Harmony from Chaos? Perceptual-Motor Delays Enhance Behavioral Anticipation in Social Interaction

Abstract

Effective interpersonal coordination is fundamental to robust social interaction, and the ability to anticipate a co-actor's behavior is essential for achieving this coordination. However, coordination research has focused on the behavioral synchrony that occurs between the simple periodic movements of co-actors and, thus, little is known about the anticipation that occurs during complex, everyday interaction. Research on the dynamics of coupled neurons, human motor control, electrical circuits, and laser semiconductors universally demonstrates that small temporal feedback delays are necessary for the anticipation of chaotic events. We therefore investigated whether similar feedback delays would promote anticipatory behavior during social interaction. Results revealed that co-actors were not only able to anticipate others' chaotic movements when experiencing small perceptual-motor delays, but also exhibited movement patterns of equivalent complexity. This suggests that such delays, including those within the human nervous system, may enhance, rather than hinder, the anticipatory processes that underlie successful social interaction.

Coordinating one's behaviors with those of another individual is fundamental to successful interpersonal interaction. One only has to consider the multitude of actions that must be performed during a daily commute to work, or the coordination that arises between members of a sports team, to be reminded that such interaction is typically effortless and efficient, even when we are faced with highly variable and often unpredictable behavioral events. Key to achieving such coordination is being able to predict or anticipate the behaviors of other individuals. The majority of research investigating the mechanisms that support behavioral anticipation has been based on hypotheses about neural simulation processes (Blakemore & Decety, 2001; Decety & Grèzes, 1999), feed-forward internal models and motor programs (Noy, Dekel, & Alon, 2011; Wolpert, Doya, & Kawato, 2003), or shared intentional and representational states (Sebanz, Bekkering, & Knoblich, 2006). These and similar constructs have been formulated to account for how the human nervous system compensates for the temporal delays that inherently occur between the production of a movement and the perception of its outcome (i.e., feedback). The traditional assumption, grounded in linear systems theory, is that perceptual-motor feedback delays present a problem for coordinating behavior because they amplify errors and lead to instability (Stepp & Turvey, 2010; Wolpert et al., 2003).

In contrast to this traditional assumption, recent work examining the dynamics of laser semiconductors (Masoller, 2001; Sivaprakasam, Shahverdiev, Spencer & Shore, 2001), electrical circuits (Voss, 2002), and coupled neurons (Toral, Masoller, Mirasso, Ciszak & Calvo, 2003) has demonstrated that small temporal feedback delays can actually enhance the ability for a system to synchronize with unpredictable, chaotic events. This counterintuitive phenomenon, referred to as self-organized anticipation or anticipatory synchronization, has been found to emerge when a “slave” system (i.e., electronic circuit) is unidirectionally coupled to a chaotically behaving “master” system (i.e., a second electronic circuit). As the slave system begins to synchronize with the chaotic behavior of the master system, small temporal delays are introduced into the feedback loop between the slave's behavior and the resulting outcomes of that behavior. Surprisingly, following the introduction of these delays, the actions of the slave system begin to anticipate the ongoing behavior exhibited by the chaotic master system. In other words, a small temporal feedback delay in these systems supports, rather than hinders, anticipatory behavior by prospectively tuning the behavior of the slave system to the evolving dynamics of the master system (Stephen, Stepp, Dixon, & Turvey, 2008; Stepp & Turvey, 2008).

Stepp (2009) investigated whether the phenomenon of anticipatory synchronization might underlie anticipatory motor control in humans. In order to examine this possibility, he designed a simple visual-motor coordination task, in which individual participants were instructed to control and coordinate a visual stimulus dot, using a hand-held pen and a touch-sensitive tablet, with a computer controlled, chaotically moving stimulus dot displayed on a computer screen. The results demonstrated that individuals were able to coordinate with the computer stimulus using real-time information about the movements of their hands relative to the stimulus, but with a significant phase lag (i.e., the participant's movements lagged behind the chaotic motion of the computer controlled stimulus dot). However, once a perceptual-motor delay was introduced between a participant's hand movements and those of the on-screen dot the participant controlled (i.e., when information about the outcome of the participants hand movements was temporally delayed with respect to the production of their hand movements), participants were not only able to coordinate with the chaotic stimulus, but could do so in an anticipatory manner. That is, the participants' hand movements started to lead the chaotic motion of the computer controlled stimulus. Furthermore, the degree of anticipation observed varied systematically with the amount of feedback delay experienced, with no anticipation exhibited for delays of less than 200 ms, a linear increase in the degree of anticipation observed for delays of 200-400 ms, and an overall decrease in coordinative stability for delays of 600 ms and higher.

In discussing a potential explanatory mechanism for anticipatory synchronization as it has been observed in a variety of physical systems, Stepp and Turvey (2010) indicate that the potential behavior states must first be similarly constrained, with the slave system sensitive to these constraints. The slave system can then be understood as embodying the constraints, and consequently the inherent dynamics, of the master system. With the introduction of a feedback delay with respect to the outcomes of its own behavior, the slave system must actually start to perform in an anticipatory manner in order to maintain synchrony with the master system. The ability of the slave system to successfully anticipate the chaotic behavior of a master system in this context can then be understood as resulting from the embodiment of the dynamics of the master system along with the need to act ahead of the master system in order to maintain synchrony when experiencing a delay. This understanding of anticipatory chaotic behavior has previously been referred to as strong anticipation (Dubois, 2001), and is thought to operate in a similar manner across physical systems. In contrast, a theory of weak anticipation depends on the existence of internal models for the achievement of anticipatory behavior in humans (Dubois, 2001). In considering the potential of these two views to account for the many observations of anticipatory synchronization of chaotic behavior, strong anticipation provides an explanation for the role of delays in facilitating anticipation across systems while weak anticipation a) does not allow for an explanation that generalizes across physical systems which necessarily do not have internal models (e.g., laser semiconductors, electronic circuits) and b) would characterize the introduction of delays as a threat to successful behavioral coordination. The current study was therefore conceptualized with the idea that strong anticipation allows for a more parsimonious explanation of the existing evidence for anticipatory synchronization of chaotic behaviors.

