Scalar counters

Learning and Motivation 33, 63–87 (2002) doi:10.1006/lmot.2001.1100, available online at http://www.idealibrary.com on Scalar Counters Peter R Killeen Arizona State University Poisson processes—random pacemakers driving accurate counters—are common models for timers Such clocks get relatively more accurate the faster they go This is not true of real clocks, where the relative error is approximately constant, an example of Weber’s law known as scalar timing This distinction was the core problem motivating Gibbon’s Scalar Expectancy Theory Since worse pacemakers cannot generate scalar timing, the necessary variance must be found elsewhere This article reviews three failure modes of counters and shows that any one (or all together) provides a mechanism for scalar timing Unique microdeviations from proportional timing in real data provide signatures of underlying machinery This paper assays the maps between these signatures and those of stochastic counters, finding family resemblances that range from kissing cousins to clones  2002 Elsevier Science (USA) Poisson systems comprise the worst possible pacemaker driving the best possible counter Poisson clocks have a simple mathematical description and benefit from the cancellation of pacemaker errors, so that the ratio of the standard deviation to the mean is always a decreasing function of the mean The more counts, the more (relative) accuracy With a fast enough pacemaker they can be as accurate as desired However, this is not true for the timing that is typical of living organisms Biological timing is most accurate for a narrow range of pacemaker speeds (Fetterman & Killeen, 1990), above which it shows a proportionally increasing error (Rosenbaum & Collyer, 1998) This proportionality is called scalar timing, which is a manifestation of Weber’s law in the realm of temporal discrimination The work of Gibbon and his colleagues and students was focused on understanding scalar timing DISCRIMINATION Consider a discrimination of stimuli such as time, weight, intensity, or length, as did Thurstone: The perceptual events (percepts) are ordered like the physical stimuli, but have additional error variance The organism deThis paper is dedicated to the memory of John Gibbon We were each other’s whetstones The research was supported by NIMH Grant K05 MH01293 I thank R Church, L Allan, and T Taylor for their contributions to the manuscript Correspondence and reprint requests should be addressed to Peter R Killeen, Department of Psychology, Arizona State University, Tempe, AZ 85287-1104 E-mail: killeen@asu.edu 63 0023-9690/02 $35.00  2002 Elsevier Science (USA) All rights reserved 64 PETER R KILLEEN cides whether the percept exceeds a criterion and responds appropriately Figure shows the process: An organism trained to say ‘‘short’’ to 200-ms stimuli and ‘‘long’’ to 1400-ms stimuli is tested with intermediate stimuli The densities of the percepts are Gaussian and scalar—that is, they are generated from normal distributions whose standard deviations (SDs) are proportional to their mean If the curves in Fig were rescaled by dividing their x-axes by their means while keeping their areas at unity, they would fall on top of one another, or superpose Superposition is a feature of Gibbon’s Scalar Expectancy Theory (SET; Gibbon, 1986) The probability of saying long—the rising ogive—is given by the area under each density to the right of the criterion, drawn here as a vertical line at 700 ms Scalar dispersions will cause an 800-ms stimulus to be called short more often than a 600-ms stimulus is called long: Compare the area of the 800-ms process to the left of the criterion with the area of the 600ms process to the right of the criterion This results in a skewed psychometric function called a pseudodistribution (Killeen, Fetterman & Bizo, 1997) It is pseudo because it does not asymptote at 1.0 and is thus not a true distribution: Linearly increasing SDs always leave a finite tail to the left of the criterion, and therefore the area to the right of the criterion never reaches 1.0 Data validate this logic (Allan, 1999, 2001a; Allan & Gerhardt, 2001) What is the nature of the perceptual continuum—the abscissae of Fig 1— and whence the error? Gibbon took the representation to be pulses from a pacemaker that were accumulated in a storage device There were various FIG Temporal stimuli are represented on a continuum with scalar variance (Weber’s law) The distributions of the percepts are drawn above the stimulus values that elicit them Thus, the percept elicited by a 200-ms stimulus will most often seem to be about 200 ms long, but on (rare) occasions may seem 300 ms long As an epoch gets longer, observers become less certain about its exact length, as seen