (Coefficient of Variation)

45 bpm 0.039

(0.002)

0.053

(0.002)

0.047

(0.003)

0.052

(0.003)

0.048

(0.002)

0.050

(0.003)

0.053

(0.004)

0.048

(0.002)

0.050

(0.002)

60 bpm 0.042

(0.002)

0.048

(0.002)

0.048

(0.002)

0.048

(0.003)

0.046

(0.002)

0.047

(0.003)

0.051

(0.004)

0.046

(0.002)

0.050

(0.002)

Total 0.041

(0.001)

0.050

(0.002)

0.047

(0.002)

0.050

(0.002)

0.047

(0.001)

0.049

(0.002)

0.052

(0.003)

0.047

(0.002)

0.050

(0.002)

* The estimated marginal means for each asymmetric coupling function (i.e., coupled, independent, and clamped conditions) would not yield the same arithmetic mean to their corresponding coupling condition’s total estimated marginal means. This inconsistency exists because the total estimated marginal means of the coupling conditions were derived from an omnibus LMM, while the estimated marginal means for each asymmetric coupling function were calculated from the separate analysis within the coupling condition.

38

Figure 2. Mean period by frequency and coupling conditions. Error bars represent standard error. Dotted lines represent the target period for each drumming frequency.

Among the coupling conditions, the partial-coupling and clamped-coupling conditions

created asymmetry of coupling function among participants in a triad, while the participants in

the no-coupling, all-coupling, and rotation-coupling conditions were considered symmetrical or

equivalent. Specifically, in each trial of the partial-coupling condition, one participant in a triad

did not receive any drumming sound from the other two participants but did send his or her

drumming sound toward the other two. Hence, the individual was considered an independent

oscillator in the partial-coupling condition. In the clamped-coupling condition, one of the three

oscillators received the drumming sound from the other two, but did not send its signal back to

the rest of the group. Therefore, the oscillator was being influenced by the two coupled

oscillators, but not vice versa. In other words, the participant in this coupling function was

‘clamped’ by drumming signals from the rest of the group. To investigate the effect of such

39

asymmetry in the coupling function, separate analyses were conducted for the partial- and

clamped-coupling conditions. The means of average periods for each coupling function are

shown in Table 1 (in regular font).

For the partial-coupling condition, the means of average period in the 45-bpm and 60-

bpm condition were 1.336 s (SE = 0.011) and 1.001 s (SE = 0.011), respectively. This main

effect of frequency was significant, F(1, 73.49) = 475.52, p < .001. Also, the main effect of

asymmetric coupling function was significant, F(1, 127.81) = 36.00, p < .001. Regardless of

drumming frequency, the independent oscillators’ average period (M = 1.194 s, SE = 0.009) was

longer than the other two coupled oscillators (M = 1.143 s, SE = 0.008). The interaction effect

was not significant, F(1, 127.81) = 0.03, p = .866. These results suggest that, in general, the

participants who did not hear the rest of the group drummed slower than the others who were

coupled to each other (see Figure 3a).

In the clamped-coupling condition, there was a significant difference of average period

by frequency (M45bpm = 1.257 s, SE45bpm = 0.009 vs. M60bpm = 0.993 s, SE60bpm = 0.009), F(1,

72.50) = 420.81, p < .001. The main effect of asymmetric coupling function was also significant,

F(1, 116.23) = 6.47, p = .012. The average period of the clamped oscillators (M = 1.137 s, SE =

0.009) was longer than that of the coupled oscillators (M = 1.113 s, SE = 0.007). Again, among

the participants in a triad, those who were bidirectionally coupled to each other demonstrated

shorter average periods or faster drumming frequency (see Figure 3b). The interaction between

the frequency and asymmetric coupling function was not significant, F(1, 116.23) = 0.001, p =

.981.

40

Figure 3. Mean period for each coupling function in (a) the partial-coupling and (b) the clamped-coupling conditions. Error bars represent standard errors. Dotted lines represent the target period for each drumming frequency.

Period instability. Instability of drumming period was measured by a coefficient of

variation, which is defined as the standard deviation of drumming period during a trial divided

by the target period (i.e., 1 s for the 60-bpm condition and 1.33 s for the 45-bpm condition). By

scaling the standard deviation to the target period, the influence of period magnitude was

neutralized, allowing for fair comparisons of instability between different oscillatory periods.

Table 1 shows the period instability for each condition.

The coefficient of variation in the 45-bpm condition (M = 0.048, SE = 0.002) was not

significantly different from the 60-bpm condition (M = 0.046, SE = 0.002), F(1, 52.05) = 0.24, p

= .624. For the main effect of coupling condition, the instability of the no-coupling condition (M

= 0.041, SE = 0.001) was lower than all other four coupling conditions (Ms = 0.047-0.050, SEs =

0.001–0.002), F(4, 59.40) = 9.89, p < .001 (see Figure 4). The interaction between frequency and

coupling conditions was not significant, F(4, 59.40) = 1.41, p = .241.

a. b.

41

Figure 4. Mean period instability (coefficient of variation) by frequency and coupling conditions. Error bars represent standard errors.

Separate analyses were conducted for the partial-coupling and clamped coupling

conditions to examine the effect of asymmetric coupling function. For the partial-coupling

condition, none of the main or interaction effects were significant, Fs = 0.22–1.76, ps = .187–

.643. Also, for the clamped-coupling condition, none of the effects were significant, Fs = 0.11–

1.84, ps = .178–.743.

Discrete relative phase of drumming periods. The frequency distributions of ϕ from

each trial were averaged for each of the frequency, coupling, and asymmetric coupling function

conditions. Figure 5 shows the mean relative phase frequency distributions by condition. To

determine whether the % occurrence in each bin was due to chance, the % occurrence within

each condition was bootstrapped to estimate a 95% confidence interval. If the lower limit of the

42

confidence interval was higher than the probability of a uniform distribution (100/18 bins =

5.56%), the mean % occurrence in that bin was considered above chance level, indicating

significant spontaneous coordination at that bin’s phase angle.

For the all-coupling condition, ϕ distributed around -40° to 40° in both the 45-bpm and

60-bpm conditions, suggesting spontaneous inphase coordination. Moreover, albeit

nonsignificant, the participants showed a trend of multi-stability in the 45-bpm condition, where

antiphase coordination might also emerge occasionally.

For the partial-coupling condition, the mean % occurrence distribution was not above

chance level in the 45-bpm condition. However, upon closer inspection, these means had large

variance because they were averages of participants who mainly coordinated inphase and those

who mainly coordinated antiphase. By averaging % occurrence distribution of these two groups,

the peaks at both 0° and 180° were attenuated, giving a false impression of no coordination. In

the 45-bpm partial-coupling condition, there were roughly equal number of time series pairs that

were inphase and antiphase2 (28 and 37 pairs, respectively). The other coupling conditions,

except the no-coupling condition, had more inphase pairs than antiphase pairs (see Table 2 for

more detail). Although the 45-bpm partial-coupling condition had a larger proportion of pairs

with antiphase than did the other coupling conditions (except the no-coupling condition), within

the condition, the proportion of antiphase pair occurrence was the same for the independent-

coupled oscillator pair and the coupled-coupled oscillator pair, c2(3, N = 93) = 0.84, p = .840.

That is, the independent oscillator was not more likely to coordinate antiphase than the rest of the

2 Inphase and antiphase time series pairs were determined by comparing the %occurrence in the 0°, -180°, and 180° bins to the chance level. If the % occurrence in the -20° to 20° bin was more than 5.56%, the pair was counted as inphase. If the % occurrence in the -180° to -160° or 160° to 180° bins was above chance level, the pair was antiphase. If the pair was both inphase and antiphase, it was counted as a separate category, the multiphase. The pair that was neither inphase nor antiphase was categorized as the no-phase.

43

group; instead, the group as whole was more likely to engage in antiphase coordination.

However, these patterns were limited to the 45-bpm condition. The proportion in the 60-bpm

partial-coupling conditions showed that the inphase pairs were more likely to occur than the

antiphase pairs, and the % occurrence of ϕ distributed around -40° to 40° phase angle, indicating

inphase coordination. Hence, the prediction that the independent oscillator is more likely to

coordinate antiphase was not supported at either drumming frequency. Nonetheless, the results

gave a partial support to Hypothesis 1.2, which also predicted an increase in antiphase

coordination at the slower frequency.

The relative phase distributions in the clamped-coupling condition were similar to the

all-coupling condition; ϕ distributed around -40° to 40° for both drumming frequencies. Also, the

clamped and the coupled oscillators did not show any difference in their coordination with other

oscillators. Overall, inphase coordination was the main mode of spontaneous coordination in the

clamped-coupling condition.

For the rotation-coupling condition, the frequency distributions were above chance

around -40° to 40° in the 45-bpm condition and around -60° to 40° in the 60-bpm condition.

Contrary to the prediction, participants did not adopt a rotation drumming pattern in this

condition. Instead, the participants showed spontaneous inphase coordination.

For the no-coupling condition, the mean % occurrence distribution was flat for both the

45-bpm and 60-bpm conditions. Although higher proportions of time series pairs were in the

antiphase than the inphase mode, more pairs were also in the multiphase mode (see Table 2).

