Dynamical Patterns in the Development of Clapping - Google Sites

Dynamical Patterns in the Development of Clapping Paula Fitzpatriek Assumption College and CESPA, University of Connecticut

R. C. Sehmidt College of the Holy Cross and CESPA, University of Connecticut

Jeffrey J. Lockman Tulane University FITZPATRICK, PAULA; SCHMIDT, R. C ; and LOCKMAN, JEFFREY J. Dynamical Pattems in the Devel-

opment of Clapping. CHILD DEVELOPMENT, 1996, 67,2691-2708. The development of a complex rhythmical behavior—clapping—is modeled using a formal, explicit model of coupled oscillator dynamics. Even though this behavior manifests a good deal of nonstationarity and high variability within and across subjects, results indicate that these properties may be dynamically modeled quantitatively as well as qualitatively. Results suggest that clapping goes through a less stable period of relative coordination between 3 and 7 years before more stable absolute coordination is achieved. Nevertheless, in that the clapping behavior is affected in highly predictable ways by inertial loading ofthe limbs, the same underlying dynamic seems responsible for the coordination of both the younger and older children. Developmentally, the behavior of the coordination variable (relative phase) changes from a nonconstant magnitude in younger clappers to a constant magnitude in older clappers. These results suggest that development of proficiency in rhythmic motor skills displays developmental changes that can be understood well in dynamical terms.

There has been a growing interest recently in using dynamical theory as a means for understanding and modeling developing systems (Fitzpatrick, in press; Thelen, 1995; Thelen & Smith, 1994; Turvey & Fitzpatrick, 1993). A defining feature of developing systems is that they change behaviorai modes. Dynamical theory is particularly well suited for modeling developing systems due to its reliance on general principies of stabiiity and change to expiain the emergence of new behavioral organizations, In broad strokes, a biological organism is a complex physical system. Its behavioral patterns can be thought of as attractive states tbat sirise from an interplay of forces and mutual influence of the very many components of this system. Under certain circumstances, particular patterns are preferred and act as attractors in the sense that they are stabie

and chosen most often. Changes in circumstances (e.g., task, intention, environment, age) can alter the stability characteristics of those behavioral patterns and resuit in the appearance of new behavioral modes, In the dynamicai modeiing of behavioral systems, rhythmic interlimb coordination patterns have been a main focus on tbe formai modeiing efforts. This is due in part to the fact that coordination patterns in which the two limbs move at a common frequency are basic to many everyday activities. Proficiency in rhythmic behaviors seems to be intimately tied to the ability to bring simpie motor rhythms (repetitive patterns with fixed temporal structure) under voluntary control to produce coordinated behaviors with complex temporai sequences (e.g., von Hofsten & Ronnqvist, 1993; Rob-

The research reported here was part of a dissertation presented to the University of Connecticut by Paula Fitzpatrick. The research was supported in part by a Doctoral Dissertation Fellowship and an Extraordinary Expense Award from the University of Connecticut, NSF grant BNS 91-09880, and LEQSF Grant 92-95-RD-A-23 awarded to Richard Schmidt. The flrst author was also supported by an NSF Predoctoral Fellowship during data collection. We would like to thank Lisa Sherman for her help in data collection. Correspondence concerning this article should be addressed to Paula Fitzpatrick, CESPA, U-20, Department of Psychology, 406 Babbidge Road, University of Connecticut, Storrs, CT 06269; e-mail [email protected]. [Child Development, 1996,67,2691-2708. © 1996 by the Society for Research in Child Development, Inc. All rights reserved. 0009-3920/96/6706-0027$01.00]

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ertson, 1993; Thelen, 1981, 1991; Thelen et al., 1993; Wolff, 1967, 1968, 1991). It has been argued (Kelso, 1994) that coupled oscillator dynamics are essentialiy the oniy coordination option for achieving such coordination patterns. In this sense, coupled oscillator dynamics are an elementary coordination dynamic—a simple organizing principle found over and over in the dynamics of behavioral systems. The generality of the these coordination principles is supported by a brief sampling of the very many different types of movements that bave succumbed to such modeling: oscillations of an individual's two index fingers (e.g., Kelso, 1984), an arm and a leg (Jeka, Kelso, & Kiemel, 1993; Kelso & Jeka, 1992), pendulums held in each hand (e.g., Schmidt, Shaw, & Turvey, 1993), pendulums held by two individuals (e.g., Schmidt & Turvey, 1994), the legs of two individuals (Schmidt, Carelio, & Turvey, 1990), juggling (Beek & Turvey, 1992), finger tapping (Peper, Beek, & van Wieringen, 1991), and clapping (Fitzpatrick, Schmidt, & Careiio, 1996).

such research efforts has been to determine whether infants use these behaviors in a discriminatory or goal-directed manner. It is only recently, however, that investigators have begun to examine how young children harness the oscillatory properties of the limbs in performing skilled actions. In this connection, investigators have attempted to model the dynamics that underlie infant interlimb coordination patterns.