Given that anticipating another individual's behavior is conducive to effective social interaction, but often challenging when these behaviors are seemingly unpredictable, a provocative hypothesis is that small feedback delays might also promote the ability of individuals to anticipate the chaotic behaviors of other people. Accordingly, the overall aim of current study was to determine whether self-organized anticipatory synchronization could account for the ability of co-acting individuals to coordinate complex aperiodic behavioral movements. More specifically, we investigated whether the introduction of small perceptual-motor feedback delays would enable, rather than inhibit, a naïve coordinator's ability to anticipate the chaotic movements of another actor.

In Stepp's (2009) work, described above, identification of anticipatory synchronization involved determining the short-term lead/lag patterns of coordination exhibited between the actor's hand movements and those of the computer stimulus—with this analysis revealing how the presence of small feedback delays facilitated the ability of participants to temporally lead the highly variable and chaotic movements of the stimulus dot (i.e., by up to 300 ms). The same method was used in order to identify instances of interpersonal anticipatory synchronization in the current study. However, theories about the underlying processes that support complex interpersonal coordination have also recently started to emphasize the existence of coordination between longer-term behavioral patterns, and address the relative importance of short-term versus long-term coordination (Marmelat & Delignieres, 2012). Thus, in addition to considering short-term behavioral synchronization, the current study also investigated whether longer-term patterns of coordination can occur between the complex movements of co-acting individuals.

Of particular relevance to the current study, is recent empirical work that has successfully demonstrated that individuals can come to embody the long-term statistical structure of others' behaviors during a variety of joint action tasks, including finger tapping (e.g., Torre, Varlet, & Marmelat, 2013), pendulum swinging (e.g., Marmelat & Delignières, 2012), treadmill walking (e.g., Marmelat et al., 2014) and conversational interaction (Abney, Paxton, Dale, & Kello, 2014). Such long-term coordinative relationships are thought to indicate ‘global coordination’ between behaviors, and can be assessed through a comparison of the patterns of behavioral variability found within each of two concurrent, coordinated behavioral sequences (i.e., requires determining whether the complexity of the two behavioral sequences match; so called complexity matching). More specifically, this type of behavioral similarity can be quantified by comparing the fractal (i.e., self-similar) structure of the behavioral variability found within each of the two actors' concurrent, coordinated behaviors to determine whether the complexity of the two behavioral sequences match. Such global coordination appears to be a signature of self-organized anticipation because it indicates that the behavioral dynamics of each actor are self-similar and long-range dependent (Delignières & Marmelat, 2014)—meaning that each actor will display recurrent patterns of sensorimotor variability over a broad range of time scales. Thus far, it appears that both local and global associations between two coordinated, complex behaviors are valuable in shaping a shared structure. Questions remain, however, about their relative importance with respect to the locally defined phenomenon of anticipatory synchronization (see Stephen & Dixon, 2011 for further details). The evaluation of global complexity matching processes can therefore provide additional information about the underlying coordinative processes that support complex interpersonal coordination beyond that provided by analyses of local coordination (e.g., Delignières & Marmelat, 2014; Marmelat & Delignières, 2012), and were incorporated here to gain further information about the anticipation of chaotic behaviors in a joint-action context.

Current Study

In order to investigate whether perceptual-motor feedback delays might provide an opportunity for the emergence of interpersonal anticipatory synchronization, participants were asked to synchronize their arm movements with the chaotic movements of a co-actor while experiencing one of a range of visual-motor feedback delays. It is worth noting that past investigations of anticipatory synchronization (Masoller, 2001; Sivaprakasam et al., 2001; Stepp, 2009; Stepp & Frank, 2009; Toral, Mirasso, Hernández-García, & Piro, 2001; Voss, 2002) have involved a unidirectional coupling between subsystems whereby the slave system gains information about the master system, but not vice versa. For example, in Stepp's (2009) study of human-environment anticipatory synchronization, this coupling was achieved via visual attention (i.e., with an individual coupled/coupling to an ongoing stimulus display). In contrast to this unidirectional coupling between subsystems, social interaction often involves a bidirectional coupling, or mutual enslavement, between actors such that both actors have information about the other's behaviors through one or more sensory modalities (e.g., in face-to-face conversation both visual and auditory coupling typically exist between co-actor behaviors). Previous research comparing the effects of unidirectional and bidirectional coupling on short-term, interpersonal synchronization in finger tapping has demonstrated that bidirectional coupling leads to significantly higher levels of synchronization, apparently as a result of continuous mutual adaptation between co-actor behaviors (Konvalinka, Vuust, Roepstorff, & Frith, 2010). In the current study, we therefore chose to test two distinct visual coupling conditions between the co-actors, both of which involved the mutual enslavement characteristic of most joint action tasks. This allowed us to test whether anticipatory synchronization can occur in a bidirectionally coupled master-slave system, and subsequently establish whether this phenomenon does indeed occur during interpersonal interaction.

Based on the previous findings of Stepp (2009), we hypothesized that local coordination analyses would reveal that participants were able to both coordinate with and anticipate the chaotic, and seemingly unpredictable, movements of a co-actor when experiencing a small perceptual-motor delay. Furthermore, the use of two visual coupling conditions provided an opportunity for us to examine how the information available to a behavioral producer (i.e., master) about a coordinating co-actor's movements affects the producer's behaviors and, subsequently, the occurrence of anticipatory synchronization. In order to also gain information about the relationship between any local and global coordinative processes associated with the production of anticipatory synchronization, resulting coordination was assessed using an analysis of complexity matching in addition to analyses designed to identify local coordination. By bringing together research on visual rhythmic coordination, anticipatory synchronization, and complexity matching, we were particularly interested in whether the complex joint-action that is critical to achieving many everyday tasks might be supported by feedback delay-enhanced anticipatory processes of coordination.

