A developmental transition in prehension modeled as a cusp catastrophe

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R. H. Wimmers G. J. P. Savelsbergh J. van der Kamp Institute for Fundamental and Clinical Human Movement Sciences Amsterdam/Nijmegen, The Netherlands and Faculty of Human Movement Sciences Vrije Universiteit Amsterdam, The Netherlands

A Developmental Transition in Prehension Modeled as a Cusp Catastrophe

P. Hartelman Department of Developmental Psychology Universiteit Amsterdam Amsterdam, The Netherlands

Received 17 May 1996; accepted 11 February 1997 ABSTRACT: The purpose of the study was to show that the change from reaching without grasping to reaching with grasping during the first 6 months of life carried the characteristics of a discontinuous phase transition (catastrophe). A cross-sectional study was carried out with 58 infants between 60 and 408 days old. The infants were seated in a specially designed seat, and presented with nine detachable balls on a black curved board within reaching distance at shoulder height. The number of reaches without and with grasping were scored from video. A cusp catastrophe model was fitted to the data. A Likelihood-Ratio test indicated that the likelihood of the cusp model was significantly higher, p , .001, than a linear regression model. The cusp model was also compared with a logistic model. Akaike’s Information criterion for the cusp catastrophe exceeded the logistic model, thus indicating a general better fit. Based on prior research, the following potential control parameters were chosen: crown–heel length, total body weight, arm length, arm circumference, ponderal index, arm volume, arm weight, and body position relative to the horizontal. The cusp model predicted that arm weight and arm circumference significantly contributed to the control parameters. It was found that these two variables had their largest contribution to the asymmetry control parameters. q1998 John Wiley & Sons, Inc. Dev Psychobiol 32: 23– 35, 1998

Keywords: development; infants; cusp catastrophe; transitions; eye–hand coordination

In the late 1920s and early 1930s a number of descriptive studies addressed the development of infant manCorrespondence to: R. H. Wimmers Contract grant sponsor: NWO Contract grant numbers: 575-63-080; 575-59-055; 575-63-081 Contract grant sponsor: Royal Netherlands Academy of Arts and Sciences

q 1998 John Wiley & Sons, Inc.

ual control (Gesell, 1929; Halverson, 1931). The classic study of Halverson (1931) provided a detailed statistical record of infants’ prehension. He observed different styles of reaching or approaches to a cube between 16 and 52 weeks of age. This early research suggests that the ability to reach becomes evident only in a rudimentary form after 3 months of age and that CCC 0012-1630/98/010023-13

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it is a gradual process of coordinating visual and prehensile functions. Bower (1972; Bower, Broughton, & Moore, 1970) changed this picture drastically when he stated that even newborns were capable of directed reaching. Although Bower’s results were disputed by several authors (e.g., Dodwell, Muir, & DiFranco, 1976; Ruff & Halton, 1978), it was confirmed by Von Hofsten (1984, 1986), who showed that newborns display reaching movements without grasping. Around 2 months of age there is a regression — indicated by a decline in the number of reaching attempts characterized by jerky movements with closed hands — heralding a change to movements with open hands, object contact, and grasping occurring after 3 months. Another example of a qualitative change in the development of reaching is the alterations between bilateral and unilateral periods of reaching which occurs in the 1st year of life (Corbetta & Thelen, 1996). With the above examples in mind, it is clear that the study of motor development, or more specifically prehension, has to involve an examination of how these developmental changes occur over time. The concepts and methods of the dynamical systems approach may be of help in investigating qualitative changes in development. With this approach, it is possible to determine how “motor milestones” and other behaviors actually develop in terms of their time evolution. Answering this question gives an indication of the mechanisms of developmental stability and change, which are still poorly understood. If we, for instance, can determine that the observed behavioral change is a nonlinear, discontinuous phase transition, it will signal an underlying developmental process of continuity (modification of the intrinsic dynamics) and self-organization. Conceptually and computationally straightforward methods to determine whether a discontinuous phase transition is present in a particular data set are provided by catastrophe theory. Catastrophe theory is a mathematical theory for describing and predicting discontinuous phase transitions in “gradient systems” (Thom, 1975). These systems, with fixed parameters, always move to an equilibrium state, i.e., to a minimum (or maximum) of a certain quantity (entropy or energy). Catastrophe theory describes the seven simplest ways for a catastrophe to occur depending on the number of control parameters (with a maximum of four). Two approaches in the application of catastrophe theory can be used to answer the question whether developmental changes constitute discontinuous phase transitions, namely, catastrophe detection and catastrophe modeling. For catastrophe detection, Gilmore (1981) distinguished eight “catastrophe flags” that can

