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FIGURE 7.4 Illustration of four potential developmental changes in temporal generalization gradients. Results come from simulations using a computer model based on the developmental version of scalar timing theory (for details, see text and Droit-Volet et al., 2001). (From Droit-Volet, S., Clément, A., and Wearden, J., J. Exp. Child Psychol, 80, 271-288, 2001.)

distribution with a coefficient of variation c. At the decision level, the subject evaluates the difference between the just-presented duration and the memory of the standard duration. When this difference is judged small, the subject responds, "Yes, it's the standard." Furthermore, the final judgment is controlled by a decision threshold b, which can be more or less conservative.

Some properties of this model are illustrated in Figure 7.4. The effect of varying the coefficient of variation of the long-term memory representation of the standard duration is shown in Figure 7.4a. Increasing the coefficient of variation makes the memory representation of the standard fuzzy, which decreases the time sensitivity, and flattens the gradient. The effect of varying the threshold b, when the coefficient of variation is constant, is shown in Figure 7.4b. Increasing b makes the decision as to whether to respond yes less conservative, and the overall proportion of yes responses increases. Figure 7.4c shows the effect of varying the proportion of random responses. Increasing p increases the proportion of random responses for each stimulus duration, even the shortest ones. Figure 7.4d shows the temporal generalization gradient shifted leftward relative to the standard, that is, related to a change in the memory representation of the standard, modeled in scalar timing theory as a change in the multiplicative memory translation constant, k* (see Gibbon et al., 1984; Meck, this volume). Therefore, McCormack et al. (1999) suggested that the leftward shifted gradient observed in their young children is due to a distortion of the representation of the standard. Thus, they added a distortion parameter k. If k was equal to 1.0, the standard value was remembered veridically; if k was less than 1.0, the standard was remembered as shorter than it really was; and if k was more than 1.0, it was remembered as longer than it really was.

In sum, our model proposes four parameters to take into account the age-related differences in the temporal generalization gradients obtained from children: (1) the variability of the long-term memory representation of the standard duration, (2) the decision threshold more or less conservative, (3) the proportion of random responses, and (4) the memory distortion of the standard duration. This developmental model fits our data very well and suggests that three of these four parameters were involved in the developmental changes observed in temporal generalization gradients (for more details, see Droit-Volet, 2002). The first is the coefficient of variation for the memory representation of the standard, which was higher in the 3- and 5-year-olds than in the 8-year-olds. Therefore, the increase with age in the steepness of the generalization gradient, namely, the increase in temporal sensitivity, was related to the decrease in the variability of the memory representation of the standard. The second relevant parameter is the proportion of random responses; the younger children emitted a greater number of random responses than did the older children. The third contributing parameter is the threshold value, b. Our model suggested that there was a small, but significant increase of the threshold value with increasing age. Thus, in their temporal judgments, the 8-year-olds were less conservative than the 3- or 5-year-olds. The real role of this decision threshold is still unclear (see Wearden, 1999). However, some studies have shown that subjects who are more confident in their knowledge are also less conservative, i.e., they take more risks and produce more errors when facing ambiguous cases (Nelson, 1996). We can therefore suppose that the 8-year-olds are less conservative because they were more confident in their temporal knowledge. We believe this for three main reasons. First, the 8-year-olds' memory representation of the standard was more precise than that of the younger children. Second, they have a better mastery of the concept of time (for reviews, see Droit-Volet, 2000a, 2000b; Droit and Pouthas, 1992; Pouthas et al., 1993; Friedman, 1990a). Consequently, the temporal generalization task may appear for them relatively easy. Third, they obtained more positive feedback than the younger children, who made more errors due to a higher level of random responding and a more variable memory representation.

In contrast, the last parameter, the distortion value applied to the memory representation of the standard, did not vary with age; it was stable throughout the three age groups at a value of 1.0. Therefore, in our studies of children as young as 3 years of age, no multiplicative distortions were observed in their memory representation of the standard (Droit-Volet, 2002). The problem now is to explain the developmental shift from a symmetrical to a rightward asymmetrical response gradient. A view consistent with scalar timing theory would postulate that the main difference in temporal generalization gradients between animals and human adults lies in the type of decision rule employed. The discrepancy between the just-

Stimulus duration (ms)

FIGURE 7.5 Illustration of the effect on temporal generalization gradients of an increase in the proportion of random responses (parameter p). Data derived from the computer simulation of individual subjects based on the developmental version of scalar timing theory. (From Droit-Volet, S., Q. J. Exp. Psychol., A, 55A, 1193-1209.)