Method

Participants

Twenty-two students (11 pairs) were recruited from the University of Cincinnati to participate in the experiment. Participants ranged in age from 18 to 27 years. Sample size was determined due to the fact that 6-12 data sets per experimental condition is the standard in comparable research examining the complex dynamics of human movement and coordination (see Marmelat & Delignières, 2012; Richardson, Campbell & Schmidt, 2009; Schmidt, Shaw, & Turvey, 1993; Stephen et al., 2008; Stepp, 2009).

Procedure and Design

Each participant was asked to sit facing their own display monitor (50” HD Plasma TV) and was equipped with a magnetic motion sensor, attached to the middle joint of the first two fingers of their right hand, in order to control the movement of a visual stimulus dot (Fig. 1). After arriving at the laboratory, one participant from each pair was randomly chosen to be a “producer” while the other was assigned the role of “coordinator”. Participants were not informed of these roles, but they were told that one of them would have the opportunity to practice briefly before they both would be asked to perform the experimental task together. The producer was then asked to come into the testing room first while the coordinator waited outside.

Fig. 1.

Experimental set-up. Producer movements were displayed on both screens as a red dot, and coordinator movements were displayed as a blue dot. Producers were asked to create chaotic, elliptical movements, and coordinators were instructed to synchronize with these movements.

During this practice period, the producer was asked to complete two trials, each lasting 100 s, in which they were instructed to coordinate with fully chaotic, simulated sequences based on the equation for a “chaotic spring” system, where the ‘x’ and ‘y’ coordinates for the stimulus dot were generated by the x₁ and x₂ dimensions of the system (Stepp, 2009)

$$\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-{\left(2\pi \right(\frac{x_3}{\alpha }+\beta \left)\right)}^2{x}_1\\ {\dot{x}}_3=-x_4-x_5\\ {\dot{x}}_4=x_3+\alpha x_4\\ {\dot{x}}_5=b+x_5(x_3-c)\end{array}$$ (1)

The x₃, x₄ and x₅ dimensions define a standard Rössler attractor, which produces the chaotic dynamics used to drive the simple harmonic oscillator specified in x₁ and x₂. Ultimately this system maintains an elliptical trajectory while varying chaotically in amplitude and frequency, resulting in movements that are possible for participants to track but difficult to predict. As previously established (Stepp, 2009), system parameters were set to a = b = 0.1, c = 14, α = 100 and β = 0.3. For each stimulus file created, an 800 s (sample rate = 120 Hz) simulated sequence was generated using the initial conditions x₁ = 1, x₂ = 0, x₃ = taken from uniform distribution of values between [18.5, 19.5], x₄ = 3.432, and x₅ = 20.9 (Stepp, 2009). From each 800 s sequence, a 200 s segment was selected for which amplitude remained relatively consistent. These 200 s segments were used as the stimulus time series.

The same two stimulus sequences were provided to all producers, but were presented in alternating order by participant pair. The producer saw these sequences displayed as a blue dot (2 cm in diameter) and their own real time, sensor-tracked movements displayed using a red dot (2 cm in diameter), on the right half of a white screen (the other half of the screen was covered). They were instructed to “keep the red dot as close to on top of the blue dot as possible” in order to complete the task. Similar to Stepp's (2009) task, and another circle drawing task previously used by Knoblich and Kircher (2004), participants were therefore explicitly required to attend to and coordinate their own intended distal events (e.g., the movement of the red dot). Once they had completed the two practice trials, the producer was told that their goal for the duration of the study would be to produce the same kind of movements they had been making: “generally circular and always in the same direction, but somewhat unpredictable in terms of the speed of movements and where they go”. They were also informed that the goal of the other participant would be to coordinate with their movements, as they had had to do with the computer stimulus during the practice trials.

The coordinator was then brought into the room and situated in front of their own display screen so that the two participants were back-to-back. They were then told that their co-participant had just practiced the kind of movement they would be making for the duration of the study and that their own goal was going to be to coordinate with that person's movements. The coordinators were informed that their co-participant's movements would be displayed using a red dot (2 cm in diameter), while their own real time, sensor-tracked movements would be reflected on the screen as a blue dot (2 cm in diameter). They were instructed to “keep the blue dot as close to on top of the red dot as possible” in order to complete the task (see Figure 2 for example movement time series). The coordinator would see these dots displayed on the left half of white screen (the other half of the screen was covered).

Fig. 2.

Bidirectional Coupling. (Top) For trials with no feedback delay, coordinator movements lagged producer movements. (Middle) In the ‘congruent’ coupling condition coordinator movements came to anticipate those of the producer. (Bottom) Coordinator movements also came to anticipate producer movements in the ‘incongruent’ coupling condition.

In order to assess whether producer and coordinator movements exhibited behavioral dynamics consistent with chaos, an analysis of the largest Lyapunov exponent (LLE) was conducted for each participant's movements (details of this analysis can be found in the appendix). The same patterns were observed in both the ‘x’ and ‘y’ dimensions for both coordinator and producer movements, and these values were averaged to establish characteristic LLE values for the producer and coordinator during each trial. Results of this analysis reveal that on average participant LLEs were positive (Table 1), indicating that participants produced chaotic movements for the majority of trials.

Table 1. LLEs for producer and coordinator movement.

	Visual Condition: Congruent
	Feedback Delay
	0 (ms)		200 (ms)		400 (ms)		600 (ms)
	M	SD	M	SD	M	SD	M	SD
LLE Producer	.075	0.05	.069	0.06	.078	0.03	.097	0.08
LLE Coordinator	.015	0.04	.027	0.04	.016	0.03	.025	0.05
	Visual Condition: Incongruent
	Feedback Delay
	0 ms		200 ms		400 ms		600 ms
	M	SD	M	SD	M	SD	M	SD
LLE Producer	.069	0.03	.071	0.05	.073	0.03	.073	0.05
LLE Coordinator	.030	0.04	.025	0.04	.033	0.02	.014	0.02

As discussed above, the unidirectional coupling between subsystems used in previous studies of anticipatory synchronization (Masoller, 2001; Sivaprakasam et al., 2001; Stepp, 2009; Stepp & Frank, 2009; Toral et al., 2001; Voss, 2002) is not representative of the relationship between individuals during the majority of social interactions. The design of the current study therefore utilized two visual bidirectional coupling conditions between the producer and coordinator participants (Fig. 2), both of which involved the mutual enslavement characteristic of most joint action tasks. That is, the producer (i.e. ‘master’ system), as well as the coordinator (i.e., ‘slave’ system) always had the opportunity to see the movements of their co-actor's dot with respect to their own movement outcomes. This gave us the opportunity to determine what arrangements of bidirectional coupling between actors might be able to support interpersonal anticipatory synchronization.

