Journal of Motor Behavior, Vol. 41, No. 6, 2009 Copyright OC 2009 Heldref Publications

Horsing Around: Spontaneous Four-Legged Coordination

Steven J. Harrison1, Michael J. Richardson2

1Department of Physical Therapy, Neag School of Education, University of Connecticut, Storrs. 2Department of Psychology, Colby College, Waterville, Maine.

ABSTRACT. Motivated by previous research suggesting that infor- mational and mechanical interlimb coupling can stabilize rhythmic movement patterns, the authors show that stable 4-legged patterns between 2 individuals, either walking or running, can emerge unin- tentionally from simple forms of coupling. Specifically, they show that the leg movements of pairs of naive individuals become spon- taneously phase locked when visually or mechanically coupled via a foam appendage. Analysis of each of the phase locked trials re- vealed distinct preferences for particular 4-legged patterns, with in- terpersonal in- and anti-phase coordination patterns (equitable with quadruped pace and trot, respectively) observed almost exclusively. Preference for either pattern depended on the strength of coupling. The authors discuss these findings in light of previous claims that the patterns of human and animal locomotion—as well as coor- dinated movements in general—can emerge from lawful coupling relations that exist between the subcomponents of perceptual-motor systems.

Keywords: coordination dynamics, interpersonal coordination, me- chanical coupling, quadruped gait, visual coupling

S

ynchronization is a hallmark of biological systems and is observed at many scales of nature, from the beating of pace-maker cells (Honerkamp, 1983) to the chirping of crick- ets (Walker, 1969) and from the flashing of fireflies (Hanson, 1969) to the synchronous applause of an enamored audience (Ne´da, Ravasz, Brechet, Vicsek, & Baraba´si, 2000). Over the past several decades, advances in the understanding of non- linear systems have revealed how these, as well as the many other rhythmic behaviors of living (and nonliving) systems, may be understood in terms of the universal dynamics of cou- pled oscillators (Collins & Stewart, 1993; Scho¨ner & Kelso,

1988; Strogatz, 2003; Strogatz & Stewart, 1993).

Not surprisingly, the phenomena of synchronization and the dynamics of coupled nonlinear oscillators have frequently been considered hallmark characteristics of human and an- imal movement systems. Legged locomotion provides the prototypical example whereby the oscillatory limb move- ments of two-, four-, and six-legged creatures exhibit a pre- dictable set of synchronous phase locked patterns (Golubit- sky & Stewart, 2006). Such multilegged coordination pat- terns have been modeled both in terms of the symmetries (and breaks in symmetry) of networks of coupled neural os- cillators (Collins & Stewart, 1993; Golubitsky & Stewart, 1999; Golubitsky, Stewart, Buono, & Collins, 1998) as well as more abstractly in terms of the dynamics of an appropri- ately chosen collective variable (e.g., Haken, 1996; Kelso, 1997). A collective variable reflects the order parameter of a system and indexes the spatiotemporal stabilities of multi- limb coordination patterns, including the form, change, and

persistence of coordination patterns as well as the transi- tions between coordination patterns (i.e., linear and nonlinear phase transitions). For multilimb coordination, the collective variable is typically the relative phase of the limbs in ques- tion (e.g., Deidrich & Warren, 1995; Haken, Kelso, & Bunz, 1985; Jeka, Kelso & Keimel, 1993a, 1993b; Kelso & Jeka, 1992; Scho¨ner, Jiang, & Kelso; 1990).

Of particular relevance to the current study is that the latter approach, known as coordination dynamics (see Haken, 1996; Kelso, 1997), does not assume any privileged scale of analysis. Rather, it examines whether the macro- scopic patterns of coordinated movement (including bipedal, quadrapedal, and hexapedal locomotion) are the result of self- organization and emerge from the nonlinear, yet lawful, mul- tilevel interactions that occur among the many perceptual- motor components of biological movement systems (Kelso, 1997; Turvey, 2007). Motivated by this possibility and ex- perimental evidence suggesting that the stable gait patterns of quadrupeds are still observed even when the central or peripheral nervous systems of the creatures being exam- ined are separated or removed (Brown, 1911; Forssberg, Grillner, Halbertsma, & Rossignol, 1980; Shik, Severin, & Orlovskii, 1967), in the present study, we examined whether two naive walkers (independent nervous systems) who were visually or mechanically coupled would spontaneously syn- chronize their leg movements and produce gaits associated with quadrupedal locomotion (e.g., pace, trot).