be tested empirically. The first five flags — bimodality, inaccessibility, sudden jump, hysteresis, and divergence — always occur in conjunction when a system undergoes a discontinuous transition. The last three flags — critical slowing down, anomalous variance, and divergence of linear response — can be found prior to a transition, and signal an upcoming transition. We will explain these criteria briefly. The first flag is the occurrence of bi(or multi)modality, that is, the presence of two or more qualitatively different behaviors on the same (behavioral) dimension. For this flag to be confirmed, the dependent variable must exhibit at least a bimodal frequency distribution. The second flag, sudden jump, involves an abrupt, qualitative change in behavior due to a small, smooth change in the control parameters. Inaccessibility, the third flag, implies that the behavioral variable cannot occupy a stable state between the two behavioral modes. Divergence, the fourth flag, refers to the fact that, at the transition, two adjacent initial conditions may diverge rapidly and result in two different behaviors. This flag is important for revealing the stability of the behavior for different initial conditions. The fifth flag is hysteresis, which means that the value of the control variable at which the transition occurs depends on the direction in which it was scaled (i.e., increasing or decreasing). Critical slowing down, the sixth flag, is a consequence of the loss of stability of the pretransition behavioral mode prior to switching to another mode. The stability of a behavioral mode is indexed by the local relaxation time, that is, the time it takes for the system to stabilize after a perturbation caused by internal or external forces. Critical slowing down refers to the increase of relaxation time as the transition is approached. In addition to the local relaxation time, the global relaxation time or equilibration time is of great relevance for understanding the underlying dynamics. It is defined as the time required to achieve a stationary distribution from a random initial distribution. In a situation in which two behavioral equilibrium states are present and no transitions take place, equilibration time is mainly determined by the time to switch from one state to another. If the two relaxation times are of a same duration, critical phenomena and phase transitions can be observed (see Kelso, 1995; Zanone, Kelso, & Jeka, 1993). The seventh flag is anomalous variance, which is associated with an increase in fluctuations of the behavior before a transition. A way to detect such fluctuations is to examine the variance of the incidence per time unit of a particular value or behavioral category prior to a transition. Another way is to study the

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degree of switching between attractor states. If, for example, two attractor states of comparable strength coexist in a particular state space, they will have competing effects on the system’s behavior. Oscillations can then be observed between the two behaviors. The last catastrophe flag, divergence of linear response, is a consequence of a perturbation near a transition; the magnitude of the variance increases and large oscillations of the behavioral variable are seen, both as a result of a loss of stability. In a longitudinal study (Wimmers, Savelsbergh, Beek, & Hopkins, in press) designed to detect the presence of some of the catastrophe flags in the change from reaching without to reaching with grasping, we found evidence for a sudden jump, bimodality, inaccessibility, and anomalous variance. Taken together, these findings indicate that the ability to reach and grasp emerges from a discontinuous phase transition, or in the parlance of catastrophe theory, a catastrophe. What still needs to be addressed is whether the transition can be modeled as a particular type of catastrophic change. The simplest discontinuous transition between two equilibrium states in catastrophe theory is the cusp catastrophe. Verifying compliance with this catastrophe then leads to the question of what might be the agents or control parameters responsible for this qualitative change.

where x is the behavioral variable of the system, t denotes time and V is the potential function. The cusp catastrophe, which is one of the elementary catastrophes in catastrophe theory, is the simplest discontinuous transition between two equilibrium states, and constitutes a starting point for modeling the transition from reaching without to reaching with grasping. A discontinuous transition is defined as a sudden change in an independent variable or control parameter. Although the control parameter is entirely unspecific as to the resulting change, it is instrumental in creating a new order at the macroscopic level by leading the system through its respective states of equilibrium. In a cusp model, the transition between two equilibrium states of an unidimensional behavioral variable is governed by two control parameters. To model the cusp, a cusp function: V(x) 5 1⁄4x 4 2 1⁄2bx 2 2 ax

Thom’s (1975) catastrophe theory is concerned with gradient systems that continually seek to minimize (maximize) some quantity, for example, entropy or energy. The behavior of a gradient system can be described by the form of a mathematical function, the potential function: dx dV 52 dt dx

FIGURE 1

(1)

(2)

is used where x is the behavioral variable and a and b are the control parameters. Variable a has been called the asymmetry or normal factor and b the bifurcation factor or splitting factor. The minima can be calculated by taking the first derivative of the cusp function and setting it to zero: V˙ (x) 5 x 3 2 bx 2 a 5 0

CUSP CATASTROPHE

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(3)

The cusp model can be seen as a generalization of a linear model in that it involves an extra cubic term. In a linear model the dependent variable is a linear function of the independent variables and there is only one possible value for the dependent variable. In the cusp model, however, there are multiple dependent-variable values for certain values of a and b. Figure 1 shows the different potential fields that belong to the cusp potential. By changing the control parameters of the potential function, the minimum of the function changes. The minima of the cusp function

Different potential fields as a result of changes in a.

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FIGURE 2 A cusp model for the development of prehension. The gray area in the control surface plane is the bifurcation set. Leaving the bifurcation set induces a bifurcation from one behavioral plane to the other. For example, following Path I with a high and constant value of the bifurcation parameter and an increasing asymmetry control parameters the jump occurs at x. No such bifurcation occurs on Path II with a low value of the bifurcation control parameter.

are called equilibrium points, and are illustrated by the position of the black ball. The equilibrium points correspond with stable equilibrium states of a behavioral variable. In Figure 2 the cusp surface is shown representing all the solutions of Equation 3 with the different control parameter values. Changes in the behavioral variable are denoted by Pathways I and II on the surface (Figure 2). If we change the control parameter on the a axis starting from a low value of a, the function with one minimum changes and eventually results in two stable behavioral states, the lower and upper sheet, respectively. If two sheets are present at the same value of a, the function is said to be bimodal. The region between the upper and lower sheets gives rise to a region of inaccessibility. By increasing the control parameter along the a axis, we see that the original sheet disappears, and a sudden jump occurs to a new state. By decreasing a, the jump occurs at a different a value. When the jump with an increasing control parameter occurs at a different value than with a decreasing control parameter, then hysteresis has been detected. These phenomena are only visible along Path I at a high value of b. If b is low, the change between behavioral equilibrium states is continuous (Path II). In behavioral terms, this means that the percentage of reaches with grasping would grow gradually from 0 to 100%. The corresponding distribution of the percentage score would be flat. What we know from our longitudinal study (Wimmers et al., in press) is that the change from reach-

ing without grasping to reaching with grasping occurs rather suddenly. Consequently, Path I has a high probability of occurrence. In a cross-sectional study, such as the one reported in this article, individual changes cannot be investigated. However, if the majority of infants change along Path II, a linear model should fit the data points satisfactorily.