Stimulus duration (ms)

FIGURE 7.5 Illustration of the effect on temporal generalization gradients of an increase in the proportion of random responses (parameter p). Data derived from the computer simulation of individual subjects based on the developmental version of scalar timing theory. (From Droit-Volet, S., Q. J. Exp. Psychol., A, 55A, 1193-1209.)

presented duration and the standard memory sample is normalized by the standard sample in animals, that is, (s* - t)/s*, and by the duration value in human adults, that is, (s* - t)/t (i.e., Church and Gibbon (1982) model vs. Wearden (1992) model). However, the reason why animals and human adults would use different decision rules is not entirely clear (Allan, 1998). Moreover, in the course of ontogenesis, there is no particular reason to suppose such changes in the decision rules. In contrast, there is a decrease in the number of random responses with increasing age in children. As suggested earlier and illustrated in Figure 7.4c, the increase in random responses increases the proportion of yes responses for each stimulus duration, but the proportion of yes responses for the shortest duration increases relatively more. In this case, the greater proportion of yes responses at the shortest durations can balance the greater proportion of yes responses at the longest durations, usually obtained in the adult-like right-skewed gradient. Consequently, the temporal generalization gradient that is rightward asymmetrical can become symmetrical when the number of random responses increases. This is precisely what we found when we simulated individual data from subjects by changing the proportion of random responses while holding the other parameters constant (Figure 7.5). When the proportion of random responses is fixed at zero percent, the gradient is rightward asymmetrical, and when the proportion of random responses is higher, the gradient significantly shifts toward symmetry. Therefore, the decrease in the proportion of random responses can, in part, explain this age-related shift from symmetrical to adult-like asymmetrical gradients.

In summary, our data on temporal generalization are consistent with those found in temporal bisection. There is an increase in the sensitivity to signal duration with increasing age, due to the decrease both in the variability of remembered time and in the probability of random responses. However, in temporal generalization, the decision threshold also changed, with the older children becoming more confident in their temporal knowledge and less conservative. In the temporal bisection procedure, this effect of age on the decision threshold was not observed. This is probably linked to the temporal bisection task, which is relatively less difficult than the temporal generalization task (McCormack et al., 1999; Wearden et al., 1997). Supporting this idea, other studies have shown that interval timing performance was worse in elderly adults than in young adults in a temporal generalization task, but not in a temporal bisection task (Wearden et al., 1997; but see Lustig and Meck, 2001).

7.3 IS AN INTERNAL CLOCK FUNCTIONAL AT AN EARLY AGE?

Beyond age differences in temporal performance, the bisection and generalization methods used in animals and human adults produced orderly data from all children. Furthermore, children's timing behavior conformed well to the scalar property of interval timing. On the whole, these findings suggest that the clock-based system underlying time perception in animals and human adults is functional at an early age (see Brannon and Roitman, this volume).

The success of the scalar timing theory comes precisely from the revival of the idea that animals and human adults might possess a sort of internal clock, and from studies derived from this theory demonstrating its existence. The clearest evidence comes from animal studies that have succeeded in speeding up or slowing down the pacemaker of the internal clock by administering drugs (Maricq et al., 1981; Meck, 1983). However, Treisman et al. (1990, 1994) have recently invented a new method that is able to alter the speed of the internal clock and is easier to use in human subjects. This method consists of accompanying or preceding an event to be timed by a short period of repetitive sensory stimulation that clicks or flashes. Using this new method, several studies have shown that adults judge stimuli to be longer if they are preceded by repetitive auditory clicks (e.g., Burle and Casini, 2001; Penton-Voak et al., 1996). In order to solve the issue of the functionality of an early internal clock, we conducted a series of experiments in children aged 3 to 8 years, based on the repetitive stimulation procedure first used by Treisman (Droit-Volet and Wearden, 2002).

In this experiment, the children were given a temporal bisection task in two different duration conditions. In the first condition, the short standard was 200 msec and the long standard 800 msec; in the second condition, the anchor durations were 400 and 1600 msec. In the training phase, the children were presented with a white circle for 5 sec, followed immediately by a blue circle, which was the stimulus to be timed (i.e., standard duration). In the testing phase, they received comparison stimulus durations either with the flicking of the white circle (flicker condition) or without flicking of the white circle (no flicker condition). As illustrated in Figure 7.6, the results showed that the flicker shifted the psychophysical functions toward the left, whatever the age group or the duration condition tested. Consequently, the bisection points were lower with flicker than without flicker, and to the same extent in the three age groups. As of the age of 3 years, the stimulus durations were thus judged longer with than without repetitive stimulation produced by the flicker.

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