The first, congruent, visual condition was designed so that both individuals had the same information about the coordinator's behavior; the producer saw the coordinator's movements at the same perceptual delay that the coordinator experienced. In the second, incongruent, condition the producer always viewed the coordinator's movements in real time while the coordinator saw his or her own movements with a feedback delay. This situation introduced the possibility that, should anticipatory synchronization occur, the producer would perceive the coordinator's movements as leading their own. If anticipatory synchronization was dependent on having a master system that operates independently of slave system behavior, we might expect that allowing producers to perceive that coordinator behaviors occur before their own would result in a breakdown of the coordination, or a switching of co-actor roles such the coordinator, rather than the producer, would begin to drive the patterns of social motor coordination. However, if anticipatory synchronization is to be useful in understanding complex, interpersonal coordination then it must still occur in the context of bidirectional coupling between co-actors, and necessarily when both actors are able to see the other's behavior in real time. Accordingly, the bidirectional visual coupling conditions employed in the present study provided a test of whether self-organized anticipation can be achieved between mutually interactive subsystems, as well as whether different forms of bidirectional coupling might differentially impact resulting coordination and anticipation. Each feedback delay for the coordinator was experienced once in each display condition for a total of eight trials, with display condition-feedback delay pairings randomized within participant pairs. Each trial lasted a total of 100 s.

The display was generated by an application written using C/C++ and OpenGL. A Polhmeus Liberty magnetic tracking sensors (Polhemus LTD, Colchester, VT) were used by participants to control their visual stimulus. The OpenGL program was also used to record the movement data collected by the Polhemus motion sensors, at a sampling rate of 120 Hz. Horizontal and vertical coordinates of the undelayed participant movements, i.e. the movements of the arm, were captured as y₁ and y₂ with respect to the master-slave system

$$\begin{array}{c}\dot{x}=f(\mathbf{x})\\ \dot{y}=g(\mathbf{y})+k(\mathbf{x}-{\mathbf{y}}_{\tau })\end{array}$$ (2)

Here the x and y vectors represent the states of the master and slave systems, respectively, so that f(x) and g(x) specify the intrinsic dynamics of these systems and k is the coupling strength between the systems. The term y_τ = y(t – τ), or the state of the slave system delayed by τ in an unspecified unit of time.

It is important to note that while the coordinator's task goal in the current study was to maintain spatiotemporal synchrony between the blue and red stimulus dots, our interest was in evaluating the underlying behavioral anticipation which would allow the coordinator to achieve this desired distal event. All analyses were therefore conducted to examine the coupling and synchronization between the coordinator's arm and the producer's dot. The delayed movements of the coordinator's dot are synonymous with the term y_τ in Equation 2. The first 10 s of each time series were discarded to remove transients.

Measures and Analyses

Cross-Correlation and Phase Lead

To determine whether anticipatory synchronization occurred between coordinators and producers, we first performed a cross-correlation analysis between the movements of the coordinator and producer (see Stepp, 2009). Initially, the time series for the ‘x’ dimension of the coordinator movement and the time series for the ‘x’ dimension of the producer movement were each low-pass filtered with a cutoff frequency of 10 Hz using a Butterworth filter, and compared

$$\mathit{\text{xcorr}}(h)=\frac{{\sum }_i[(x(i)-\overline{x})(y(i-h)-\overline{y})]}{\sqrt{{\sum }_i{(x(i)-\overline{x})}^2}\sqrt{{\sum }_i{(y(i-h)-\overline{y})}^2}}$$ (3)

Here the x and y variables correspond to coordinator and producer positions, respectively, and xcorr(h) represents the normalized cross-correlation function of the two time series taken at a phase shift of the participant with respect to the stimulus equal to h. For each trial, the value of the cross-correlation between the two time series was calculated for each of a range of phase shifts of the participant with respect to the stimulus, extending 1 s ahead of and 1 s behind perfect synchrony (h = [-120, 120]). The following equation was then used in order to establish both the highest level of synchrony and the associated degree of phase shift for the two time series.

$$\mathrm{\text{degree of synchrony}}=\mathit{\text{xcorr}}(\text{phase lead})=max\mathit{\text{xcorr}}(h)$$ (4)

The values for maximum cross-correlation and phase lead were taken to be representative of the relationship between coordinator and producer movements for a given trial. This process was then repeated to compare the time series for the ‘y’ dimension of the coordinator movement to the ‘y’ dimension of the producer movement. Maximum cross-correlations between the coordinator and producer time series were calculated separately for the ‘x’ and ‘y’ dimensions. As the same patterns were observed in both dimensions, these values were then averaged across the ‘x’ and ‘y’ dimensions to establish a characteristic maximum cross-correlation and phase lead for each trial.

