Differentiating The Models

6.7.1 Are Animals and Infants Merely Subitizing?

When adult humans name the number of elements in a visual display, the slope in the function relating reaction time to numerosity is approximately 57 msec per item for values 1 to 4 and 300 msec per item for values 5 to 20 (Klahr, 1973). This apparent elbow in the slope of the reaction time function for labeling the number of items in visual displays is typically offered as evidence that two distinct processes are engaged for small and large numbers. However, there is controversy over whether a true discontinuity actually exists (e.g., Gallistel and Gelman, 1991; Trick and Pylyshyn, 1994). Balakrishnan and Ashby (1992) conducted a quantitative analysis of reaction time distributions and found no evidence for a discontinuity.

Although there are few data that can directly address whether animals show a discontinuity in the reaction time function for estimating numerosity, there have nevertheless been suggestions in the literature that animals are limited to subitizing (e.g., Davis and Perusse, 1988; Rumbaugh et al., 1987). Given that animals are capable of discriminating the large values 8 vs. 9, or even double-digit values when the ratio is large enough, the idea that animals are limited to a subitizing mechanism seems untenable. It is possible that animals use subitizing for small values and a counting-like process for larger values (Matsuzawa et al., 1991; Murofushi, 1997). If separate processes support small and large number representations, we might expect that rules learned in one numerical range would not be extrapolated to the whole spectrum of values. However, as discussed previously, monkeys trained to respond in descending order to 6-5-4 performed at higher accuracies on pairs of the small values 1, 2, and 3 (1 vs. 2, 2 vs. 3, and 1 vs. 3) than they did on pairs of familiar values such as 4 vs. 6 (Brannon et al., unpublished data).

6.7.2 Are Animals and Infants Limited to Object-Files?

The object-file model can in some sense be thought of as a more specified version of the subitizing hypothesis. Both seek to explain why small numbers are treated differently from large numbers. However, the object-file model proposes that the ability to perceive small numbers is a by-product of the visual system. If animals or infants were limited to the object-file model as a means of representing number, then they should be limited by absolute set size. In keeping with this idea, Starkey and Cooper (1980) found that 4- to 7-month-old infants could discriminate 2 vs. 3 in a visual habituation paradigm, but failed to discriminate 4 vs. 6. More recently, Feigenson et al. (2002a) found that 10- and 12-month-old infants could reliably choose the larger set in a 2 vs. 3 comparison, but not a 2 vs. 4 comparison. Both of these studies suggest that infants' numerical abilities are limited by absolute set size.

However, the object-file model is not sufficient to explain all of the available data on infants' numerical abilities. The object-file model cannot explain how infants could discriminate sounds or events because object-files are limited to the visual system (e.g., Bijeljac-Babic et al., 1993; Lipton and Spelke, 2001, 2002; Wynn, 1996). In addition, if limited to object files, infants should not be able to discriminate large values such as 8 vs. 16 (Xu and Spelke, 2000). Also, the object-file and subitizing models have no built-in mechanism for comparing ordinal relations (Bran-non, 2002a). The available data suggest that in at least some contexts, infants represent number as mental magnitudes and are not limited to object-file representations. Carey (1998, 2001) and Spelke (2000) have suggested that infants may possess both an object-file and an accumulator mechanism, and that these processes may operate over different numerical ranges. Future work should investigate whether there are really two distinct processes and what conditions invoke each system.

There is much stronger evidence that animals possess a mechanism for representing number that is more powerful than subitizing or the object-file model. The main lines of evidence are that (a) the values that animals can discriminate greatly exceed the range than can be handled by object-files; (b) animals can perform computations with their numerical representations, such as ordering; and (c) animal number discrimination follows Weber's law, implicating an analog representational format. Thus animals and human infants both appear to have a mechanism for representing number that is more powerful than subitizing or the object-file model. The remaining models are attempts to specify a mechanism for representing number nonverbally and predict the distance and magnitude effects found in animals.

6.7.3 Are Animals and Infants Counting?

The arbitrary numeron model and the mode-control model are the only models described above that conform to Gelman and Gallistel's (1978) counting principles. In the event mode, each to-be-counted event results in a fixed increment (200 msec for rats) in the accumulator and thus abides by the one-to-one correspondence principle, which states that only one numeron can be assigned to each event. The stable-order principle is upheld because the states of the accumulator have a fixed order (you cannot get four pulses without first having three pulses). Finally, the resulting value in the accumulator represents the numerosity of events or objects and thus conforms to the cardinal principle. A more detailed description of why the mode-control model of temporal integration is functionally equivalent to counting has been provided by Broadbent et al. (1993) and Meck (1997).