Buttressed by research that demonstrates that both me- chanical (e.g., Bennett, Schatz, Rockwood, & Wiesenfeld, 2002; Koditschek, Full, & Buehler, 2004; Sternad, Duarte, Katsumata, & Schaal, 2001) and informational (e.g., Kelso, Fink, Delaplain, & Carson, 2001; Schmidt, Carello, & Tur- vey, 1990; van Ulzen, Lamoth, Daffertshofer, Semin, & Beek, 2008) couplings can act to stabilize interlimb coordination patterns, we expected that spontaneous four-legged coordina- tion would be observed and that the form and stability of the observed coordination would depend on the overall strength of the interpersonal coupling (e.g., Richardson, Marsh, Isen- hower, Goodman, & Schmidt, 2007; Schmidt, Richardson, Arsenault, & Galantucci, 2007). Specifically, we anticipated that the combination of visual and mechanical coupling to- gether would result in more stable coordination than visual or mechanical coupling alone. We also expected that a change in

Correspondence address: Steven J. Harrison, Department of Physical Therapy, Neag School of Education, University of Con- necticut, 358 Mansfield Road, Storrs, CT 06269, USA. e-mail: ste[email protected]

 

519

locomotion speed would be found to act as a control param- eter1 for the two-person, four-limb system, whereby increas- ing the pace of locomotion would alter the stability of the observed four-legged coordination patterns or result in pref- erences for certain four-legged coordination patterns (e.g., Jeka et al., 1993a; Diedrich & Warren, 1995).

Method

Participants included six pairs of height-matched male un- dergraduate students (M age = 19.9 years, SD = 1.0 years) from Colby College who were naive to the purpose of the study and walked or jogged a 35-m long path under four conditions of coupling (see Figure 3A). Participant pairs first completed a no coupling (NC) control condition to obtain a measure of chance level coordination. For these NC trials, each participant walked and jogged alone at his own com- fortable pace. For the visual coupling (VC) condition, partic- ipants walked in line, 1 participant behind another, with the back participant 0.75 m behind from the front participant. The participants were instructed to move at their own com- fortable pace but were also asked to maintain a fixed distance of separation over the course of the trial. Although we did not objectively measure or record the pairs ability to maintain this fixed distance of separation, an experimenter performed a real-time visual inspection of the between-participant sep- aration during the completion of each VC trial. A different experimenter also inspected video recordings of the VC trials. In both cases, the subjective impression was that participants had little trouble maintaining a separation distance close to the instructed separation of 0.75 m. For the mechanical cou- pling (MC) condition, participants were physically connected through a large foam block. The block was 0.45 m in width and height and 0.75 m long. Participants were strapped to the ends of the blocks and thus maintained a constant 0.75 m separation. The foam was relatively firm and was minimally deformed by pushes and pulls of the 2 participants in a pair. For MC trials, we eliminated the influence of visual coupling by having the back participant wear a blindfold. For the fi- nal coupling condition, the blindfold was removed and the combined influence of visual and mechanical coupling was investigated (VMC). Each participant completed 4 trials at a walking pace and 4 trials at a light jogging pace in each of the four coupling conditions for a total of 32 trials. Trials were blocked by pace and condition, with the order of VC, MC, and VMC conditions randomized.