in the proportionately increasing spreads of the distributions If an observer responds long for all percepts that exceed 700 ms, the psychometric function drawn against the right ordinate results Understanding the statistical properties of the representations forms the core of SET and other theories of timing SCALAR COUNTERS 65 sources of error in the system (Gibbon & Church, 1984, 1992): the stochastic nature of the pacemaker (which was largely discounted; Gibbon, 1992), errors in gating the pulses to the counter, and most importantly, proportional error in the counter Why should a counter suffer proportional error? The present article dilates SET’s treatment of counter error into a set of models that provide mechanisms that incur proportional error It is based on the Killeen and Taylor (2000a) theory of stochastic counters (TSC) A single neuron accumulates, concatenating thousands of inputs and firing when excitation exceeds inhibition by some threshold However, this is not the same as counting; more than a single logical element is necessary to represent an ordered continuum The common counting algorithm—1, 2, , 9, 10, 11, —exemplifies one architecture for such extended accumulators: Events are named until they exceed the base (10 in the denary system, in the binary system), and then a higher-order element is incremented and the lower order reset We name events through 9, and for the next event increment the ‘‘tens’’ element and reset the units In the binary system there are two names, and 1, and successive elements represent powers of Binary counters are taken as the basic architecture of neural accumulators here This is because they are simple: A single threshold is required, with elements being in only one of two states—off or on, firing or quiescent Evidence will be presented that the binary system is used in human temporal discrimination Perfectly functioning devices are opaque; inaccuracies reveal structure Three inaccuracies are key to the thesis of this argument: (1) proportional error in many sensory discriminations, known as Weber’s law; (2) minuscule deviations from Weber’s law; and (3) channel capacities in absolute identifications These signatures were used to validate the Killeen and Taylor (2000a) theory Stochastic counters are counters that suffer probabilistic failures of any kind The Killeen and Taylor TSC located the error in the probabilistic transmission of an increment from one element, or stage, of a binary counter to the next Their theory accounted for many but not all of the microdeviations from Weber’s law This paper investigates other models of counter failure in the hope that they will account for more of the data STOCHASTIC COUNTERS The architecture of a binary counter is shown in Fig The first input sets the element in Stage The second input resets it, and that sets the element in Stage The third sets the first stage, and the fourth resets the first stage, which resets the second, which sets the third, and so on ad infinitum Taps from each of the elements (not shown), weighted as powers of the base, lead to an accumulator with threshold set to a criterion count This is how computers count It is likely that biological organisms must usually learn how to weight the outputs from the stages, by coincidence of some profile of activated stages with regularities in the environment (Church, Broadbent, and Gibbon, 1992; Killeen & Taylor, 2000a; Rodriguez, Wiles, & 66 PETER R KILLEEN FIG States of three stages of a counter (top) and corresponding stochastic transition diagram (bottom) The stages are like bits in a binary counter and contribute to the accumulated count as k , where k is the number of the stage The first stage contributes only units (2 ϭ 1), the second twos (2 ϭ 2), the third fours (2 ϭ 4), and so on Thus, after five inputs, the accumulated value is ϩ 2 ϭ If, upon the 6th input, the second stage fails to latch on, the count will revert to This happens with probability p(1 Ϫ p), the probability that the first stage will be activated but the second will not This may also be read from the transition diagram on the arrow carrying the system from to Some failures are worse than others Note that if at a count of all first three stages are successfully activated (which happens with probability p 3), but the fourth is not (with probability Ϫ p), then the counter will not increment to 8, but rather will decrement to Elman, 1999) With varied exposure, it is possible that the weights will converge on those of a binary counter In the present paper, the weights are assumed binary to give an ideal case in which to study the consequences of counter error Fallible Binary Counters The most obvious kind of error is a failure to set the element in the next stage when a