Journal of Experimental Psychology: Human Perception and Performance

Transcription

1 Journal of Experimental Psychology: Human Perception and Performance Self-Organized Complementary Joint Action: Behavioral Dynamics of an Interpersonal Collision-Avoidance Task Michael J. Richardson, Steven J. Harrison, Rachel W. Kallen, Ashley Walton, Brian A. Eiler, Elliot Saltzman, and R. C. Schmidt Online First Publication, March 9, CITATION Richardson, M. J., Harrison, S. J., Kallen, R. W., Walton, A., Eiler, B. A., Saltzman, E., & Schmidt, R. C. (2015, March 9). Self-Organized Complementary Joint Action: Behavioral Dynamics of an Interpersonal Collision-Avoidance Task. Journal of Experimental Psychology: Human Perception and Performance. Advance online publication.

2 Journal of Experimental Psychology: Human Perception and Performance 2015, Vol. 41, No. 2, American Psychological Association /15/$ Self-Organized Complementary Joint Action: Behavioral Dynamics of an Interpersonal Collision-Avoidance Task Michael J. Richardson University of Cincinnati Rachel W. Kallen, Ashley Walton, and Brian A. Eiler University of Cincinnati Steven J. Harrison University of Nebraska Elliot Saltzman Boston University and Haskins Laboratories, New Haven, Connecticut R. C. Schmidt College of the Holy Cross Understanding stable patterns of interpersonal movement coordination is essential to understanding successful social interaction and activity (i.e., joint action). Previous research investigating such coordination has primarily focused on the synchronization of simple rhythmic movements (e.g., finger/forearm oscillations or pendulum swinging). Very few studies, however, have explored the stable patterns of coordination that emerge during task-directed complementary coordination tasks. Thus, the aim of the current study was to investigate and model the behavioral dynamics of a complementary collision-avoidance task. Participant pairs performed a repetitive targeting task in which they moved computer stimuli back and forth between sets of target locations without colliding into each other. The results revealed that pairs quickly converged onto a stable, asymmetric pattern of movement coordination that reflected differential control across participants, with 1 participant adopting a more straight-line movement trajectory between targets, and the other participant adopting a more elliptical trajectory between targets. This asymmetric movement pattern was also characterized by a phase lag between participants and was essential to task success. Coupling directionality analysis and dynamical modeling revealed that this dynamic regime was due to participant-specific differences in the coupling functions that defined the task-dynamics of participant pairs. Collectively, the current findings provide evidence that the dynamical coordination processes previously identified to underlie simple motor synchronization can also support more complex, goal-directed, joint action behavior, and can participate the spontaneous emergence of complementary joint action roles. Keywords: joint action, movement coordination, task-dynamics, perception-action Many of our physical actions and movements are performed together with other individuals and can be characterized as interpersonal or joint actions. 1 Although there is a growing body of Michael J. Richardson, Center for Cognition, Perception and Action, Department of Psychology, University of Cincinnati; Steven J. Harrison, School of Health, Physical Education and Recreation, University of Nebraska; Rachel W. Kallen, Ashley Walton, and Brian A. Eiler, Center for Cognition, Perception and Action, Department of Psychology, University of Cincinnati; Elliot Saltzman, Department of Physical Therapy and Athletic Training, Sargent College of Health and Rehabilitation Sciences, Boston University and Haskins Laboratories, New Haven, Connecticut; R. C. Schmidt, Department of Psychology, College of the Holy Cross. The research was supported by National Institutes of Health, R01GM Special thanks to Ryan May for help with data collection. Thanks also to Charles Coey and Manu Varlet for methodological suggestions and pilot testing. Correspondence concerning this article should be addressed to Michael J. Richardson, Center for Cognition, Action and Perception, Department of Psychology, University of Cincinnati, ML 0376, 4150 B, Cincinnati, OH michael.richardson@uc.edu research investigating the neural and cognitive mechanisms that play a role in joint action (e.g., Bekkering et al., 2009; Gallese, 2003; Graf, Schütz-Bosbach, & Prinz, 2009; Newman-Norlund, Noordzij, Meulenbroek, & Bekkering, 2007; Vesper, Butterfill, Knoblich, Sebanz, 2010), identifying the dynamical processes of motor coordination by which individuals are mutually responsive to one another in time and space is also crucial to understanding such behavior (Schmidt, Fitzpatrick, Caron, & Mergeche, 2011; van der Wel, Knoblich, & Sebanz, 2011; Vesper, van der Wel, Knoblich, Sebanz, 2013). Indeed, the dynamics of interpersonal motor coordination appear to provide the fundamental structure for joint activity by forming the embodied context for shared intentions (e.g., Coey, Varlet & Richardson, 2012; Marsh, Richardson & Schmidt, 2009; Oullier & Basso, 2010; Sebanz & Knoblich, 2009). Moreover, stable motor coordination can increase rapport and cooperation between individuals (Hove & Risen, 2009), re- 1 Throughout this article the term joint action is used to label the cooperative physical activity displayed between two individuals participating in a task together. It does not refer to movement coordination at the level of individual joints in the musculoskeletal system. 1

3 2 RICHARDSON ET AL. duce perceived social differences and prejudice (Miles, Lumsden, Richardson, & Macrae, 2011), and has been shown to break down in pathologies such as premature birth, autism, and schizophrenia (Goldfield, Richardson, Lee, Margetts, 2006; Fitzpatrick, Diorio, Richardson & Schmidt, 2013; Varlet et al., 2012). Given the importance of understanding the dynamics of joint coordination, it is perhaps surprising that the majority of the existing research has only investigated the presence of these dynamic processes in tasks in which intentional or unintentional coordination was observed between highly stereotyped or nonfunctionally directed oscillatory limb or body movements (e.g., finger/forearm oscillations, pendulum swinging; e.g., Oullier, De Guzman, Jantzen, Lagarde, & Kelso, 2008; Richardson, Marsh, & Schmidt, 2005; Schmidt, Carello, & Turvey, 1990; Schmidt & O Brien, 1997). 