Following Scho¨ner et al. (1990), relative phase was adopted as the candidate observable for indexing pattern- ing and stability of interlimb coordination in each of the experimental trials. To obtain an estimate of relative phase, knee flexion and extension were recorded using electrogo- niometers (Biometrics, Gwent, UK) that were attached to the lateral surfaces of the left and right upper and lower legs of each participant in a pair. To eliminate transients, movement recording started at the 5-m mark of the 35-m long path. Knee angle was sampled at 50 Hz, and peak knee flexion events were identified in the digitized data. To assess inter- limb coordination, successive peak knee flexion events from an elected reference limb were used to establish a reference cycle (Diedrich & Warren, 1995). We assessed the relative phase (¢) of each of the considered target limbs by evalu- ating at what fraction of the reference cycle (expressed in the range 0–360◦) peak knee flexion in the target limb cy- cle occurred (see Figure 1). Here, the front-right limb (FR) was used as the reference limb and the front-left (FL), back- right (BR), and back-left (BL) limbs were treated as target limbs. For each trial, mean relative phase () and stan- dard deviation of relative phase (SD¢) were used to assess

1לכידה

2לכידהלכידה3

the form and stability of the coordination, respectively. Two limbs were classified as phase locked at when the cor- responding SD¢ was less than 20◦. Figure 2C and D show examples of trials in which the coordination between two limbs was classified as phase locked (SD¢ < 20◦). Figure 2A illustrates an example of no coordination (no phase lock- ing) and Figure 2B illustrates an example of intermittent phase locking (SD¢ > 20◦) in which periods of both phase wandering (no coordination) and phase locking were exhib- ited. Although the latter form of synchrony was observed in the VC, MC, and VMC conditions and is characteristic of weakly coupled systems (Kelso, 1997; Schmidt & Richard- son, 2008), we present only the analysis of the phase locked trials.

Determining the form of the four-legged patterns in trials identified as phase locked was simplified by the fact that the coordination between the left and right limbs of both the front and back participants were found to be, without exception, phase locked in an anti-phase coordination ( = 177.1◦, SD¢ = 4.23◦). This meant that all observed four-legged pat- terns could be considered symmetrical (Abourachid, 2003) and that it was possible to determine the coordination pattern using only the mean relative phase between the back- and front-right limbs (BR). Further justification for this de-

cision came from a consideration of Scho¨ner et al.’s (1990) four component model of four-legged coordination dynam- ics, which revealed that if the assumption of a fixed relative phase relation between the left and right limbs of both partic- ipants is applied as a constraint, then the order and stability of the dynamics of this four-legged system can be captured by the single collective variable, (¢BR). Accordingly, BR and SD¢BR were used to capture the collective variable dy- namics of the system, where BR reflected the order of the system and captured the overall pattern of coordination and SD¢BR indexed the dynamic stability of a particular coordination pattern.

Each phase locked trial was classified as one of four sym- metrical four-legged patterns—namely, pace (PACE), trot (TROT), lateral walk (LATW), and diagonal walk (DIAW).

We classified the four patterns on the basis of the order param- eter (BR) as follows, PACE: BR = 0◦ ±45◦, DIAW: BR = 90◦ ±45◦, TROT: BR = 180◦ ±45◦, and LATW: BR   = 270◦ ±45◦.

Results and Discussion

As predicted, the visual and mechanical couplings did re- sult in the spontaneous synchronization of the leg movements of the front and back participants, with the VC, MC, and VMC couplings each resulting in an increase in the observed frequency of phase locked trials relative to the NC control. Specifically, the leg movements of the front and back par- ticipants were phase locked on 40% of VC trials (M SD¢BR

= 8.3◦), 63% of MC trials (M SD¢BR = 9.2◦), and 77% of

VMC trials (M SD¢BR = 7.7◦), compared with only 8% of NC trials (M SD¢BR = 13.8◦). Note, phase locking in the NC condition revealed how similarities in the preferred walking and jogging paces of the height-matched pairs led to chance occurrences of synchrony in a very limited number of cases. Given our analysis, we assumed that the conditions under which phase locking was observed more frequently to be the most stable. The increase in the number of phase locked

trials from VC to MC to VMC and the observed decrease in SD¢ were consequently taken to suggest that coupling affected the dynamic stability of the observed four-legged coordination patterns. A 4 (Coupling: NC, VC, MC, VMC)