For exampie, von Hofsten and Ronnqvist (1993) have shown that the spontaneous arm movements of infants 3-5 days old are clearly coupled, demonstrating either a strong positive or strong negative correlation. Corbetta and Thelen (1994), in an investigation of the development of reaching, report that initially bimanual reaches predominate and are characterized by a strong coupling between the arms. Later, when control ofthe contralateral arm is gained, the two arms can be controlled independently, and unimanual reaches appear in the behaviorai repertoire. Additionally, Thelen and Fisher (1983) have shown that early leg If coupled osciliator dynamics are in- movements, aithough not generaily goal dideed elementary, then developmentally one rected, are well coordinated. In particular, should expect to see evidence of coupled os- they demonstrate that the leg joints of a sincillator dynamics underlying the ontoge- gle limb move in synchrony and the movenetic change of interlimb coordination pat- ments between limbs are weli-coordinated, tems. Given the generality and power ofthe usually in an alternating pattern. Later, inprinciples of coordination dynamics estab- fants begin to decouple the whole ieg movelished in the adult literature, the present ments to aiiow the leg joints within a singie study proposes to use interlimb patterns as iimb to move independentiy to accompiish an entry point for the formal dynamical mod- such tasks as crawling or walking (Jensen, eling of coordination in a developing sys- Uirich, Thelen, Schneider, & Zernicke, in tem. Since the behavior of children is very press). Similarly, Clark, Whitail, and Phillips noisy and variable by its nature, a robust co- (1988) demonstrate that the interlimb coorordination task or modei system is essential dination between the two legs is very tight for successful modeling. Elsewhere, the soon after the onset of walking. Indeed, very robust dynamical basis of a naturally given the rather severe task constraints inoccurring rhythmic behavior—clapping— volved in walking, this strong coupling may has been established (Fitzpatrick et al., be important in defining the gait pattern as 1996). Clapping is a behavior exhibited be- walking rather than some other iocomotory ginning at a relativeiy young age (between pattern. 8 and 12 months; Kaye & Marcus, 1981) and In sum, these studies suggest that baone known to adhere, in adults, to coupled oscillatory dynamics. This basic behavior bies are born with strong interlimb and inhas been chosen as a window through wbich tralimb coupling in spontaneous movements to view the development of coordination dy- and demonstrate a preference for moving the arms in an inphase pattern and the legs in namics. an antiphase. Further, the stability of the In deveiopmentai work, coordination coupling within and between the limbs does research has predominantly been limited to not appear to be static but seems to wax and investigations of infant behaviors that can wane throughout the first year, and perhaps be characterized as rhythmic or oscillatory. into childhood. Understanding the circumFor example, manual behaviors such as stances surrounding these changes in inbanging, shaking, and waving have been in- terlimb coupling could be important in unvestigated (Lockman & McHale, 1989; derstanding how interlimb coordination Palmer, 1989; Ramsay, 1985). The goai of unfoids in developmental time. Extending

Fitzpatriek, Schmidt, and Loekman

Coupling of e, and 02 FIG. 1.—One cycle of each of two oscillators is represented as closed orbits in a coordinate system defined by position and velocity. 6 represents the position of an oscillator in its cycle. In the figure, the oscillator on the right is slightly ahead in its cycle ofthe oscillator on the left, as indicated by a larger angle at Oj than at 6i. In this case, relative phase ((j>) would be less than 0.

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iocation of an oscillator in its cycle is represented as a phase angle 6 which changes from 0 to 2iT rad from the start to the end of an oscillation cycle. The variable that has been used to capture the coordination ofthe two oscillators (the collective variable or order parameter) is the relative phase angle, 9i — 02, namely, the difference between the two oscillators' individual phase angles. Using this coordination measure, one can determine whether the two oscillators are in the same parts of their cycles at the same time (i.e., Bi = 62), or whether one oscillator is ahead of the other. If the two oscillators are perfectly in phase, then = IT rad.