2 The significance of this research is that it demonstrates that the rhythmic movements of informationally coupled individuals are dispositionally constrained to inphase (a stable 0 relative phase relation) and antiphase (a stable 180 relative phase relation) patterns of one-to-one behavioral synchrony, 3 and can be understood and modeled using the same coupled oscillator dynamic known to underlie intrapersonal interlimb coordination (see Kelso, 1995; Schmidt & Richardson, 2008 for a review). Although pertinent to understanding the coordination dynamics of the spontaneous movement synchronization that occurs, say between the leg movements or full-body movements of two individuals while walking or sitting side-by-side (e.g., Harrison & Richardson, 2009; Richardson, Marsh, Isenhower, Goodman & Schmidt, 2007; van Ulzen, Lamoth, Daffertshofer, Semin, & Beek, 2008; Varlet, Marin, Lagarde, & Bardy, 2011), the extent to which the conclusions drawn from these interpersonal coordination studies can be generalized to more natural joint actions remains to be determined. Everyday joint actions typically require that coactors coordinate goal-directed movements in a complementary manner, such that each actor adopts a different action role to jointly accomplish the task at hand (Bosga, Meulenbroek, & Cuijpers, 2010; Graf et al., 2009). In other words, the mutual responsiveness or spatialtemporal patterning of coactors movements (i.e., who does what and when) is often asymmetric rather than symmetric. 4 Furthermore, natural joint actions are often nonstationary and adaptive, with a temporal and spatial patterning evolving in real time as the interaction between the coacting agents and their environment unfolds (Sebanz & Knoblich, 2009). Accordingly, the current study investigated the dynamics of a more complex joint action rhythmic coordination task, in which individuals had to perform goal-directed movements without colliding into one another. A continuous repetitive targeting task with a collision-avoidance requirement was chosen and is illustrated in Figure 1. The task required a pair of participants to continuously and repetitively move computer stimuli (visual stimulus dots) back and forth between diagonally opposed target locations (i.e., bottom-left to top-right for one participant, bottom-right to top-left for the other participant) at a comfortable, self-paced tempo without the stimuli colliding into each other. The stimulus display (i.e., targets and controlled stimuli) was presented to participants on large computer monitors and participants used a hand-held motion-tracking sensor to control and move the stimuli back and forth between their respective target locations (see the Method section below for more details). This task was chosen because many joint actions involve the continuous production of repetitive movements over time. For instance, the same or similar movements are performed in a repetitive manner when two individuals are loading a dishwasher, stacking a pile of blocks or magazines, dancing, or fighting together. These social activities, however, do not involve the incidental inphase or antiphase movement synchronization that has been the focus of the majority of previous studies of social coordination. On the contrary, they require that individuals explicitly avoid colliding into each other and establish a more complex or complementary pattern of movement coordination since prototypical inphase or antiphase coordination patterns can often be detrimental to task performance. In a similar fashion, the current task was designed so that prototypical inphase or antiphase patterns of movement coordination would result in task failure. More specifically, if both participants were to move in a straight-line path between targets, then their stimulus dots would collide if they coordinated in an inphase or antiphase manner because they would pass the position located equal distance from their respective target locations at the same time (i.e., their stimulus dots would cross the center of the task space at the same time). This can be discerned from an inspection of Figure 1c. It is important to appreciate that because the target locations that each participant had to move back and forth between were positioned near the ends of the workspace s major diagonal axis (for one participant) and minor diagonal axis (for the other participant), inphase and antiphase coordination can only be defined arbitrarily. That is, synchronously arriving at the top and bottom targets together or synchronously arriving at the left and right targets together can both be considered to define either inphase or antiphase patterns of coordination. The implication of designing the task such that straight-line inphase coordination would result in failure (i.e., collision) was that the task entailed a conflict between the natural tendency of coactors to coordinate their movements in a synchronous manner (e.g., Richardson et al., 2007; Richardson, Campbell, & Schmidt, 2009; Schmidt & O Brien, 1997) and the task goal of avoiding collision. A natural question, therefore, is how would the task be completed? What, if any, stable patterns of movement coordination would ensure task success? One task solution would be for participants in a pair to maintain straight-line trajectories between their respective targets, but coordinate their movements with a non-0 or non-180 phase relation. For example, a 90 relative phase relation, with one participant moving a ¼ cycle in front or 2 Research investigating the dynamics of interpersonal and mutliagent coordination in sport is the exception (see, e.g., Lagarde, Peham, Licka, & Kelso, 2005; Hristovski, Davids, & Araujo, 2006; Okumura et al., 2012; Passos et al., 2009). Also see Mottet, Guiard, Ferrand, and Bootsma (2001) for an example of research examining an interpersonal Fitts Law task. 3 Inphase or 0 relative phase synchronization refers to the movements that oscillate in the same direction at the same time. Antiphase or 180 relative phase synchronization refers to the movements that oscillate in opposite directions at the same time. 4 Throughout this article we use the term symmetry to refer to an equivalence of some kind. This equivalence could be a spatial or temporal equality or an invariant correspondence under some form of functional transformation. The term symmetry can also apply to mathematical objects, operations or functions that result in an equivalence in the state(s) or changes in state(s) of a mathematical variable or system. An asymmetry or break in the symmetry of an object or system, therefore, refers to a relationship of nonequivalence (or one that has fewer equalities or invariant relations) with respect to some form of spatial, temporal, functional or mathematical transformation.