FITTING DATA TO A CUSP MODEL Cobb (1978, 1980, 1981; Cobb & Zacks, 1985) developed a method for fitting a catastrophe model to observed data. By using stochastic differential equations, he connected a probability density function e2V(x) to the potential function V(x). For the cusp catastrophe this becomes fs(z; a, b) 5 j eaz 1 1/2b z2 2 1/4z4

(4)

z 5 (x 2 l)/s.

(5)

where

In Equation 4, j is a constant depending on a and b, which “normalizes the probability distribution function so that it has unit integral over its range” (Cobb, 1980), and a and b are the control parameters. Z is the standardized behavioral variable involving l, the location parameter, and s, the scale parameter. These

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parameters play a role similar to that of the mean and standard deviation in calculating the Z scores of a normal distribution. Equation 4 represents a cusp catastrophe in terms of a probability density function. Because the probability functions of the cusp model are known, a and b can be estimated by a maximum likelihood procedure. As an illustration, we plotted Equation 4 with a 5 0, b 5 2, and j 5 1 in Figure 3. From this figure, we see that the stable and unstable equilibrium points now correspond to a mode of high probability and an antimode of low probability, respectively. For the control parameters, Cobb (1981) made the simplifying assumption that they are linear functions of a number of independent variables in the following way

a 5 (a 0 1 a 1 I1 1 · · · 1 a n In ), b 5 ( b0 1 b1 I1 1 · · · 1 bn In )

(6)

The as 1 to n and bs 1 to n are weights given to each independent Variable I. Together, they represent the control parameters a and b. These weights provide a means of evaluating each independent variable used in the model in terms of its contribution to the control parameter. Thus, the question arises: What are the relevant independent variables for the cusp model? Two global factors in the development of coordination are organismic and environmental in nature. For example, the effect of changes in physical composition (organismic) on the development of leg movements has been elegantly demonstrated by Thelen and her colleagues (e.g., Thelen, 1983, 1985; Thelen, Fisher, & Ridley-Johnson, 1984). Their results suggested that the disappearance of newborn stepping

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movements was influenced by asynchronous change between muscle strength and mass increase (as shown by the ponderal index). In the development of reaching, Savelsbergh and van der Kamp (1994) found that arm growth was correlated with the number of reaches, and suggested arm strength (operationalized by arm volume) as an important factor in the development of reaching from the 3rd to 5th month. Based on these findings, it is expected that physical changes may contribute to the asymmetry control parameter. The role of different test conditions as a bifurcation parameter was put forward by Maas and Molenaar (1992). They argued that the influence of optimal test conditions for detecting spurts in longitudinal data as found in a study by Fischer, Pipp, and Bullock (1984), complies with the role of the bifurcation control parameter in a cusp model. Significant developmental spurts will only be found if the performance on the developmental test is optimized. This suggests that the optimality of environmental conditions controls the discontinuity similar to the way that the bifurcation parameter controls the discontinuity in catastrophe theory (Maas & Molenaar, 1992). A candidate variable for the bifurcation parameter similar to the above in the development of prehension comes from a study by Savelsbergh and van der Kamp (1994). They manipulated the angle of the chair relative to the horizontal. For 12- to 19-week-old infants, significantly more reaches occurred in the vertical position in comparison to the supine position. A similar effect was also found in a study by Kamm (1994) who concluded that reaching from a more-upright position preceded reaching from supine or prone. She also showed that infants reached more frequently and were more likely to grasp rather than touch toys from upright positions.

FIGURE 3 The probability density function for the cusp with a 5 0 and b 5 2.

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In the present study, the variables, ponderal index, arm volume, and the body orientation of each infant were tested in order to examine whether they contributed to the control parameters in the cusp model. We also examined some more-specific body and arm growth measures. At this point we want to emphasize that in catastrophe models time is implicit, and is expressed in the changes of the value of the control parameters (Maas & Molenaar, 1992). These changes will lead to different paths through the surfaces. Consequently, age as indicator for time cannot be an explicit variable. Therefore, age is not used as an independent variable. This is in agreement with the notion that time is not a genuine causal factor in development (Lewis, 1990; Wohlwill, 1973).

