Infant Bouncing: The Assembly and Tuning of Action Systems

Eugene C. Goldfield Children's Hospital, Boston, and Harvard Medical School

Bruce A. Kay and William H. Warren, Jr. Brown University

GOLDFIELD, EUGENE C.; KAY, BRUCE A.; and WARREN, WILLIAM H., JR. Infant Bouncing: The Assembly and Tuning of Action Systems. CHILD DEVELOPMENT, 1993,64,1128-1142. We outline a theory of infant skill acquisition characterized by an assembly phase, during which a task­specific, low-dimensional action pattern emerges from spontaneous movement in the context of task constraints, and a tuning phase, during which adjustment of the system parameters yields a more energetically efficient and more stable movement. 8 infants were observed longitudinally when bouncing while supported by a harness attached to a spring. We found an initial assembly phase in which kicking was irregular and variable in period, and a tuning phase with more periodic kicking, followed by the sudden appearance of long bouts of sustained bouncing. This "peak" behavior was characterized by oscillation at the resonant frequency of the mass-spring system, an increase in amplitude, and a decrease in period variability. The data are consistent with a forced mass-spring operating at resonance.

Consider the situation faced by a 6-month-old when first placed in a "Jolly Jumper" infant bouncer. The infant is hang­ing in a harness from a linear spring, with the soles of the feet just touching the floor. What is the "task"? What limb movements will make something interesting happen? There are no instructions or models-the behavior of the system must be discovered. The infant may tryout various movements before finding that kicking against the floor has interesting consequences. Over the next several sessions in the bouncer, the infant will go from a few sporadic kicks and irregu­lar bouncing to stable, sustained oscillation. What has occurred to yield this coordinated, task-specific organization of the action sys­tem?

This situation is not unlike those con­fronting infants in the development of many other motor skills, such as reaching, shaking a rattle, balanced sitting, crawling, or walk­ing. In each case, infants are discovering and

refining task-specific regimes of organization within given task constraints. From the fetal period onward, humans are capable of spon­taneously moving the articulators (Smoth­erman & Robinson, 1988). Many of these spontaneous movements are high-dimen­sional, characterized by high variability and disorganization. Fortunately, there are a number of constraints that conspire to re­duce this dimensionality so that organized low-dimensional action patterns emerge, re­ferred to variously as "coordinative struc­tures," "synergies," or "dynamical regimes" (Turvey, 1988). These constraints include properties of the actor and the environment, such as the architecture of the nervous and musculoskeletal systems, the masses and lengths of the limbs, the material properties of environmental surfaces and objects, and an omnipresent gravitational field. We sug­gest that organized action patterns emerge from spontaneous activity in the context of such task constraints, as infants explore and exploit the physical properties of their bod-

The authors wish to thank Catherine Eliot for assistance in data collection, PlOfessors Thomas Ammirati and Michael Monee for help in measuring the damping and stiffness of the spring, and two anonymous reviewers for their evaluations of a previous version. This research was supported by National Research Service Award IF32MH09056 to Eugene C. Goldfield from the National Institute of Mental Health, and by Grant AG05223 from the National Institutes of Health to William H. Warren. We would like to point out that the authors contributed equally to the paper and that order of authorship is arbitrary. Correspondence to Eugene C. Goldfield, Department of Psychiatry, Children's Hospital, 300 Longwood Ave., Gardner 5, Boston, MA 02115.

[Child Development, 1993,64, 1128-1142. © 1993 by the Society for Research in Child Development, Inc. Al! rights reserved. 0009-3920/93/6404-0008$01.00]

ies and environments. These task dynamics provide the landmarks around which behav­ior is organized.

In this article we wish to pursue the no­tion that concepts from the field of dynami­cal systems that have recently been applied to studies of rhythmic movement in adults (Kay, Saltzman, & Kelso, 1991; Kelso & SchOner, 1988; Kugler & Turvey, 1987) may also illuminate problems of motor learning (Schmidt, Treffner, Shaw, & Turvey, 1992; Schoner, Zanone, & Kelso, 1992) and motor development (Goldfield, in press; Thelen, 1989). Our purpose here is twofold: to sketch an approach to motor learning and develop­ment motivated from a dynamical perspec­tive and to report initial results on infant bouncing that illustrate part of this approach. We confess at the outset that the data will only partially address the theory, but we present some theoretical background to es­tablish the motivation and direction of our research.

Specifically, we propose that two pro­cesses are involved in the developmental transformation of spontaneous activity into a task-specific action pattern: assembly of an action system with low-dimensional dynam­ics and tuning the system to refine and adapt the movement. AssemblY'is a process of self­organization that establishes a temporary re­lationship among the components of the musculoskeletal system, transforming it into a task-specific action system such as a kicker, a walker, or a shaker (Bingham, 1988; Saltz­man & Kelso, 1987). Once assembled, the parameters of this dynamical system are tuned in order to adapt the movement pat­tern to particular conditions-kicking in an infant bouncer, for example, as opposed to supine kicking. These notions can be made more concrete through a characterization of the task dynamics.

