Modelling Chronometric Counting

in Timing & Time Perception
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Participants performed on a temporal generalization task with standard durations being either 4 or 8 s, and comparison durations ranging from 2.5 to 5.5, or 5 to 11 s. They were required to count during all stimulus presentations, and counts were recorded as spacebar presses. Generalization gradients around both standard values peaked at the standard, but the gradient from the 8-s condition was steeper. Measured counts had low variance, both within trials and between trials, and a start process, which was different from the counting sequence, could also be identified in data. A computer model assuming that a comparison duration was identified as the standard when the count value for the comparison was one that had previously occurred for a standard fitted the temporal generalization gradients well. The model was also applied to some published data on temporal reproduction with counting, and generally fitted data adequately. The model makes a distinction between the variance of the count unit from one trial to another, and the counts within the trial, and this distinction was related to the overall variance of behaviours resulting from counting, and the ways in which variability of timing measures change with the duration timed.



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  • Diagrams of the counting process during stimulus presentation. The upper panel shows a trial without a post-stimulus count, the lower panel a trial with a post-stimulus count. ON/OFF = onset and offset of stimulus to be timed. ICI n = inter-count interval n. Also indicated are the start time, the time from the start of the stimulus to the first count, and in the lower panel the downward-pointing arrow after stimulus offset shows the post-stimulus count.

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  • Upper panel: Mean (and standard error) of inter-count intervals from the 4-s (4 s ICI) and 8-s (8 s ICI) conditions, and start times (4 s ST and 8 s ST). Lower panel: Coefficients of variation (standard deviation/mean) from count intervals and start times. BET = between-trials cv; WITH = within-trials cv, ST = start time. Vertical bars indicate standard errors of the mean.

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  • Performance on the temporal generalization task. The proportion of YES responses (identifications of a comparison duration as the standard) is plotted against comparison duration. Values shown are the means, with the standard error shown by vertical bars. Upper panel: data from the condition with a 4-s standard; lower panel: data from the condition with an 8-s standard.

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  • Upper panel: Data from Fig. 2 plotted on the same relative scale, with the proportion of YES responses plotted again the comparison duration divided by the standard value. Other details as Fig. 2. Lower panel: Data from Fig. 2 plotted in terms of proportion of potential trials that were misses (failures to identify the standard duration when presented), and false alarms (responding YES after a non-standard duration). miss4 and miss8 are miss proportions from the 4 and 8-s condition, respectively, and fa4 and fa8 show false alarm rates from the two conditions.

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  • Results of simulation of temporal generalization. Upper panel: 4-s condition; lower panel: 8-s condition. Data points are show as unconnected filled circles, and the line shows the simulation values.

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  • Standard deviations of reproductions of 11 s, derived from the simulation, with mean inter-count interval varied over values of 150, 350, and 1200 ms. Each point shows five simulations of 100 reproduction trials. The line connects the average standard deviation from each inter-count interval used.

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  • Schematic diagram of the counting model used to simulate temporal generalization performance. See text for details.

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  • Results from the simulation plotted as the data in Fig. 3. Upper panel: simulated proportion of YES responses plotted on the same relative scale. Lower panel: miss and false alarm analysis of simulation results. Other details as Fig. 3.

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  • Simulated generalization gradients resulting from varying between- and within-trials cv. Upper panel: effect of varying the within-trials cv (values are shown in the small panel); Lower panel: effect of varying the between-trials cv (values are shown in the small panel). See text for other details.

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  • Simulations of temporal reproduction performance. Upper panel: mean time produced plotted against target time. Lower panel: standard deviation of times produced plotted against target time. Results in each panel come from two simulations described in the text. Straight lines show best-fitting linear regressions.

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  • Coefficient of variation of reproductions resulting from varying between- and within-trials cv. Upper panel: effect of varying within-trials cv (values [w] are shown in the small panel); Lower panel: effect of varying the between-trials cv (values [b] are shown in the small panel). See text for other details.

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