× 2 (Pace: walk, jog) repeated-measurements analysis of variance conducted on percentage of phase locked trials con- firmed this. The analysis revealed significant main effects for both coupling, F(3, 15) = 8.42, prep = 1.00, η2 = .63, and pace F(1, 5) = 14.41, prep = .99, η2 = .74. A Fisher’s protected least significant difference post hoc analysis re- vealed that percentage of phase locked trials for VC, MC, and VMC were all significantly higher than for NC (all ps

< .05). Moreover, the percentage of phase locked trials for VMC was found to be significantly higher than for VC (all ps < .05), and the differences between MC and VMC and MC and VC were not found to be significant (p > .05). No interaction was found between pace and coupling, indicating that for all of the coupling conditions, jogging (32%) resulted in fewer phase locked trials that walking (62%). This result is consistent with our hypothesis that the speed of locomotion is a control parameter of the system and would affect the dynamic stability of the four-legged coordination patterns observed.

A basic fact of rhythmic coordinated movement is that in a particular context of constraints, some modes of coordina- tion tend to be performed more frequently than others (Hoyt & Taylor, 1981). Figure 3B shows the distributions of for each target-by-referent limb combination for the phase locked trials. Although it is apparent from these distributions that there was notable variability in the exact form of coor- dination patterns observed from trial to trial, it remains clear that distinct preferences for a subset of quadrupedal locomo- tion patterns were present in phase locked trials. An analysis

of ¢BR revealed that PACE (¢BR = 0◦ ±45◦) and TROT (¢BR

= 180◦ ±45◦) were observed almost exclusively and that the relative frequency of these patterns varied as a function of coupling (Figure 3C). Overall, the interpersonal in-phase coordination of the PACE pattern was observed most fre- quently. For the phase locked VC trials, PACE was the only pattern observed, with the participants in each pair falling into a coordinated unintentional march. For the phase locked MC trials, both PACE and TROT patterns were observed. This suggests that the pushes and pulls passed back and forth through the homogeneous foam block acted to stabilize both coordination patterns, although the in-phase coordination of the PACE pattern would appear to be the more dynamically stable of the two. In the VMC condition, the presence of visual coupling added to the MC condition acted to stabilize the anti-phase coordination of the TROT pattern relative to PACE.

Interestingly, the influence of visual coupling on the ob- served coordination patterns changed markedly over the ex- perimental conditions. In the VC condition, visual coupling acted to dynamically stabilize a PACE pattern; in contrast, in the VMC coupling condition, visual coupling appeared to stabilize a TROT pattern. We conclude that this was due to changes in the details of the visual coupling over the VC and VMC condition. In the VC condition, the back participant was able to see the full movements of the front participant. In this condition, we may reasonably assume that sight of the front participant’s oscillating lower limbs is the source of the informational coupling. In contrast, in the VMC condi- tion, the foam block occluded sight of the front participant’s limbs, and only the movements of the front participants head and shoulders were visible. In this condition, we conclude that sight of the movements of the shoulders, which move 180◦ out of phase with legs in stable walking and jogging, was the source of the informational coupling.

Last, whereas walking trials resulted in a greater frequency of phase locking than jogging trials, the relative frequencies of the TROT and PACE patterns in each of the coupling con- ditions was preserved over the manipulation of pace. Impor- tantly, this suggests that although an increase in locomotion speed was found to decrease the dynamic stability of the ob- served coordination patterns, moving from walk to jog did not result in observable changes in the relative stability of the TROT and PACE coordination patterns. Thus, no clear evi- dence was found for a systematic preference across changes in locomotion speed (cf. Jeka et al., 1993a). One possibil- ity is that identifying relative changes in the persistence or stability of the observed coordination patterns may require a more dramatic manipulation of locomotion speed. That is, here we only considered a walk and light jog, whereas the differences in coordination across a walk and run may be more appropriate. Moreover, examining whether pairs ex- hibited a transition between gait patterns would also seem pertinent and could be addressed in future research by hav- ing the speed of locomotion increased and decreased across a period of locomotion.