4 SELF-ORGANIZED COMPLEMENTARY JOINT ACTION 3 behind their coparticipant (here the term cycle refers to a single back and forth movement; e.g., a participant going from their bottom target to their top target and then back to their bottom again). Thus, when one participant was arriving at a target location the other participant would be midway between target locations. It is well known, however, that for rhythmic or repetitive movement tasks a 90 relative phase relation (or any non-0 or non-180 relative phase relation) is inherently unstable and therefore quite difficult to learn and maintain (e.g., Haken, Kelso, & Bunz, 1985; Kelso, 1995; Zanone, & Kelso, 1992). We therefore expected that it would be more likely for one or both participants to deviate from a strict straight-line path and converge on a set of more elliptical movement trajectories that minimized the chance of a collision (i.e., created a path of safe travel) but that, at the same time, also allowed subjects to move between or arrive at the target locations in a synchronous (i.e., inphase) manner. In truth, the novelty of the repetitive collision avoidance task employed here precluded making a specific a priori prediction about the kinematic details of this movement pattern (i.e., because the task solution space is too large). However, we did expect that participant pairs would quickly learn to perform the task effectively and would converge onto a stable pattern of behavior. Of particular interest was (a) whether all participant pairs converged on the same spatiotemporal pattern of behavioral coordination (task solution) and (b) what, if any, complementary roles emerged during stable task performance. Method Participants Twelve pairs (24 participants in total) of undergraduate students from the University of Cincinnati participated for partial course credit. All participants were right-handed and had normal or corrected-to-normal vision. All of the procedures and data collection tools employed for the current study were approved by the University of Cincinnati IRB. Materials Figure 1. Representations of the experimental setup for each (a) individual in a pair and (b) for the pair together, as well as for the (c) task display. Each participant in a pair stood facing a 50-inch computer monitor (operating at a 60-Hz refresh rate), with the screens positioned so that participants were back-to-back and could not see each other (see Figure 1a and 1b). The stimulus controlled by each participant was a small red dot 3.5 cm in diameter. The targets were 10 cm squares and were positioned in each of the four corners of the monitor, with one participant moving their stimulus between the bottom-left and top-right target set, and the other participant moving their stimulus between the bottom-right and top-left target set (Figure 1c). A Polhemus FASTRACK magnetic motion tracking system was used to record and track the movements of each participant at 60 Hz. Participants held a cm motiontracking sensor in their dominant (right) hand to control their stimulus. Each monitor displayed the real-time motion of the participant s own stimulus and targets, as well as the real-time motion of their coparticipant s stimulus and targets. That is, the data from the Polhemus motion tracking sensors was used to control the stimulus in real-time. A custom C/C program, using the OpenGL graphics library, was written to present the stimuli and record the data from the motion tracking sensors. The data collection stream was synchronized with the refresh rate of the monitor (i.e., 60 Hz); hence, the delay between the participants arm/hand movements and the controlled visual stimulus dot was less than or equal to ms. Stimulus collisions were deemed to have occurred if any part of the participants stimuli would have overlapped as a function of the current and previous two sample locations. Extensive pilot testing was completed to verify the accuracy of data collection and collision detection. Procedure Participants were informed that the experiment was investigating multiperson rhythmic coordination and that each of them would be required to move a computer stimulus continuously between two sets of targets while not letting their stimuli collide. Following these brief instructions, the participants were positioned in front of their respective monitor, handed their motion sensors, and randomly assigned to a target pair along either the major or minor diagonal axis of the monitor. Each participant then completed a 40-s practice trial alone, in which they were instructed to move their stimulus back and forth between their target locations continuously and at a comfortable speed. During these practice trials, the participant who was not moving their stimulus closed their eyes and placed their hands by their sides.

