Timescale Invariance in the Pacemaker-Accumulator Family of Timing Models

in Timing & Time Perception
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Pacemaker-accumulator (PA) systems have been the most popular kind of timing model in the half-century since their introduction by Treisman (1963). Many alternative timing models have been designed predicated on different assumptions, though the dominant PA model during this period — Gibbon and Church’s Scalar Expectancy Theory (SET) — invokes most of them. As in Treisman, SET’s implementation assumes a fixed-rate clock-pulse generator and encodes durations by storing average pulse counts; unlike Treisman’s model, SET’s decision process invokes Weber’s law of magnitude-comparison to account for timescale-invariant temporal precision in animal behavior. This is one way to deal with the ‘Poisson timing’ issue, in which relative temporal precision increases for longer durations, contrafactually, in a simplified version of Treisman’s model. First, we review the fact that this problem does not afflict Treisman’s model itself due to a key assumption not shared by SET. Second, we develop a contrasting PA model, an extension of Killeen and Fetterman’s Behavioral Theory of Timing that accumulates Poisson pulses up to a fixed criterion level, with pulse rates adapting to time different intervals. Like Treisman’s model, this time-adaptive, opponent Poisson, drift–diffusion model accounts for timescale invariance without first assuming Weber’s law. It also makes new predictions about response times and learning speed and connects interval timing to the popular drift–diffusion model of perceptual decision making. With at least three different routes to timescale invariance, the PA model family can provide a more compelling account of timed behavior than may be generally appreciated.

Timescale Invariance in the Pacemaker-Accumulator Family of Timing Models

in Timing & Time Perception



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    (A) Diffusion without any drift: Particle ensembles densities plotted against position at ten successive time steps (left) and particle trajectories plotted against time for ten time units (right). (B) Diffusion with drift, forcing these ensembles toward the right (top) end of the tube.

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    The drift–diffusion model of two-choice decision making. The drift, A, determines how rapidly particles rise upward on average; the noise level, c, determines how much random perturbation of the particle occurs per unit time; the starting point is x0; the upper boundary z serves as a criterion for one type of response in two-alternative decision making, while the lower boundary, z, serves as a criterion for the other. In the case of timing, z is assumed to be much larger relative to the noise than in perceptual decision making, and z is assumed to be at negative infinity, though in practice, as long as it is more than 0.1z below the starting point, lower-boundary crossings are extremely rare and can be ignored. T0 represents a non-decision latency for sensory encoding and motor response that is added to the decision time of the diffusion model to yield a total response time.

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    (A) A simulation of a 1000 trials of a drift–diffusion process, in which a normal random variable with mean 0.1 and std 1 is accumulated over 10 000 time steps, until a first-passage across the threshold value of 500 occurs. No trajectories ever descended below −31.3, so whether or not a lower boundary (reflecting or absorbing) is included at, say, −30 or less, the first-passage-time histogram would look almost identical. (B) The exact same set of trajectories, magnified, starting at position 0, with an upper threshold at 10 and a lower threshold at −10. The histogram of upper-threshold crossing times is shown above; the histogram of lower-threshold crossing times is shown below. Note that there are far fewer lower-threshold crossings, but that the histogram has the same shape as the one above.

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    Comparison of inverse Gaussian (solid), gamma (dashed), and normal (dotted) probability densities with equivalent CVs. Densities with larger CVs (on the left) show greater deviation from normality and larger positive skewness. These densities were computed with greater noise for shorter durations.

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    Comparison of trajectories and response time distributions for 500 simulated trials of an opponent Poisson process (left panels) and the Euler–Maruyama simulation of the corresponding DDM parameterization (right panels). CVs of the RT distributions (shown in titles of bottom panels) are identical out to two decimal places; skewness is similar; and the ratio of skewness-to-CV (‘Ratio’), predicted to be 3 on average for the DDM, is close to 3 in both cases.

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    Simulations of the TopDDM for five different interval durations (2, 8, 32, 64, and 128 s). Histogram bin widths increase as the durations increase. The CVs of these distributions are all approximately 0.1.

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    Peak interval performance under two different fixed interval schedules (10 s and 30 s). The top panel shows 10-s responses as o’s; 30-s responses as x’s. The middle panel shows the binned response rates (solid = 10 s; dashed = 30 s). The bottom panel shows trajectories of the accumulation process. 100 trials were simulated. CVs for start times of periods of high rates responding are 0.121 and 0.116 for short and long durations respectively; CVs for stop times are 0.090 and 0.085 respectively. Thus start times are timescale invariant, stop times are scale invariant, but start times are more variable than stop times.

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