METHOD Subjects Subjects were 58 normal, full-term infants (33 boys and 25 girls) ranging in age from 60 to 407 days. They had no known sensory or motor impairments. Procedure The same method as outlined by Savelsbergh and van der Kamp (1994) was used for this experiment. The infants were seated in a chair that could be oriented in different positions relative to the horizontal plane. Trunk and head supports could be adjusted in accordance with the size of the child. The infants were secured by belts, but they were free to reach and grasp objects. Three different angles to the horizontal plane were used: vertical (90 degrees), recline (60 degrees), and supine (0 degrees). A black curved board (30 3 40 cm) with nine detachable balls was presented within reaching distance (2/3 of the arm length) at shoulder height. The red plastic foam balls, each with a diameter of 2.5 cm, were covered with white and yellow spots to make them more attractive to the infant. The balls were equally spaced on the board in three columns and three rows, with 5 cm between adjacent balls. Each infant was presented with the objects once in all three body positions. The sequence of the body positions was randomly varied across infants. Each trial lasted 1 min. Grasped balls were removed from the infant’s hands and replaced on the board. The trials were recorded with a Super-VHS videocamera, placed approximately 1 m from the top of the infant’s head. Camera position was changed for each body orientation to provide a clear view of the infant. After the trials, five anthropometrical measures

were obtained twice: total body weight (g), arm length (distance of the shoulder to wrist; cm), crown – heel length (cm), circumference of mid – upper arm, and the maximum circumference of the forearm (cm). The average of the two measurements was used.

Data Analysis To obtain the behavioral variable, two observable categories were scored from the videorecordings, namely, reaching without grasping in which one of the hands contacts a ball, and reaching with grasping in which one of the hands touches the ball followed by a closure of two or more fingers around it. The categories were extracted by the first and third author. The first 5 infants were scored together, and another 5 infants separately. There were no differences in the number or type of category between the two judges. The categories were converted into one variable, percentage of occurrence of grasping. This variable was calculated by taking the number of occurrences of grasping as a percentage of the total number of reaching behaviors in each condition. As a result of having three conditions, three percentage of occurrence scores for each infant were obtained. The following measures were used to derive potential control parameters: a. Ponderal index (PI). This is a measure for chubbiness or stockiness calculated by the formula: (total body weight/crown – heel length3) 3 100 The two contributions to the PI were tested separately: b. Crown – heel length (cm), and c. Total body weight (g); d. Arm volume ({arm circumference2/4p} 3 arm length); Again, each contributory factor to arm volume was tested separately: e. Arm length (cm), and f. Arm circumference (cm): (mid-upper arm circumference 1 maximum lower arm circumference)/2; g. Arm weight (the sum of the estimated masses of the upper and lower arm derived from the regression equations of Schneider & Zernicke, 1992). h. Position relative to the horizontal plane. On the basis of these measures, the control parameters can be written as:

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a 5 (a0 1 a1a 1 a2b 1 a3c 1 a4d 1 a5e 1 a6 f 1 a7g 1 a8h) b 5 ( b0 1 b1a 1 b2b 1 b3c 1 b4d 1 b5e 1 b6 f 1 b7g 1 b8h) where a to h refer to the measures listed above. The final form of the cusp surface in raw score form is:

a1b

S D S D x2l s

2

x2l s

3

50

(7)

The analysis program (Hartelman, 1996), which uses the algorithm of Cobb, performs the calculation by means of the maximum likelihood method necessary to provide the values for Equation 7. The program also calculated the estimated values (modes) of the behavioral variable for each value of the control parameters. To select the variables for the cusp model, a stepwise selection procedure was used based on the Likelihood-Ratio (LR) test, which is chi-square distributed with n degrees of freedom, where n is the difference between the number of parameters of the two models. A good model is one that results in a high likelihood for the observed results. Because the likelihood is a small number, it is customary to use 2 2 3 the log of the likelihood as a measure of how well the estimated model fits the data. The stepwise selection procedure involves the following steps: (1) deriving the basic model, that is, the model that contains only the constants (a0 and b0) and no variables; (2) calculating the log likelihood for all variables independently; (3) putting the variable with the smallest log likelihood into the model, and comparing it with the basic model. To determine whether the variable has to be entered, the difference between 2 2 3 the log likelihood of the basic model and the model with the variable is compared by means of the LR test. This tests the null hypotheses that the coefficient (weight factor) for the independent variable in the current model, excluding the constants, is equal to 0; (4) assessing improvements between successive steps by means of the LR test. It tests the null hypothesis that the coefficient for the variable added at the last step is equal to 0; (5) withdrawing each variable from the model and then testing the change in the 2 2 3 the log likelihood. If the change is significant, the variable remains in the model, otherwise it is removed. Steps 3 to 5 continue until either a previously considered model is encountered or no variables are available anymore. There is no single definitive statistical test for acceptability of a cusp model (Cobb, 1992). According

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to Cobb, this is partly due to the fact that the statistical model is not linear in its parameters, and that there are more than one predicted value possible for the independent variable. This makes it difficult to define the prediction error, and therefore the goodness-of-fit measures are nearly useless because these measurements are heavily based on the concept of prediction error. However, a correlation coefficient is calculated for the cusp as the proportion of the variance accounted for by the model if the nearest mode is used as the predicted value of x. As pointed out, it is still possible that the behavioral variable x belongs to the other mode. Thus, the correlation coefficient is only an approximation and is therefore called pseudo-R 2. The main test to confirm a cusp model is based on a comparison of the likelihood of the cusp model with that of a linear model. The log likelihood statistic can be used to compare the linear and cusp model by means of a x2 test. The degrees of freedom for this test is the difference in the degrees of freedom for the two models being compared. The linear model is calculated with a multiple regression analysis using a stepwise selection method. Two additional tests were used to confirm a cusp model (Cobb, 1992). The first one, already part of the stepwise selection procedure, namely, finding a cusp model in which the coefficients of the control parameters are significantly different from zero, thus indicates that the control parameters are not just the Constants a0 and b0. The second test is that 10% of the data points are within the bifurcation set of the estimated model. It could happen that the cusp model was indicated as the best model, but none of the data points are located within the bifurcation set. Then the location of the bifurcation set is only known by extrapolation from the given data. Therefore, for a reliable fit it is necessary to have observations within and outside the bifurcation set. Due to the fact that a cusp model also fits well with a developmental process where the changes occur with a rigid acceleration instead of a jump, the cusp model was compared with the following logistic model (Hartelman, 1996): y5l1

s 1 1 e2(a/b 2)