Instantaneous Relative Phase

To confirm the cross-correlation results, an analysis of the relative phase between the movements of the coordinator and producer in each participant pair was conducted (Haken, Kelso & Bunz, 1985; Lopresti-Goodman, Richardson, Silva & Schmidt, 2008; Pikovsky, Rosenblum & Kurths, 2003; Schmidt, Shaw & Turvey, 1993). Here, the time series for the ‘x’ dimension of the coordinator movement and the time series for the ‘x’ dimension of the producer movement were each submitted separately to a Hilbert transform in order to compute continuous phase angle series corresponding to each of the movement time series

$$\varphi (t)=arctan\left(\frac{s(t)}{Hs(t)}\right)$$ (5)

This process is based on the concept of the analytic signal (Gabor, 1946), with s(t) corresponding to the real part of the signal and Hs(t) corresponding to the imaginary part of the signal (Pikovsky, Rosenblum & Kurths, 2003). The instantaneous relative phase between the movements of the two actors can then be calculated as

$$\varphi (t)={\varphi }_1(t)-{\varphi }_2(t)$$ (6)

with φ₁(t) and φ₂(t) representing the continuous relative phase angles of coordinator and producer behaviors, respectively. The resulting instantaneous relative phase time series was used to create a frequency distribution of relative phase relationships visited over the course of a trial for each of 37 relative phase regions (-180°-180°, in 5° increments for the regions closest to 0° and 10° increments for all other regions). This process was then repeated to compare the time series for the ‘y’ dimension of the coordinator movement to the ‘y’ dimension of the producer movement.

The instantaneous relative phase between coordinator and producer movements was calculated separately for the ‘x’ and ‘y’ dimensions. As the same patterns were observed in both dimensions, these values were then averaged across the ‘x’ and ‘y’ dimensions to establish relative phase measures for each trial.

Box Counting

As mentioned above, recent research has demonstrated that the movements of interacting individuals do not only become entrained (matched) on a local or synchronous time-scale but can also become matched with respect to their long-term statistical structure and behavioral complexity (Coey et al., 2014; Delignières & Marmelat, 2014; Marmelat & Delignières, 2012; Torre et al., 2013). Such global coordination or complexity matching provides further evidence of self-organized anticipation by demonstrating that the behavior of co-actors is self-similar and long-range dependent (Coey et al., 2014; Delignières & Marmelat, 2014). As these characteristics are directly related to patterns of behavioral variability exhibited across several timescales, we were interested in measuring the long-term global coordination that occurred between the co-actors as a compliment to the assessments of the local (short-term) coordination provided by the previously discussed cross-correlation and relative phase analyses. To determine whether the movements of the coordinator were globally coordinated with the chaotic dynamics of the producer's movements, we first assessed the spatial self-similarities between producer and coordinator movements during each trial by calculating the fractal dimension (FD) of producer and coordinator movement time series (Grassberger & Procaccia, 1983; Kay, 1988). This method provides a measure of scale invariance and self-similarity over time through the analysis of a visual representation of behavior. Analysis involved covering the image with a grid of boxes, and obtaining a count of how many of the boxes contained part of the time series. This was done repeatedly for a single time series using a larger scaling factor, for which the number of boxes was equal to the square of the scaling factor. In the current study, a total of nine iterations were conducted with each successive scaling factor being the square of the previous, progressing from two to 512. The slope of a regression line fit to a plot of the logarithm of the inverse of the box size vs. the logarithm of the number of filled boxes yielded the measure of FD. For a fractal image these values should be non-integers between 1 and 2 (this upper limit of this dimension cannot be higher than that of the box used during measurement), with higher non-integer values indicating greater ‘fractality’. This provided an opportunity for us to compare behavioral complexity across trials and between concurrent movement time series.

In order to calculate the overall FD of each coordinator and producer movement series, we first created a two-dimensional image of each time series with the dimensions 512 × 512 pixels (using both the ‘x’ and ‘y’ dimensions). Images were then submitted to a box counting procedure which returned a unique FD value for each image.

Results & Discussion

Cross-Correlation and Phase Lead

Cross-correlation analysis demonstrated that the coordinators were able to successfully coordinate their movements with those of the producers throughout the experiment. A 2 (visual condition) × 4 (feedback delay) factorial ANOVA for the maximum cross-correlation between coordinator and producer movements revealed a significant main effect of feedback delay, F (3, 30) = 50.69, p < .001, η_p² = .84, but no main effect of visual condition or interaction between the two variables. As can be seen in Figure 3a, maximum cross-correlation generally decreased with an increase in feedback delay. Fisher's LSD post hoc comparisons revealed that all differences in average maximum cross correlation between feedback delay conditions were significant (p < .005).

Fig. 3.

Interpersonal Anticipatory Synchronization. (a) Average maximum cross-correlation between coordinator and producer movements for both visual conditions, in each of the feedback delay conditions. (b) Average temporal lead between coordinator and producer movements for both visual conditions, in each of the feedback delay conditions. Error bars show standard error. *p < .05, **p < .01. (c) Average distribution of instantaneous relative phase (IRP) values between coordinator and producer movements for the congruent visual condition (top) and incongruent visual condition (bottom) over the course of a trial, for each feedback delay condition.

A 2 (visual condition) × 4 (feedback delay) factorial ANOVA for the phase lead of the coordinator to the producer movements revealed a significant main effect of feedback delay, F (3, 30) = 6.65, p = .001, η_p² = .40, but no main effect of visual condition or interaction between the two variables. When coordinators did not experience delayed feedback about their own movements no anticipation (as measured by the time lag/lead at which the maximum cross correlation coefficient was found) was observed. Consistent with the phenomenon of anticipatory synchronization, however, in the 400 ms feedback delay condition the movements of the coordinator began to lead those of the producer, indicating that the coordinator was in fact anticipating the producer's chaotic (i.e., fundamentally deterministic, yet unpredictable) movements. A smaller degree of anticipatory synchronization was also observed for the 600 ms feedback delay condition, but overall the stability of coordination at this delay was poor in comparison to the other delay conditions, with the coordination becoming highly unstable, such that co-actor movements were no longer closely synchronized. Consistent with our observation of participants performing the task, it appears that the 600 ms delay simply makes the coordinator's goal of synchronizing with the producer so difficult that coordination in general is no longer well supported. It therefore appears that the emergence of anticipatory synchronization is sensitive to the length of feedback introduced such that longer delays allow for greater temporal lead by the coordinator, but only so long as high levels of coordination between the coordinator and producer are achievable. Fisher's LSD post hoc comparisons revealed significant differences in phase lead between the 0 ms feedback delay condition and both the 200 ms and 400 ms delay conditions (p < .001), as well as between the 200 ms delay condition and the 400 ms delay condition (p < .05).