Clapping is an inphase coordination pattern by definition since homoiogous muscie dynamical treatments beyond the infancy groups are extending or flexing at the same period to other behaviors of chiidren with time during the ciapping movements. (For osciiiatory components such as banging, exampie, when the palms are together, the shaking, ciapping, and jumping could also flexor muscle group of each arm is at peak be important in increasing understanding of flexion.) Relative phasing in clapping can be the generai principies underiying deveiop- iliustrated with a simpie demonstration. mentai cbange. Stand facing a partiy open door and clap each hand on either side of the door at a The Dynamical Motieling comfortable tempo. The hands wiil contact of Clapping Behavior As a result of the mass-spring and oscil- the door at precisely the same time and one latory properties of rhythmic coordination bang or "clap" will be heard. In other words, patterns (Kandei, Schwartz, & Jessel, 1991; each hand will achieve peak flexion (when Kugler, Kelso, & Turvey, 1989), rhythmi- hitting the door) at the same time, indicating cally moved limb segments have been a relative phase of 0 rad between the two treated as self-sustained physical oscillators. hands. In tiiis case, the phasing between the Such an oscillator has it own preferred fre- iimbs is very stabie and wouid be constant quency of oscillation, or eigenfrequency from one ciap to the next. Now, clap at a ((o)—it prefers to complete one cycle of its faster tempo or strap a light wrist-weight on behavior at a certain rate (Kadar, Schmidt, & one arm. You will notice that one hand will Turvey, 1993; Kay, Saltzman, & Kelso, 1991; hit the door slightly before the other. In Kelso, Holt, Rubin, & Kugler, 1981). In- other words, one hand will lag slightiy beteriimb coordination patterns have been for- hind the other. If one continues to increase mally modeled by two such oscillators inter- the clapping tempo or add more and more acting across a coupling medium (Haken, mass to the hand, the lag between the two Kelso, & Bunz, 1985). If the oscillators have hands contacting the door will continue to different preferred frequencies (to), the cou- increase. This demonstration illusfrates that pling requires that each osciilator modify its it becomes more difficuit to maintain a relaintrinsic frequency so the two oscillators can tive phase relation of precisely 0 when clapbecome coordinated at a common frequency. ping frequency increases or the mass imbalance between the iimbs increases. Under The dynamics of an oscillator can be these circumstances, the inphase mode begraphically understood by representing its comes less stable and less precise.^ change of position and velocity during its cycle. As seen in Figure 1, each oscillator These kinds of systematic changes in traces a closed orbit in such a coordinate sys- the behavior of relative phase from * body mass for the "light" mass condition 0 increases with increasing frequency. and 0.015 * body mass for the "heavy" conPrediction 4. SD i> increases with increasing dition. Table 1 lists a range of mean body frequency. masses appropriate for the children particiPrediction 5. SD 4> increases as the magni- pating in the study (Brandt, 1979; Haywood, 1986) as well as the corresponding dimentude of Ato deviates from 0. sions ofthe appropriate wrist weights. Wrist weights were selected on the basis of body Relative Coordination Predictions mass. Six Styrofoam balls wrapped in reflecPrediction 6. (}> should be distributed across all relative phases but concentrated near 0 and u tive tape (3M) were used as joint markers for a video digitizing system (described below) rad. required to analyze the unconstrained clapPrediction 7. The distribution of should be more disperse as Ato deviates from 0. Prediction 9. (j) should increase with increasing frequency. Prediction 10. 0 indicates that the right limb is loaded, while Ato < 0 indicates that the left limb is loaded. The inertial loadings and ACQ values are provided in Table 2. Children were asked to clap at one of three tempos—at a self-cbosen pace, or in time to a metronome set to .88 Hz or 2.09 Hz. Children in the youngest age group (3year-olds) did not complete the metronomepaced trials because pilot testing indicated (a) they could not keep the metronome beat, and (fo) they would become tired and distracted before the end of the session. All children clapped very quietly and therefore had no difficulty hearing the metronome during the metronome-paced trials. For the 3-year-olds, the experimental design consisted simply ofthe five Ato condi-

