Timescale invariance in the pacemaker accumulator family of timing models
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Timescale invariance in the pacemaker accumulator family of timing models
Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models 163'22134468-00002018Patrick Snnen1. Francois Rivest2. Elliot A. Ludvig3, Fuat Balci4 and Peter Killeen5Oberlin College. Department of Neuroscience 119 Woodland St.. Oberlin. OH 44074'Royal Military College of Canada. Department of Mathematics & Computer Science PO Box 17000. Station Forces. Kingsto Timescale invariance in the pacemaker accumulator family of timing models n. Ontario CANADA. K7K 7B4 andCentre for Neuroscience Studies. Queen's University. Kingston. ON. Canada Princeton University, Princeton Neuroscience ITimescale invariance in the pacemaker accumulator family of timing models
nstituteGreen Hall. Washington Rd.. Princeton. NJ 085404Koẹ University. College of Social Science & Hiunamties Riunelifeneri Yolu. 34450. Sariyer - IsTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models en@ober lin.edu 119 Woodland St.Oberlin. OH 44074Short Title: The PA familyHord Count: 9042Figures: 5Email: lpsimen@oberlin.edu. 'francois.rivest@nnc.ca. fiancois.rivest@mail mcgill ca 3ehidvig@princeton edn. 4 fbalci@kuedu.tr. 5killeen@asu edu1. IntroductionPerhaps the most intuitive model of an an Timescale invariance in the pacemaker accumulator family of timing models imal's internal clock IS a simple pacemakeraccumulator (PA) system: Discrete units of some physical quantity accumulate at a constant rate over the coTimescale invariance in the pacemaker accumulator family of timing models
urse of an interval. When the total sum reaches a critical level, the animal behaves as if the interval is over The PA approach seems intuitive becausTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models (1963) PA model (hereafter denoted TPA) is the prototype PA model of timing. It uses a pacemaker, whose pulses are accumulated by a counter and sent to a store to encode durations. Critically, in TPA. the inter-pulse durations within trials are correlated, with shorter-than-average durations in some Timescale invariance in the pacemaker accumulator family of timing models trials, and longer-than-average durations 111 others (Postulate 2. Treisman. 1963). In other words, the pace of the pacemaker varies randomly acrossTimescale invariance in the pacemaker accumulator family of timing models
trials around a fixed average. As shown in Tieisman. this property of the TPA accounts for the strict form of "Weber's law for timing", a temporal anaTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models hich of two non-temporal stimuli has greater intensity (e.g.. heavier, brighter, etc.). Although Weber investigated perceptual representations by finding the just noticeable difference between very similar- stimuli, the law can be restated as holding that accuracy is constant whenever the two compar Timescale invariance in the pacemaker accumulator family of timing models ison stimuli are proportionally strengthened or weakened 111 intensity. This relationship suggests a level of perceptual imprecision tliat IS intensitTimescale invariance in the pacemaker accumulator family of timing models
y-scale-invariant: specifically, the intensity estimates across repeated trials of a task are distributed so that the standard deviation s of the estiTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models Treisman (1963). the cv of human behavioral response times in timing tasks was indeed found to be roughly constant across different durations in temporal production, reproduction, decision and estimation tasks, although a collection factor a was required 111 a generalized form of Weber's law: s = k Timescale invariance in the pacemaker accumulator family of timing models • (M + a).For durations ranging from 0.25 to 9 seconds. Treisman found k in the range 0.05-0.1. a around 0.5. and M accurate but subject to some biasTimescale invariance in the pacemaker accumulator family of timing models
es toward shorter or longer estimates, depending on3the procedure. One of the main empirical goals of Treisman (1963) was to address the diversity of Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models e was little evidence for any local minimum of (he cv al any particular duration, as had been previously hy pothesized.Since rrcisman (1963), experiments with non-human animals, for which verbal and cognitive strategics such as counting would likely be minimized, suggested even stronger support for Timescale invariance in the pacemaker accumulator family of timing models lire strict form of the law, s k A (c.g.. Gibbon. 1977. among marry subsequent replications, blit see also Rizo et al.. 2006. for some exceptions). FuTimescale invariance in the pacemaker accumulator family of timing models
rthermore. Gibbon and colleagues frequently observed that the entire distribution of response times, when divided by rhe mean response rime, typicallyTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models er al., 1997). fins superiinposition property is sometimes dubbed scalar invariance'. for consistency with the general use of the similar phrase scale invariance across disciplines, we w ill use timescale invariance to refer to the same property.The TPA route to timescale invariance.In the TPA model Timescale invariance in the pacemaker accumulator family of timing models , a duration T is timed by counting the pulses emitted by a pacemaker. Treisnian (1963) does not specify precisely what kind of pulse train is emittedTimescale invariance in the pacemaker accumulator family of timing models
from the pacemaker, other than to stale that the inter-pulse durations are highly regular witliin trials. The TPA pacemaker has a fixed average rate,Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models hat the inter-pulse times are essentially identical and the pacemaker is essentially periodic (Postulate I), bur with a different period in each trial (Postulate 2). All inter-pulse durations within any given trial are almost the same. Under this assumption, the standard deviation, across trials, of Timescale invariance in the pacemaker accumulator family of timing models rhe sum of n inter-pulse times, all of which equal the same random duration A' equals n lunes the sliuidaid deviation of A'. Illis follows from die vTimescale invariance in the pacemaker accumulator family of timing models
ariance formula for a constant, n. and a random variable A; Vai(nA) n~ VarfA). Taking the square root gives Std(nA) n Sld(A). Note the contrast from tTimescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models s have identical4variances Var(.Y) across trials, then the variance of the stun of M independent pulses is n Var(A); the n in this case is not squared.For (he nearly-periodic TPA. different durations T are timed by counting different numbers. w. of the fixed-rate pacemaker's pulses, which occur with Timescale invariance in the pacemaker accumulator family of timing models average inter-pulse duration equal to Mean(A). This process yields an estimate, r. of T. as follows: T' = 2"=1 Xịi- For this estimate. CV(T') = Std(TTimescale invariance in the pacemaker accumulator family of timing models
')/Mean(T') = n Std(A) / ( n Mean(A) ). The n cancels out of the numerator and denominator, and the cv is thus constant for all T. A similar argument Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11 Timescale invariance in the pacemaker accumulator family of timing models Theory (SET: Gibbon. 1977) and its information processing implementation (1PI. (Gibbon. Church. & Meek, 1984; Gibbon & Church. 1984). emphasized pacemakers with Poisson characteristics, which emit highly irregular pulse trains Inter-pulse durations in such models are independent and exponentially di Timescale invariance in the pacemaker accumulator family of timing models stributed For such models with independent inter-pulse durations, the variance of the pulse counts is n • Var(A). which implies:CV(n=VHStd(«/(nMean(X)Timescale invariance in the pacemaker accumulator family of timing models
) = ;^ỆJ•••= Std(X)/(ựMean(X) • Ợn • Mean(X)) = [Std(X)/ựMean(X)] • 1/vT.Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11Timescale invariance in the pacemaker-accumulator family of timing modelsTo appear in Timing & Time Perception 1 (2013) 159-188 http dx.doi.org TO. 11Gọi ngay
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