Interestingly, the absence of a main effect between visual coupling conditions indicates that this difference had no influence on the behavioral patterns of coordination observed for the different feedback delay conditions (see Fig. 3). That is, when the coordinator was experiencing one of the manipulated feedback delays, allowing the producer to have information about the coordinator's movements in real time (i.e., instead of at the feedback delay that the coordinator was experiencing) did not appear to have any substantial effect on the occurrence of anticipatory synchronization. Moreover, compared to what has been observed in the context of unidirectional actor-environment coupling (Stepp, 2009), the bidirectional nature of the visual coupling employed in the current study appeared to have little effect on the emergence of anticipatory synchronization. This finding is critical to the understanding of anticipatory self-organization as an interpersonal coordinative process, as many complex social behaviors inherently involve mutual enslavement and information flow between actors.

Instantaneous Relative Phase

Consistent with the results of the maximum cross-correlation analysis, instantaneous relative phase distributions indicated that the coordinator did indeed anticipate the aperiodic movements of the producer for the 400 ms feedback delay condition (Fig. 3c). Additionally, although a modest degree of anticipatory behavior was observed for the 600 ms condition, the stability of the phase relationship between the coordinator and the producer at this feedback delay was again found to be very low. More interestingly, however, the distributions of relative phase not only revealed anticipatory behavior on the part of the coordinator for the 200 ms condition, but that for this condition, as well as for the 400 and 600 ms feedback delay conditions, coordinators both lagged and led producer movements. That is, substantial peaks in the relative phase distributions were observed at both negative (phase lag) and positive (phase lead) relative phase relationships, suggesting that the anticipatory synchronization observed was intermittent. Intermittent, or relative, coordination is a known characteristic of weakly coupled physical or biological limit-cycle oscillators (Kelso & Ding, 1993; von Holst, 1973), including visually coupled rhythmic limb movements of co-acting individuals (Schmidt & Richardson, 2008). However, it has never before been demonstrated with respect to anticipatory synchronization, nor has this phenomenon been observed between more complex, visually coupled movements of co-acting individuals.

Box Counting

The results of a box counting analysis on contemporaneous coordinator and producer movements revealed significant evidence of complexity matching in that there was a very strong positive relationship between the FD of producer movements and the FD of the associated coordinator movements (Fig. 4). Correlations between producer FD and coordinator FD at each feedback delay and within each coupling condition demonstrate that this was the case irrespective of visual coupling and feedback delay, although the relationship was weakest for the 600 ms feedback delay condition and highest for the 200 ms feedback delay condition (see Table 2).

Fig. 4.

Complexity Matching. Scatterplots show average fractal dimensions (FD) for producer and coordinator movements, by participant pair, for each combination of visual conditions and feedback delay conditions.

Table 2. Box counting correlations.

	Feedback Delay
	0 (ms)	200 (ms)	400 (ms)	600 (ms)
Congruent Visual Condition	.94***	.95***	.95***	.76**
Incongruent Visual Condition	.93***	.96***	.76**	.76**

^**p < .01

^***p < .001

The observation that complexity matching occurred for the current coordination task is consistent with existing research indicating that complexity matching can occur between actors during a wide variety of socially situated behaviors (Coey et al., 2014; Marmelat & Delignières, 2012; Stephen et al., 2008; Torre et al., 2013). However, although the occurrence of complexity matching is argued to result from the same lawful processes that support self-organized anticipation (Delignières & Marmelat, 2014; Stepp & Turvey, 2010), it is not yet clear that anything other than short-range coupling and adaptation processes are responsible for the emergence of such global coordination phenomena (Marmelat & Delignières, 2012; Torre et al., 2013). There also remains some debate as to whether local and global coordinative processes are mutually exclusive, or whether one or the other is more likely to occur in specific contexts (see Stephen & Dixon, 2011 for further details).

To address this question for the current task, correlations between the average maximum cross-correlation and the average difference between coordinator and producer FDs were calculated for the different delay and visual coupling conditions. This analysis revealed no consistent relationship (Table 3), indicating that the degree of complexity matching and global coordination observed within a given trial was not entirely dependent on the level of local or synchronous behavioral coordination achieved. Generally speaking, coordinators appear to have embodied the long-term structure of the producers' chaotic movements regardless of the strength of the local coordination observed. The one potential exception to this conclusion comes from the incongruent visual condition at the 400 ms feedback delay, for which there was a significant, negative association between the cross-correlation of co-actor movement and the degree of complexity matching between behaviors. In other words, more short-term synchrony between coordinator and producer movements was actually associated with lower levels of complexity matching between the two actors. The fact that this occurred only during the incongruent coupling condition suggests that the coordination processes associated with the observed anticipatory synchronization may in fact be somewhat affected by the nature of the bidirectional coupling between actors. In other words, while the two visual coupling conditions employed here were equivalent with respect to the actual levels of local and global coordination that were supported, it appears that the relationship between these phenomena may be affected by the consistency of information available to the two co-actors during performance. However, further work beyond the scope of the current study is needed gain a more in-depth understanding of the factors influencing a potential relationship between the local and global coordinative processes associated with anticipation.

Table 3. Relationship between local and global coordination measures.