Thus the mode-control model predicts that infants and animals serially enumerate. In contrast, the Dehaene and Changeux neural network model and the connec-tionist timing model predict that infants and animals perceive number in parallel (i.e., all at once). Unfortunately, there are few or no data that can help determine whether animals or infants use an iterative or parallel process. However, it is interesting to note that given multiple switches and a single accumulator, the mode-control model could readily be modified to accumulate multiple simultaneously presented events or objects in parallel. This idea has not been explicitly described; however, the assumption that there are multiple switches is inherent to the model in that animals are thought to count (event mode) and time (run or stop mode) simultaneously (e.g., Meck and Church, 1983; Roberts et al., 2000).

6.7.4 How Do the Models Handle Development?

A comprehensive model of nonverbal numerical abilities should address the development of cardinal and ordinal numerical knowledge and the development of numerical abilities in relation to nonnumerical abilities. The Dehaene and Changeux neural network model posits that the initial state of the organism consists solely of numer-osity detectors and that the system learns the relationship between quantities and motor outputs from external reward input or from an autoevaluation loop. Thus, the ability to make ordinal numerical judgments arises after the ability to make cardinal discriminations. Furthermore, a specific prediction made by the Dehaene and Changeux neural network model is that ordinal judgments come on line with the maturation of the frontal lobes at about 10 months of age in humans. This prediction is supported by the finding that 11- but not 9-month-old infants detect a reversal in the ordinal direction of a numerical sequence (Brannon, 2002a).

6.7.5 Weber's Law

The mode-control model, the connectionist timing model, and the Dehaene and Changeux neural network model all predict that numerical discriminations will follow Weber's law. The mode-control model specifically posits that the variability in the memory distributions for each numerical value increases proportionally to the mean of that value. Confusion results from increasing overlap in the memory distributions, which increases the probability that a value pulled randomly from two distributions with means X and Y will be equal or reversed in their ordinal relations (e.g., Y1 < X1). Thus, the mode-control model posits that the internal number line is linearly spaced, but that the variability in the representations of each number is proportional to the value represented. An alternative hypothesis is that the internal number line might be logarithmically scaled (Dehaene, 1992).

The linear-with-scalar-variability subjective number line hypothesis and the logarithmic subjective number line hypothesis have been treated as functionally equivalent because of their similar empirical predictions (Dehaene, 1992). However, Gibbon and Church (1981) developed a clever experimental paradigm to address whether time is subjectively represented in linear or logarithmic coordinates (time-left). This paradigm makes use of the simple fact that if the subjective time scale is logarithmically scaled, behavior based on the difference between two values will be the same whenever the two values have the same ratio, no matter how far apart these values are objectively. In other words, subtraction in a logarithmic scale is equal to division in a linear scale. Brannon et al. (2001) adapted the time-left paradigm to investigate the subjective scaling of number. The task required pigeons to compare a numerical difference that varied from trial to trial to a constant value (e.g., 8 - 2 vs. 4, or 8 -6 vs. 4). The pigeons should choose whichever value seemed smaller, the difference or the constant value. If the number scale is logarithmically compressed, then the comparison of the difference with the constant value will depend not on the objective difference between the two numbers, but rather on their objective ratio. Thus, the number that is judged as equal to a given subjective difference will not increase if the two numbers being compared are made objectively bigger, as long as the same ratio is maintained between them. In contrast, when the two numbers are scaled up proportionally, the objective difference between them increases on a linear scale. If number is represented in linear coordinates, the number judged as equal to the subjective difference should increase linearly with objective number.

Results indicated that changing the absolute values of the numerical constant and the numerical difference had a linear effect on the number that was subjectively equal to a given subjective difference, even though the ratio between the constant and the difference was maintained. The reward schedule insured that number rather than time controlled the pigeons' behavior. These results are contrary to the predictions of the logarithmic compression hypothesis and indicate that subjective number is linearly related to objective number. However, the Dehaene and Changeux neural network model can be implemented with linear spacing of the mental number line and scalar variability (Dehaene, 2001).