5 4 RICHARDSON ET AL. After completing the practice trials, participants were informed that they were to begin the joint coordination task. Participants were instructed to start each trial at their bottom target and that the goal of the task is to move your stimulus back and forth between your target locations continuously and at a comfortable speed and to coordinate your movements with each other so that your stimuli do not collide, hit, or bump into each other. No other instructions as to the pattern or form of the coordination were provided. Participants were also told to not converse during the experiment. Once participants indicated that they understood the task instructions, they were informed that the trial-by-trial procedure was designed as a game. Specifically, they were told that if they continuously moved between their respective targets without their stimuli colliding into each other for the entire length of a trial 40 seconds they would receive 1 point and that the experiment would end when they reached a score of 15 points. After completing 15 successful trials, participants were debriefed and thanked for their cooperation. Data Reduction and Processing One pair was unable to complete the task within the allotted time for the experimental session (40 minutes) only reaching a score of 7 of 31 trials and was therefore dropped from the analysis. The data from the last successful trial (i.e., Trial 15) of another pair was also dropped due to equipment malfunction. Thus, there were a total of 164 successful trials completed by participant pairs. On average, the 11 pairs retained for analysis took 21 trials to reach a score of 15 (range trials, median 19), with 8 of the 11 pairs reaching a score of 15 in less than 22 trials (i.e., 7 or fewer failed trials). Given the relatively small number of failed trials, compared to successful trials, and the fact that 69% of failures occurred within the first 20 s of a trial (mean failure time of s), only data from successful trials were analyzed. Prior to analysis, movement time-series from successful trials were low-pass filtered using a 10 Hz Butterworth filter and the first 5 s of each trial was removed to eliminate startup movement transients. The movement and coordination patterns that emerged during successful task completion were determined using (a) movement frequency analysis, (b) principal components analysis, (c) measures of relative phase; and (d) coupling directionality (see below for more details). Movement Frequency Results For each participant on each trial, average movement frequency was computed separately for the x (horizontal/frontal plane) and y (vertical/sagittal plane) movement time-series as the average of the sequence of inverse peak-to-peak durations in the respective x and y time series. An inspection of these movement frequencies revealed that there was no meaningful difference between the movement frequencies of participants x and y time series (Mean difference Hz), nor was there a meaningful difference between the movement frequencies of coparticipants on a given trial (Mean difference Hz). Thus, the mean and COV (i.e., the coefficient of variation of movement frequency SD of movement frequency normalized by the mean movement frequency) for each pair on each trial was calculated as the average of the coparticipants pooled sets of movement frequencies from both the x (horizontal/frontal plane) and y (vertical/sagittal plane) movement time-series. As can be seen from an inspection of Figure 2, in which the across-pair mean and COV of frequency are plotted as a function of trial, participant pairs increased their frequency of movement across trials from approximately 0.4 Hz for the first trial to approximately 0.58 Hz in the 15th trial, while maintaining the same magnitude of frequency variability. To determine whether the changes in the mean and COV of movement frequency over trials were significant, the difference between the data averaged within the first, middle, and last three trials (i.e., Trials 1 3, 7 9, and 13 15, respectively) was compared using a one-way repeated measures ANOVA. This analysis revealed a significant difference in mean frequency, F(2, 20) 47.22, p.01, 2 p.83, across trial blocks, but no difference in the COV of frequency, F(2, 20) 1. With respect to the former mean frequency analysis, Bonferroni post hoc tests revealed that the first, middle, and last trial blocks were all significantly different from each other (ps.02). Principal components analysis (PCA). PCA was conducted on the movement data from each trial as a whole (i.e., not to each individual movement cycle) and was used to calculate the normalized spread or width ( ) of each participant s cyclic movement paths about the principal axis of movement. equals the ratio of the eigenvalues ( ) from the covariance matrix between a participant s x (horizontal /frontal) and y (vertical/sagittal) data (i.e., Figure 2. Across-pair mean and COV of movement frequency as a function of trial. Error bars indicate standard error of the mean ( p.05; p.01; ns. p.05).

6 SELF-ORGANIZED COMPLEMENTARY JOINT ACTION 5 2 / 1 ). As a normalized ratio of the excursions perpendicular to the principal axis of motion, provides a measure of movement straightness or spread relative to the angular direction of motion (Duarte & Zatsiorsky, 2002). For the current data, the larger the value of, the greater the deviation from a straight-line trajectory between targets (i.e., the more elliptical a participant s movements). The importance of for understanding the movement patterns adopted by participants to successfully perform and complete the task can be seen from an inspection of Figure 3, in which the movement time-series trajectories for Trials 1, 8, and 15 for five different pairs of participants are displayed. The data is a representative sample of the movement patterns observed across all pairs and captures the characteristic changes in the trajectories of participants from the first to the last successful trial. Two distinct features of the movements displayed in Figure 3 should be noted. First, one or both participants in a pair tended to adopt a more elliptical or oval movement trajectory over the course of the experiment, particularly within the first ½ of the successfully completed trials. A one-way repeated measures ANOVA comparing pairs mean for the first, middle, and last blocks of trials was significant, F(2, 20) 8.40, p.01, p 2.47, with Bonferroni post hoc tests revealing that the first trial block was significantly lower (i.e., less elliptical movement paths) than the middle and last trials blocks (both p.05; see Figure 4, left panel). Figure 3. pairs. Example movement data. Trials 1, 8, and 15 for five different Second, one participant in a pair tended to adopt a greater degree of movement ellipticality than