	Feedback Delay
	0 (ms)	200 (ms)	400 (ms)	600 (ms)
Congruent Visual Condition	-.07	-.19	.09	-.27
Incongruent Visual Condition	.19	-.24	-.72*	-.48

^*p < .05

Conclusions

The current study was designed to demonstrate that anticipatory synchronization of chaotic behaviors could occur during human social interaction. Our findings demonstrate that the short perceptual feedback delays necessary for an actor to achieve anticipatory synchronization in an intrapersonal (i.e., non-social) actor-environment context (Stepp, 2009; Voss, 2000) also appear to be critical for anticipatory synchronization in the context of social interaction. It seems likely that the reactions of a co-actor may serve as one source of delayed feedback during naturally occurring social interaction. More generally, evidence that feedback delays act to facilitate anticipatory synchronization similarly in a variety of physical systems indicates that a consistent mechanism is likely to underlie the phenomenon across contexts. Within the current task, feedback delays around 200 ms to 400 ms were observed to be most effective in facilitating anticipation, with a significant breakdown in coordination being observed for the longer delay of 600 ms. These results further suggest that the very short temporal delays known to exist within the human sensorimotor system may also aid in the production of anticipatory behaviors (Thorpe, 2002; Wallot & Van Orden, 2012). In other words, the presence of these sensorimotor feedback delays may facilitate the production of stable intrapersonal and interpersonal coordination patterns, rather than destabilize behavioral performance as is often assumed. In fact, for a single individual coordinating multiple aspects of their behavior (e.g., bimanual movements) we would not only expect to see a similar pattern of results with respect to anticipation as that presented here, but also increased levels of both local and global coordination due to the fact that the kinematic constraints of a single person at two different time points, or the two simultaneously moving hands of one individual are likely to be more similar than those experienced by two unique individuals. This possibility calls into question the need to always posit theoretical constructs such as internal forward models (Noy et al., 2011; Wolpert et al., 2003) in order to explain how behavioral production and interaction can be accurate and stable despite the supposedly deleterious effects of perceptual-motor feedback delays.

It is important to note that the experimentally introduced delays used here are superimposed on top of the delays already inherent within the human sensorimotor system, and likely provide an exaggerated view of the naturally occurring anticipatory synchronization processes that result from existing delays. In fact, the ability of an actor to coordinate with the chaotic behaviors of their co-actor, at a very short temporal lag, and in the absence of any experimentally introduced feedback delay, may be evidence for naturally occurring anticipatory synchronization (Stepp & Turvey, 2010; Stepp & Turvey, 2008). Indeed, successful anticipatory behavior does not necessarily require that an actor consistently lead a co-actor's behavior, as is indicated by the intermittency of the anticipatory synchronization observed here. Rather, the outcome may be constrained to the maintenance of a functional level of synchronization, but with increases in feedback delay (up to around 400 ms) supporting an exaggerated expression of naturally occurring behavioral anticipation.

Notably, our findings also demonstrate that anticipatory synchronization can be achieved not only for unidirectionally coupled “master-slave” systems, as has been previously observed (Masoller, 2001; Sivaprakasam et al., 2001; Stepp, 2009; Toral et al., 2001; Voss, 2002), but also within a system made up of two bidirectionally coupled, human co-actors. Even within the incongruent visual coupling condition, in which the producer could have noticed that coordinator movements were ahead of their own, the same anticipatory relationship between coordinator and producer movements was achieved. Our findings therefore indicate that it is not necessary for one to be “ahead” in order to be driving coordinated joint-action. On the contrary, it appears that functional coordination of such complex behaviors is flexible, and resilient to small fluctuations in the phase relationship between movements. This intermittency might make naturally occurring anticipatory synchronization less obvious during everyday behaviors, as people often move ahead of or behind one another.

The self-organized anticipation underlying such local, interpersonal coordinative behaviors also appears to have resulted in global coordination between co-actors. Interestingly, while feedback delays around 200-400 ms seem to promote high levels of local behavioral anticipation, the occurrence of global coordination does not appear to be influenced by the introduction of these short delays (the 600 ms delay appears to disrupt the stability of coordination at all levels examined). Previous studies have demonstrated, however, that complexity matching for one system with respect to another is dependent on pre-existing statistical self-similarity of the behavior that a coordinating system or individual is trying to match (Delignières & Marmelat, 2014). Ultimately, this allows the coordinating system to exploit the existing complexity of this ongoing behavior, in order to produce more adaptive and efficient behavior with respect to any task goal (Delignières & Marmelat, 2014; Marmelat & Delignières, 2012). For a bidirectionally coupled system, the chaotic or self-similar complexity within each component subsystem provides an opportunity for mutual adaptation and bidirectional anticipation (Delignières & Marmelat, 2014; Konvalinka et al., 2010; Marmelat & Delignières, 2012) that is essential for the development of corresponding fluctuations in behavioral variability (e.g., changes in movement trajectory and velocity). Therefore, the chaotic behaviors produced in the current task are understood as supporting self-organization of both local and global coordinative phenomena.

The absence of a consistent relationship between the level of local coordination and the degree of similarity in global behavioral structure for coordinated movements, however, demonstrates that the level of local coordination between co-actors does not directly predict similarity in large-scale behavioral complexity. There is, therefore, necessarily some dissociation between local and global coordinative processes for the aperiodic behaviors being generated. As required by the task, participants in the current study were always locally coordinated to their co-participant, but it seems possible that even very low levels of local anticipatory synchronization may be associated with substantial global anticipation. It is also possible that the occurrence of complexity matching could be derived from similarities in the intrinsic physical and dynamical properties of perceptual-motor systems and that these similarities might act to shape the global structure of the behavioral dynamics without requiring behavioral synchrony on short time scales.

The local anticipatory behavior and global complexity matching of interacting individuals observed here is understood to be a natural consequence of the universal, lawful dynamics that shape and constrain the time-evolving structure of behavior. Although many biological and human behaviors may be chaotic (Canavier, Clark & Byrne, 1990; Chay & Rinzel, 1985; Mitra, Riley & Turvey, 1997; Newell, Challis & Morrison, 2000; Newell, Deutsch & Morrison, 2000), they are still lawful and deterministic. This implies that the behavioral dynamics of all human, perceiving-acting agents are constrained by the same physical laws of energy dissipation and information flow, and that these intrinsic commonalities in behavioral order allow for the self-organized emergence of anticipatory coordination (Kelty-Stephen & Dixon, 2012; Kugler & Shaw, 1990; Stepp & Turvey, 2008; Stepp & Tuvey, 2010). The dynamics of delay-induced anticipatory synchronization might therefore provide a lawful explanation for how and why we can achieve the robust social anticipation and coordination that underlies everyday activities. The implication is that explanations of how an individual is able to navigate a busy sidewalk, load a dishwasher with a friend or family member, or coordinate their movements with others during a dance or music performance, while necessarily shaped by the dynamics of the brain and nervous system, might not require recourse to a set of internal, ‘black-box’ compensatory neural simulations, representations, or feed-forward motor programs.