The Assembly and Tuning of Low-dimensional Dynamics

Dynamics in the classical sense is the study of how the forces in a system evolve over time to produce motions. Recently the notion has been expanded to include situa­tions in which forces and motions are ab­stract concepts that merely express relation­ships among the variables of interest (e.g., Abraham & Shaw, 1982). The resulting ab­stract dynamics is a general science of how systems evolve over time; where possible, however, we will seek a physical interpreta­tion grounded in the physical task con-

Goldfield, Kay, and Warren 1129

straints. Whereas the aim of such an analysis is to place motor behavior in the context of natural physical law, we believe biological systems differ from garden-variety mechani­cal systems in two important respects. First, they are intentional systems whose actions are goal directed. We will consider a goal to be a boundary constraint on the assembly of an action system, which thence behaves as a dynamical system. The problem is to as­semble an action system that yields a stable movement pattern, or attractor, which cor­responds to the intended action. Second, biological systems are regulated by infor­mation in Gibson's (1966) sense-visual, au­ditory, haptic, somatosensory, and vestibular patterns of stimulation-that informs the or­ganism about the state of the environment and the body. In particular, Kugler and Tur­vey (1987) proposed that haptic information about the underlying dynamics can specify the form of stable movement patterns. Thus, we will talk about the active exploration of task dynamics on the basis of such informa­tion as an essential part of the assembly and tuning process. Given a particular goal or task, and assuming that action is regulated by information about dynamics, the behavior of the system can be analyzed as determinis­tic and predictable.

There are at least three levels at which task dynamics may be considered, differenti­ated mainly by the time scale at which their variables change (Farmer, 1990; Saltzman & Munhall, 1992; SchOner et aI., 1992). These are referred to as the graph level, the param­eter level, and the state level, each with its own associated dynamics. At the graph level, a task-specific action pattern is assembled from the many components of the musculo­skeletal system-abstractly, a function for a dynamical system. At an intermediate level, the parameters on the function are tuned to yield a movement adapted to the task at hand. When performing the task, the dynam­ical system "runs" or evolves through a se­ries of states to an attractor or stable move­ment pattern.

For illustrative purposes, let us repre­sent the set of all possible limb configura­tions ina high-dimensional state space, which characterizes the state of all 100 de­grees of freedom of the joints. An action pat­tern then corresponds to a set of trajectories in a restricted region of this space, reflecting a particular relationship among the many de­grees of freedom. However, a description of the system at this level of detail would be inordinately complex. Rather than repre-

1130 Child Development

senting all the microscopic degrees of free­dom of an organized system, we can provide a simplifying macroscopic description of its behavior in terms of a function for a dynami­cal system. The function expresses the sys­tem's low-dimensional dynamics, character­ized by the preferred state or attractors toward which the system tends. Although such attractors may be abstract mathematical objects, in physical systems they also tend to correspond with energetic minima (see below). This graph level provides a macro­scopic description of the components of the action system and their relations, which are implicated in a particular task.

As a familiar example, rhythmic move­ments may behave like a linear mass-spring system, described by the following equation (French, 1971):

mx + hi: + kx = 0, (1)

where x is the position of the mass m, k is the spring stiffness, and b is a damping coef­ficient or friction term. When set into mo­tion, an ideal undamped linear system (b = 0) will oscillate indefinitely at its natural fre­quency

(2)

but its amplitude is unstable because it will change when the mass is perturbed. Thus, the system has no preferred states; its possi-

3

2

ble steady-state behavior fills up the phase plane, which plots position against velocity. However, when a linear mass-spring is damped (b > 0), the system acquires a single preferred end state, as the mass eventually comes to a stop at the resting length of the spring. This corresponds to a point attractor in the phase plane (x = 0, x = 0) and has been developed as a model of discrete limb movements (Feldman, 1986; Saltzman & Kelso, 1987).

Such a linear mass-spring may be driven by an external forcing function:

mx + bx + kx = Fo cos (oot), (3)

where the driving force is sinusoidally mod­ulated at a frequency 00 with an extrinsic time scale. In this case the phase plane tra­jectory is a limit cycle attractor, correspond­ing to stable oscillatory movement. When such a system is forced at its natural fre­quency (00 = 000), a "resonance peak" results yielding the largest amplitude for the mini­mum force (Fig. 1).

A nonlinear system (one with a nonlin­ear damping term) that is fed from a constant energy source rather than a periodic driver may also exhibit such limit-cycle oscillation autonomously, as in the case of a van der Pol oscillator:

mx - ax + bx2x + kx = O. (4)

2

CJ) I CJ) 0

FIG. I.-An example of a resonance curve for the sinusoidally forced linear damped mass-spring. The damping is such that the quality (Q) factor of the response is 3.

Here the damping term with coefficient a delivers energy to the mass, and the damp­ing term with coefficient b takes energy away from the mass, yielding a self-sus­tained oscillation. The critical distinction between nonautonomous and autonomous systems is that the former require an exter­nal clock to provide the timing of the forcing function, whereas the timing of the latter is intrinsically determined, contingent on the state of the system itself. It follows that, if a clock-driven system is perturbed, it will return to be in phase with its oscillation prior to perturbation, because the clock is unaffected by the disturbance. On the other hand, the phase of an autonomous system will be shifted after the perturbation, a phenomenon known as "phase resetting." Only nonlinear systems can behave au­tonomously.