Also, the categorization of the phase mode did not take into account the magnitude of %

occurrence, allowing fairly low % occurrence, yet higher than 5.56%, to represent the phase

mode. Nonetheless, the higher proportion of antiphase and lower proportion of inphase might

44

contributed to an unexpected significant % occurrence in the -180° to -160° and 160° to 180° bin

in the 45-bpm condition. The higher chance of antiphase coordination might due to imperfect

noise isolation between the participants. The physical ‘thud’ sound occurring when a drumstick

hit a pad might have leaked through the white-noise background and influenced participants into

the antiphase mode. Nonetheless, % occurrence in those bins was not high, and the overall

picture still suggested a flat distribution of ϕ, suggesting no main mode of coordination.

Table 2

Number of time series pairs categorized as different phase modes by coupling condition (in bold)

and by asymmetric coupling function (in regular)

Coupling Conditions

No

All

Partial

Clamped

Rotation

Frequency Coupled-

Coupled

Independent-

Coupled

Total Coupled-

Coupled

Clamped-

Coupled

Total

45 bpm

Inphase 15 51 9 19 28 19 28 47 50

Antiphase 29 27 11 26 37 5 14 19 27

Multiphase 34 15 7 12 19 4 15 19 18

No-phase 12 3 4 5 9 2 3 5 13

60 bpm

Inphase 21 51 20 32 52 20 37 57 63

Antiphase 44 11 5 12 17 5 13 18 14

Multiphase 22 8 8 17 25 4 11 15 14

No-phase 12 11 3 11 14 5 7 12 11

45

Figure 5. Mean frequency distribution of relative phase by frequency and coupling conditions. Error bars represent bootstrapped 95% confidence intervals.

45 bpm 60 bpm

46

Coordination Stability. The stability of coordination or synchrony was measured with

r, an inverse of circular variance of the relative phase. The value ranges from 0 to 1, where 1

indicates perfect synchrony and 0 indicates no synchrony. The means and standard errors of r by

the conditions are shown in Table 3. The LMM revealed a significant main effect of coupling

condition, F(4, 61.45) = 69.00, p < .001. Regardless of the drumming frequency, r in the no-

coupling condition (M = .32, SE = .01) was lower than in all other conditions (Ms = .50–.64, SEs

= .02–.03), ps < .001. The coordination in the partial-coupling (M = .50, SE = .02) was less stable

than in the all-coupling (M = .64, SE = .03) and the clamped-coupling conditions (M = .60, SE =

.02), ps < .01. Also, the mean r in the rotation-coupling condition (M = .55, SE = .02) was lower

than in the all-coupling condition, p = .013 (see Figure 6; all pair-wise comparisons were

subjected to a Bonferroni correction). The main effect of frequency, F(1, 56.72) = 0.03, p = .863,

and the interaction effect of frequency and coupling condition were not significant, F(4, 61.45) =

1.46, p = .224.

Figure 6. Means coordination stability (r) by frequency and coupling conditions, sorted in ascending order. * denotes significant difference at a = .05 with a Bonferroni adjustment.

47

Table 3

Estimated marginal means of coordination stability (r) for each coupling condition (in bold),

asymmetric coupling function (in regular*), and drumming frequency. Standard errors are in

parentheses.

Coupling Conditions

No

All

Partial

Clamped

Rotation

Frequency Coupled-

Coupled

Independent-

Coupled

Total Coupled-

Coupled

Clamped-

Coupled

Total

45 bpm .33

(.02)

.65

(.03)

.63

(.06)

.51

(.04)

.53

(.03)

.58

(.05)

.54

(.04)

.57

(.03)

.54

(.03)

60 bpm .32

(.02)

.63

(.04)

.52

(.05)

.45

(.04)

.47

(.03)

.62

(.05)

.62

(.03)

.62

(.03)

.55

(.03)

Total .32

(.01)

.64

(.03)

.58

(.04)

.48

(.03)

.50

(.02)

.60

(.04)

.58

(.03)

.60

(.02)

.55

(.02)

* The estimated marginal means for each asymmetric coupling function (i.e., coupled, independent, and clamped conditions) would not yield the same arithmetic mean to their corresponding coupling condition’s total estimated marginal means. This inconsistency exists because the total estimated marginal means of the coupling conditions were derived from an omnibus LMM, while the estimated marginal means for each asymmetric coupling function were calculated from the separate analysis within the coupling condition.

Separate analyses were conducted for the partial-coupling and clamped-coupling

conditions to determine the effect of asymmetric coupling function within the two conditions.

For the partial-coupling condition, the main effect of frequency was marginally significant, F(1,

84.79) = 2.95, p = .090. Mean r in the 45-bpm condition (M = .57, SE = .04) tended to be higher

than mean r in the 60-bpm condition (M = .49, SE = .03). The high r in the 45-bpm condition

might due to the fact that more occurrences of antiphase coordination in the partial-coupling

48

condition contributed to a higher average coordination stability. The main effect of the

asymmetric coupling function was significant, F(1, 130.54) = 4.58, p = .034. The coordination

stability between the independent oscillator and the rest of group (M = .48, SE = .03) was lower

than the coordination stability among the other two oscillators themselves (M = .58, SE = .04; see

Figure 7). The interaction effect was not significant, F(1, 130.54) = 0.29, p = .593.

In the clamped-coupling condition, none of the main nor interaction effects were

significant, Fs = 0.17–1.96, p = .165–.683.

Figure 7. Mean coordination stability (r) for asymmetric coupling in the partial-coupling condition. Error bars represent standard errors.

Discussion

The purpose of Study 1 was to test whether asymmetry in informational coupling could

lead the corresponding asymmetry in coordination patterns during the three-person drumming

task (i.e., Hypothesis 1.1). However, the results suggested that inphase coordination was the

main mode of coordination among the four coupling conditions (i.e., all-, partial-, clamped, and

49

rotation-coupling), while the participants showed little to no coordination in the no-coupling

condition3 (see Figure 5). The results also suggested an influence of drumming frequency on the

emergence of spontaneous coordination in the partial-coupling condition. That is, the participants

in the 45-bpm partial-coupling condition coordinated inphase and antiphase at about the same

proportion. However, in the 60-bpm condition, inphase coordination dominated even in the

partial-coupling constraints. Consistent with the HKB model (Haken et al., 1985), the slower

oscillation seems to allow an emergence of multi-stable system, where both inphase and

antiphase coordination could be a solution of the system. The trend of % occurrence distribution

around the -180° and 180° bins in the all-coupling and partial-coupling conditions lend support

to that notion. When the oscillation frequency increased to 60 bpm, antiphase coordination was

apparently no longer a stable solution of the system and the spontaneous coordination became

inphase. Nonetheless, the multi-stability due to the drumming frequency was not associated with

asymmetric coupling (i.e., independent vs. coupled oscillators). In the partial-coupling condition,

both independent and coupled oscillators were equally like to coordinate antiphase with the other

oscillators. In sum, asymmetry of the coupling function in the drumming task context was not

enough to induce symmetry breaking of the coordination pattern per se. However, as discussed in

the next paragraph, the symmetry-breaking of the coupling function did influence the stability of

spontaneous coordination.

Overall, there was no effect of drumming frequency on coordination stability across all

five coupling conditions. As expected, the participants’ coordination was the least stable in the

3 The significant %occurrence distribution around antiphase bin in the 45-bpm condition might result from a leakage of physical drumming sound through the background white noise. This effect was small and only occurred in the slower frequency, where the system might be multi-stable. As the drumming frequency was increased to 60 bpm, the participants did not show any spontaneous coordination.

50

no-coupling conditions and strongest in the all-coupling condition (see Figure 6). Also, mean

coordination stability for the rotation-coupling condition was in between the no-coupling and the

all-coupling conditions, which suggested the variation of coordination strength as a function of

the coupling configurations.

The coupling condition did not only affect spontaneous coordination but also the

drumming frequency. In the slower drumming frequency of 45 bpm, the participants were

drumming slower and closer to the target period of 1.33 s in the no-coupling and partial-coupling

than in the other coupling conditions (see Figure 2). In the no-coupling condition, it seems that

the absence of a positive feedback loop from bidirectional coupling allowed the participants to

maintain their target periods without being interfered by other participants’ drumming

frequencies. In the case of the partial-coupling condition, asymmetric coupling modulated the

participant’s drumming period. Since the independent oscillator was unaffected by the group, it

could keep its cycles slower than the coupled oscillators. In addition, by maintaining a lower

frequency, the independent oscillator influenced the coupled oscillators via its unidirectional

coupling to them, resulting in a slower average period for the whole group than the three other

coupling conditions (i.e., all-, clamped-, and rotation-coupling). The notable difference between

the unidirectional and bidirectional coupling was also demonstrated in the clamped-coupling

condition. Although the clamped oscillator was unidirectionally affected by the increased

frequency of the two bidirectionally coupled oscillators, its frequency was still slower than the

rest of the group. These results suggested that a positive feedback loop created by bidirectional

coupling had a stronger effect in modulating the oscillation frequency than the unidirectional

coupling. This conclusion was also consistent with the frequency increase in the rotation-

coupling condition. Despite having only unidirectional coupling with adjacent oscillators, the

51

positive feedback loop operated via an intermediate oscillator in the rotation coupling, resulting

in a faster mean period. Nonetheless, most of the effects were attenuated in the 60-bpm

condition. It is possible that the participants were drumming at about half the frequency of what

people typically prefer (approximately 2 Hz or 120 bpm; MacDougall & Moore, 2005; Moelants,

2002), which might allow a better frequency maintenance. The similarity between the exogenous

and endogenous frequency might facilitate stable coordination (Zamm, Wellman, & Palmer,

2016), which helped the participants maintain their periods. Nonetheless, the effect of

asymmetric coupling function (i.e., independent or clamped vs. coupled oscillators) on the

periods still held in the 60-bpm partial-coupling and clamped-coupling conditions, suggesting a

persistent difference between unidirectional and bidirectional coupling. Finally, being coupled

with others also increased period instability (see Figure 4), which suggested that variability

increased as the participants influenced each other.