their coparticipant. The differentiation of participants with respect to the degree of elliptical motion can also be identified from an examination of the data in Table 1, in which the mean and SD of for each participant in a pair (averaged across trials) is reported along with the percentage of trials a given participant displayed a greater degree of elliptical motion than their coparticipant. Interestingly, not only did the participants within a pair adopt differing degrees of movement ellipticality across trials, but for nine of the 11 pairs, this movement trajectory differentiation remained constant or increased across trials. An inspection of the data revealed that this interparticipant movement differentiation occurred for most pairs within the first five trials and remained that way for the remainder of the experimental session. For the two pairs in which the participants exhibited an approximately equal number of trials with a greater magnitude of curvature (i.e., pair 7 and 10) that is they switch between roles over the course of successful trials it was still the case that on any given trial one participant adopted a more curvilinear motion between targets. It is also important to note that once differentiation occurred very few collision trials occurred for the remainder of the experimental session. In fact, nine of the 11 pairs had no more than five collisions (5 pairs had 0 or 1 collisions) after differentiation occurred. The other two pairs had six and 15 collision trials after differentiation respectively. To further determine whether the increased interparticipant differentiation in movement ellipticality was a significant factor across trials, a difference score was calculated for each trial by taking the difference in between the participant with the straighter (less elliptical, smaller ) trajectory ps and the participant with the more elliptical (larger ) trajectory pe, (i.e., pe ps ). The resulting scores are displayed in Figure 4 (right panel) and, although there was an increase in the between-pair variability of across trials, there was also a clear increase in. Although this increase in average was only found to be marginally significant between the first, middle, and last trial blocks, F(2, 20) 3.02, p.07, 2 p.23, a subsequent one-way ANOVA comparing for the single Trials 1, 8, and 15 was found to be significant, F(2, 20) 4.552, p.05, 2 p.31. For this latter analysis, Bonferroni post hoc tests revealed that for Trial 1 was significantly lower compared to Trial 8 (p.01) and marginally lower compared to Trial 15 (both p.095). There was no difference between Trial 8 and 15 (p.95). Relative phase analysis. Given the PCA results, the relative phase,, of a pair s movement was calculated with respect to the principal axis of each participant s movements in the x-y plane (i.e., at each point in time, a participant s actual cursor position was projected onto the diagonal between his or her designated targets) where the phase of the participant with the smaller value of (more straight-line: ps ) for the corresponding trial was subtracted from the phase of the participant with the greater value of (more elliptical: pe ), that is, pe ps. For each trial of each participant pair, we first computed a relative phase timeseries between the principal axis movement of pairs. This was done by calculating the difference between the instantaneous phase angles of each participant s movement along their principal axis using the Hilbert transform (see Pikovsky, Rosenblum, & Kurths, 2001 for details about this transformation). For each relative phase time-series, we constructed a frequency distribution of relative

7 6 RICHARDSON ET AL. Table 1 Statistics for Each Pair Averaged Across Trials Pair no. Figure 4. Mean and as a function of trial. Error bars indicate standard error of the mean ( p.05; p.01; ns. p.05). Participant A Participant B Mean SD % Mean SD % phases, or DRP, using all samples in the trial. The binning procedure for constructing each trial s DRP consisted of assigning each sample to one of wide bins between 0 and 360 according to its relative phase value, and then dividing the number of samples in each bin by the total number of samples in the trial (see Richardson et al., 2005; Schmidt & O Brien, 1997). We then computed a mean DRP across all pairs and trials in order to index the overall patterning of relative phase in the experiment. Phaseentrainment is indicated by a concentration of relative phase angles in a portion of the distribution. Given the way in which the relative phase time-series were calculated (i.e., pe ps ; see above), negative relative phases correspond to the participant with the greater ( pe associated with the participant with a more elliptical trajectory) following the phase of the participant with the lower ( ps associated with the participant with a straighter or less elliptical trajectory). What is apparent from an inspection of Figure 5 (left) is that, as expected, most participants did not coordinate in a canonical inphase pattern. Rather, the peak in the relative phase histogram was at 30, indicating that, for the largest number of participant pairs, the participant who adopted the more elliptical movement trajectory ( pe ) typically phase lagged the participant with the straighter or less elliptical movement trajectory ( ps ) by approximately 10 to 40. In other words, the participant who adopted the straighter movement path tended to be the phase-leader. Directionality of coupling. Finally, we estimated the directionality of the coupling between the participants using the evolution map approach proposed by Rosenblum and Pikovsky (2001; see also Rosenblum, Cimponeriu, Bezerianos, Patzak, & Mrowka, 2002). Briefly stated, the method provides a measure of the directionality of the coupling between two self-sustained oscillatory movements by estimating the ratio of the coupling terms from the phase angle time series of the movements (i.e., from 1 and 2 ). This coupling or directionality index, d (1,2), varies from 1inthe case of unidirectional coupling ( 1 2 only) to 1 in the opposite case ( 2 1 only), with d (1,2) 0 corresponding to symmetric bidirectional coupling ( 2 1 ). As with the calculations of relative phase above, the directionality index, d, was calculated for each trial with respect to the value of for that trial, with the phase angle time series of the participant with the greater value of (i.e., pe ) labeled as pe, and the phase angle time series of the participant with the lower value of (i.e., ps ) labeled as ps. The average d (pe,ps) values across participant pairs as a function of trial are displayed in Figure 5 (right). Interestingly, there was a small but consistent asymmetry in the coupling between pe and ps, with slightly positive d (pe,ps) values for each trial. There was no difference in