Finally, forcing a nonlinear system at different frequencies and amplitudes can give rise to a variety of other attractors, in­cluding quasiperiodic and nonrepeating chaotic attractors (Thompson & Stewart, 1986). This suggests the beginnings of a bes­tiary of dynamical regimes with characteris­tic attractors, on which action-system func­tions may be modeled.

An unresolved problem in motor coordi­nation is the process by which such a func­tion is assembled. How do the microscopic degrees of freedom of the legs self-organize to become a periodic "kicker" with macro­scopic limit-cycle behavior? This is a very difficult problem that we can only begin to address here. We suggest that spontaneous activity in the context of task constraints re­sults in the formation of stable action pat­terns, akin to morphogenesis in biological systems or pattern formation in physical sys­tems (Haken, 1977; Murray, 1989). The first part of such an answer would be that the solution space is restricted by the task con­straints-the intrinsic dynamics of a system of pendular limbs and spring-like muscles and the extrinsic dynamics of the environ­mental context. These define a layout of pos­sible attractors that yield specific classes of movement, such as point-attractor or limit­cycle behaviors. The second part of an an­swer would be that the actor explores this layout to locate an attractor-sometimes ran­domly, as in the global flailing often exhib­ited by infants, or in more directed ways, such as probing the space with an existing repertoire of skills or reflexes. The discovery of a low-dimensional attractor would serve to index a possible configuration of the ac-

Goldfield, Kay, and Warren 1131

tion system's components for the task, which could be evaluated based on information about the effectiveness and stability of the resulting movement. As Haken (1977) has ar­gued, the emergence of an attractor from the interplay of the system's many degrees of freedom reciprocally acts to select that par­ticular combination of degrees of freedom. This implies that one of the functions of spontaneous activity in infancy is to explore possible organizations by allowing the free interplay of components and evaluating the attractors that emerge.

At an intermediate level of analysis, we can consider the effects of varying the pa­rameters of this function, such as the mass, stiffness, damping, and forcing frequency of a mass-spring, on the behavior of the system. This can be represented as exploring "pa­rameter space," whose dimensions are the system parameters. The effects of different parameter combinations can be evaluated via some measure of the resulting behavior, such as a cost function. A parameter surface or "landscape" would then reflect the cost of various parameter settings, with a global minimum in the surface indexing an effi­cient set of parameter values for the task. Some parameters are often fixed by task con­straints, such as the stiffness of the spring and the mass of the infant in an infant bouncer. However, other free parameters, such as forcing frequency, may be modu­lated to refine the oscillation. Such parame­ter tuning adjusts movement to the local con­ditions of the task and adapts to changing conditions. The developing infant's percep­tual system thus becomes sensitive to pro­prioceptive information specifying minima in the landscape, which can configure the parameters for a given task.

Most parameter changes are benign in that they do not qualitatively affect the sys­tem's behavior, and the system is considered "structurally stable" under such changes. But for nonlinear systems certain parameter changes can alter the system's behavior qualitatively, giving rise to a different num­ber, type, or layout of attractors. At such criti­cal points the system is said to "bifurcate" or, in physical terms, undergo a phase transi­tion. Exploring the parameter range assesses the structural stability of a particular organi­zation and may be critical in learning to con­trol transitions from one action mode to an­other, such as rocking to crawling or walking to running.

At the lowest level, once parameter val-

1132 Child Development

ues are set, we can consider how the state of the system evolves over time from various initial conditions. The states of a mass­spring system are the possible positions and velocities of the mass, represented by points in the phase plane; for the action system, they may be a subset of limb positions and velocities. As noted above, damped systems will be drawn to attractors, represented as stable trajectories in the phase plane and corresponding to stable movement patterns. The properties of the resulting behavior may be used to evaluate the action system func­tion and tune its parameters.

There are two possible advantages to operating at an attractor for a given task. First, it is often noted that preferred move­ments are energetically efficient. Indeed, there is a large literature showing that, for actions as various as walking to using a bicy­cle pump, actors freely adopt "optimal" movement patterns that require minimum energy expenditure within the constraints of the task (Corlett & Mahaveda, 1970; Holt, Hamill, & Andres, 1990; Hoyt & Taylor, 1981; Ralston, 1976). On this interpretation, the dynamics can be described in terms of a landscape with hills (maxima) representing high energy cost and valleys (minima) repre­senting low energy cost. Such landscapes can be defined at each level of analysis. Min­ima in function space correspond to effective musculoskeletal organizations for a given task. Minima in parameter space correspond to efficient parameter configurations. With fixed parameters, the attractor toward which the system evolves is the minimum energy state. The evidence supports the view that the action system tends toward energetic minima defined within the given task con­straints.

