Optimal timing and the Weber function.

Abstract

How is it that counting to ourselves helps us to estimate an interval of time? To address this question, we develop a generalized clock-counter model of duration discrimination that allows error in both the timing and the counting processes. We show that in order to minimize variability in temporal judgments, it is usually to the subject's advantage to segment the interval to be judged into subintervals. The optima] duration of the subintervals will depend on the parameters of the fundamental error equations that relate variance to the duration and number of the subintervals; in most cases, however, the optimal duration will be independent of the duration of the interval to be timed. The canonical form of the Weber function derived from our analysis takes as special cases the forms predicted by various other models of temporal discrimination. For long intervals it reduces to Weber's law, with the constant in that law solely a function of counting error.

How do changes in speed affect the perception of duration?

“…Many models assume that time perception is based on some kind of counting or accumulation process (e.g., Creelman, 1963;Gibbon et al, 1984;Killeen & Weiss, 1987;Treisman, 1963;Treisman et al, 1990). There is usually assumed to be an internal pacemaker or oscillator which emits counts or pulses; stimulus timing involves counting the number of pulses which occur during the to-be-timed interval.…”

Production of time intervals from segmented and nonsegmented inputs

“…To explain the usefulness of this strategy, Killeen and Weiss (1987) reported a general theoretical framework. Indeed, their analysis holds for empirical reports, whether or not they involve explicit counting.In brief, the component analysis of Killeen and Weiss (1987) states that given the average duration of an entire interval t (or P.T), the average number of subintervals n (or P.N), and the average duration of a subinterval d (or p.n), …”

“…These reproductions of intervals involved a series of taps on a key until the target duration was reached; the number of responses (n), the duration between successive taps (d), and the total duration of their reproductions (t) were recorded. They mainly reported that although the rate of responding-that is, the length of d-differed considerably between subjects, it remained approximately the same for one given subject for different values of t; the variability in counting (n) was mainly responsible for explaining the total variability of estimates; there were less counting errors when subjects used explicit counting; and the variability increased when subjects were asked to deviate from their preferred rates of responding.The series of experiments by Fetterman and Killeen (1990) was a major attempt to test Killeen and Weiss's (1987) analysis, but some important questions remained open or have arisen following these experiments. One question regards the optimal tradeoff between n and d for getting the best timing performance.…”

Processing of temporal duration information in working memory after frontodorsal tumour excisions

“…Indeed, there is evidence that acquisition, representation, and processing of temporal information in neural systems presumably differ from the processing of object or spatial stimulus features. For example, unlike stimulus dimensions like color and shape, temporal representations are acquired dynamically subserved by the use of effective encoding and monitoring strategies (Gilliland & Martin, 1940;Kileen & Weiss, 1987) and are thus specifically sensitive to attentional manipulations (Casini & Ivry, 1999;Mangels et al, 1998;Zakay & Block, 1996). The involvement of these strategic operations might have rendered the executive monitoring of duration information more sensitive to frontal lobe damage than the respective operations on spatial information.…”

“…Yet, acquisition and representation of temporal information in neural systems presumably differ fundamentally from the processing of object or spatial stimulus features. Representations of temporal duration are acquired dynamically, demanding attentional gating until stimulus off-set and thus are specifically sensitive to the use of effective encoding and attentional allocation strategies (Gilliland & Martin, 1940;Kileen & Weiss, 1987;Niki & Watanabe, 1979;Zakay & Block, 1996). Since the classification task demanded additional strategic monitoring of attentional resources, a higher degree of impairment in frontal lobe patients is predicted for temporal duration classification as compared to the spatial task.…”