In sum, the hypothesis that asymmetry in coupling configuration results in a

corresponding asymmetry in coordination pattern was not supported. Nonetheless, the coupling

conditions were found to affect the average drumming period. The directionality of the coupling

function played an important role in modulating the drumming frequency. The positive feedback

loop found in bidirectional coupling might be responsible for the increased frequency of the

entire group, whereas unidirectional coupling was less likely to bring the other oscillators to the

same level. However, depending of the direction of the information flow, an oscillator may help

maintain the entire group frequency at a target period (as the independent oscillator did) or

follow the faster periods by the rest of the group (as the clamped oscillator did).

52

Chapter 3

Study 2

Overview

The goal of Study 2 was to determine the effect of social information on the symmetry of

spontaneous coordination during the three-person drumming task. The participants’ coordination

was predicted to be constrained by asymmetry of their group memberships. The minimal group

paradigm was utilized to manipulate the participants’ group identity. The notion that asymmetry

in group membership would lead to asymmetry in coordination was tested during spontaneous

drumming. In addition, the explicit drumming task was used to test the predictions that the

participants would choose their role in the task according to the symmetry of their group

membership and that congruency between group membership and task symmetry would lead to

stable coordination.

Participants

Twenty-nine triads of same-sex participants were randomly assigned into either the

homogenous (i.e., the control condition; triad n = 14) or the heterogeneous (i.e., the experimental

condition; triad n = 15) minimal group conditions. The average age of participants was 21.63 yrs

(SD = 6.75). There were 17 triads of female participants and 12 triads of the males. Most of the

participants were Caucasian (62.1%), followed by African American (16.1%), Other (12.6%),

Asian (5.7%), and Hispanic/Latino (3.4%).

Instruments

The drum set and the motion capture sensors were as described in Study 1. However,

there was no manipulation of the auditory and visual information in this study; participants could

hear and see each other without any constraints during the experiment. In addition, due to a

53

technical limitation, only movement data from the drumsticks were collected; the MIDI audio

was not recorded during this study.

Minimal group manipulation. At the beginning of the experiment, each group of three

participants was ostensibly tested for their music preference as a cover story for the minimal

group assignment. In this ‘music affiliativity’ test, the participants listened to nine short audio

clips, guessed a music genre of each clip from multiple choices, and ranked their preferences for

those clips. An experimenter pretended to grade their responses and gave the results back to the

participants. Unbeknownst to the participants, they were randomly assigned into one of two

bogus musical preferences: isorimi-firmus or plureriti-firmus. Participants who were assigned

into the former group would wear red caps during the experiment, while the latter group would

wear blue caps. To increase credibility of the cover story, a brief description of each musical

group was told to the participants; i.e., isoritmi-firmus describes people who can identify and

differentiate different patterns of pitches within a specific repeating rhythmic pattern, while

plureriti-firmus describes people who can identify and differentiate different patterns of pitches

in complementary repeating rhythmic patterns.

In a group of three participants, there are four permutations of possible minimal group

assignment: All three red caps, two red and one blue caps, one red and two blue caps, or all three

blue caps. However, because the minimal group identities are essentially equivalent and

symmetrical, half of the possible combinations are interchangeable with the other half. To

simplify the experimental conditions, only one homogeneous minimal group condition—a group

with all three red-cap participants—and one heterogeneous minimal group condition—a group

with two red-cap and one blue-cap participants—were utilized in this study.

54

Spontaneous and explicit drumming tasks. Two variations of the three-person

drumming task were used in this study. The first task was the spontaneous drumming task, which

was similar to Study 1’s task. Each of the three participants listened and drummed to slightly off-

set metronome beats at either 75, 80, or 85 bpm for 10 s. After the metronomes stopped, the

participants continued drumming at their given tempo for 80 s. The slight offsets of the

metronome were within ±10% of the middle tempo (i.e., 80 bpm). This spontaneous drumming

task was used to test the hypothesis that asymmetry in social identity would lead to asymmetry in

rhythmic coordination.

The second task was the explicit drumming task. The three participants were asked to

drum in the partial-inphase pattern (i.e., two participants drummed inphase and one participant

drummed antiphase with the first two). At the beginning of each trial, the participants discussed

and decided their drumming role. An 80-bpm metronome was played through audio speakers in

order to set the tempo for the drumming task. The continuation paradigm was utilized; the

metronome was played for first 10 s, and the trial continued for another 80 s. The second task

was designed to investigate two questions. First, do the decided drumming roles correspond to

each participant’s minimal group identity? Second, do participants who drum in a congruent role

with respect to their group identity coordinate better than those who adopt an incongruent role?

The symmetry of the participant’s drumming role was considered congruent with the symmetry

of his or her group identity when the red participant (i.e., a majority member) chose to drum

inphase and the blue participant (i.e., a minority member) chose to drum antiphase. In contrast,

the incongruent symmetry occurred when the blue member adopted the inphase drumming and

the red member drummed in the antiphase pattern.

55

Questionnaire. At the end of the experiment, the participants were asked to rate each of

the other participants on liking, similarity, and perceived coordination. The participants rated

each person to their left and right on a corresponding item.

Liking. The participants rated an item read, ‘Based on your impression, how much do

you like the participant sitting to your LEFT/RIGHT?’, on a scale of 1 (Not at all) to 7 (A lot).

Similarity. The participants rated an item read, ‘How similar are you and

the participant sitting to your LEFT/RIGHT?’, on a scale of 1 (Not at all like me) to 7 (Just like

me).

Perceived synchrony. The participants rated an item read, ‘How good was the

coordination between you and the participant sitting to your LEFT/RIGHT?’, on a scale of 1

(Very poor) to 7 (Very good).

Procedure

For each session, three participants were seated facing each other, forming a triangle,

with a drum pad in front of each of them. In contrast to the first experiment, the participants

could see each other. They were given the ‘musical affiliativity’ test, which was designed as a

cover story for the minimal group assignment. In the homogeneous minimal group condition, all

participants were assigned to the ‘isoritmi-firmus’ type and wore a red cap during the

experiment. The heterogeneous minimal group had two participants assigned into the ‘isoritmi-

firmus’ type, the red group, and one participant into the ‘plureriti-firmus’ type, the blue group.

After that, the participants first performed the spontaneous drumming task for six trials. Then

they performed the explicit drumming task for another six trials. During each trial of the second

task, the experimenter took note of the drumming roles that the participants decided among

themselves. Upon completion of the drumming tasks, the participants completed a questionnaire

56

and manipulation check. The participants were probed for any suspicion with funnel debriefing

questions. After that, the participants were debriefed about the actual purpose of the study.

Data Preparation, Reduction, and Analysis

Manipulation check. After completing the questionnaire, the participants were asked to

recall their cap’s color and identify any other participants who were in the same ‘musical

affiliativity’ group. One of the 87 participant incorrectly recalled his cap’s color. However, the

removal of this triad did not alter the results. In addition, among the 87 participants, seven of

them incorrectly identified the other participants who shared the same minimal group

assignment. After the funnel debriefing, seven participants reported suspicion that the cover story

for minimal group assignment was either fake or random. Nonetheless, the removal of these

participants and the triads to which they belonged did not change the results. These participants

were retained in the analysis.

Drumstick movement. The three-dimensional position time series of the drumstick

movement were trimmed to include the data from the 15th to the 85th second of the trial, resulting

in 70 s of time series data. There were five out of 1,044 trials that contained anomalies (e.g., the

participant accidentally dropped a drumstick) toward the beginning or the end of the remaining

70 s. Those five time series were further trimmed to remove the anomalous portion and resulted

in the shortest time series length of 60 s. Any time series that contain anomalies that resulted in a

reduction to less that 60 s were removed from the analysis. In addition, anomalies caused by

errors from the motion capture system (e.g., a blip in recording values) were removed and

interpolated using a cubic spline procedure. Twenty-three time series that could not be reliably

reconstructed were dropped from the analysis.

57

The remaining time series were subjected to the principal component analysis (PCA) to

determine the primary axis of drumstick movement. The primary axis was arranged so that the

peaks of the drumming cycle corresponded to the drumming event (i.e., when the drumstick hit

the pad). In 41 cases, from which the PCA could not successfully extract the primary movement

axis (e.g., due to nonstationary oscillatory movement), the primary axis was chosen from either

the x-, y-, or z-axis that best represented the drumming motion. Of these time series, 15 were

removed from the analysis because the drumming events could not be identified from the

movement. In total, 38 out of 1044 time series (3.6%) were excluded from the analysis.

The drumming events were calculated from the peaks of oscillation and were further used

in the calculation of drumming periods. Any periods that were larger than 1.5 times the target

periods (i.e., the period of 80 bpm is 0.75 s, and 0.75 ´ 1.5 = 1.125 s) were excluded from the

analysis.

Relative phase analysis. Both discrete and instantaneous relative phase could be derived

from the position data. The discrete relative phase (DRP) was calculated from the periods of the

drumming events, while the instantaneous relative phase was calculated from the drumstick

position time series. Both method of relative phase calculation yielded very similar results. For

the sake of consistency with Study 1, only the DRP is reported here.