d (pe,ps) between the first, middle, and last trial blocks, F(2, 20) 1, and d (pe,ps) averaged across all trials was significantly different from zero, t(10) 3.46, p.01. This indicates that on any given trial the participant with the straighter trajectory, ( ps ), drove the movements of the participant with the more elliptical trajectory, ( pe ), more strongly than vice versa. Dynamical Modeling The results of the collision avoidance task investigated here revealed that pairs converged onto a stable relative phase relationship that averaged 30, with one participant spontaneously adopting and maintaining a more straight-line trajectory between targets, and with the other participant in the pair adopting a more elliptical trajectory between targets. The deviation from straightline trajectories (i.e., the appearance of ellipticality) can be interpreted as the recruitment of a movement degree-of-freedom orthogonal to the straight-line principal motion axis between targets. Indeed, if pairs attempted to move using straight-line trajectories, they nearly always collided or had to perform the task very slowly. As discussed in the Introduction, such ellipticality would be sufficient to avoid collisions while still allowing a canonical inphase relationship to be maintained between the participants principal motion axes. Therefore, why a noncanonical and typically

8 SELF-ORGANIZED COMPLEMENTARY JOINT ACTION 7 Figure 5. (Left) Mean frequency distribution of relative phases (DRP). (Right) Index of coupling directionality, d (pe,ps). Error bars indicate standard error of the mean ( p.05; p.01; ns. p.05). difficult-to-perform noninphase relation emerged in addition to ellipticality as part of the interpersonal dynamic synergy that implemented the complementary task solution in which the symmetry of participants roles was broken. The reason for the asymmetry and phase shift becomes apparent when one attempts to model the steady-state behavioral dynamics (Warren, 2006) for this collision avoidance task. Here we modeled the behavioral dynamics observed in this task using a simple task-dynamic model (Saltzman & Kelso, 1987), which captured the terminal movement objectives (goals) of the actors by means of a functionally defined task space that included the minimal number of relevant task dimensions (i.e., task variables or task-space axes) and a minimal set of task-dynamic equations of motion. We developed the model in three steps. First, we defined the task space for a single participant. For this task, the actor was instructed to control and move a stimulus between two targets along a primary axis of motion (i.e., diagonally between two designated targets) using a motion sensor held to their hand. We modeled this task as one that involved oscillating a point-mass within a two-dimensional plane; the corresponding two-dimensional task space is illustrated in Figure 6. Axis x 1 corresponds to the axis of instructed oscillation for the participant, using limit cycle dynamics to generate a self-sustained oscillation of the point-mass (endeffector) along this axis between the two targets. The orthogonal axis, y 1, corresponds to deviations away from the oscillatory motion axis. Given that the most efficient path between a participant s two targets is a straight-line trajectory along the primary motion axis, x 1, a simple point-attractor dynamics (damped mass-spring equation) was used along y 1 to minimize deviations away from the primary motion axis. Assuming a point-mass of 1 (for simplicity), the dynamics in this functionally defined task space can be defined by the following set of motion equations: ẍ 1 b x1 ẋ 1 c x1 x 2 1 ẋ 1 k x1 x 1 0 (1a) ÿ 1 b y1 ẏ 1 k y1 y 1 0 where x 1 and y 1, ẋ 1 and ẏ 1, ẍ 1 and ÿ 1 correspond to the position, velocity and acceleration of the end effector along each axis, respectively, k x1 and k y1 are stiffness coefficients for axis x 1 and y 1, respectively, b x1 and b y1 are the damping coefficients for axis x 1 and y 1, respectively, and (c x1 x 2 1 ẋ 1 ) is the van der Pol (limit cycle oscillator) escapement function for axis x 1. It is important to note that Equation 1a has been previously employed as an abstract task-dynamic model of rhythmic reaching between two target locations (Saltzman & Kelso, 1987). Our second step was to extend Equation 1a to an interpersonal rhythmic coordination situation. This required: (a) the addition of a second system of equations to describe the task-dynamics of the second participant ẍ 2 b x2 ẋ 2 c x2 x 2 2 ẋ 2 k x2 x 2 0 (1b) ÿ 2 b x2 ẏ 2 k x2 y 2 0 where x 2 and y 2, ẋ 2 and ẏ 2, ẍ 2. and ÿ 2 correspond to the position, velocity and acceleration of the point-mass along each axis for the second participant; and (b) a set of coupling functions to link the point-mass movements of the coactors. For the generalized rhythmic coordination task being captured here, this was achieved by adding the linear, dissipative coupling functions and x1 (ẋ 2 ẋ 1 ) x2 (ẋ 1 ẋ 2 ) (2a) (2b) to the equations that define each actor s instructed axes of motion (i.e., x 1 and x 2 ). The reader should note that some previous models of rhythmic interpersonal coordination have employed a more complex coupling function, namely one that captures situations in which inphase and antiphase coordination reflect two nonarbitrarily defined, stable modes of coordination that differ in stability (i.e., the function captures the fact that inphase coordination is typically more stable that antiphase coordination) as is the case in the Haken et al. (1985) model of rhythmic coordination (commonly referred to as the HKB model). Given the arbitrary nature of defining inphase and antiphase coordination for the current task (as discussed in the Introduction), we felt that the more complex HKB coupling function was not necessary to model the behavior under consideration in the present study. However, such coupling functions could easily be incorporated for other collision avoidance tasks in which inphase and antiphase movements can be nonarbitrarily defined. The combination of Eqations 1a and 1b with Equations 2a and 2b results in the coupled system