However, there are many small-motor tasks for which the energetic consequences of moving away from the preferred state are biologically insignificant in the context of daily metabolic fluxes, and it is hard to ratio­nalize them by traditional optimality crite­ria. But there is a second advantage for operating at an attractor-its stability. A minimum in an energy landscape provides a qualitative point that is easily detected, pre­cisely recovered after perturbation, and re­producible on separate occasions. Contrast this with trying to maintain a state on a slope in the landscape, for which there is no intrin­sic information in the surface itself. It has recently been shown that human actors de­liberately operate near but not locked onto attractors in order to continuously sense the

gradient for the attractor's location (Beek, 1989; DeGuzman & Kelso, 1991), for at the minimum itself the gradient disappears. Thus, even if the energetic advantages are irrelevant, exploiting the dynamics may yield more stable, organized movements than fighting them.

In sum, we suggest that motor develop­ment involves a process of exploring the task dynamics at these three levels of analysis. In the course of learning a task, the infant experiments with different combinations of musculoskeletal components, in effect adopting different functions over the de­grees of freedom of the action system, and explores the attractor layout that emerges. Parameter tuning optimizes the configura­tion of parameters that is most efficient for a given task. Within a particular parameter setting, the state of the system evolves to an attractor, yielding a stable movement pat­tern. Thus, we would expect that, when an infant learns a task, the large-scale variabil­ity in movement trajectories would be re­duced as the infant becomes sensitive to information specifying the task dynamics, lo­cates an attractor, and tunes the parameters. Conversely, we would expect initially rigid "reflex" movements to become more flexible and adaptive under varying conditions as sensitivity to the nuances of the landscape develops. Tasks that possess relatively sim­ple dynamics-for example, a function land­scape dominated by a large basin of at­traction with few local minima-would presumably be easier to learn and would be mastered earlier in development.

A Model of the Bouncing Infant As an example, let us return to the

bouncing infant. Presumably, initial explora­tion reveals that kicking produces entertain­ing effects. A task-specific periodic kicking system is assembled, with macroscopic be­havior akin to that of a forced mass-spring. As a first approximation, we can model this function by Equation (3):

mx + bX + kx = Fo cos (wt).

Here, the mass parameter (m) represents the mass of the infant, and the stiffness (k) and damping (b) parameters represent the char­acteristics of the infant bouncer's spring. The infant's kicking is represented by the driver of the right-hand side; the actual ver­tical motion of the infant is represented by x and its time derivatives. The free parameters that may be regulated by information appear to be the driving force Fo and the forcing

frequency w: how much force to apply and when to apply it.

This equation has a very clear opti­mality property, resonance. For any given driving force, the amplitude of the mass's oscillations is maximal at a specific fre­quency (see Fig. 1); conversely, a given am­plitude requires minimum driving force at this frequency (Kugler & Turvey, 1987). This value is termed the resonant fre­quency, and it is close to the natural fre­quency of the undriven system (Eq. [2]). One possibility is that infants search fre­quency space until they find the resonant frequency.

However, several limitations are imme­diately apparent. First, Equation (3) repre­sents a continuous sinusoidal forcing func­tion, but during sustained bouncing the infant's feet are actually in contact with the ground for less than half a cycle. Further­more, the baby can exert force in only one direction, that is, can only push against the floor with her muscles and not pull. Thus, it may be more appropriate to replace the sinusoidal driver with one having a some­what more complicated form:

mx + bi + kx = Fof(t), (5)

where f(t) = 0 when the feet are off the ground and fit) = 1 - sin(wt) (which is al­ways greater than or equal to 0, i.e., upward against the mass) when the feet are on the ground. This does not entail a drastic alter­ation to the model: it alters the shape of the resonance curve but does not move the fun­damental resonance peak's location away from w (Thomson, 1981, pp. 77-78). Prior to sustained bouncing, the feet were observed to be on the ground most of the time, so the simpler sinusoidal forcing function may be an adequate model during that time.

Second, during ground contact, it is un­likely that the infant's legs are acting as pure force applicators (as expressed in the right­hand sides of Eqq. [3] and [5]), but more like springs, with the joints and muscles having stiffness and damping characteristics of their own. That is, the infant's legs are contribut­ing their own stiffness and damping to the situation, and we need to add such terms to Equation (3). Unfortunately, little is known about the damping characteristics of muscle, but it is known that stiffness can be modified (Hogan, 1979; Oguztoreli & Stein, 1991). Thus, we add a stiffness term kL for the legs to the term ks for the spring:

mx + hi + (ks + kL)x = Fof(t). (6)

Goldfield, Kay, and Warren 1133

In effect, the muscles act as a transmis­sion between the actual force-generating mechanism (sliding filaments) and the load that they are moving (the infant's body). It is well known from engineering theory (e.g., Ogata, 1970) that the maximum power can be transmitted to the load if the impedance properties of the transmission (the muscles) are matched to the impedance properties of the load. That is, the infant contributes to the stiffness of the entire system, and she will transfer maximum power to her body's mass if she matches her legs' stiffness to that of the attached spring, kL = ks. This means that the stiffness of the legs averaged over the full cycle is equivalent to the spring stiff­ness, even though ground contact is only for half a cycle. For the impedance-matched condition, then, the total stiffness of the sys­tem is twice that of the infant bouncer's spring (k = ks + kL). The natural frequency of the total system is v'2 times that of the mass-spring alone, and the resonant fre­quency is also v'2 times that of the simpler driven system.