(8)

a 5 (a0 1 a1 I1 1 · · · 1 anIn ), b 5 ( b0 1 b1 I1 1 · · · 1 bn In ) The logistic model is S-shaped and represents the “best nondiscontinuous transitional” model in the sense that it is “closest” to a cusp model. With large values of b

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FIGURE 4 The logistic model with l 5 0, s 5 1, and b respectively 1⁄2 and 1⁄4.

the change is smooth, with low values of b the change is rapid, but there will never be a bifurcation (see Figure 4). Because the cusp model and the logistic model are not nested, the difference between the low likelihood gives some information about what to select. For comparison, we used Akaike’s Information Criterion (AIC), which is defined as 2 2 3 the log likelihood plus 2 3 the number of parameters estimated from the data (Bozdogan, 1987). The advantage of this criterion is that it takes into account the number of parameters used in the model. Doing so safeguards against the selection of an overly complex model with too many parameters. The idea is to select the model with the smallest AIC.

RESULTS Due to loss of the antrophometric data of 2 infants, and the fact that 7 infants did not reach in either position, data was scored for 49 infants who reached for the object at least in one of the three positions. Across all infants and positions, there were 810 reaching movements of which 401 ended with grasping. Due to the fact that not all infants reached in each position, the total amount of percentage of occurrence resulted in 128 data points. The percentage of occurrence values distributed over the three positions as follows: 38 at 0 degrees, 44 at 60 degrees, and 46 at 90 degrees, with the average number of reaches per infant being, respectively, 5.33 (SD 5 3.37), 6.23 (SD 5 3.84), and 6.25 (SD 5 3.69). The percentage of infants who showed only reaching without grasping was 18.3%,

reaching with grasping 14.3%, and both types of reaching 67.4%. This indicates that a large group is situated in the bifurcation set. If we look at the percentage of occurrence, 44.9% of the infants always have percentage of occurrence values smaller than the 50% level, i.e., most of the reaches are without grasping, with 30.6% higher than 50%, i.e., most of the reaches are with grasping, and 24.9% with both low and high percentage of occurrence values. Figure 5 shows the relation of the percentage of occurrence values with age and position. An increase in reaching without grasping to reaching with grasping can be seen between the 3rd and 5th month after birth. From Table 1, it can be seen that all variables significantly effected the cusp model when they were tested against the basic cusp model (i.e., Step 2 in the stepwise selection procedure). The best-fitting cusp plane, derived by the following steps of the stepwise selection procedure, and by use of the maximum likelihood algorithm of Cobb, is described by Equation 7:

a1b

S

x 2 51.04 26.98

D S 2

x 2 51.04 26.98

D

3

50

where

a 5 5.98 2 1.07f 1 31.92g b 5 5.56 2 .25f 1 .38g These outcomes reveal that arm circumference ( f ) and arm weight (g) made significant contributions to the control parameters. The pseudo-R2 for this model

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FIGURE 5 The behavioral variable “percentage of occurrence” in the three positions plotted against age.

was .81, the log likelihood was 2 83.48, and the AIC was 182.95. The weight factor coefficients of the independent variables in the control parameter show that both variables have their strongest influence on the a control parameter. A stepwise multiple regression analysis revealed three significant variables, arm weight (g), arm volume (d ), and arm length (e) (see Table 2). The resultant best fitting equation is y 5 1070.60g 2 .82d 1 10.42e 2 206.22 The R2 for this model was .57, the log likelihood was 2 235.61 and the AIC was 481.23. Comparing the relevant log likelihoods yielded a statistical better fit for the cusp model, x2(3) 5 304.26, p , .001.

The values for the logistic model obtained with the maximum likelihood of Equation 8 were: y 5 2 22.35 1

115.83 1 1 e2(7.22 2 1.30f 1 40.23g/(2.74 1 .22f 2 4.91g)2)

The R2 for this model was .60, the log likelihood was 2123.75 and the AIC was 261.50. Given that it cannot be statistically compared with the cusp model, the AIC was used in order to select for the best-fitting model. Based on the AIC, the best-fitting model is again the cusp model. This best-fitting cusp equation can be used to calculate the estimated values of the behavioral variable x in relation to the control parameters. Stable solutions, having high probabilities, are called modes and correspond to the lower (reaching without grasping) and

Table 1. Summary of the Cusp Model for the Individual Variables model x 2 test

Table 2. Summary of the Linear Regression

Variable

Log likelihood

a) Ponderal index b) Crown– heel length c) Weight d) Arm volume e) Arm length f) Arm circumference g) Arm weight h) Body position (Constant)

2131.87 2101.56 291.48 2105.25 292.25 2102.70 290.33 2132.45 2135.92

8.10* 68.72*** 88.88*** 61.34*** 87.34*** 66.44*** 91.18*** 6.94*

a) Ponderal index b) Crown– heel length c) Weight d) Arm volume e) Arm length f) Arm circumference g) Arm weight h) Body position (Constant)

Note. The x 2 is the difference between the basic model and the model with the independent variable. For all x 2s the df are 2. *p , .05. **p , .01. ***p , .001.