9 8 RICHARDSON ET AL. Figure 6. (a) 2-dimensional task space for an individual instructed to perform a simple point-mass (end-effector) rhythmic movement task. (b) The 2-dimensional task space embedded at a 45 angle within a shouldercentered, body-space coordinate system. For the task space, x 1 corresponds to the axis of instructed direction of oscillation, with movement along this axis defined by a limit cycle oscillator. x 2 corresponds to orthogonal deviations away from the principal movement axis, such that movement along the orthogonal axis can be defined by a simple damped mass-spring function. 1 and 2 correspond to the horizontal (frontal) and vertical (sagittal) dimensions of the task movements with respect to shouldercentered body-space. ẍ 1 b x1 ẋ 1 c x1 x 2 1 ẋ 1 k x1 x 1 x1 (ẋ 2 ẋ 1 ) ÿ 1 b y1 ẏ 1 k y1 y 1 0 ẍ 2 b x2 ẋ 2 c x2 x 2 2 ẋ 2 k x2 x 2 x2 (ẋ 1 ẋ 2 ) ÿ 2 b y2 ẏ 2 k y2 y 2 0 (3) and reflects an idealized model of synchronous inphase interpersonal rhythmic coordination. For clarification purposes, an abstract representation of this preliminary system (Equation 3), as well as simulated time-series are provided in Figure 7. For the sake of illustration, the implementation of Equation 3 in Figure 7a models the task of two actors instructed to rhythmically coordinate inphase with each other at a fixed amplitude and at a fixed diagonal angle of 45 with respect to their shoulder/body plane. The simulated time-series presented in Figure 7c and their embedding as movement trajectories in body-space in Figure 7b together reveal how Equation 3 results in the synchronized movement of x 1 and x 2 after an initial transient period. During the transient period, initial deviations away from perfect inphase coordination along the x 1 and x 2 axes are damped out, as are initial deviations away from the instructed axis of motion (i.e., deviations along y 1 and y 2 ) that resulted from the arbitrarily chosen initial conditions. Note that although in the current experiment both participants started with the same initial conditions on each trial (at their respective bottom targets with zero velocity), different initial conditions were chosen for each participant in the simulation to illustrate the equifinality of the dynamical solution (i.e., the solution is attained independently of initial conditions). In many respects, Equation 3 captures the dynamics observed between many of participant pairs at the beginning of the experimental session. Indeed, most pairs typically failed in the first few trials because they were naturally attracted toward a pattern of inphase synchronization along straight-line trajectories on each participant s instructed motion axis. As we detailed in the introduction, this natural tendency of interacting individuals to synchronize their movements when visually (informationally) coupled is, of course, well known (for a review, see Schmidt & Richardson, 2008) and has been modeled in a similar manner by numerous researchers in the past (e.g., Mörtl et al., 2012; Varlet et al., 2012), using systems of equations similar to those employed by Kelso and colleagues to capture intrapersonal interlimb synchrony (e.g., Haken et al., 1985; Schöner, Haken, Kelso, 1986; see Kelso, 1995 for a review). A third step was needed, however, to capture the behavioral dynamics that would produce behavior qualitatively similar to that produced by participant pairs at the end of an experimental session. Given that the task instructions were to avoid bumping or colliding into each other, the simplest modification was to create a repelling coupling force that acted on participant-i s end-effector, and whose strength was a function of the body-space displacement vector from participant-j s ( origin ) end-effector to participant-i s ( target ) end-effector (see Figure 8). This was accomplished using repeller functions partially derived from the obstacle avoidance couplings previously employed by Fajen and Warren (see, e.g., Fajen & Warren, 2003; Warren & Fajen, 2008; Warren, 2006). The repeller functions took the form and x1 (x 1 y 2 )e x 1 y 2 x2 (x 2 y 1 )e x 2 y 1 for the primary task axes, x 1 and x 2, respectively, and and y1 (y 1 x 2 )e y 1 x 2 y2 (y 2 x 1 )e y 2 x 1 (4a) (4b) (4c) (4d) for the secondary task axes, y 1 and y 2, respectively. There are three terms in these coupling functions. The leftmost terms are participant-dependent constants that scale the strength of the repelling force along each participant s task-axes. The middle terms in parentheses are the projections of the relevant body-space

10 SELF-ORGANIZED COMPLEMENTARY JOINT ACTION 9 Figure 7. (a) Abstract representation of the coupled rhythmic coordination system captured by Equation 2 for a task in which two actors are instructed to coordinate a point-mass (end-effector) at the 45 angle with respect to a shoulder-centered coordinate system. (b) Example rhythmic movement trajectories projected in body-space (shoulder-centered) coordinates. (c) Simulated time-series for arbitrary initial conditions using parameter settings: b x1 b x2 1, b y1 b y2 2, k x1 k x2 k y1 k y2 2, c x1 c x2.5 and x1 x2.5. displacement vector onto the task-space axes of the vector s target participant; these projections specify the direction, and contribute to the magnitude, of the displacement-dependent repelling force along these axes (see Figure 8 and its caption for details regarding these projections and the sign conventions used to compute these terms). The leftmost terms scale the magnitude of the repelling forces as a decreasing exponential function of the length of these projections onto the task-axes, that is, small projection lengths specify stronger repelling forces. Accordingly, if i for a given participant is set to zero or close to zero, the effect of the repeller coupling terms is null or minimal, and a straight-line trajectory will be created that is aligned along the systems principal axis the participant will behave in a manner similar to Equation 1. If i is set to a value greater than zero, however, the repeller coupling terms will add forces along both task-axes of participant-i, resulting not only in greater ellipticality (due to forces added along the point-attractor task axis), but also in a greater phase lag relative to participant-j (due to forces added along the limit cycle axis). It is important to note that the increased ellipticality should not be considered to result from the recruitment of a previously quiescent kinematic degree of freedom (e.g., recruitment of a biomechanical degree of freedom, as in studies by Kelso, Buchanan, de Guzman, & Ding, 1993; Fink, Kelso, Jirsa & de Guzman, 2000). Rather, both the increased ellipticality and increased phase lag result from the recruitment of repeller dynamics onto both axes of the participant s two-dimensional task-space, such that these dynamics are added to the point-attractor and limit cycle dynamics of Equation 3 that was used to define a synchronization task with no collision-avoidance constraint. Combining the coupling functions in Equations 4a d with Equation 3 resulted in the following system of equations ẍ 1 b x1 ẋ 1 c x1 x 1 2 ẋ 1 k x1 x 1 x1 (ẋ 2 ẋ 1 ) x1 (x 1 y 2 )e x 1 y 2 ÿ 1 b y1 ẏ 1 k y1 y 1 y1 (y 1 x 2 )e y 1 x 2 ẍ 2 b x2 ẋ 2 c x2 x 2 2 ẋ 2 k x2 x 2 x2 (ẋ 1 ẋ 2 ) x2 (x 2 y 1 )e x 2 y 1 ÿ 2 b y2 ẏ 2 k y2 y 2 y2 (y 2 x 1 )e y 2 x 1 which can successfully capture the stable task solution adopted by participants when successfully performing the composite task (rhythmic synchronization collision avoidance). An example of the movement trajectories and time-series data generated from Equation 5 are displayed in Figure 9, using various settings for x1 y1 and x2 y2. Consistent with the proposal that participants adopted an asymmetric relation in coupling in order to avoid collisions while simultaneously synchronizing their between target movements, when x1 y1 x2 y2 an asymmetry in the movement trajectory emerges, as well as a phase lag between the moreelliptical and the more-straight-line trajectory. This asymmetry is qualitatively similar to that observed in the experimental data reported above. In fact, by modulating the magnitudes of x1 y1 and x2 y2 and the ratio x1 y1 x2 y2, Equation 5 can generate a range of movement trajectory patterns that match the (5)