Statistical analysis. Similar to Study 1, the marginal LMM was utilized to investigate the

effect of minimal group manipulation. The minimal group condition was a nested term with three

levels: The majority group, the minority group, and the control group. The majority and minority

group were nested in the experimental condition, while the control group was nested in the

control condition. Because the factors were not crossed, they could not be analyzed with the

ANOVA framework. However, the LMM framework allows a specification of such nested

58

effects. Estimated marginal means and standard errors were reported in the results section unless

otherwise mentioned.

Results

Spontaneous Drumming Task

Drumming periods. The means periods of the control condition (M = 0.73 s, SE =

0.006), the majority group (M = 0.72 s, SE = 0.007), and the minority group (M = 0.71 s, SE =

0.010) were not significantly different from one another, F(2, 83.14) = 1.42, p = .247. These

averages were slightly below the 80-bpm period of 0.75 s.

Period instability. The drumming period’s coefficient of variation served as a measure

of period instability. There were no significant differences in period instability between the

control condition (M = 0.07, SE = 0.003), the majority group (M = 0.07, SE = 0.003), and the

minority group (M = 0.06, SE = 0.004), F(2, 82.73) = 1.07, p = .346.

Discrete relative phase. The relative phase analysis was conducted for each pair within

the triad. The pairing fell into three categories: the pairs within the control conditions, the

ingroup-ingroup (red-red) pairs in the experimental condition, and the ingroup-outgroup (red-

blue) pairs in the experimental condition. The frequency distributions of relative phase for each

minimal group condition are shown in Figure 8. Overall, there was no substantial difference in

the distribution of % occurrence among the minimal group pairs. Most of the participants

demonstrated spontaneous inphase coordination with some degree of phase lag (~30°) with each

other.

59

Figure 8. Mean frequency distribution of relative phase for the spontaneous drumming task. Error bars represented bootstrapped 95% confidence intervals.

Coordination stability. The r values among the three pairings (Mcontrol = .56, SEcontrol =

.03; Mingroup-ingroup = .59, SEingroup-ingroup = .05; Mingroup-outgroup = .62, SEingroup-outgroup = .04) were not

significantly different from each other, F(2, 82.15) = 0.71, p = .494. The means values suggested

moderate degree of spontaneous coordination.

Explicit Drumming Task

In the second drumming task, the interaction between minimal group conditions and the

participant’s drumming role (i.e., inphase or antiphase) was investigated.

Drumming period. The main effect of minimal group condition and its interaction with

the drumming role on average period were not significant; F(2, 84.63) = 1.49, p = .232, and F(2,

257.49) = 0.01, p = .908, respectively. However, the main effect of drumming role was

significant, F(1, 257.34 = 14.89, p < .001. That is, the participants who chose to drum inphase

had shorter periods (M = 0.70 s, SE = 0.004) than those who drummed antiphase (M = 0.71 s, SE

= .004). The mean periods for each condition are shown in Table 4.

Period instability. A significant main effect of drumming role was also found on period

instability, F(1, 238.16) = 27.50, p < .001. The mean coefficient of variation was higher in the

antiphase drummers (M = 0.08, SE = 0.003) than in the inphase drummers (M = 0.06, SE =

60

0.003). The main effect of minimal group condition and the interaction effect were not

significant, F(2, 83.56) = 0.72, p = .489 and F(2, 239.10) = 0.36, p = .697, respectively.

Table 4

Estimated marginal means (and standard errors) of average period and period instability

Control Experimental

Majority Minority

Spontaneous Drumming Task

Period Mean 0.73 (0.006) 0.72 (0.007) 0.71 (0.010)

Period Instability 0.07 (0.003) 0.07 (0.003) 0.06 (0.004)

Explicit Drumming Task

Inphase Antiphase Inphase Antiphase Inphase Antiphase

Period Mean 0.71 (0.005) 0.72 (0.005) 0.70 (0.005) 0.71 (0.006) 0.70 (0.008) 0.71 (0.008)

Period Instability 0.07 (0.004) 0.08 (0.004) 0.07 (0.004) 0.08 (0.005) 0.06 (0.006) 0.07 (0.007)

Discrete relative phase. In the second drumming task, two modes of coordination—

inphase and antiphase—were expected. Figure 9 shows the frequency distributions of relative

phase by the minimal group pairs and the drumming role. The participants were able to drum

according to their chosen role. The inphase pairs showed stable inphase coordination (around -

40° to 40° bins), while the antiphase pairs demonstrated stable antiphase coordination (around -

180° to -140° bins and 140° to 180° bins). The minimal group pairing did not influence the

coordination during this task.

61

Figure 9. Mean frequency distribution of relative phase for the explicit drumming task. Error bars represented bootstrapped 95% confidence intervals.

Coordination stability. The stability of coordination was higher among the inphase

drummers (M = .86, SE = .018) than the antiphase pairs (M = .76, SE = .015), F(1, 284.83) =

33.53, p < .001. However, the effect pairing types and its interaction with the drumming role

were not significant; F(2, 86.98) = 1.06, p = .352, and F(2, 290.39) = 0.42, p = .652,

respectively. The means and standard errors are shown in Table 5.

Table 5

Estimated marginal means (and standard errors) of coordination stability (r) for each minimal

group pairing

Control Experimental

Ingroup-Ingroup Ingroup-Ingroup Ingroup-Outgroup

Spontaneous Drumming Task

.56 (.03) .59 (.05) .62 (.04)

Explicit Drumming Task

Inphase Antiphase Inphase Antiphase Inphase Antiphase

.87 (.02) .79 (.02) .83 (.04) .73 (.03) .86 (.03) .75 (.02)

62

Drumming role choice. Table 6 shows a cross tabulation of total drumming roles

choices from all trials. Most of the time, the participants followed the rotation symmetry. That is,

because the six total trials of the second task was a multiple of three, the participants decided to

rotate their drumming role and let everyone have an equal chance to drum in the antiphase role.

Table 6

Frequency (and expected frequency) of total chosen drumming roles from all trials during the

explicit partial-inphase drumming task

Minimal Group

Drumming Role Control Majority Minority Total

Inphase 168 (168) 124 (120) 56 (60) 348

Antiphase 84 (84) 56 (60) 34 (30) 174

Total 252 180 90 522

Questionnaire

The means of the questionnaire variables were reported in the actual group means instead

of the estimated marginal means (see Table 7).

Liking. Each participant rated their liking for the other two participants in the triad.

Hence, in the control condition, all participants rated the other two ingroup members (M = 5.29,

SD = 1.24). On the other hand, in the experimental condition, the majority group (i.e., red

participants) rated one ingroup member (M = 5.17, SD = 1.58) and one outgroup member (M =

5.50, SD = 1.41), and the minority group (i.e., blue participants) rated two outgroup members (M

= 5.43, SD = 1.10). The comparison of this nested effect showed that none of the liking means

were significantly different from each other, F(3, 84.65) = 0.55, p = 0.650.

63

Similarity. Each participant also rated their similarity to each of the other two

participants in the triad. Ratings of different target groups were not significantly different from

one another, F(3, 84.65) = 1.03, p = .383.

Perceived coordination. There was no significant effect of the target’s group

membership on perceived coordination, F(3, 84.64) = 1.09, p = .356.

Table 7

Means (and standard deviations) of liking, similarity, and perceived coordination of ingroup and

outgroup targets by minimal group conditions

Control Experimental

Majority Minority

Ingroup Ingroup Outgroup Outgroup

Liking 5.29 (1.24) 5.17 (1.58) 5.50 (1.41) 5.43 (1.10)

Similarity 4.18 (1.42) 4.43 (1.50) 4.50 (1.33) 3.83 (1.60)

Perceived Coordination 4.94 (1.36) 4.50 (1.48) 4.60 (1.57) 4.97 (1.52)

Discussion

Overall, the results suggested that the minimal group manipulation did not influence the

coordination among the participants in the drumming tasks. During the spontaneous drumming

task, inphase coordination emerged and was equally stable across the minimal group conditions.

The hypothesis that asymmetry in social identity would break the symmetry of spontaneous

coordination (Hypothesis 2.1) was not supported. In addition, beside the differences in periods

and coordination stability between inphase and antiphase coordination during the explicit partial-

inphase drumming task, no difference due to the minimal group manipulation was observed. The

64

participants successfully drummed to their respective drumming role (i.e., inphase or antiphase),

and their coordination was not effected by their minimal group identity.

Hypothesis 2.2, which suggested the outgroup participant would be more likely to choose

the antiphase drumming role during the second task, was not supported. The most common

solution to the role choice during the six trials of the second task was a role rotation. Such

solution matched the symmetry of the task constraints, and the minimal group manipulation was

not enough to break the attraction toward such symmetry.

Also, there was no difference in coordination of the congruent or incongruent symmetry

between the drumming role and the minimal group status. The participants in the minority group

did not show more stable coordination when they adopted the antiphase role than when they

adopted the inphase one. Although the participants in the majority group showed less stable

coordination during the antiphase role, antiphase coordination was less stable in general and was

not due to their minimal group status. Hence, Hypothesis 2.3 was not supported.

In sum, the minimal group manipulation did not produce a symmetry-breaking effect in

the three-person drumming task. It is worth noting that traditional interpersonal measures such as

liking and similarity did not vary as a function of the minimal group manipulation. Although one

plausible explanation is a failure of the manipulation to elicit any effect, an alternative

explanation is that the coordination during the experiment might neutralize any minimal group

effect, rendering its effect null. Further experiments are needed to test this notion.