Conclusion

The current study used relative phase (cf. Haken et al., 1985; Scho¨ner et al., 1990) to quantify whether an informa- tional or mechanical coupling could bring about spontaneous four-legged coordination between two independent nervous systems. An analysis of the symmetries of the two-person, four-legged system revealed that the order and stability of observed coordination were effectively captured using the collective variable (¢BR). Consistent with our hypotheses, both informational and mechanical couplings were found to affect the observed collective variable dynamics. In general, only PACE (¢BR = 0◦) and TROT (¢BR = 180◦) patterns were found to be stable states for this system.

Our results also revealed that the relative stability of   the PACE and TROT coordinative states systematically de- pended on the manipulations of visual and mechanical cou- pling. Although developing a complete understanding of how these couplings act to stabilize such coordination patterns is a matter for future research, the paradigm presented here, along with further and more refined manipulations of infor- mational and mechanical coupling, will no doubt aid in this investigation. In the case of visual coupling, there is already some indication that the activity of the visual system (i.e., eye and head movements) when attending to the kinematics of environmental stimuli or the limb movements of another individual may be the mediating factor behind spontaneous visual coordination (Schmidt & Richardson, 2008; Schmidt et al., 2007). In addition, certain informational properties of the observed kinematics appear to be more important for stylizing perceptual-motor coordination than others. Specif- ically, the end-points of motion (e.g., peak extension and flexion), which operate to anchor movements during coor- dination (e.g., Byblow, Carson, & Goodman, 1994; Fink, Foo, Jirsa, & Kelso, 2000), and relative direction informa- tion, which specifies relative phase, and can account for the differing stabilities of visuomotor coordination given that the detection of relative direction information is dependent on the speed of movement (for more details, see Bingham, 2004; Wilson, Collins, & Bingham, 2005). In the case of me- chanical coupling, some insights have also been developed regarding how specific dynamically stable organizations can emerge out of the physical dynamics of mechanically inter- acting rhythmic systems (Katsumata, Zatsiorsky, & Sternad, 2003; Sternad et al., 2001).

Perhaps the most interesting question posed by the current findings regards the degree to which the pairs of participants formed a functional quadrupedal system. If they did so, can it be considered and understood as a single perception–action system—a synergy or coordinative structure (Kelso, 1997; Kugler & Turvey, 1987)? Although it is clear that infor- mational and mechanical couplings have the potential to dynamically stabilize the coordination patterns of naturally evolved quadrupeds, the extent to which such coupling oper- ates alongside the hypothesized role of central pattern gener- ators or centralized control processes (Delcomyn, 1980; Ivry

& Richardson, 2002; Keel, 1982; Marden & Bucher, 2001) requires further inquiry and thus precludes us from draw- ing any definitive conclusions. However, it seems apparent that finding that stable multilegged coordination patterns can emerge without direct neural-muscular coupling between all of participating limbs is consistent with the theory of coordi- nation dynamics, providing support for the claim that the sta- ble patterns of coordinated movement reflect self-organizing processes and can emerge from the free interplay of the neu- ral, muscular-articulator, mechanical, and informational de- grees of freedom that characterize biological movement sys- tems.

ACKNOWLEDGMENTS

The authors thank Veronica Romero and Michele Chu for help with data collection and Damian Stephen for help in developing and running a pilot experiment. This research was supported in part by grants from the Provost’s Office at the University of Connecticut and by the National Science Foundation (award #BCS-0750190).

NOTE

  1. A control parameter is a system parameter that when changed beyond some critical value induces a qualitative change in the order of the system. Note that the phrase control parameter does not refer to a system property that prescribes the observed pattern or change in pattern. It simply refers to a parameter that leads the system through the variety of possible patterns (Kelso, 1997).

REFERENCES

Abourachid, A. (2003). A new way of analysing symmetrical and asymmetrical gaits in quadrupeds. Comptes Rendus Biologies, 326, 625–630.