11 10 RICHARDSON ET AL. Figure 8. Illustration of how the displacement vector between participants on-screen end-effector locations is projected onto the appropriate task-dynamic axes for defining the repeller coupling functions in Equations 4a d. The target locations (gray squares) are located at the corners of the screen. The solid gray lines along the screen s intertarget diagonals represent the primary (limit cycle) task-space motion axes (x 1 and x 2 ) and orthogonal (point-attractor) motion axes (y 1 and y 2 ) of the participants, P 1 and P 2. The task axes of P 1 are rotated CCW 45 from the screen s right-horizontal axis, and the task axes of P 2 are rotated CCW 135 from the screen s right-horizontal axis; thus, the labels x i and y i at the ends of the intertarget on-screen diagonals denote the positive-going directions for motions along the corresponding x i and y i axes of the ith participant. The gray dashed lines reflect idealized on-screen movement trajectories of the participants, with the gray disks representing the current on-screen locations of P 1 and P 2 along these movement trajectories. In this example, the solid black line with open arrow heads, D, is the displacement vector from P 1 to P 2, and the dashed black lines with closed arrow heads, d A and d B, represent the projections of D onto the relevant task axes of P 2, that is, d A y 2 ( x 1 ) y 2 x 1 as in Eq. 4d, and d B x 2 y 1 as in Equation 4b. Note that the displacement vector from P 2 to P 1 would be represented by D, and its projections onto the task axes of P 1 would be defined by d A x 1 ( y 2 ) x 1 y 2 as in Eq. 4a and d B y 1 x 2 as in Eq. 4c. range of coordinated movement patterns exhibited by participants and displayed in Figure 3. Note also that: a) if x1 y1 x2 y2 0, then no motion is created along y 1 or y 2 (i.e., ẏ 1 ẏ 2 0), which makes the behavior of Equation 5 equivalent to the behavior of Equation 3; and b) if x1 y1 x2 y2 0, then equivalent motion patterns are created along y 1 and y 2 resulting in elliptical trajectories that are symmetric across participants with oscillations along x 1 and x 2 that are synchronized with zero phase lag. The potential significance of such symmetric acrossparticipant values will be discussed in more detail below, as this situation also results in a stable collision avoidance solution, especially for x1 0, but one that does not include a phase lag along the principal motion axes (i.e., trajectories are coordinated inphase with a 0 relative phase relationship along these axes). Discussion The experimental study presented here was designed to investigate the dynamics of the interpersonal movement coordination that occurred between a pair of participants completing a rhythmic collision avoidance task. Participants were instructed to control and move a computer stimulus back and forth between sets of diagonally positioned targets without colliding into one another. Although no specific instructions were provided to participants about how they should coordinate their movements, the results revealed that nearly all pairs converged onto the same stable pattern of coordination. More specifically, pairs converged onto a stable relative phase relationship of 10 to 40 between their primary motion axes, with one participant maintaining a straightline trajectory between targets and the other participant tending to exhibit a more elliptical trajectory. There also appeared to be an asymmetry in coupling strength between coactors, with the movements of the participant who adopted a more elliptical trajectory being more coupled to (driven by) the movements of the participant who adopted a more straight-line trajectory. Of particular significance, task success was dependent on the participants discovering this complementary task solution. If pairs tried to move using equally straight movement trajectories, they nearly always collided into one another or, alternatively, had to perform the task exceptionally slowly. This is because pairs were unable to maintain a phase difference large enough to avoid collision when adopting similarly straight-line shaped betweentarget movement trajectories. Essentially, participants faced a conflict between the natural attraction toward synchronizing their movements inphase and the fact that such synchronization would result in task failure (i.e., a collision). Both the asymmetry in participant path shapes and the nonzero relative phase between participant s principal motion axes enabled pairs to overcome this conflict. Our simulations support the hypothesis that both of the asymmetries that contribute to task success (i.e., asymmetry in path ellipticality, and deviations from 0 relative phase) are the result of interparticipant asymmetries in the strength of repeller dynamics between the participants end-effectors. Consistent with these modeling results is the experimental finding that, in a participant pair, the participant who adopted a more elliptical trajectory tended to display a greater phase lag relative to the participant who adopted a more straight-line trajectory. Interestingly, a correlational analysis between the path shape asymmetry measure,, and the coupling directionality index, d (pe,ps ), for the first, middle, and last trial blocks revealed an increasing positive relationship between these two measures across trials, going from r.08, to r.37, to r.85, respectively (with only the latter correlation significant at p.05). With regard to the coupling directionality index, d (pe,ps), the interparticipant directional asymmetry in coupling strength was by no means large, and future research will be required to determine whether this could have been a result of attentional differences between participants in a pair or a functional consequence of adopting a more elliptical versus straightline movement trajectory. With regard to the task dynamic model that was developed to capture the steady-state dynamics observed between pairs, this model produced a similar pattern of asymmetric behavior when the repeller coupling weights of the experimentally motivated repel-