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FIGURE 3.7 Multiple local maxima in sensitivity to time are observed in the discrimination of time across several orders of magnitude. The existence of a local maximum near a circadian oscillator (rightmost peak; square symbols) and other local maxima in the short-interval range (left side; filled and unfilled circles) is consistent with the hypothesis that timing is mediated by multiple oscillators. Intervals in the blank region in the center of the figure have not been tested. Left side: Rats discriminated short and long durations, with the long duration adjusted to maintain accuracy at 75% correct. Short durations were tested in sequential order (filled circles; N = 26) or independent order (unfilled circles; N = 20). Circles represent relative sensitivity using d' from the signal detection theory and are plotted using the y-axis on the left side of the figure. Right side: Rats received food in 3-h meals with fixed intermeal intervals by breaking a photobeam inside the food trough. The rate of photobeam interruption increased before the meal. Squares represent sensitivity, which was measured as the width of the anticipatory function at 70% of the maximum rate prior to the meal, expressed as a percentage of the interval (N = 29). The interval is the time between light offset and meal onset in a 12-12 light-dark cycle (leftmost two squares) or the intermeal interval in constant darkness (all other squares). Squares are plotted with respect to the reversed-order y-axis on the right side of the figure. Y-axes use different scales, and the x-axis uses a log scale. (Adapted from Crystal, J.D., J. Exp. Psychol. Anim. Behav. Process, 25 , 3-17, 1999; Crystal, J.D., J. Exp. Psychol. Anim. Behav. Process, 27, 68-78, 2001a; and Crystal, J.D., Behav. Process, 55, 35-49, 2001b.)

nonlinearities in sensitivity to time is in conflict with current versions of scalar timing theory.

### 3.5.2 Multiple-Oscillator Theory of Timing

The multiple-oscillator theory of timing proposes that time is represented by a set of oscillators, each with a unique period (e.g., 100, 200, 400, 800 msec, etc.); rewarded times are stored in an associative matrix and current times are compared with remembered times to render a response decision. The multiple-oscillator theory predicted the existence of nonlinearities in sensitivity to time. The proposal that short-interval timing is mediated by multiple short-interval oscillators provides a basis for the development of a unified theory of timing that can accommodate data ranging from milliseconds to days.

### 3.5.3 Broadcast Theory of Timing

The broadcast theory of timing proposes that timing of behavior is based on the time required for neural signals to travel different distances in the nervous system; the variance of delays is proportional to the square of the distance. Therefore, if the delay is subdivided (i.e., time two short intervals instead of one relatively long interval), then the variance of time estimation will be characterized by multiple local peaks. This is consistent with nonlinear sensitivity of time. However, the observation of a local peak in sensitivity at 24 h suggests the involvement of a circadian oscillator rather than the transmission of neural signals across a relatively long distance.

### 3.5.4 Stochastic Counting Cascades

The model of stochastic counting cascades proposes that counting events (e.g., pulses from a pacemaker) is characterized by failures to set the element in the next stage of a binary counter when a lower element is reset; while counting up, a set failure results in an underestimation of the number of events. Similarly, the resetting of a bit in a binary counter to its zero state may fail to occur; a reset failure results in an overestimation of the number of events. Fallible binary counters can produce nonlinearities in sensitivity to time. Furthermore, the model of stochastic counting cascades may be inserted into other theories of timing as a counting module (Church, 1997; Killeen, 2002; Killeen and Taylor, 2000). Indeed, there is some similarity between the role played by oscillators in the multiple-oscillator theory and Killeen's binary counters; the periods of the oscillators in the multiple-oscillator theory increase by powers of 2, and the behavior of elements in a binary counter is to oscillate with a period corresponding to each element of the counter.

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