TABLE 2 INDEX FOR FIVE INERTIAL LOADINGS OF THE LIMBS

Left Limb

Right Limb

AM Index

"Heavy' "Light"

0 0 0

-2 -1 0

'/Light" "Heavy"

+1 +2

0 0 0

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tions at a self-chosen pace (a total of 5 experimental conditions). The experimental design for the three older groups was a repeated-measures design in which the three frequencies (self-chosen pace, 0.88 Hz, 2.09 Hz) were completely crossed with the five Ato conditions, for a total of 15 experimental conditions. Each condition was given only once. Presentation of trials occurred in blocks, with the self-chosen pace block always occurring first; whether a block of slow metronome trials or a block of fast metronome trials was the second or third block was randomized across subjects. Order of presentation of Aco conditions was randomized within each frequency block, with the exception that the unweighted selfchosen pace trial was always first. This trial was repeated again atthe end ofthe first block to reduce any biasing due to practice. Each session lasted approximately 20-30 min. Data Reduction Digitized movement trajectories (i.e., the three-dimensional coordinates ofthe image space) were obtained post hoc from each videotaped experimental session. The Peak Performance Automatic Data Capture module was used to track contrasting bright markers (with image pixel values greater than a user-defined minimum) against a dark background, using a cubic spline search algorithm. Only the markers on the fingertips and elbows were tracked. The experimenter had to intervene manually whenever the Automatic Module failed to keep the fingertip markers distinct.

FIG. 2.—Reflective markers were attached to each arm as indicated in the figure. A is a veetor along the longitudinal axis of the forearm; B is a vector aeross the body. Together, they determine the primary angle of excursion for a limb.

The frequency of oscillation of each limb and the ^ times series of the two limbs were determined using software routines. The time of maximum extension of each limb was calculated using a peak picking algorithm. The frequency of oscillation was calculated using the peak extension times: /„ = l/(time of peak extensionn — time

/Q%

These cycle frequencies were used to calculate the mean frequency of oscillation of each limb for each trial and condition. Coupled frequency of the two-limb system was calculated 'as the mean of the frequency of oscillation of each limb. The phase angle of each limb (Bj) was calculated at each sample (60/sec) to get a time series of 9;. The phase angle of limb i at sample j (6,.,) was calculated as

The 3-D raw data were filtered using a Butterworth filter with the optimal filtering parameter determined by the Jackson Knee (e,j) = arctan (4) Method (Jackson, 1989). The threedimensional coordinates were then trans- where x,-,- is the velocity of limb i at sample formed so that the origin was at the left el- j divided by mean trial frequency, and Ax,-,bow and the primary angular excursion for is the position of the time series of limb i at each limb was calculated. The primary angle sample J minus the average trial position. ^ of excursion (a^) was defined as the angle between the right and left limbs was calcuformed between two vectors originating at lated as Bieft; — Ojighy. Thus, if the right limb the biceps: A extending to the fingertip, and is ahead ofthe left in its cycle, tj) will be B extending to an arbitary fixed point on the negative (4) < 0); if the left limb is ahead of the right in its cycle, ^ will be positive {^ > body as demonstrated in Figure 2. 0.). In clapping, time series allows an evaluation of In other words, the primary angle represents the stability of the coordination across the the angle formed between the forearm seg- different conditions. This evaluation was acment ofthe arm and the vertical plane ofthe complished by calculating the mean .05). The significant main effect of hand (right = 1.79 Hz, left = 2.02 Hz) indicates a tendency for the left hand to oscillate slightly faster than the right. The left hand is about 4 msec ahead of the right, on average. Visual inspection of representative angular excursion time series (Fig. 4) further demonstrates the difficulty the young children had in tracking the slow metronome. This figure suggests that the 4-year-olds solve the coordination problem of tracking the metronome by (a) keeping the hands together for a significant portion of the movement cycle (as evidenced by plateaus at the valleys during hand contact), and (b) moving the hands faster during the fiexion portion of the cycle than the extension portion (as evidenced by the sparsely spaced points). Thus, they adopt a strategy of speeding up as the hands come together and compensate by slowing down (or stopping) at contact and as the hands move apart. As seen in Figure 4, this strategy is more prevalent in younger

children than older children and for slow metronome tracking as opposed to fast metronome tracking. Evaluation of Absolute Coordination Predictions Relative phase.—An ANOVA with a within factor of Aw was performed on mean 0), it lags in phase (mean ^ > 0), and when the left limb is loaded (Aw s 0), it lags in phase (mean .05), indicating that overall variability was rather high but did not change in a systematic manner under manipulation of Aw. In other words. Prediction 5 was not supported for the 3-year-olds. A 3 x 3 x 5 ANOVA as above was performed on SD4> for the 4-, 5-, and 7-year-olds and resulted in a