65

Chapter 4

General Discussion

Asymmetric Informational Coupling Constraints

The aim of Study 1 was to demonstrate symmetry breaking of interpersonal coordination

as a function of asymmetry in perceptual coupling. However, the participants predominantly

showed spontaneous inphase coordination across the four coupling conditions (i.e., all-, partial-,

clamped- and rotation-coupling) with an exception of the 45-bpm partial-coupling condition,

where antiphase coordination occurred at about the same rate as inphase coordination. In other

words, the asymmetric auditory coupling constraints were not sufficient to induce corresponding

symmetry-breaking in the three-person interpersonal drumming task. Although the results did not

fully support the hypothesis, the informational coupling constraints had an effect of participants’

drumming period and their coordination when the target drumming frequency was at 45 bpm. At

this drumming frequency, directionality of informational coupling seems to play an important

role in explaining the effect of asymmetric coupling configurations.

First, the bidirectionally coupled oscillators had faster frequency than the unidirectionally

coupled oscillators, which in turn were faster than the non-coupled oscillators. As a group, triads

in the all-coupling, clamped-coupling, and rotation-coupling condition drummed faster than

those in the no-coupling and partial-coupling conditions, which drummed closer to the target

frequency (see Figure 2). The average period data in the no-coupling condition suggest that, even

at the slower 45 bpm, the participants were able to maintain their target frequency. However, this

drumming frequency was much slower than what people typically prefer (Moelants, 2002), and

the participants drifted toward the faster frequency when they were influenced by other

oscillators in the triad. The period shift might result from anticipation of signals from other

66

drummers, and the bidirectional coupling allowed the period shift to accumulate among the

oscillators over time via a positive feedback loop. The unidirectional coupling also influenced

period shift, although this influence was weaker than that of the bidirectional coupling. For

example, in the clamped-coupling condition, a clamped oscillator, which was unidirectionally

driven faster by the other two coupled oscillators in a triad, was still slower than the coupled

ones, suggesting weaker influences under the unidirectional coupling. Furthermore, in the

partial-coupling condition, an independent oscillator, which unidirectionally drove the other two

coupled oscillators, seemed to slow down the period shift among the coupled pairs. The

independent oscillator could maintain its period without being influenced by the rest of the group

and, at the same time, pulled the other two oscillators slower through its unidirectional influence.

As a result, participants in the partial-coupling condition, as a group, drummed at about the target

frequency. Nonetheless, the effect of unidirectional coupling on period shift was weaker than

that of the bidirectional coupling. Thus, in the partial-coupling condition, the coupled oscillators

were still faster than the independent oscillator (see Figure 3). As for the rotation-coupling

condition, which consisted of unidirectionally coupled oscillators in a ring formation, none of the

oscillators were totally uninfluenced. For example, Oscillator 1 might be unidirectionally

influenced by Oscillator 2, but, at the same time, Oscillator 1’s influence on Oscillator 3 would

indirectly influence Oscillator 2’s behavior. Therefore, the period shift could still occur via the

indirect positive feedback loop. In sum, the presence of bidirectional coupling could allow a

stronger phase shift than unidirectional coupling, which had a stronger effect than no coupling at

all. Nonetheless, the divergence of drumming periods in the partial-coupling and clamped-

coupling conditions illustrated that the collective behavior of the triads still depended on the

asymmetric coupling configuration.

67

Second, bidirectional coupling resulted in stronger coordination stability than

unidirectional coupling or no coupling. The coordination was strongest when the coupling

functions were bidirectional as in the all-coupling condition and weakest in the no-coupling

condition. The clamped-coupling condition also showed strong coordination stability because

both coupled oscillators were driving the clamped one. On the other hand, the partial-coupling

condition had weaker coordination stability than the all-coupling and clamped-coupling

conditions. As mentioned in the previous section on the period shift, asymmetric coupling in the

partial-coupling condition allowed the independent oscillator to perturb the coupled oscillators

and weaken the coordination stability. In addition, by being unidirectionally coupled to the rest

of the group, the coordination stability of the independent oscillator and the other two oscillators

was weaker than the stability among the coupled oscillators themselves. Moreover, the results

suggested that although an indirect positive feedback loop via unidirectional coupling in the

rotation-coupling condition could induce period shift, the direct bidirectional coupling as in the

all-coupling condition was more stable than the indirect coupling in the rotation-coupling

condition. The mutual direct influence between oscillators resulted in an emergence of a stronger

phase-locked pattern.

Third, to state the obvious, the effect of asymmetric coupling was not a sum of its pair-

wise coupling function, but a result of contextual constraints provided by the symmetry group.

The partial-coupling and clamped-coupling condition had the same pair-wise coupling functions:

One bidirectional and two unidirectional couplings. Nonetheless, the independent and the

clamped oscillators played different roles in the group coordination because of the imposed

constraints on the direction of informational flow. In the partial-coupling condition, the

independent oscillator perturbed the coordination of the bidirectionally coupled oscillators and

68

slowed down their frequency. In contrast, the clamped oscillator in the clamped-coupling

condition was driven by the rest of group, which resulted in faster periods and stable

coordination. Despite the linear increase in strength from unidirectional to bidirectional coupling,

the complex interaction between multiple oscillators must take into account the contextual

configuration of the group.

Drumming Frequency

Although the effect of asymmetric coupling on drumming periods was found, this was

only the case for the 45-bpm drumming frequency and did not occur in the 60-bpm condition.

While it may seem as if the 45-bpm drumming was less stable and, hence, being affected by

asymmetric coupling more than the 60-bpm drumming, there was no evidence from the analysis

that coordination in the 60-bpm condition was stronger than in the 45-bpm condition. While

spontaneous coordination at the 60-bpm did not require any period shift to be stable under

different coupling constraints, the period shift in the 45-bpm conditions might be a solution to

achieve a stable rhythm under some coupling conditions (i.e., all-coupling, clamped-coupling,

and rotation coupling) at the slower frequency. The larger deviation from human’s preferred

frequency of ~120 bpm combined with the positive feedback loop of influences seemed to

exacerbate the period shift in the drumming task. To determine a system’s stable solution in such

a situation, we must consider both frequency and coupling constraints.

Hypothesis 1.2 was partially supported as a higher proportion of antiphase coordination

was found in the 45-bpm partial-coupling condition than in other conditions. This result was also

consistent with the HBK model (Haken et al., 1985), which suggests bi-stability (i.e., inphase

and antiphase) when frequency is low. When the frequency increased to 60 bpm, the system

solutions seemed to collapse into a single mode, inphase.

69

The moderation effect of drumming frequency in Study 1 may shed light on the null

effect of social asymmetry in Study 2. If the effect of informational coupling started to disappear

even at 60 bpm, the participants in Study 2 who performed the task at 80 bpm might have even

smaller chances to break the symmetry of their coordination. Given the strength of the system’s

solution at this frequency, a minimal social constraint might not have been strong enough to have

an influence.

Social Constraints

Spontaneous coordination. While the goal of Study 2 was to test the hypothesis that

asymmetry in social identity would break the symmetry of interpersonal coordination during the

spontaneous drumming task, the null results suggested that the minimal group manipulation

could not induce symmetry-breaking of the coordination. Several possibilities or the combination

of them may be responsible for the null effects. First, as mentioned in the previous section, the

frequency constraints on the task might render other modes of coordination unstable. Given that

each participant could also see and hear all other participants in a triad, spontaneous coordination

in such a situation with no perceptual constraints might likely default to inphase coordination. In

other words, the constraints imposed by the minimal group manipulation could not break the

symmetry of the coupling between individuals in a triad. Second, the minimal group effect might

be overridden by the reciprocal nature of the drumming task. A typical minimal group paradigm

involves minimal or no interaction with other individuals (e.g., DiDonato, Ullrich, & Krueger,

2011; Gaertner & Insko, 2000; Tajfel, 1970). In a similar study on the effect of minimal group on

spontaneous coordination (Miles et al., 2011), the participants only unidirectionally coordinated

with a video stimulus on a pretense that it was a video link of another participant from a different

location. Nonetheless, the effective coupling was only unidirectional. The existence of

70

bidirectional coupling through perceptual information and co-present social interaction in the

current study might create a stronger attractor, regardless of the minimal group conditions. Third,

the synchronicity nature of the task might create a feeling of cooperation and entitativity (Lakens

& Stel, 2011) that diminished the minimal group effect. Such psychological effects might also

explain why none of the liking, similarity, and perceived coordination ratings varied as a

function of minimal group status. A fourth possibility also exists—the current minimal group

manipulation failed to produce any effect in the first place. Unfortunately, the current study’s

design did not provide a test to distinguish these possibilities. Nonetheless, considering the

established effect of the minimal group paradigm and spontaneous coordination, it is more

plausible that the current null effects were a combination of the frequency and coupling

constraints, which created a deep basin of attractor of inphase coordination.

Explicit coordination. Two hypotheses were tested in the second task of Study 2. First,

the minimal outgroup participants were expected to adopt the antiphase drumming role more

often than did the ingroup participants. Second, the participants who adopted a drumming role

that was not congruent with the symmetry group of their minimal group identity were expected

to be less stable than those who adopted a congruent role. None of these hypotheses were

supported.

For the first prediction, although asymmetry of minimal group status might prime the

participants to settle for roles that were congruent with the partial-inphase symmetry, !" # , the

symmetrical solution the role selection on six trials was to rotate the antiphase drummer role in

two rounds. As the results suggested, the minimal group manipulation was not able to break that

symmetry. A recent study showed that a systems’ solution tend to correspond with the symmetry

of highest order isotropy subgroup (Kijima, Shima, Okumura, Yamamoto, & Richardson, 2017).