A third limitation is that this model as­sumes an external forcing function with ex­trinsic timing. However, the appropriate fre­quency and phasing of kicking may be intrinsically specified for the infant by the moment of foot contact with the ground or some equivalent property such as the mo­ment of maximum foot pressure or maximum leg flexion. We can symbolically represent this intrinsic timing by

mx + hi + (ks + kL)x = F(<I», (7)

where F is some function of the phase (<I» of the mass's motion. This is a function descrip­tion of the form that F should take, mer,ely specifying that it should have <I> as its princi­pal argument, although we are not in a posi­tion to hypothesize any concrete form of F at this time. The important point is that this haptic closing of the loop turns a linear ex­ternally driven mass-spring into an autono­mous limit-cycle system with the intrinsic timing determined by foot contact, a charac­teristic of nonlinear oscillators. An analo­gous example is learning how to "pump" on a playground swing, where the timing of leg flexion and extension is intrinsically speci­fied by the peaks of swinging, perhaps via vestibular information.

Once this kicking system is assembled, tuning its parameters yields other etTects. The ratio of bounce height for a given kick­ing force increases as the frequency of kick­ing approaches the resonant frequency of

1134 Child Development

the system (Fig. 1). As we have just seen, this resonant frequency depends on both the stiffness of the spring and the stiffness of the legs: when leg stiffness matches spring stiff­ness, the infant achieves maximum ampli­tude for minimum force. Thus, parameter tuning involves relaxing to a minimum in frequency-stiffness space, where the ratio of force to height provides a cost function. In principle, this cost function could be sensed via somatosensory information about muscle force and visual information about ampli­tude. The task dynamics are relatively sim­ple, with a single basin of attraction in a two­dimensional space. However, given that the resonant frequency is intrinsically specified by foot contact, this may be simplified even further to a one-dimensional search of stiffness.

These considerations allow us to make the following predictions:

1. There should be an early "assembly" phase characterized by sporadic, irregular kicking without sustained bouncing.

2. Emerging from this should be a "tun­ing" phase with more periodic kicking, during which forcing frequency and leg stiffness vary, yielding high variability in period.

3. Once bouncing is optimized at a sta­ble attractor, a sustained bouncing phase should occur with the following characteris­tics: (a) oscillation at the resonant period; (b) a decrease in the variability of period; (c) an increase in amplitude, due to operating at resonance; (d) a possible increase in the variability of amplitude, due to the fact that at resonance small fluctuations in the forcing frequency yield larger variations in ampli­tude (see Fig. 1); however, this would de­pend on the comparative range of variation in forcing frequency during tuning; (e) 1:1 phase locking of kicking and bouncing, due to operating at resonance; (f) stable limit­cycle behavior in the face of perturbation; and (g) phase resetting in response to pertur­bation, if the system is autonomous.

4. If the infant has learned the low­dimensional dynamics of the task rather than a specific forcing frequency and leg stiffness, there should be rapid adaptation to changes in task conditions. Specifically, manipula­tions of the system mass or spring stiffness that shift the system's resonant frequency should elicit corresponding changes in forc­ing frequency and leg stiffness, yielding os­cillations at the new resonant frequency.

The following experiment is a first at­tempt to test predictions 1 to 3d in a longitu­dinal study of infants learning to use an in­fant bouncer.

Method Subjects.-Eight infants (two girls and

six boys) served as subjects. These infants were part of a group of 15 participating in a longitudinal study of locomotor develop­ment (see Goldfield, 1989). The data from seven infants in this group were not in­cluded, either because they exhibited dis­tress when placed in the harness or because they did not bounce during all of the ses­sions. The mean age of infants at which they exhibited the longest string of successive bounces (see "Results") was 244.4 days (SD = 26.7). All infants were recruited through notices distributed in the office of a group pediatric practice in the Boston metropolitan area. Participating parents signed an in­formed consent at the beginning of the study and received a copy of the videocassette re­cordings made of their infant. The infants were all white and came from predomi­nantly middle- and upper-middle-income homes.

Apparatus and procedure.-Each infant was observed once each week for at least 6 weeks in his or her home with one or both parents present. Each infant was weighed at the first and last observations using a Seca pediatric scale. A portable color television camera (Panasonic WV 3170) mounted on a tripod and a videocassette recorder (JVC BR-6200) were used to record infant behavior. A time signal was simultaneously recorded on the videotape to facilitate later scoring. A commercially available spring-mounted har­ness (Jolly Jumper) secured to a door frame was used to support the infant while he or she bounced. Each infant was placed so that the harness supported them between their legs and around their chest and back. Care was taken to position the infant so that when he or she was still, the knees were slightly flexed while the soles of the feet were touch­ing the floor. The infant was always bare­footed when tested and most often wore a shirt, diaper, and short pants. Each infant was allowed to become comfortable in the harness, and then the camera recorded bouncing for a minimum of 4 min. The cam­era was set up in approximately the same position at each visit.