Variable

r

Beta

t(weight)

2.16* .63** .67** .67** .69** .62** .70** .08

0.01 20.07 20.29 22.09 0.66 0.20 2.15 0.08

0.20 20.52 20.93 24.42** 3.64** 0.55 5.00** 1.34 25.68**

Note. r 5 correlation between behavioral data and independent variables. t(weight) 5 t test on the regression weight. *p , .05. **p , .001.

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FIGURE 6 The predicted modes and antimodes of the cusp surface, together with the behavioral data points. (a) The modes, antimodes, and behavioral data are plotted against the estimated asymmetry control parameters; (b) The modes, antimodes, and behavioral data are plotted against the estimated bifurcation control parameters.

upper surface (reaching with grasping) of the cusp. The middle sheet, which is a region of low probability, is termed an antimode. The predicted modes and antimode of the best-fitting cusp model are shown in Figure 6. In addition to the modes and antimodes, the behavioral data is plotted against the estimated asymmetry control parameters (Figure 6a), and the estimated bifurcation control parameters (Figure 6b). As the modes and antimode are predicted from a stochastic cusp model, it is possible that behavioral values are situated in the antimode area. As we already pointed out, the antimode area corresponds with the low probability mode. As a result that a three-dimensional surface is plotted in two dimensions, the predicted modes and antimodes of the cusp surface (Figure 6) is a bit scattered; however, they all belong to the same cusp surface.

GENERAL DISCUSSION Our findings confirm earlier results on the development of prehension, namely, an increase in reaching without grasping to reaching with grasping between the 3rd and 5th month after birth (Clifton, Muir, Ash-

mead, & Clarkson, 1993; McDonnell, 1975; Savelsbergh & van der Kamp, 1994; Von Hofsten, 1979; White, Castle, & Held, 1964). However, the main goal of this study was not to show this well-known phenomenon, but to answer the question whether the developmental change in prehension can be modeled as a cusp catastrophe. In addition, we wanted to find those variables that can serve as potential control parameters for this change. The findings demonstrated that the cusp model fitted the behavioral data better than a linear or a logistic model. These findings, collectively with the results of the longitudinal study (Wimmers et al., in press), hint in the direction of an affirmative answer to the question of whether a discontinuous phase transition occurs in the early development of prehension. Furthermore, using the stepwise selection procedure, it became clear that the combination of arm weight and arm circumference had the strongest influence on the control parameters. Arm weight was the most influential of all variables as it had the smallest log likelihood. However, changes in arm weight can arises from different tissues, for instance, bone, fat, or muscle mass, or any combination of these. It is possible that the amount of muscle mass increases, which may involve an increase in the

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active power of the arm muscles relative to arm mass. This is in line with Thelen’s (1985) hypothesis that the control parameter for the disappearance of infant stepping was an increase in leg mass relative to muscle power. A question could be raised: How do arm weight and arm circumference influence the addition of the grasping component? Trevarthen (1974) suggested that fisted reaching can be a consequence of exerting high levels of force. It can be postulated that exerting high levels of force often co-occur with coactivation of agonist and antagonist muscles, which could result in flexion of the fingers. If the ratio changes and more muscle power is available relative to arm weight, reaching with grasping becomes prominent. More insight and evidence for the role of a variable as part of a control parameter can be obtained with experimental manipulations. Based on our results, a possible manipulation could be the scaling up and down of the arm weight by putting weights on the arm, or making it lighter by lifting or putting the arm in water. Such an experiment would also provide information as to whether hysteresis will occur. Hysteresis is one of the strongest indications of a transition as it can only be seen in a discontinuous phase transition. These kinds of bifurcation experiments are the most appropriate method for identifying the role of different variables as part of the control parameter(s) responsible for the reported developmental change. In contrast to what was expected, the influence of body position did not contribute significantly to the other two variables, x2(2) 5 5.04, p , .09. This indicates that the candidate variable for the bifurcation parameters was not an appropriate variable. How might this be explained? In the Savelsbergh and van der Kamp (1994) study, the body position had only an effect on the 12- to 19-week-old infants and not on the 20- to 27-week-old infants. Because their dependent variable was the number of reaches, body position had an effect on the emergence of reaching. It is possible that body position does not have a similar effect on the transition from reaching to grasping. However, as test condition could influence prehension, it is relevant to evaluate different studies on prehension in regard to the body position used. A relevant literature review showed that body position of the infants differed greatly between studies reported (Pijpers & Savelsbergh, 1995; Savelsbergh & van der Kamp, 1993). The differences in body position could be responsible for the results obtained between studies and the conclusions about the development of prehension. Are there other possible variables that can contribute to the control parameters? For instance, one could be the degree of postural control (see Rochat, 1992;