71

In order words, when a higher order subgroup is plausible, the system is more likely to converge

to such symmetry. This phenomenon could be observed when biases or prejudices could not be

justified (Crandall & Eshleman, 2003). For example, when only given information on

socioeconomic status (SES) of a child, participants were reluctant to rely on the stereotype of

low and high SES to rate the child’s ability. However, when the SES data was accompanied by

additional information (i.e., a videotape of the child taking a test), participants who were led to

believe the child came from lower SES rated the child’s ability as below grade level, and the

opposite was true for participants who believed the child was from higher SES (Darley & Gross,

1983). Similarly, when the situation did not provide a cover for biases or prejudice, participants

would not avoid sharing a theater with a disable confederate (Snyder, Kleck, Strenta, & Mentzer,

1979). In the explicit task of Study 2, the highest order symmetry subgroup of the role choice

(not the coordination) was the !$ rotation symmetry. A break of symmetry with an outgroup

participant being chosen for the antiphase role more than twice would clearly reveal the bias due

to his or her outgroup status. Because the situational context did not justify such prejudice, the

discriminatory behavior was suppressed and the role assignment followed the most symmetrical

solution.

As for the second hypothesis, beside the fact that antiphase coordination tended to be less

stable regardless of the minimal group status, the congruency of the drumming role and minimal

group status did not affect coordination stability. The same explanation that the basin of

attraction was deep and stable at this drumming frequency might also apply here.

In summary, the main hypotheses that the asymmetry of informational coupling and

social context would induce the corresponding asymmetry in the interpersonal coordination were

not fully supported. However, asymmetric information coupling could shift the system’s

72

behavior, such as periods, when the system was far from its preferred frequency. Also, the

behavior of the system was not just a linear sum of its coupling function; the configuration or

structure of the coupling functions must also be considered.

Limitations & Future Research

One of the reason why the results were not as predicted might stem from the nature of the

three-person drumming task itself. While the task was chosen because it was free of hard

constraints and, hence, could embody all symmetry subgroups, the highest order subgroup, D3,

posed a challenge for a symmetry-breaking process. Softer constraints such as perceptual and

social information might only be effective when the system is in a multi-stable regime (e.g.,

lower frequency). Other researchers have avoided this issue by adopting a task that has built-in

constraints to avoid the D3 symmetry, such as a ball-possession task (Yokoyama & Yamamoto,

2011). Future research may incorporate hard constraints to contain the system within certain

symmetry subgroups and investigate the effect of low-energy constraints, such as perceptual or

social information, on the stability of coordination. Nonetheless, the three-person drumming task

was not without merit. Because the task embodies all isotropy subgroups, the symmetry-breaking

process could be brought about by a different approach to the task. For example, in a previous

study, Ariyabuddhiphongs and Richardson (2015, April) found that the triads of participants

were able to voluntarily coordinate in patterns from each symmetry subgroup. In other words,

intentions to perform an instructed pattern functioned as a constraint that breaks the symmetry of

the coupling between individuals in a triad. The current limitation of the spontaneous version of

the drumming task suggests that future research may focus on the symmetry breaking process as

a function of intentional constraints (Shaw, Kadar, Sim, & Repperger, 1992; Washburn, Coey,

Romero, Malone, & Richardson, 2015).

73

Another limitation of the current study was an inability to determine the cause of the

ineffective minimal group manipulation. Future studies should include another control condition

where the participants independently perform a task. That way, it could be determined whether

mutual synchrony overrides the effect of minimal group manipulation. In addition, for a face-to-

face coordination task, a stronger social manipulation may be needed. Future research may

explore the asymmetry related to more controversial topics such as political affiliation. Also, as

the justification-suppression model (Crandall & Eshleman, 2003) suggests, asymmetric task

constraints may provide a cover or justification for acting out prejudice, while symmetric task

constraints created ambiguity and attenuate it. Therefore, researchers may use the higher order

isotropy subgroup to identify task constraints that are congruent with the desired behavior.

In closing, Richardson and Kallen (2016) suggested that symmetry and symmetry-

breaking principles provide a useful framework to understand the organization of interpersonal

behavior. The application of symmetry theory and dynamical systems approaches in social

psychology is still relatively new, so a lot of work remains, and the field is filled with uncharted

territories. That said, the approach holds vast potential and will reward those who yearn for

discoveries in the frontier.

74

References

Adolphs, R. (2009). The social brain: Neural basis of social knowledge. Annual Review of

Psychology, 60, 693-716. doi: 10.1146/annurev.psych.60.110707.163514

Ajzen, I. (1991). The theory of planned behavior. Organizational Behavior and Human Decision

Processes, 50, 179-211. doi: 10.1016/0749-5978(91)90020-T

Ariyabuddhiphongs, K., & Richardson, M. J. (2015, April). Coordination patterns of three

coupled oscillators on a drumming task. Poster presented at the CAP Center Guy Van

Orden Research Conference, Cincinnati, OH.

Bargh, J. A., & Chartrand, T. L. (1999). The unbearable automaticity of being. American

Psychologist, 54, 462-479.

Bargh, J. A., Chen, M., & Burrows, L. (1996). Automaticity of social behavior: Direct effects of

trait construct and stereotype-activation on action. Journal of Personality and Social

Psychology, 71, 230-244.

Blaszczyk, J. W., & Klonowski, W. (2001). Postural stability and fractal dynamics. Acta

Neurobiologiae Experimentalis, 61, 105-112.

Buck, J., & Buck, E. (1976). Synchronous fireflies. Scientific American, 234(5), 74-85. doi:

10.1038/scientificamerican0576-74

Buckner, R. L., & Carroll, D. C. (2007). Self-projection and the brain. Trends in Cognitive

Sciences, 11, 49-57. doi: 10.1016/j.tics.2006.11.004

Chartrand, T. L., & Bargh, J. A. (1999). The chameleon effect: The perception-behavior link and

social interaction. Journal of Personality and Social Psychology, 76, 893-910.

Chartrand, T. L., Maddux, W. W., & Lakin, J. L. (2005). Beyond the perception-behavior link:

The ubiquitous utility and motivational moderators of nonconscious mimicry. In R. R.

75

Hassin, J. S. Uleman, & J. A. Bargh (Eds.), The new unconscious (pp. 334-361). New

York: Oxford University Press.

Coey, C. A., Hassebrock, J., Kloos, H., & Richardson, M. J. (2015). The complexities of keeping

the beat: Dynamical structure in the nested behaviors of finger tapping. Attention,

Perception, & Psychophysics, 77, 1423-1439. doi: 10.3758/s13414-015-0842-4

Collins, J. J., & Stewart, I. N. (1993). Coupled nonlinear oscillators and the symmetries of

animal gaits. Journal of Nonlinear Science, 3, 349-392. doi: 10.1007/BF02429870

Correll, J. (2008). 1/f noise and effort on implicit measures of bias. Journal of Personality and

Social Psychology, 94, 48-59. doi: 10.1037/0022-3514.94.1.48

Crandall, C. S., & Eshleman, A. (2003). A justification-suppression model of the expression and

experience of prejudice. Psychological Bulletin, 129, 414-446. doi: 10.1037/0033-

2909.129.3.414

Cuddy, A. J. C., Wilmuth, C. A., Yap, A. J., & Carney, D. R. (2015). Preparatory power posing

affects nonverbal presence and job interview performance. Journal of Applied

Psychology, 100, 1286-1295. doi: 10.1037/a0038543

Curie, P. (1894). Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique

et d'un champ magnétique. Journal de Physique Théorique et Appliquée, 3, 393-415. doi:

10.1051/jphystap:018940030039300

Darley, J. M., & Gross, P. H. (1983). A hypothesis-confirming bias in labeling effects. Journal of

Personality and Social Psychology, 44, 20-33. doi: 10.1037/0022-3514.44.1.20

Decety, J., & Chaminade, T. (2003). When the self represents the other: A new cognitive

neuroscience view on psychological identification. Consciousness and Cognition, 12,

577-596. doi: 10.1016/s1053-8100(03)00076-x

76

DiDonato, T. E., Ullrich, J., & Krueger, J. I. (2011). Social perception as induction and

inference: An integrative model of intergroup differentiation, ingroup favoritism, and

differential accuracy. Journal of Personality and Social Psychology, 100, 66-83. doi:

10.1037/a0021051

Dijksterhuis, A., & Bargh, J. A. (2001). The perception-behavior expressway: Automatic effects

of social perception on social behavior. In P. Z. Mark (Ed.), Advances in Experimental

Social Psychology (Vol. 33, pp. 1-40). San Diego: Academic Press.

Donders, F. C. (1969). Over de snelheid van psychische processen [On the speed of

psychological processes] (W. Koster, Trans.). In W. Koster (Ed.), Attention and

performance: II. Amsterdam: North-Holland. (Original work published 1868)

Eiler, B. A., Kallen, R. W., Harrison, S. J., & Richardson, M. J. (2013). Origins of order in joint

activity and social behavior. Ecological Psychology, 25, 316-326. doi:

10.1080/10407413.2013.810107

Gaertner, L., & Insko, C. A. (2000). Intergroup discrimination in the minimal group paradigm:

Categorization, reciprocation, or fear? Journal of Personality and Social Psychology, 79,

77-94. doi: 10.1037/0022-3514.79.1.77

Gallese, V. (2003). The roots of empathy: The shared manifold hypothesis and the neural basis

of intersubjectivity. Psychopathology, 36, 171-180. doi: 10.1159/000072786

Gallese, V., & Goldman, A. (1998). Mirror neurons and the simulation theory of mind-reading.