6.7.6 Representation of Time and Number

A prediction of the mode-control and connectionist timing, but not the Dehaene and Changeux neural network model, is that time and number are represented by the same mechanism. A great deal of evidence supports this claim in animals (Church and Meck, 1984; Fetterman, 1993; Meck and Church, 1983; Meck et al., 1985; Roberts, 1995; Roberts and Boisvert, 1998; Roberts et al., 1995, 2000, 2002; Roberts and Mitchell, 1994; Santi and Hope, 2001). Here we will review four relevant findings.

First, Meck and Church (1983) trained rats in the duration bisection procedure to make one response to a 2-sec two-cycle stimulus and another distinct response to an 8-sec eight-cycle stimulus. Rats were then tested with duration held constant at 4 sec and number varied, or number held constant at 4 and duration varied. In both cases, the rats' behavior was modulated by the stimulus dimension that varied, showing that the rats had encoded both number and time when the two were confounded. Second, when the probability of making a "long" or "many" response was plotted against stimulus duration or number, the point of subjective equality (PSE) was equivalent for time and number and the functions were virtually identical. The PSE is the value at which the animals were equally likely to categorize the stimulus as long or short, or many or few. Third, when the rats were administered methamphetamine, they showed the same leftward shift in the psychophysical curve that relates the probability of a "long" and "many" response to the actual times or counts with which the animal was presented (Meck and Church, 1983). This suggests that the drug manipulation may function to increase the speed of the pacemaker and implicates a single pacemaker for timing and counting in the seconds range (Church and Meck, 1984). Fourth, studies that evaluate working memory for time and number also suggest that a single mechanism is used to time and count. Spetch and Wilkie (1983) trained pigeons in a delayed-match-to-sample procedure to make one response after a short stimulus and a second response after a long stimulus. As the retention interval between the sample presentation and the choice was increased, the 2-sec sample retention curve was unaffected while the 8-sec retention curve suffered substantially. Thus, as the retention interval increased, the pigeons' bias to choose the small response increased. A parallel effect was found when pigeons discriminated small and large numbers of responses (Fetterman and MacEwen, 1989) or sequences of flashes (Roberts et al., 1995).

In the mode-control model, animals are thought to encode time and number automatically. However, recent studies question whether time may be more salient than number for animals under some situations. In one study, a key color cued pigeons of reward after a fixed number of flashes and a second color cued reward after a fixed duration. Midway through a trial, the experimenters changed the cue and found that when the pigeons had originally been cued to count, they had also kept track of time; however, when cued to time, the pigeons did not count (Roberts et al., 2000). It is thus possible that there is an asymmetry in the mode-control model whereby timing can occur automatically and counting does less so (see also Breu-kelaar and Dalrymple-Alford, 1998; Roberts et al., 2002).

Unfortunately, there are no data that can address whether infants use a single mechanism to time and count. In fact, in contrast to the large number of studies that have investigated what infants know about number, relatively few studies have asked about infants' ability to represent time intervals (for a review, see Pouthas et al., 1993). Wynn (2000) recently tested 6-month-old infants in a habituation paradigm and found that they discriminated intervals with a 1:2 ratio but not a 2:3 ratio. This suggests that infants can represent time and that their discriminations may be controlled by the ratio of the intervals juxtaposed. Similarly, in preliminary studies in our lab (Brannon et al., 2002), 10-month-old infants showed a mismatch negativity (MMN) to a time deviation when 500-msec interstimulus intervals (ISIs) were occasionally and unexpectedly intermixed with 1500-msec ISIs. The MMN is elicited by deviant auditory stimuli in a train of standard stimuli and is a negative deflection, between 130 and 200 msec poststimulus, in the difference wave derived from subtracting the event-related potential (ERP) elicited to the standard from the ERP elicited to the deviant. Collectively, the behavioral data from Wynn's lab and our ERP results suggest that human infants are capable of interval timing in the seconds range. Future studies must test whether time discrimination in human infants follows Weber's law and what the relationship is between timing and number discrimination in infancy.

In summary, there is a great deal of evidence that shows that many animal species possess a mechanism for representing number that is more powerful than a subitizing process or the object-file model. A handful of studies also implicate analog magnitude representations of number in human infancy. In support of the mode-control and connectionist timing models, a large literature substantiates the idea that animals represent number and time with a single currency, but no data test this hypothesis for human infants. More research is needed to test predictions of these three hypotheses in both nonhuman animals and human infants and to investigate how any of these models might be implemented in the nervous system.

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