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Fontaine & Pieraut-le Bonniec, 1988). Fontaine and Pieraut-le Bonniec examined the effect of differences in postural control on reaching. Infants between 4 and 10 months of age participated in the study and were divided into four groups: infants without control of the sitting position, infants able to sit while supporting themselves with their hands, infants able to sit without supporting themselves, and infants able to stand up with assistance. While there was considerable overlap in the age ranges of the four groups, the best predictor of reaching performance was the degree of postural control rather than age. At ages younger than those studied by Fontaine and Pieraut-le Bonniec, head control may serve as the agent for the emergence of reaching. In the 3rd month, the control of the head increases and, therefore, the level of postural control of the neck muscles could be considered as a control parameter. Visual abilities could also contribute to the control parameters during the period between the 2nd and 4th month. For example, visual acuity (van Hof-van Duin, 1989), binocular vision (van Hof-van Duin, Heersema, Groenendaal, Baerts & Fetter, 1992) and binocular convergence (Von Hofsten, 1977) undergo related periods of change. Another candidate could be the relationship between object size and hand size. In real time, this ratio determines how an object is grasped (Newell, Scully, McDonald, & Baillargeon, 1989; Savelsbergh, Davis, van der Kamp, & Singh-Badhan, 1994; Savelsbergh, van der Kamp, & van der Peijl, 1995; Van der Kamp, Savelsbergh, & Davis, in press). Large objects are grasped with two hands and small objects are grasped with one hand. In development, the ratio between object size and hand size changes because of growth of the hand. Therefore, it is plausible to consider this ratio as a part of the control parameters with respect to the transition examined. What is the overall contribution of this study to motor development? Traditionally, motor development is assumed to be an essentially discontinuous process in that new behaviors are deemed to correspond to new structures (e.g., genetically determined neural structures or motor programs). An example is the view that limb movements first occur under the dominance of the subcortical nuclei, but later come to be controlled by the cortex when it has become sufficiently mature and able to inhibit certain phyletic functions of the nuclei (McGraw, 1945). Another example is provided by Zelazo (1983) who accounted for the shift to walking in terms of the completion of a major cognitive change for walking. The main problem with this assumption is its circular nature: Development is deemed discontinuous because the underlying structures are discontinuous, and because the

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underlying structures are treated as exclusive categories, behavioral development is deemed discontinuous. Hence, there is the danger of eliminating apriori the possibility of a single underlying structure that changes continuously. On this view, new behaviors occur “out of the blue” without any reference to earlier behaviors. Alternatively, motor development may be seen as a continuous process in that new behaviors are reducible to old behaviors. Accordingly, development would be a process of modification of the underlying structure, resulting in transitions from one behavioral mode to another. The transition from reaching without grasping to reaching with grasping reported here, together with the data of the longitudinal study (Wimmers et al., 1998), provide support for this hypothesis. Our results show empirically that the change from reaching without grasping to reaching with grasping is discontinuous. Now the challenge is to uncover further the corresponding structural changes at the different levels that are tightly coupled to this behavioral change.

NOTES We are grateful to the parents and their infants who participated in these studies. We wish to thank Brian Hopkins and Beatrix Vereijken for helpful comments on an earlier draft of this article.

REFERENCES Bower, T. G. R. (1972). Object perception in infants. Perception, 1, 15– 30. Bower, T. G. R., Broughton, J. M., & Moore, M. K. (1970). Demonstration of intention in the reaching of neonate humans. Nature, 228, 679– 681. Bozdogan, H. (1987). Model selection and Akaike’s information criteria (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370. Clifton, R. K., Muir, D. W., Ashmead, D. H., & Clarkson, M. G. (1993). Is visually guided reaching in early infancy a myth? Child Development, 64, 1099– 1110. Cobb, L. (1978). Stochastic catastrophe models and multimodal distribution. Behavioral Science, 23, 360– 373. Cobb, L. (1980). Estimation theory for the cusp catastrophe model. Proceedings of the Section on Survey Research Methods (pp. 772–776). Washington, D.C.: American Statistical Association. Cobb, L. (1981). Parameter estimation for the cusp catastrophe model. Behavioral Science, 26, 75–78. Cobb, L. (1992). Cusp surface analysis users’s guide. Unpublished manuscript. Cobb, L., & Zacks, S. (1985). Applications of catastrophe

theory for statistical modeling in the biosciences. Journal of the American Statistical Association, 80, 793– 802. Corbetta, D., & Thelen, E. (1996). The developmental origins of bimanual coordination: A dynamic perspective. Journal of Experimental Psychology: Human Perception and Performance, 22, 502–522. Dodwell, P. C., Muir, D., & DiFranco, D. (1976). Responses of infants to visually presented objects. Science, 194, 304– 308. Fischer, K. W., Pipp, S. L., & Bullock, D. (1984). Detecting developmental discontinuities: Methods and measurement. In R. N. Emde & R. J. Harmon (Eds.), Continuities and discontinuities in development (pp. 95– 122). New York: Plenum Press. Fontaine & Pierault-le Bonniec (1988). Postural evolution and integration of prehension gesture in children aged 4 to 10 months. British Journal of Developmental Psychology, 6, 223– 233. Gesell, A. (1929). Maturation and infant behavior pattern. Psychological Review, 36, 307–319. Gilmore, R. (1981). Catastrophe theory for scientists and engineers. New York: Wiley. Halverson, H. M. (1931). An experimental study of prehension in infants by means of systematic cinema records. Genetic Psychological Monograph, 10, 107– 283. Hartelman, P. (1996). Cusp analysis program. Internal manuscript. Department of Psychology, Universiteit van Amsterdam. Kamm, K. (1994). The influence of postural development on the development of skilled reaching. Infant Behavior and Development, 17, 389. Kelso, Y. A. S. (1995). Dynamic patterns: The self-organization of brain and behavior. London: MIT Press. Lewis, M. (1990). Development, time, and catastrophe: An alternative view of discontinuity. In P. Baltes, D. L. Featherman, & R. Lerner, (Eds.), Life span development and behavior series (Vol. 10, pp. 325–350). Hillsdale, NJ: Erlbaum. Maas, van der, H. L. J., & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395– 417. McDonnell, P. (1975). The development of visually guided reaching. Perception and Psychophysics, 19, 181– 185. McGraw, M. B. (1945). The neuromuscular maturation of the human infant. New York: Columbia University Press. Newell, K. M., Scully, D. M., McDonald, P. V., & Baillargeon, R. (1989). task constraints and infant grip configurations. Developmental Psychobiology, 22, 817– 832. Pijpers, J. P., & Savelsbergh, G. J. P. (1995). Houding en de ontwikkeling van het reiken en grijpen van baby’s (Posture and the development of prehension of babies). Tijdschrift voor Ontwikkelingspsychologie, 22, 77– 103. Rochat, P. (1992). Self-sitting and reaching in 5- to 8-monthold infants: The impact of posture and its development on early eye–hand coordination. Journal of Motor Behavior, 24, 210– 220. Ruff, H. A., & Halton, A. (1978). Is there directed reaching in the human neoate? Developmental Psychology, 14, 425–426.