Trends in Cognitive Sciences, 2, 493-501. doi: 10.1016/S1364-6613(98)01262-5

Gilbert, D. T., & Malone, P. S. (1995). The correspondence bias. Psychological Bulletin, 117,

21-38.

77

Goldman, A. I. (2006). Simulating minds : The philosophy, psychology, and neuroscience of

mindreading. New York, NY: Oxford University Press.

Golubitsky, M., & Stewart, I. (1985). Hopf bifurcation in the presence of symmetry. Archive for

Rational Mechanics and Analysis, 87, 107-165. doi: 10.1007/BF00280698

Golubitsky, M., & Stewart, I. (1986). Hopf bifurcation with dihedral group symmetry: Coupled

nonlinear oscillators. Contemporary Mathematics, 56, 131-173.

Golubitsky, M., & Stewart, I. (2003). The symmetry perspective: From equilibrium to chaos in

phase space and physical space. Basel, Switzerland: Birkhäuser Verlag.

Golubitsky, M., Stewart, I. N., & Schaeffer, D. G. (1988). Singularities and groups in

bifurcation theory (Vol. II). New York: Springer-Verlag.

Gottsdanker, R., & Shragg, G. P. (1985). Verification of Donders' subtraction method. Journal of

Experimental Psychology: Human Perception and Performance, 11, 765-776. doi:

10.1037/0096-1523.11.6.765

Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human

hand movements. Biological Cybernetics, 51, 347-356.

Hausdorff, J. M. (2007). Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride

fluctuations of human walking. Human Movement Science, 26, 555-589. doi:

10.1016/j.humov.2007.05.003

Hausdorff, J. M., Mitchell, S. L., Firtion, R., Peng, C. K., Cudkowicz, M. E., Wei, J. Y., &

Goldberger, A. L. (1997). Altered fractal dynamics of gait: Reduced stride-interval

correlations with aging and Huntington's disease. Journal of Applied Physiology, 82, 262-

269.

78

Herrmann, E., Call, J., Hernàndez-Lloreda, M. V., Hare, B., & Tomasello, M. (2007). Humans

have evolved specialized skills of social cognition: The Cultural Intelligence Hypothesis.

Science, 317, 1360-1366.

Holden, J. G., & Rajaraman, S. (2012). The self-organization of a spoken word. Frontiers in

Psychology, 3, 209. doi: 10.3389/fpsyg.2012.00209

Hommel, B., Müsseler, J., Aschersleben, G., & Prinz, W. (2001). The theory of event coding

(TEC): A framework for perception and action planning. Behavioral and Brain Sciences,

24, 849-937.

Horner, V., & Whiten, A. (2004). Causal knowledge and imitation/emulation switching in

chimpanzees (Pan troglodytes) and children (Homo sapiens). Animal Cognition, 8, 164-

181. doi: 10.1007/s10071-004-0239-6

Iacoboni, M. (2009). Imitation, empathy, and mirror neurons. Annual Review of Psychology, 60,

653-670. doi: 10.1146/annurev.psych.60.110707.163604

Iacoboni, M., Molnar-Szakacs, I., Gallese, V., Buccino, G., Mazziotta, J. C., & Rizzolatti, G.

(2005). Grasping the intentions of others with one's own mirror neuron system. PLoS

Biology, 3(3), e79. doi: 10.1371/journal.pbio.0030079

Jensen, H. J. (1998). Self-organized criticality. Cambridge, England: Cambridge University

Press.

Jordan, J. S., & Ghin, M. (2006). (Proto-) consciousness as a contextually emergent property of

self-sustaining systems. Mind & Matter, 4, 45-68.

Katsuta, Y., & Kawakami, H. (2006). Four-phase oscillations observed in two oscillators coupled

with nonlinear inductors. Electronics and Communications in Japan, 89, 15-21. doi:

10.1002/ecjc.20196

79

Kelley, H. H., & Michela, J. L. (1980). Attribution theory and research. Annual Review of

Psychology, 31, 457-501. doi: 10.1146/annurev.ps.31.020180.002325

Kelso, J. A. S. (1984). Phase transitions and critical behavior in human bimanual coordination.

American Journal of Physiology - Regulatory, Integrative and Comparative Physiology,

246, R1000-R1004.

Kijima, A., Shima, H., Okumura, M., Yamamoto, Y., & Richardson, M. (2017). Effects of agent-

environment symmetry on the coordination dynamics of triadic jumping. Frontiers in

Psychology, 8(3). doi: 10.3389/fpsyg.2017.00003

Knoblich, G., & Sebanz, N. (2008). Evolving intentions for social interaction: From entrainment

to joint action. Philosophical Transactions of the Royal Society B: Biological Sciences,

363, 2021-2031. doi: 10.1098/rstb.2008.0006

Lakens, D., & Stel, M. (2011). If they move in sync, they must feel in sync: Movement

synchrony leads to attributions of rapport and entitativity. Social Cognition, 29, 1-14.

Lakin, J. L., Chartrand, T. L., & Arkin, R. M. (2008). I am too just like you: Nonconscious

mimicry as an automatic behavioral response to social exclusion. Psychological Science,

19, 816-822. doi: 10.1111/j.1467-9280.2008.02162.x

Lopresti-Goodman, S. M., Richardson, M. J., Silva, P. L., & Schmidt, R. C. (2008). Period basin

of entrainment for unintentional visual coordination. Journal of Motor Behavior, 40, 3-

10. doi: 10.3200/JMBR.40.1.3-10

Lumsden, J., Miles, L. K., Richardson, M. J., Smith, C. A., & Macrae, C. N. (2012). Who syncs?

Social motives and interpersonal coordination. Journal of Experimental Social

Psychology, 48, 746-751. doi: 10.1016/j.jesp.2011.12.007

80

Macdonald, J. H. G. (2008). Pedestrian-induced vibrations of the Clifton Suspension Bridge,

UK. Proceedings of the Institution of Civil Engineers - Bridge Engineering, 161, 69-77.

doi: 10.1680/bren.2008.161.2.69

Macdonald, J. H. G. (2009). Lateral excitation of bridges by balancing pedestrians. Proceedings

of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465, 1055-

1073. doi: 10.1098/rspa.2008.0367

MacDougall, H. G., & Moore, S. T. (2005). Marching to the beat of the same drummer: the

spontaneous tempo of human locomotion. Journal of Applied Physiology, 99, 1164-1173.

doi: 10.1152/japplphysiol.00138.2005

Maner, J. K., DeWall, C. N., Baumeister, R. F., & Schaller, M. (2007). Does social exclusion

motivate interpersonal reconnection? Resolving the "porcupine problem.". Journal of

Personality and Social Psychology, 92, 42-55. doi: 10.1037/0022-3514.92.1.42

Meltzoff, A. N. (2007). ‘Like me’: A foundation for social cognition. Developmental Science,

10, 126-134. doi: 10.1111/j.1467-7687.2007.00574.x

Miles, L. K., Griffiths, J. L., Richardson, M. J., & Macrae, C. N. (2010). Too late to coordinate:

Contextual influences on behavioral synchrony. European Journal of Social Psychology,

40, 52-60. doi: 10.1002/ejsp.721

Miles, L. K., Lumsden, J., Richardson, M. J., & Macrae, C. N. (2011). Do birds of a feather

move together? Group membership and behavioral synchrony. Experimental Brain

Research, 211, 495-503. doi: 10.1007/s00221-011-2641-z

Miles, L. K., Nind, L. K., & Macrae, C. N. (2009). The rhythm of rapport: Interpersonal

synchrony and social perception. Journal of Experimental Social Psychology, 45, 585-

589. doi: 10.1016/j.jesp.2009.02.002

81

Mischel, W., & Shoda, Y. (2008). Toward a unified theory of personality. In O. P. John, R. W.

Robins, & L. A. Pervin (Eds.), Handbook of personality: Theory and reserach (3 ed.).

New York, NY: Guilford Press.

Moelants, D. (2002). Preferred tempo reconsidered. Paper presented at the Proceedings of the

7th International Conference on Music Perception and Cognition, Sydney.

Muijres, F. T., & Dickinson, M. H. (2014). Bird flight: Fly with a little flap from your friends.

Nature, 505, 295-296. doi: 10.1038/505295a

Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization : A universal concept in

nonlinear sciences. Cambridge, NY: Cambridge University Press.

Premack, D. (2010). Why humans are unique: Three theories. Perspectives on Psychological

Science, 5, 22-32.

Premack, D., & Woodruff, G. (1978). Does the chimpanzee have a theory of mind? Behavioral

and Brain Sciences, 4, 515-526.

Read, S. J., Monroe, B. M., Brownstein, A. L., Yang, Y., Chopra, G., & Miller, L. C. (2010). A

neural network model of the structure and dynamics of human personality. Psychological

Review, 117, 61-92. doi: 10.1037/a0018131

Richardson, M. J., Garcia, R. L., Frank, T. D., Gregor, M., & Marsh, K. L. (2012). Measuring

group synchrony: A cluster-phase method for analyzing multivariate movement time-

series. Frontiers in Physiology, 3, 405. doi: 10.3389/fphys.2012.00405

Richardson, M. J., Harrison, S. J., Kallen, R. W., Walton, A., Eiler, B. A., Saltzman, E., &

Schmidt, R. C. (2015). Self-organized complementary joint action: Behavioral dynamics

of an interpersonal collision-avoidance task. Journal of Experimental Psychology:

Human Perception and Performance, 41, 665-679. doi: 10.1037/xhp0000041

82

Richardson, M. J., & Kallen, R. W. (2016). Symmetry-breaking and the contextual emergence of

human multiagent coordination and social activity. In E. Dzhafarov, J. S. Jordan, R.