Fitzpatriek, Sehmidt, and Loekman

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4 Year, Slo-w MetronDme O Right Hani + Left Hand

3.75

4

^.35

^.5

-1.75

5

5.35

5.5

5.75

6.25

Time (sec)

\^

V-X -•40

3.75

4

4.Z5 4.5

4.75 5 5.25 Time (sec)

5.5

5.75

6.25

4 Year, Fast Metranome O Right HaM + Left HaM

-30

3.75

4

4.25

4.5

4.75 5 5.25 Time (sec)

5.5

5.75

6.25

7 Year, FastMetoonomje O Right Hand + LeftHaad

; 15 105 0-5 -10j-15

-20 4.75

5

5.25

5.5

5.75 6 6.25 Time (sec)

6.5

6.75

7.25

FIG. 4.—Representative angular excursion time series from individual 4-year-old and 7-year-old subjects for slow (top two panels) and fast (bottom two panels) trials. significant effect of Aco only, F(4, 48) = 2.55, MSe = .06, p < .05. Mean variability for the five Ao) conditions was .77, .70, .60, .70, and .71 rad, respectively. This indicates that as A(i) deviates from 0 variability increases, and the magnitude of this variability increases with the magnitude of Aw as Prediction 5 states. There was no evidence for the pre-

dicted changes in variability as a function of increasing frequency (Prediction 4). Again, this is most likely due to the difficulty children had in tracking the metronome. Summary of absolute coordination results.—Ofthe five predictions used to evaluate absolute coordination patterns of the

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Child Development TABLE 3 REGRESSION COEFFICIENTS OF near 0 rad supports Prediction 6 that should be concentrated closer to the points of weak attraction (0 rad in the case of an inphase coordination pattern like clapping). Additionally, the developmental trend for the concentration of (|) nearer to 0 rad to increase suggests that the attractor at 0 is increasing in strength and, developmentally, clapping patterns are becoming more like absolute coordination patterns. The proportion of the distribution of : As the distribution of ^ moves outside of the stable region (). A significant effect of frequency, F(2, 24) = 13.32, MSe = .01, p < .01 indicates that increases with increasing frequency (.09, .13, and .18 for the .88 Hz [slow], 2.09 Hz [fast], and 2.86 Hz [self-chosen] conditions, respectively) as predicted by the coupled oscillator model (Prediction 9). Post-hoc t tests revealed that 4> at the slow tempo is significantly different from fast and self-chosen (p < .01), but fast and self-chosen are not significantly different from each other (p > .05). A significant effect of Aw, F(4, 48) = 5.08, MSg = .002, p < .01, indicates that ^ increases as Aw deviates from 0 (.14, .12, .11,

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.13, .13, for Aw = - 2 to 2, respectively) also as expected of a system abiding by coupled oscillatory dynamics (Prediction 10). This interpretation was confirmed by post-hoc t tests—(j) at Aw = 0 is significantly different from (j) at all other Aw conditions (p < .05), except Aw = — 1 (p < .05). A simple regression with mean 4> as the dependent variable and age group as the independent variable resulted in a significant regression, r^ = .48, F(l, 18) = 16.65, MSe = •0003, p = .0007, with a negative slope ( — .01). This suggests (albeit modestly given the small sample) that developmentally 4> is decreasing. An inspection of mean (j) confirms this interpretation (M = .14, .15, .12, .11 for the 3-, 4-, 5-, and 7-year-olds, respectively). In sum, these results support the model predictions that i should increase as attractor strength decreases (increasing frequency, increasing deviation of Aw from 0). Additionally, a developmental trend hints at a decrease in i>, which suggests that strength of the attractor is increasing developmentally and moving in the direction of absolute coordination.

pled oscillator regime (Equation 1) is a viable model of the interlimb phasing found in children's clapping at all stages of development. The clapping patterns of younger children can be characterized as relative coordination and are suggestive of an oscillatory system with weak attractors (i.e., a weakly coupled system) and considerable noise. In the younger children, the clapping patterns displayed more variability and did not demonstrate characteristic changes in relative phasing under manipulation of frequency and inertial imbalance. By about 7 years, however, clapping patterns more closely approximate absolute coordination patterns and are suggestive of a more strongly coupled oscillatory system. These clapping movements were less variable, more stable, and displayed changes in relative phasing characteristic of stable coupled oscillators. Thus, clapping behavior appears to demonstrate a developmental progression from relative coordination to absolute coordination.