Bennett, M., Schatz, M.F., Rockwood, H., & Wiesenfeld, K. (2002). Huygens’ clocks. Proceedings of the Royal Society of London. Series A, 458, 563–581.

Bingham, G. P. (2004). A perceptually driven dynamical model of bimanual rhythmic movement (and phase perception). Ecological Psychology, 16, 45–53.

Brown, T. G. (1911). The intrinsic factors in the act of progression in the mammal. Proceedings of the Royal Society of London. Series B, Containing Papers of a Biological Character (1905–1934), 84, 308–319.

Byblow, W. D., Carson, R. G., & Goodman, D. (1994). Expressions of asymmetries and anchoring in bimanual coordination. Human Movement Science, 13, 3–28.

Collins, J., & Stewart, I. (1993). Coupled nonlinear oscillators and the symmetries of animal gaits. Journal of Nonlinear Science, 3, 349–392.

Delcomyn, F. (1980). Neural basis of rhythmic behavior in animals.

Science, 210, 492–498.

Diedrich, F. J., & Warren, W. H. (1995). Why change gaits? Dy- namics of the walk-run transition. Journal of Experimental Psy- chology: Human Perception and Performance, 21, 183–202.

Fink, P. W., Foo, P., Jirsa, V. K., & Kelso, J. A. S. (2000). Local and global stabilization of coordination by sensory information. Experimental Brain Research, 134, 9–20.

Forssberg, H., Grillner, S., Halbertsma, J., & Rossignol, S. (1980). The locomotion of the low spinal cat. II. Interlimb coordination. Acta Physiologica Scandinavica, 108, 283–295.

Golubitsky, M., & Stewart, I. (1999). Symmetry in locomotor cen- tral pattern generator and animal gaits. Nature, 401, 693–695.

Golubitsky, M., Stewart, I., Buono, P.-L., & Collins, J. (1998). A modular network for legged locomotion. Physica D, 115, 56–72. Haken, H. (1996). Principles of brain functioning. Berlin, Germany:

Springer.

Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand movements. Biological Cy- bernetics, 51, 347–356.

Hanson, F. E. (1978). Comparative studies of firefly pacemakers.

Federation Proceedings, 37, 2158–2164.

Honerkamp, J. (1983). The heart as a system of coupled nonlinear oscillators. Journal of Mathematical Biology, 18, 69–88.

Hoyt, D. F., & Taylor, C. R. (1981). Gait and the energetics of locomotion in horses. Nature, 292, 239–240.

Ivry, R. B., & Richardson, T. C. (2002). Temporal Control and Coordination: The Multiple Timer Model. Brain & Cognition, 48, 117–132.

Jeka, J. J., Kelso, J. A. S., & Kiemel, T. (1993a). Pattern switching in human multilimb coordination dynamics. Bulletin of Mathe- matical Biology, 55, 829–845.

Jeka, J. J., Kelso, J. A. S., & Kiemel, T. (1993b). Spontaneous transitions and symmetry: Pattern dynamics in human four limb coordination. Human Movement Science, 12, 627–651.

Katsumata, H., Zatsiorsky, V., & Sternad, D. (2003). Control of ball-racket interactions in rhythmic propulsion of elastic and non- elastic balls. Experimental Brain Research, 149, 17–29.

Keel, S. W. (1982). Learning and control of coordinated motor patterns: the programming perspective. In J. A. S. Kelso (Ed.), Human motor behavior (pp. 161–186). Hillsdale, NJ: Erlbaum.

Kelso, J. A. S. (1997). Dynamic patterns: The self-organization of brain and behavior. Cambridge, MA: MIT Press.

Kelso, J. A. S., Fink, P. W., DeLaplain, C. R., & Carson, R. G. (2001). Haptic information stabilizes and destabilizes coordi- nation dynamics. Proceedings. Biological Sciences/The Royal Society, 268, 1207–1213.