Zhang, & V. Cervantes (Eds.), Contextuality from quantum physics to psychology (pp.

229-286). Singapore: World Scientific.

Richardson, M. J., Marsh, K. L., Isenhower, R. W., Goodman, J. R. L., & Schmidt, R. C. (2007).

Rocking together: dynamics of intentional and unintentional interpersonal coordination.

Human Movement Science, 26, 867-891. doi: 10.1016/j.humov.2007.07.002

Richardson, M. J., Marsh, K. L., & Schmidt, R. C. (2005). Effects of visual and verbal

interaction on unintentional interpersonal coordination. Journal of Experimental

Psychology: Human Perception and Performance, 31, 62-79. doi: 10.1037/0096-

1523.31.1.62

Riley, M. A., Kuznetsov, N., & Bonnette, S. (2011). State-, parameter-, and graph-dynamics:

Constraints and the distillation of postural control systems. Science & Motricité, 74, 5-18.

doi: 10.1051/sm/2011117

Rosen, J. (1995). Symmetry in science: An introduction to the general theory. New York, NY:

Springer-Verlag.

Saltzman, E. L., & Munhall, K. G. (1992). Skill acquisition and development: The roles of state-,

parameter-, and graph-dynamics. Journal of Motor Behavior, 24, 49-57. doi:

10.1080/00222895.1992.9941600

Schelling, T. C. (1971). Dynamic models of segregation. Journal of Mathematical Sociology, 1,

143-186. doi: 10.1080/0022250X.1971.9989794

Schmidt, R. C., Bienvenu, M., Fitzpatrick, P. A., & Amazeen, P. G. (1998). A comparison of

intra- and interpersonal interlimb coordination: coordination breakdowns and coupling

83

strength. Journal of Experimental Psychology: Human Perception and Performance, 24,

884-900.

Schmidt, R. C., Carello, C., & Turvey, M. T. (1990). Phase transitions and critical fluctuations in

the visual coordination of rhythmic movements between people. Journal of Experimental

Psychology: Human Perception and Performance, 16, 227-247. doi: 10.1037/0096-

1523.16.2.227

Schmidt, R. C., Fitzpatrick, P., Caron, R., & Mergeche, J. (2011). Understanding social motor

coordination. Human Movement Science, 30, 834-845. doi: 10.1016/j.humov.2010.05.014

Schmidt, R. C., & O'Brien, B. (1997). Evaluating the dynamics of unintended interpersonal

coordination. Ecological Psychology, 9, 189-206. doi: 10.1207/s15326969eco0903_2

Schmidt, R. C., & Richardson, M. J. (2008). Dynamics of interpersonal coordination. In A.

Fuchs & V. K. Jirsa (Eds.), Coordination: Neural, behavioral and social dynamics (pp.

281-308). Berlin, Heidelberg: Springer Berlin Heidelberg.

Schmidt, R. C., & Turvey, M. T. (1994). Phase-entrainment dynamics of visually coupled

rhythmic movements. Biological Cybernetics, 70, 369-376. doi: 10.1007/bf00200334

Shaw, R. E., Kadar, E., Sim, M., & Repperger, D. W. (1992). The intentional spring: a strategy

for modeling systems that learn to perform intentional acts. Journal of Motor Behavior,

24, 3-28. doi: 10.1080/00222895.1992.9941598

Sherif, M., Harvey, O. J., White, B. J., Hood, W. R., & Sherif, C. W. (1961). Intergroup conflict

and cooperation: The Robbers Cave experiment. Norman, OK: University of Oklahoma

Book Exchange.

Shockley, K., Richardson, D. C., & Dale, R. (2009). Conversation and coordinative structures.

Topics in Cognitive Science, 1, 305-319. doi: 10.1111/j.1756-8765.2009.01021.x

84

Simon, H. A. (1973). The organization of complex systems. In H. H. Pattee (Ed.), Hierarchy

theory: The challenge of complex systems (pp. 1-27). New York, NY: Braziller.

Snyder, M. L., Kleck, R. E., Strenta, A., & Mentzer, S. J. (1979). Avoidance of the handicapped:

An attributional ambiguity analysis. Journal of Personality and Social Psychology, 37,

2297-2306. doi: 10.1037/0022-3514.37.12.2297

Stel, M., van Baaren, R. B., Blascovich, J., van Dijk, E., McCall, C., Pollmann, M. M. H., . . .

Vonk, R. (2010). Effects of a priori liking on the elicitation of mimicry. Experimental

Psychology, 57, 412-418. doi: 10.1027/1618-3169/a000050

Sternberg, S. (1969). The discovery of processing stages: Extensions of Donders' method. Acta

Psychologica, 30, 276-315. doi: 10.1016/0001-6918(69)90055-9

Strogatz, S. H., Abrams, D. M., McRobie, A., Eckhardt, B., & Ott, E. (2005). Theoretical

mechanics: Crowd synchrony on the Millennium Bridge. Nature, 438, 43-44.

Strogatz, S. H., & Stewart, I. (1993). Coupled oscillators and biological synchronization.

Scientific American, 269(6), 102-109.

Tajfel, H. (1970). Experiments in intergroup discrimination. Scientific American, 223(5), 96-102.

doi: 10.1038/scientificamerican1170-96

Takamatsu, A., Tanaka, R., Yamada, H., Nakagaki, T., Fujii, T., & Endo, I. (2001).

Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum

plasmodial slime mold. Physical Review Letters, 87, 078102. doi:

10.1103/PhysRevLett.87.078102

Tesser, A. (1980). When individual dispositions and social pressure conflict: A catastrophe.

Human Relations, 33, 393-407.

85

Tesser, A., McMillen, R., & Collins, J. (1997). Chaos: On making a convincing case for social

psychology. Psychological Inquiry, 8, 137-143. doi: 10.1207/s15327965pli0802_10

Tomasello, M., & Carpenter, M. (2007). Shared intentionality. Developmental Science, 10, 121-

125. doi: 10.1111/j.1467-7687.2007.00573.x

Turvey, M. T. (1990). Coordination. American Psychologist, 45, 938-953. doi: 10.1037/0003-

066X.45.8.938

Valdesolo, P., Ouyang, J., & DeSteno, D. (2010). The rhythm of joint action: Synchrony

promotes cooperative ability. Journal of Experimental Social Psychology, 46, 693-695.

doi: 10.1016/j.jesp.2010.03.004

van Baaren, R., Janssen, L., Chartrand, T. L., & Dijksterhuis, A. (2009). Where is the love? The

social aspects of mimicry. Philosophical Transactions of the Royal Society B: Biological

Sciences, 364, 2381-2389. doi: 10.1098/rstb.2009.0057

Van Orden, G. C., & Holden, J. G. (2002). Intentional contents and self-control. Ecological

Psychology, 14, 87-109.

Van Orden, G. C., Holden, J. G., & Turvey, M. T. (2003). Self-organization of cognitive

performance. Journal of Experimental Psychology: General, 132, 331-350. doi:

10.1037/0096-3445.132.3.331

Van Orden, G. C., Holden, J. G., & Turvey, M. T. (2005). Human cognition and 1/f scaling.

Journal of Experimental Psychology: General, 134, 117-123. doi: 10.1037/0096-

3445.134.1.117

Varlet, M., & Richardson, M. J. (2011). Computation of continuous relative phase and

modulation of frequency of human movement. Journal of Biomechanics, 44, 1200-1204.

doi: 10.1016/j.jbiomech.2011.02.001

86

Washburn, A., Coey, C. A., Romero, V., Malone, M., & Richardson, M. J. (2015). Interaction

between intention and environmental constraints on the fractal dynamics of human

performance. Cognitive Processing, 16, 343-350. doi: 10.1007/s10339-015-0652-6

Wheat, J. S., & Glazier, P. S. (2006). Measuring coordination and variability in coordination. In

K. Davids, S. Bennett, & K. Newell (Eds.), Movement system variability (pp. 167-182).

Champaign, IL: Human Kinetics.

Williams, K. D., Cheung, C. K. T., & Choi, W. (2000). Cyberostracism: Effects of being ignored

over the Internet. Journal of Personality and Social Psychology, 79, 748-762. doi:

10.1037/0022-3514.79.5.748

Wiltermuth, S. S., & Heath, C. (2009). Synchrony and cooperation. Psychological Science, 20,

1-5. doi: 10.1111/j.1467-9280.2008.02253.x

Wong, E. A., Vallacher, R. R., & Nowak, A. (2014). Fractal dynamics in self-evaluation reveal

self-concept clarity. Nonlinear Dynamics, Psychology, and Life Sciences, 18, 349-369.

Yokoyama, K., & Yamamoto, Y. (2011). Three people can synchronize as coupled oscillators

during sports activities. PLoS Computational Biollogy, 7(10), e1002181. doi:

10.1371/journal.pcbi.1002181

Yu, H., Russell, D. M., & Sternad, D. (2003). Task-effector asymmetries in a rhythmic

continuation task. Journal of Experimental Psychology: Human Perception and

Performance, 29, 616-630. doi: 10.1037/0096-1523.29.3.616

Zamm, A., Wellman, C., & Palmer, C. (2016). Endogenous rhythms influence interpersonal

synchrony. Journal of Experimental Psychology: Human Perception and Performance,

42, 611-616. doi: 10.1037/xhp0000201