Dynamically speaking, such a progression suggests that the strength of the inSummary of relative coordination re- terlimb coupling is increasing developmensults.—Ofthe five predictions used to evalu- tally. One way to model this change is to ate relative coordination patterns ofthe cou- assume that age (or more more likely a varipled oscillator dynamic, strong support was able associated witb age—one that quantifound for four, and one prediction was mod- fies skill level, for instance) acts like a conestly supported. In particular, 4> increased trol parameter and alters the dynamical with increasing frequency, and as Aw devi- landscape, thereby modifying the stability of ated from 0, 4> was shown to be distributed the interlimb coordination attractors. In paracross all phase relations with a concentra- ticular, changes in age may result in changes tion near 0, and the distribution of 4) became ofthe magnitudes ofthe coupling processes more dispersed (moved farther away from 0) which bring the limbs into a stable phasing as Aw deviated from 0. While the change in pattem. To estimate the coupling strength of the distribution of 4> with increasing fre- the dynamic, Ato was regressed on sin (4)) for quency was modest, it did demonstrate a each participant. The slope of such regrestrend toward increasing dispersion with in- sions is an estimate of the coupling strength creasing frequency as expected. Further, the of a coupled oscillatory dynamic (Schmidt regression analyses revealed several devel- & Turvey, 1995). Inspection ofthe coupling opmental trends—4> decreased developmen- estimate means (M = 5.94, 6.59, 7.87, and tally, and the distribution of 4> changed such 9.54 for the 3-, 4-, 5-, and 7-year olds, respecthat the distribution of 4> near 0 increased tively; SD = 3.09, 3.79, 3.29, and 3.41, rewhile all other phase relations became much spectively) suggests a modest tendency for less common. Concomitantly, these results interlimb coupling to increase developmensuggest that the clapping patterns of chil- tally. dren (particularly the younger groups) can Another modeling alternative to account be characterized as relative coordination abiding by coupled oscillator dynamics. Ad- for the developmental changes that occur in ditionally, the results suggest that develop- the progression from relative coordination to mentally the coordination patterns are be- absolute coordination is to add an additional coming more stable and less variable and dynamical term to the coupled oscillator in Equation 1 that changes its begin to more closely approximate absolute model strength with age. Such a developmental dycoordination around 7 years. namic would reconfigure the initial clapping dynamic (without fixed points) to have an Discussion attractor state at the to-be-acquired coordinaGenerally, both absolute and relative tion pattern. This is analogous to the modelcoordination measures indicate that a cou- ing strategy adopted by Schoner and col-

Fitzpatrick, Schmidt, and Lockman leagues (Schoner, 1989; Schoner, Zanone, & Kelso, 1992; Zanone & Kelso, 1992) in modeling the dynamics involved in adult acquisition of a novel bimanual phasing pattern.^ This developmental dynamic may not only alter the strength ofthe attractors but also the location of the attractors. Further, such an additional term is likely one that accounts for the intentional aspects of clapping and the fact that the intentional resources needed to maintain coordination in all probability change with age. For example, it is possible that a stable coordination dynamic is assembled initially but cannot be maintained throughout a bout of clapping. This could occur if the intention to maintain the coordination pattem was not constant throughout a trial (i.e., the intention was "turned on and off"). A young child with a weak intentional term may initially concentrate on the clapping task, become distracted by something, and then return his or her attention to the task of clapping. Older children with a stronger intentional term, on the other hand, may assemble the dynamic and not have to actively monitor their intentional goals during the activity. Such an intentional term could have a value of 0 when attention is other-directed and intermediate values scaled to the magnitude of concentration on the task.

