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FIGURE 6.11 (a) Diagram of the mode-control model, which consists of a pacemaker, switch, accumulator, working memory, reference memory, and comparator. (b) Diagram of the connectionist timing model, which consists of multiple oscillators rather than a single pacemaker, status indicators that provide information about the phase of the oscillators rather than an accumulator value, a working memory, a reference memory, and a comparator. (c) Diagram of the workings of the Dehaene and Changeux neural net model, which consists of an input retina, a topographical map of object locations that normalizes for size of the objects, and a map of numerosity detectors that sums the outputs from the location maps and provides a representation of number.

accumulator with the value(s) in reference memory to determine what type of response to make. Although the mode-control model was developed to explain timing and counting in rats, it has since been used to explain numerical behavior in human infants (e.g.,Wynn, 1995).

6.6.4 Connectionist Timing Model

A fourth proposal is the connectionist timing model (Church and Broadbent, 1990, 1991) (see Figure 6.11b). This model posits a set of oscillators with different periods ranging from milliseconds to hundreds of seconds. The oscillators are reset (entrained) at the onset of a stimulus. When the stimulus terminates, the duration of the interval is represented by a status indicator, which is a vector that provides information about the phases of multiple oscillators. Memory for intervals is represented by a matrix of connection strengths between elements on the status indicators. Thus, temporal information is represented in a distributed fashion. Like the mode-control model, the connectionist timing model posits a comparison process between working and reference memory. In this model, number is derived by dividing the cumulative duration of a sequence of events by the duration of one interval (see Figure 6.10d).

Both the connectionist timing model (Church and Broadbent, 1990, 1991) and the mode-control model (Meck and Church, 1983) propose that the representation of time and number are inextricably linked; however, there are some important differences. The mode-control model uses a single pacemaker as the basis for interval timing and counting. The connectionist timing model instead proposes that multiple oscillators with a wide range of periodicities (from hundreds of milliseconds to hundreds of seconds) serve as the basis for timing. Number is represented in the mode-control model as the magnitude of an accumulation process. In the connectionist timing model, the relationship between durations represents number. Therefore, the connectionist timing model, in contrast to the mode-control model, cannot uniquely identify number for arrhythmic presentations (e.g., Breukelaar and Dalrymple-Alford, 1998; Davis et al., 1975). Finally, both the connectionist timing and mode-control models process information serially from a clock to working memory to reference memory to a comparison. The variables that represent time and number in the mode-control model are scalars, which are associated with some uncertainty. In the connectionist timing model, temporal variables are encoded by vectors and matrices, resulting in parallel distributed representations of time, and therefore number.

6.6.5 Dehaene and Changeux Neural Network Model

A fifth model is the Dehaene and Changeux (1993) neural network model. This model posits that numerosity detectors represent the abstract number of objects independently of the size and configuration of the stimuli. Figure 6.11c shows the three layers to this model: an input "retina," a map of object locations, and an array of numerosity detectors. The map of object locations converts stimuli from the retina to a representation of each stimulus irrespective of object size (an echoic auditory memory also allows the system to enumerate sounds as well as objects). The location map sends its output to numerosity detectors, which consist of summation units and numerosity units. Each summation unit has a set threshold of a particular value. When the total activity from the output of the location map (which is proportional to numerosity) exceeds the summation unit's threshold, it will be activated. These units differ from the event mode of the mode-control model, as they are active only when the number of events exceeds some level, for example, greater than or equal to 3. Finally, the summation clusters project to numerosity clusters, which represent the numerosities 1 through 5. A given numerosity cluster will be activated if the corresponding summation cluster is active (e.g., 3), but those representing higher values (e.g., 4 and 5) are not. Therefore presentation of stimuli with the same numerosity, despite differences in size, location, and modality, results in the activation of the same numerosity detectors (see Figure 6.10e).

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