Properties of the spring.-The stiffness and damping coefficients of the spring were

determined by the dynamic method (Thom­son, 1981). This involved suspending the Jolly Jumper spring from a ceiling and add­ing weights at 2 kg increments. Reflective markers were attached to the spring and mass, and a two-camera Elite motion analy­sis system was used to measure the period of oscillation of the mass-spring system. The computed spring stiffness was 523 N/m. The logarithmic decrement method (Thomson, 1981, p. 30) was used to measure the damp­ing ratio (i.e., observed/critical damping) of the spring. For small amplitudes, the damp­ing ratio was .00l4 and for large amplitudes, .005. Because this value was so small, we adopted the assumption that the spring's damping did not contribute appreciably to the observed oscillation.

Scoring and dependent measures.-The first 4 min of the videotape recordings were scored by a coder who first counted the num­ber of bounces. A bounce was scored as a complete cycle of vertical displacement in which the knees flexed so that the body moved toward the floor, and then extended so that the body moved away from (and sometimes off) the floor, and then back to­ward the floor. A bout was defined as a con­tinuous series of bounces with no pauses during any part of a cycle; the number of bounces in a bout is termed bout length. The session during which the longest bouts oc­curred was defined as the peak of bouncing. Scorers then analyzed the detailed kinemat­ics of the first minute of the recording, ob-

12

10

..s 8 bO C:: .

.3 6 ..... = 0 ~ 4

2

0 -4 -3

Goldfield, Kay, and Warren 1135

taining bounce period from the times of suc­cessive minimum vertical displacement, and bounce amplitude from the displacement between minimum and maximum vertical displacement. Within-bout variability of the latter two measures was defined as the stan­dard deviation computed on the first three or four bounces within a bout, to allow com­parison across bouts of different lengths. We analyzed these dependent measures for three sessions: two sessions prior to peak bouncing (session - 2), one session prior to peak bouncing (session - 1), and at peak bouncing. In earlier sessions the mean bout length was less than three bounces, and we could not reliably compute the remaining measures.

Results

Bout length.-The mean number of bounces per bout appears in Figure 2 as a function of session, with individual subject data aligned by the session in which peak bouncing occurred. (Note that Fig. 2 in­cluded data from all sessions, but sessions - 4, - 3, and + 1 were not included in the remaining analyses). A one-way repeated­measures ANOV A on bout length for ses­sions - 2 to peak revealed a significant ef­fect, F(2, 14) = 28.134, p < .00l, accounting for 80% of the total sum of squares (SS). Post hoc Tukey tests showed that the peak bout length was significantly different from both preceding sessions, HSD = 3.32, p < .01, but that they were not different from each

-2 -1 Peak +1

Session FIG. 2.-Mean number of bounces per bout for sessions - 4 to peak. Error bars indicate ± 1 SE

1136 Child Development

other. In the early sessions, infants kicked irregularly, with only one or two bounces per bout. Bout length increased gradually over the next several sessions up to a mean of 4.2 bounces, until it suddenly doubled in the peak session to 8.7 bounces. Finally, after reaching a peak the bout length began to decline in subsequent sessions.

Amplitude.-The mean bounce ampli­tudes for sessions - 2 to peak appear in Fig­ure 3. Again there was an increase in ampli­tude over sessions, F(2, 14) = 15.251, p < .001, accounting for 69% of the total SS. The only statistical difference was the increase between session - 1 and peak, HSD = 2.89, p < .01. Although there was also a 50% in­crease in within-bout variability in ampli­tude in the peak session (Fig. 4), it was not statistically significant, F(2, 14) = 0.703, p > .5, and accounted for only 11% of the total SS. Thus, there was a large increase in amplitude in the peak session but no change in amplitude variability.

Period.-Mean period did not vary over sessions - 2 to peak, F(2, 14) = 0.569, p > .5, and accounted for only 8% of the total SS (see Fig. 5). However, as shown in Figure 6, within-bout variability in period decreased significantly, F(2, 14) = 11.958, p < .001, accounting for 63% of the total SS, and drop­ping by 50% in each successive session. Tukey tests showed a significant difference between session - 2 and the peak session, HSD = .055, p < .01. Thus, whereas the

15

12 ........ e u -- 9 .g a

J 6

3

average behavior of the system remained in the same ballpark over the last several ses­sions, its variability declined dramatically.

To determine whether this preferred frequency corresponded to the resonant fre­quency of the system, we first compared the observed period in the peak session to the period predicted by the external spring alone, with no stiffness contribution from the infant's legs. Predicted period was com­puted from Equation (2), using the infant's mass, the empirically determined spring stiffness constant, and a damping coefficient of 0; the results for each of the eight infants are presented in Figure 7. In all cases, each infant bounced with a shorter period than predicted by the inert mass-spring, with a mean error of .206, t(7) = 8.30, p < .001.

We next added a second spring to the model on the hypothesis that the infant's legs act like a spring that matches the imped­ance of the external spring (Eq. [6]). Ac­cording to this model, the value of total stiffness doubles and the -predicted period decreases by a factor of V2. The results ap­pear in Figure 8. The observed and pre­dicted periods for the two-spring model are in close agreement for each of the subjects, with a mean error of .016, a statistically in­significant difference, t(7) = 1.70, p > .1. Thus, the preferred bouncing frequency is predicted by the resonant frequency of the system, assuming an impedance matching strategy.

o~---.---------.--------,------2 - 1 Peak

Session FIG. 3.-Mean bounce amplitude (in centimeters) for sessions - 2 to peak. Error bars indicate

±l SE.

