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 6month-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]
Goldfield, Kay, and Warren
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-
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-
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 +
bx2 x + kx = O.
(4)
3
2
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.
Goldfield, Kay, and Warren
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-
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
Goldfield, Kay, and Warren
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 + (k s + kL)x = Fof(t).
(6)
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 + k L). 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 + (k s + 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 BR6200) 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
Goldfield, Kay, and Warren
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-
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 ANOVA 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
12
10
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-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
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.
15
12
........
eu
--a
.g
J
9
6
3
o~---.---------.--------,-----
-2
-1
Peak
Session
FIG.
±l SE.
3.-Mean bounce amplitude (in centimeters) for sessions - 2 to peak. Error bars indicate
Goldfield, Kay, and Warren
.-
1137
2.5
e
Co)
0"
2.0
CI')
'-'
0
......
......
~
......
;;
>
.g
1.5
1.0
a
......
0.5
<
0.0
-S'
-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 infants 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
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
0.75
0.70
.en
~
en
0.65
'-'
~
0
.~
~
0.60
0.55
0.50
-2
-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.)
CIl
0.08
0"
t;f)
'-'
-.....0...
0.06
~
tIS
0.04
0
0.02
'5
>
-e
'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:
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-
1.0
0.8
....--.en
U
Q.)
0.6
en
'-'
-e
0
·c
Q.)
0.4
tl.
0.2
0.0
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
----u
CIj
Q)
0.6
CIj
'-'
'"0
0
.i::
0.4
Q)
0..
0.2
0.0
2
4
Subject
8.-0bserved period for each infant at the peak session, and the period predicted from the
two-spring model.
FIG.
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-
Goldfield, Kay, and Warren
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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