Appendix. Largest Lyapunov Exponent Analysis

The largest Lyapnuov exponent (LLE) can be calculated for a single time series as a characterization of the attractor dynamics (Eckmann & Ruelle, 1985), with a positive LLE being indicative of chaotic dynamics. For this analysis, the time series for the ‘x’ dimension of the coordinator movement and the time series, the ‘y’ dimension of the coordinator movement, the ‘x’ dimension of the producer movement, and the ‘y’ dimension of the producer movement were each treated separately. A pre-existing algorithm (Rosenstein, Collins & De Luca, 1993) was used as the basis for establishing the LLE of a time series in the current study. The first step of this process is to reconstruct the attractor dynamics of the series. This necessitated the calculation of a characteristic reconstruction delay or ‘lag’, and embedding dimension.

Average Mutual Information (AMI), a measure of the degree to which the behavior of one variable provides knowledge about the behavior of another variable, was used here to establish the appropriate lag for calculation of the LLE. This process involves treating behaviors of the same system at different points in time as the two aforementioned variables (Abarbanel, Brown, Sidorowich & Tsmring, 1993). As a preliminary step to the use of this algorithm, each time series was zero-centered. The calculation for AMI within a single time series was conducted using

$$I(T)={\sum }_{n=1}^{N}P(s(n),s(n+T)){log}_2\left[\frac{P(s(n),s(n+T))}{P(s(n),P(s(n+T))}\right]$$ (1)

where P represents the probability of an event, s(n) is one set of system behaviors and s(n + T) are another set of behaviors from the same system, taken at a time lag T later. In other words, I(T) will return the average amount of information known about s(n + T) based on an observation of s(n). The AMI, I(T), can then be plotted as a function of T in order to allow for the selection of a specific reconstruction delay, T, that will define two sets of behaviors that display some independence, but are not statistically independent. Previous researchers (Fraser & Swinney, 1986) have previously identified the first local minimum (T_m) of the plot as an appropriate choice for this value. In the current study a plot for each time series was evaluated individually, and the characteristic T_m selected by hand.

In order to find an appropriate embedding dimension for the reconstruction of attractor dynamics, the False Nearest Neighbors algorithm was used (Kennel, Brown & Abarbanel, 1992). As a preliminary step to the use of this algorithm, each time series was zero-centered. The idea behind this process is to project the attractor in an increasing number of dimensions, each time assessing whether apparent crossings of the attractor orbit with itself are an artifact of being projected within too few dimensions, until no ‘false neighbors’ remain. In practice, the square of the Euclidean distance between a point in a vector time series, y(n), and its nearest neighbor y^(r)(n) within a given dimension, d, is computed using

$$R_d^2(n,r)={\sum }_{k=0}^{d-1}{[x(n+kT)-x^{(r)}(n+kT)]}^2$$ (2)

where T is the time delay used for embedding. When the embedding dimension is increased, to d+1, a new square of Euclidean distance between points can be calculated using

$$R_{d+1}^2(n,r)=R_d^2(n,r)+{[x(n+dT)-x^{(r)}(n+dT)]}^2$$ (3)

If the distance between neighbors appears to change significantly with the increase in embedding dimension, an embedding error has likely occurred. This change was quantified using

$${\left(\frac{[R_{d+1}^2(n,r)-R_d^2(n,r)]}{R_d^2(n,r)}\right)}^{1/2}>R_{\mathit{\text{tol}}}$$ (4)

with any value of R_tol greater than 15 being classified as a false neighbor. A plot of the percentage of false neighbors as a function of embedding dimension was then used to identify the minimum embedding dimension, d_min, for which the percentage of false neighbors is very close to 0. Similar to the identification of an appropriate reconstruction delay from AMI, a plot for each time series was evaluated individually, and the characteristic embedding dimension selected by hand.

Using the minimum embedding dimension, d_min, and reconstruction delay, T_m, the reconstructed attractor trajectory was defined by the matrix

$$X_i=[x_i{x}_{i+T_m}\dots x_{i+(dmin-1)T_m}]$$ (5)

where X_i is the state of the system at a discrete time, i. The LLE algorithm was then used to find the ‘nearest neighbor’ of each point in the series (Rosenstein et al., 1993). In selecting the nearest neighbor, the closest point, X_j′, was taken with respect to a given reference point, X_j, using

$$d_j(0)={min}_{X_{j\prime }}\Vert X_j-X_{j\prime }\Vert$$ (6)

where d_j (0) represents the initial distance between the j^th point and the closest neighbor, and ‖..‖ denotes the Euclidean norm. The temporal difference between points was also taken into account, with the requirement that it be greater than the mean period of the time series (also calculated previously as the average time between peaks within the original movement time series). Each neighbor point was then taken to be an initial condition for a unique trajectory, with the mean rate of separation between them constituting an estimate of the LLE for the time series. In other words, the j^th pair of nearest neighbors is thought to diverge at a rate approximately equivalent to the LLE, λ₁, as defined by

$$~C_j{e}^{{\lambda }_1(i\ast \mathrm{\Delta }t)}$$ (7)

with C_j being the initial separation between points. The logarithm of both sides of Equation 12 were then taken to produce

$$ln{d}_j(i)~ln{C}_j+{\lambda }_1(i\ast \mathrm{\Delta }t)$$ (8)

which represents a series of approximately parallel lines for increasing values of j, each with a slope roughly equivalent to λ₁. The LLE can then be calculated by creating a least-squares fit to the “average” line defined by

$$y(i)=\frac{1}{\mathrm{\Delta }t}\langle ln{d}_j(i)\rangle$$ (9)

where 〈..〉 represents the average over all values of j. This analysis allowed us to detect chaotic dynamics in participant movement.