Goldfield, Kay, and Warren 1137

2.5 .-e

Co)

0" 2.0 CI') '-'

0 1.5 ...... -...... ~ ...... ;;

1.0 > .g a 0.5 ...... -S' <

0.0 -2 -1 Peak

Session FIG. 4.-Mean bounce amplitude variability for sessions - 2 to peak. Error bars indicate ± 1 SE

Discussion The results provide evidence that in­

fants assemble and tune a periodic kicking system akin to a forced mass-spring, homing in on its resonant frequency. Let us evaluate each of the tested predictions.

1. Assembly phase.-In the earlier ses­sions kicking was sporadic and irregular, with only one or two kicks per bout, as though infants were probing the system and observing the resulting behavior. This is

0.75

0.70

.-en

~ 0.65 en '-' ~ 0

0.60 .~

~

0.55

0.50 -2

consistent with an early assembly phase in which. the dynamics of the system are ex­plored through spontaneous activity.

2. Tuning phase.-Bout length in­creased over the next several sessions, con­sistent with a parameter tuning phase. Most important, there was a steady decline in the variability of period, as would be expected from a process of relaxing to a minimum in frequency-stiffness space. The theory pro­poses that this results from increasing sensi­tivity to the foot contact information which

- 1 Peak

Session FIG. 5.-Mean bounce period (in seconds) for sessions - 2 to peak. Error bars indicate ± 1 SE

1138 Child Development

0.10

,.-.. CIl U Q.) 0.08 CIl

0" t;f) '-' 0.06 0 .... -.... ~ tIS 0.04 '5 > -e

0.02 0 'C ~

0.00 -2 -1 Peak

Session FIG. 6.-Mean bounce period variability for sessions - 2 to peak. Error bars indicate ± 1 SE

intrinsically specifies frequency, and aq­justing leg stiffness to match spring stiffness, although these specifics remain to be deter­mined.

3. Resonance.-The sudden onset of sustained bouncing strongly suggests that the system has been optimized by matching the impedance of the spring to maximize en­ergy transfer, driving the system at its reso­nant frequency, and homing in on a stable attractor. This peak behavior exhibits the fol­lowing characteristics:

1.0

0.8

....--.-en U 0.6 Q.) en '-' -e 0

0.4 ·c Q.)

tl.

0.2

0.0

a) As predicted, the preferred period of oscillation closely approximated the reso­nant period of the system, assuming imped­ance matching. The fact that the average pe­riod did not change over sessions suggests that an appropriate periodic attractor was as­sembled early on and its behavior refined but not qualitatively altered by parameter tuning.

b) Variability in period descreased dur­ing sustained bouncing, as would be ex-

Subject FIG. 7.-0bserved period for each infant at the peak session, and the period predicted from the

single spring model.

Goldfield, Kay, and Warren 1139

1.0

0.8

-----CIj

u 0.6 Q) CIj '-' '"0 0

0.4 .i:: Q)

0..

0.2

0.0 2 4

Subject FIG. 8.-0bserved period for each infant at the peak session, and the period predicted from the

two-spring model.

pected if the system had settled on a mlmmum in frequency-stiffness space. Operating at resonance is thus not only ener­getically efficient but also more stable in the frequency domain. Such stability is not to be expected from a linear mass-spring system but can be explained by haptic information about the resonant frequency acting to reg­ulate the forcing frequency. This sort of proprioceptive "feedback" characteristic of biological systems thus renders a linear mass-spring into a nonlinear autonomous system.

c) As predicted, the amplitude ofbounc­ing increased significantly. Such a "reso­nance peak" is exactly what would be ex­pected for a system operating at resonance.

d) Variability in amplitude also in­creased by 50% in the peak session, as ex­pected, although it was not statistically sig­nificant. One possible interpretation is that whatever variation in amplitude may occur due to small fluctuations in the forcing fre­quency at resonance is not significantly greater than that produced by larger adjust­ments in the forcing frequency during tuning.

In sum, over sessions the behavior moves to an optimized attractor state that corresponds to the resonant frequency of the system. This is evidenced by the close pre­diction of preferred period by the resonant period of the impedance-matched system, by the increase in amplitude at the reso-

nance peak, and by the reduction in period variability. A similar result has recently been reported by Hatsopoulos and Warren (1992) for arm swinging in adults. When joint stiffness was measured directly under various conditions of mass and spring load­ing, the preferred frequency of arm swinging was precisely predicted by the resonant fre­quency of the system. Further, joint stiffness increased with the stiffness of the exter­nal spring, consistent with an impedance­matching strategy.

