The accumulator model, briefly described in Section I, was originally developed by comparative psychologists Warren Meck and Russell Church to account for a variety of numerical abilities that they found in rats. Meck and Church noticed a number of similarities in rats' ability to determine number and their ability to measure temporal duration. To account for these similarities, they proposed that a single mechanism underlies both abilities and expanded an existing model for measurement of temporal intervals so as to incorporate a counting component. Their proposed accumulator mechanism works as follows: a pacemaker puts out pulses of energy at a constant rate, which can be passed into an accumulator by the closing of a mode switch. In its timing mode, the switch closes at the beginning of the temporal interval being timed and remains closed for its duration, passing energy into the accumulator continuously at a constant rate. Thus, the amount of energy in the accumulator varies in direct proportion to the length of the timed duration. The fullness of the accumulator after timing of some duration can be compared with fullness values previously stored in memory to determine whether the just-timed duration is longer, shorter, or the same as a duration associated with some event. In its counting mode, when an entity is experienced that is to be counted, the switch closes for a brief, fixed interval and then opens again. Thus, when counting, the accumulator fills up in equal-sized increments, one increment for each entity counted, and its final fullness value varies in direct proportion to the number of entities so counted (see Fig. 3). Note that there will be differences in the exact fullness of the accumulator on different counts of the same number of items, resulting from inherent variability in the rate at which the pacemaker is generating pulses and in the amount of time the switch closes for each increment. This variability is normally distributed around a mean fullness value for each number, and the variance increases in proportion to the numerosity represented. This mechanism contains numerous accumulators and switches, so that the animal can count different sets of events and measure several durations simultaneously.
B. Evidence for Similarity between Counting and Timing Processes in Animals
Evidence for functional similarity between rats' timing processes and their counting processes comes from several experiments. First of all, methamphetamine increases rats' perceptions of duration and numerosity by exactly the same factor, suggesting that the same mechanism is affected in both cases. This effect could
be explained in the model by the drug causing an increase in the rate of pulse generation by the pacemaker, leading to a proportionate increase in the final value of the accumulator regardless of the mode in which it was operating. Second, both numerical and duration discriminations transfer to novel stimuli equally strongly when rats trained on auditory stimuli were then tested on mixed auditory and cutaneous stimuli. Finally, an experiment tested the following prediction: If the animal's decision is based on a comparison of the final value of the accumulator with a previously stored value of the accumulator, then one might expect there to be transfer from making an evaluation on the basis of the output of the timing process to making an evaluation on the basis of the output of the counting process, so long as the final output value of the accumulator in the two cases was identical. For example, a count that yielded the same final fullness value in the accumulator as for a previously trained duration might be responded to as if it were that duration. This prediction was confirmed: when rats were trained to respond to a specific duration of continuous sound, they immediately generalized their response when presented with a certain number of 1-sec sound segments that had been calculated by the experimenters to fill up the accumulator to the same level as that for the duration on which the rats had been initially trained. Transfer was equally strong from number to duration. Meck and Church concluded that the same mechanism underlies both counting and timing processes in rats.
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