short standard long

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Cusp Catastrophe Savelsbergh, G. J. P., Davis, W. E., van der Kamp, J., & Singh-Badhan, M. (1994, March). Body scale and prehension in children with Down syndrome. Proceedings of the Second International Conference on Motor Control in Down Syndrome (pp. 90– 96). Chicago. Savelsbergh, G. J. P., & van der Kamp, J. (1993). The coordination of infant’s reaching grasping, catching, and posture: A natural physical approach. In G. J. P. Savelsbergh (Ed.), The development of coordination in infancy (pp. 289– 317). Amsterdam: North-Holland. Savelsbergh, G. J. P., & van der Kamp, J. (1994). The effect of body orientation to gravity on early infant reaching. Journal of Experimental Child Psychology, 58, 510–528. Savelsbergh, G. J. P., van der Kamp, J., & van der Peijl, I. (1995). The effect of destabilization of the infant’s head on the coordination of prehension. Journal of Sport & Exercise Psychology, 17, 91. Schneider, K., & Zernicke, R. F. (1992). Mass, center of mass, and moment of inertia estimates for infant limb segment. Journal of Biomechanics, 25, 145– 148. Stewart, I. N., & Peregoy, P. L. (1983). Catastrophe theory modeling in psychology. Psychological Bulletin, 94, 336– 362. Thelen, E. (1983). Learning to walk is still an ‘old’ problem: A reply to Zelazo. Journal of Motor Behavior, 15, 139– 161. Thelen, E. (1985). Developmental origins of motor coordination: Leg movements in human infants. Developmental Psychobiology, 18, 1–22. Thelen, E., Fisher, D. M., & Ridley-Johnson, R. (1984). The relationship between physical growth and a newborn reflex. Infant Behavior and Development, 7, 479–493. Thelen, E., & Smith, L. B. (1994). A dynamic systems approach to the development of cognition and action. Cambridge, MA: MIT Press. Thom, R. (1975). Structural stability and morphogenesis. Reading, MA: Benjamin. Trevarthen, C. (1974). The psychobiology of speech development. Neurosciences Research Program Bulletin, 12, 570– 585. Van der Kamp, J., Savelsbergh, G. J. P., & Davis, W. E. (in press). Body-scaled information as control parameter for

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prehension in 5- to 9-year-old children. Developmental Psychobiology. Van Hof-van Duin, J., Heersema, D. J., Groenendaal, F., Baerts, W., & Fetter, W. P. F. (1992). Visual field and grating acuity development in low-risk preterm infants during the first 2 1⁄2 years after term. Behavioural Brain Research, 49, 115–122. Von Hofsten, C. (1977). Binocular convergence as a determinant of reaching behavior in infancy. Perception, 6, 139–144. Von Hofsten, C. (1979). Development of visually directed reaching: The appraoch phase. Journal of Human Movement Studies, 5, 160– 178. Von Hofsten, C. (1984). Developmental changes in the organization of prereaching movements. Developmental Psychology, 20, 378–388. Von Hofsten, C. (1986). The emergence of manual skills. In M. G. Wade & H. T. A. Whiting (Eds.), Motor development in children: Aspects of coordination and control. (pp. 167– 186). Dordrecht: Nijhoff. Von Hofsten, C. (1993). Prospective control: A basic aspect of action development. Human Development, 36, 253– 270. Von Hofsten, C., & Lindhagen, K. (1979). Observation on the development of reaching for moving objects. Journal of Experimental Child Psychology, 28, 158– 173. Wimmers, R. H., Savelsbergh, G. J. P., Beek, P. J., & Hopkins, B. (in press). Evidence for a phase transition in the early developmental of prehension. Developmental Psychobiology. White, B., Castle, P., & Held, R. (1964). Observations on the development of visually directed reaching. Child Development, 35, 349–364. Wohlwill, J. F. (1973). The study of behavioral development. New York: Academic Press. Zanone, P. G., Kelso, J. A. S., & Jeka, J. J. (1993). Concepts and methods for a dynamical approach to behavioral coordination and change. In G. J. P. Savelsbergh (Ed.), The development of coordination in infancy (pp. 89– 135). Amsterdam: North-Holland. Zelazo, P. R. (1983). The development of walking: New findings and old assumptions. Journal of Motor Behavior, 15, 99–137.

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