These results are admittedly prelimi­nary and leave some obvious questions open. For example, to determine whether kicking frequency is intrinsically timed and how leg stiffness is adjusted during tuning (prediction 2), and to examine the phase locking of kicking and bouncing (prediction 3e), it would be necessary to make detailed kinematic, force plate, and EMG measure­ments longitudinally. To test predictions 3f and 3g, we would have to evaluate the stabil­ity and autonomy of the system by physically perturbing the infant bouncer and measur­ing the limit-cycle and phase-resetting be­havior. Perhaps most important, to test pre­diction 4, we would like to determine the infant's adaptability by adding mass or changing spring stiffness between bouts, thereby displacing the minimum in fre­quency-stiffness space and shifting the sys­tem's resonant frequency. If the infant has learned only a fixed driving frequency and leg stiffness, it would require a long period

1140 Child Development

of adaptation. But if, as hypothesized, the infant has learned the low-dimensional dy­namics of the task, it should adapt quickly to scale changes in the task conditions. This would provide direct evidence that infants are locating the resonant frequency and rule out a coincidental correspondence between the observed and resonant periods.

We believe such an approach bears on both motor learning and motor develop­ment. One may then ask why perform a de­velopmental study rather than a method­ologically easier adult learning task. The most obvious reason is to offer a perspective on long-standing developmental phenom­ena, such as learning to suck, shake, and crawl, that hold out the promise of being consistent with adult motor behavior. How­ever, there are at least two aspects that are unique to development. First, because de­velopment occurs on a longer time scale than adult learning, the assembly and tuning processes can be observed in an extended fashion, particularly the emergence of sud­den changes. For example, longitudinal ob­servations made it clear that learning to bounce required considerable experience on the part of infants. It was our discovery of "peak" bouncing in these observations that led us to test the hypothesis that behav­ior is optimized over time to settle on the resonant frequency of the system. Now that this has been examined, related questions can be asked of skilled bouncers.

A second distinction is that, while simi­lar general principles may apply to develop­ment and adult learning, the task constraints may be quite different, and maturational changes in constraints may account for some developmental sequences. For example, properties of the action system such as the masses of the limbs, the strength of the mus­cles, and the ability to control muscle stiff­ness provide constraints that change with development, yielding different organiza­tions at different developmental stages (Thelen & Fisher, 1982). Uneven rates of growth make action components available at different times, such that different forms of behavior emerge at different ages. Goldfield (1989) has shown that the particular charac­ter of crawling in infancy emerges from the way that three components are combined at different points in development-orienting to the support surface, using the legs for pro­pulsion, and steering with the hands. Some components cannot yet be coordinated with others, such as the transport and grasp

phases of reaching, while other components cannot yet be differentiated, such as the two limbs in early bimanual reaching (Goldfield & Michel, 1986). Such maturational influ­ences do not appear to affect the develop­ment of bouncing.

The approach presented here bears sim­ilarities to recent dynamical theories of learning in adults. Schoner et al. (1992; Schoner, 1989; Zan one & Kelso, 1992) con­ceive oflearning as a process of competition and cooperation between the "intrinsic dy­namics," or an abstract control structure gov­erning behavior, and "behavioral informa­tion," or the movement pattern required by the task. Learning is then a process of change in the intrinsic dynamics to converge on a stable solution or attractor at the re­quired pattern and can involve qualitative changes in coordination. When the two com­pete, there is increased variability in behav­ior, but when they cooperate and arrive at a common solution, variability is minimized. Schmidt et al. (1992) propose that learning a new action pattern involves organizing and parameterizing a dynamical control struc­ture by perceptually exploring its behavior. When subjects learned to coordinate two handheld pendula in a I: 2 phase-locking re­gime, they approached the 1: 2 attractor over trials with a concomitant decrease in the variability of relative phase. The rate of learning was taken as an index of the steep­ness of the landscape and thus the stability of the parameterization (Schoner, 1989).

Both of these views are quite similar to the present approach, with which they share common antecedents. They both propose two levels of analysis, each with its own dy­namics: the behavior of a control structure governing a particular movement (our state level), and the behavior of a learning process that optimizes the control structure (our tun­ing level). To this we add a third level of graph dynamics, describing the self-organi­zational process by which a control structure arises.

In sum, our theory of the develop­ment of action systems asserts that low­dimensional organizations of the musculo­skeletal system emerge from the infant's spontaneous movements within a task con­text, and the parameters of the system are tuned to optimize the resulting behavior. Consider again the role of exploration at the three levels of task dynamics and its depen­dence on information specific to the dy-

namics. At the graph level, the infant may experiment with different musculoskeletal organizations, in effect adopting different functions over the components of the action system, and explore the attractor layout that emerges. This process is essential to the self­organization of a task-specific action pattern and implies that one of the functions of spon­taneous activity in infancy is to explore pos­sible organizations by allowing the free interplay of components and sensing infor­mation for the attractors that emerge. At the tuning level, parameter space is explored by varying the parameters of a particular func­tion. This process involves becoming sensi­tive to information for the landscape and relaxing to a minimum that specifies the op­timal configuration of parameter values. At the state level, the system evolves to a par­ticular attractor, corresponding to a stable preferred action pattern. The organization of rhythmic movement around such qualitative points in the task dynamics as resonances may be a fundamental property of motor be­havior.

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