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as task, enyironmental, and intentional constraints change. Given the findings presented here, the developmental course of rhythmic motor ability appears to have a rather protracted time scale. That is, even though infants begin clapping late in the first year and are capable of other forms of auditory-manual coordination, such as reaching to a sounding object (Clifton, Muir, Ashmead, & Clarkson, 1993), stable and highly predictable absolute coordination clapping pattems suggestive of strong interlimb coupling do not appear until well into childhood. This appears to contradict other findings that reveal very tight interlimb coupling in spontaneous movements of infants, for instance (Corbetta & Thelen, 1994; Thelen & Fisher, 1983). However, a complete account ofthe development of coordination may reveal a developmental progression from stable absolute coordination to less stable relative coordination (Goldfield, 1995; Thelen & Smith, 1994) back to stable absolute coordination. It seems likely that while initial movements may be tightly coupled, such tight coupling may necessarily weaken as the infant explores new intentional behavioral pattems in producing goal-directed actions (e.g., reaching, walking). As proficiency is gained in producing the goaldirected actions, the coupling likely becomes strong again. This type of weakening and strengthening of the coupling may indeed be a process that continues throughout childhood as body dimensions, motor competence, perceptual acuity, and task and intentional constraints continually vary. Coming to terms with the relation between rudimentary interlimb coupling (e.g., in spontaneous infant arm and leg movements) and the interlimb coupling involved in more complicated rhythmical behaviors with complex spatial and timing constraints and more explicit intentional goals (e.g., expressive or communicative functions, percussive functions) is important for a more complete dynamical account ofthe development of coordination.

A combination of both modeling strategies is probably necessary in which there is an interplay between the intentional dynamic that "sets up" the attractor layout of the movement dynamic and the coupled oscillator movement dynamic itself. For example, initially the intentional dynamic may be weak and the oscillatory dynamic may have weak attractors associated with it. This combination of dynamics could result in the types of relative coordination patterns observed here in younger children. During a transitional period, one could imagine the intentional dynamic becoming stronger but the fixed points of the oscillatory dynamic remaining weak as optimal parameterization of coupling strength is explored. A period of skilled absolute coordination may result from a combination of an autonomous intentional dynamic coupled with an oscillatory It seems likely that the variability undynamic with strong attractors. Such a pro- derlying relative coordination pattems as gression from relative to transitional to abso- obseved here in clapping may also be relute coordination may occur a number times lated to other variables, for example, exploComparisons between modeling strategies adopted in the learning domain are appropriate given the dynamical system's proposal that general principles (e.g., change in stability) are responsible for changes in behavior. What distinguishes leaming from development is the time scale over which the behavior unfolds (e.g., Saltzman & Munhall, 1992; Thelen, 1989; Zanone & Kelso, 1990); learning proceeds on the order of hours and days, while development unfolds on the order of months and years.

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ration of different clapping styles. Transforming the initially rigid, stereofyped claps of very young children into the fluid, effortless applauding style seen in the absolute coordination of adults may require exploring how cupping the hands in different ways, aligning the hands in various positions, or altering the position ofthe limbs at contact affect the quality and loudness ofthe sound produced. Other factors may also be important contributors to the achievement of absolute coordination in clapping. One possibilify is that children may need to master the coordination of each arm individually before they are able to couple both arms absolutely. Kaye and Marcus (1981), for instance, report that it is easier for babies to shake a toy twice than to clap four times in succession. Developmentally, little is known about the relation between skills that have both single oscillator and coupled oscillator components. Another possibility is that children may need to control not only the temporal but also the spatial aspects of clapping to achieve absolute coordination. For instance, absolute coordination may be more easily accomplished if the hands always collide atthe same location than if they collide at different ones during a clapping bout. Although temporal measures were used primarily in the present work, future research might employ a combination of temporal and spatial measures (as well as single and coupled oscillator measures) to understand how absolute coordination develops. Concluding Remarks This research has made inroads into extending understanding of the development of coordination dynamics in several ways. First, an explicit, formal model has been applied and tested in examining the development of a complex interlimb rhythmical behavior. In this case, we have demonstrated the applicability of the Haken et al. (1985) coupled oscillator model to clapping of children and have further shown developmental changes in the parameterization of this model. Additionally, even though this behavior manifests a good deal of nonstationarity and high variability within and across young children, we have presented evidence indicating that quantitative dynamical modeling is possible nevertheless. Finally, the approach we have used suggests that clapping goes through a period of relative coordination before absolute coordination is achieved. These results suggest that developing systems can be understood in dynami-

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