Kelso, J. A. S., & Jeka, J. J. (1992). Symmetry breaking dynam- ics of human multilimb coordination. Journal of Experimental Psychology: Human Perception and Performance, 18, 645–668. Koditschek, D. E., Full, R. J., & Buehler, M. (2004). Mechanical aspects of legged locomotion control. Arthropod Structure   &

Development, 33, 251–272.

Kugler, P. N., & Turvey, M. T. (1987). Information, natural law, and the self-assembly of rhythmic movement. Hillsdale, NJ: Lawrence Erlbaum.

Marden, E., & Bucher, D. (2001). Central pattern generators and the control of rhythmic movements. Current Biology, 11, R986–R996.

Ne´da, Z., Ravasz, E., Brechet, Y., Vicsek, T., & Baraba´si, A.

  1. (2000). The sound of many hands clapping. Nature, 403, 849–850.

Richardson, M. J., Marsh, K. L., Isenhower, R., Goodman, J., & Schmidt, R. C. (2007). Rocking together: Dynamics of intentional and unintentional interpersonal coordination. Human Movement Science, 26, 867–891.

Schmidt, R. C., Carello, C., & Turvey, M. T. (1990). Phase tran- sitions and critical fluctuations in the visual coordination of rhythmic movements between people. Journal of Experimen- tal Psychology: Human Perception and Performance, 16, 227– 247.

Schmidt, R. C., & Richardson, M. J. (2008). Dynamics of inter- personal coordination. In A. Fuchs & V. K. Jirsa (Eds.), Coordi- nation: Neural, behavioral and social dynamics (pp. 281–307). Berlin, Germany: Springer-Verlag.

Schmidt, R. C., Richardson, M. J., Arsenault, C., & Galantucci,

  1. (2007). Visual tracking and entrainment to an environmental rhythm. Journal of Experimental Psychology: Human Perception and Performance, 33, 860–870.

Scho¨ner, G., Jiang, W. Y., & Kelso, J. A. S. (1990). A synergetic theory of quadrupedal gaits and gait transitions. Journal of The- oretical Biology, 142, 359–391.

Scho¨ner, G., & Kelso, J. A. S. (1988). Dynamic pattern genera- tion in behavioral and neural systems. Science (New York), 239, 1513–1520.

Shik, M. L., Severin, F. V., & Orlovskii, G. N. (1967). Brain stem structures responsible for evoked locomotion. Fiziologicheskii Zhurnal SSSR, 53, 1125–1132.

Sternad, D., Duarte, M., Katsumata, H., & Schaal, S. (2001). Bounc- ing a ball: Tuning into dynamics stability. Journal of Exper- imental Psychology: Human Perception and Performance, 27, 1163–1184.

Strogatz, S. H. (2003). Sync: The emerging science of spontaneous order. New York: Hyperion.

Strogatz, S. H., & Stewart, I. (1993). Coupled oscillators and bio- logical synchronization. Scientific American, 269, 102–109.

Turvey, M. T. (2007). Action and perception at the level of synergies.

Human Movement Science, 26, 657–697.

van Ulzen, N. R., Lamoth, C. J. C., Daffertshofer, A., Semin,   G.

  1. R., & Beek, P. J. (2008). Characteristics of instructed and uninstructed interpersonal coordination while walking side-by- side. Neuroscience Letters, 432, 88–93.

Walker, T. J. (1969). Acoustic synchrony: Two mechanisms in the snowy tree cricket. Science, 166, 891–894.

Wilson, A., Collins, D. R., & Bingham, G. P. (2005). Perceptual coupling in rhythmic movement coordination: Stable percep- tion leads to stable action. Experimental Brain Research, 164, 517–528.

Submitted December 9, 2008

Revised May 1, 2009

Accepted May 7, 2009

Copy r ight of Journal of Motor Behavior is the property of Taylor & Francis Ltd. and i ts content may not be copied or emai led to mul ti ple sites or posted to a l istserv wi thout the copy r ight holder ‘s express w r i tten permission. However. users may pr i nt. download. or emai l articles for

i nd i vidual use.