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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