lower element resets (the Y2K glitch) It is as though the aged odometer on the Chevy got to 99999, and, while able to reset the lowerordered stages, could not quite get the time-encrusted highest-order dial to turn Call the probability of a successful transmission of a signal from one stage to the next p, Ͻ p Յ 1, and assume it is the same value for every stage in the counter Figure (bottom) shows the resulting state transition diagram; if p ϭ 1, the diagram predicts the progress of the error-free binary counter above it If p ϭ 9, there is a probability of that upon receipt of an input, the counter will increment from to (Fig 2, top, second column) and a probability of that it will remain at If the first input is successful, SCALAR COUNTERS 67 upon the next input there is a p ϭ probability that the first element will be activated and a p ϭ probability that the first element will in turn succeed in activating the second element, leading to a count of with a probability of p ϭ 81; a probability of (1 Ϫ p) that the first element was never activated, leaving the count at with a probability of 1; and probability of a p(1 Ϫ p) that the first element ( p) was reset without setting the second (1 Ϫ p), moving the count back down to In some cases failure to set the next stage merely causes or dropped counts; other cases are worse The worst-case scenarios can happen as the counter is attempting to move to a new power of the base: for instance from to 4, to 8, and so on Figure shows that at these critical points there is a possibility of resetting to zero Figure shows the progress of counts registered as a function of input for 20 sequences In this series the counts were accurate up to 8, when failures started to occur In two cases when registering a count of 31 (11111), the next input reset the first stages but failed to activate the fifth, causing a failure that dropped the count to 16 In two other cases when registering a count of 31, the next input reset the first stages but failed to set the sixth, causing catastrophic failures that dropped the count to The trajectories exit the graph after 100 inputs with a distribution of counts registered If this process were continued for hundreds of runs, the distribution would be densest around a count of 88, falling off exponentially at lower counts The shape of the distribution is approximately that of a Weibull density (Killeen & Taylor, 2000a), which looks like a negatively skewed normal density and which constitutes a discriminal process such as those shown in Fig In the case of a temporal production task the participant estimates the passing time and responds when a criterion is met This is represented by drawing a horizontal criterion line in Fig and asking how many inputs it requires for the trajectories to cross that line Fallible counters usually require FIG Twenty trajectories of a 0.98 counter receiving 100 input signals Notice that large failures are most likely when the ordinates approach a power of 68 PETER R KILLEEN more inputs than the criterion to meet the criterion: To get to a criterion of 88 requires an average of 100 inputs in this example The distribution of inputs for a given criterion (the hitting times) are also Weibull in form but in this case positively skewed Reproduction tasks, a combination of the discrimination and production, are predicted by a convolution of these two Weibull densities This yields distributions that are approximately Gaussian in form but with a slight positive skew Such distributions are common in the timing literature CONSEQUENCES OF SET FAILURE Discrimination Figure displays three key properties of this system Figure 4A shows that the average count attained increases approximately as a power function of the number of pulses input The exponent of the power function is approximately log (2p) Figure 4B shows that the standard deviation of the counts registered grows in an approximately linear fashion with the input (Weber’s law), for counters of mediocre accuracy ( p Ͻ 95) For more accurate counters the standard deviation grows in steps, with large jumps at the points where a set-failure has the chance to reset the counter to zero Catastrophic failures are in fact more common with lower-accuracy counters, but their locations are diffused by the larger numbers of subcritical failures, so their Weber function is steeper but smoother Figure 4C shows the systematic small deviations from the power functions shown in A that are manifest by plotting the residuals from a regression Notice that the residuals are selfsimilar: The residuals from to 128 are a scaled-up version of those to 64, which are a scaled-up version This reflects the self-similar geometry of the transition matrix, also visible in the transition diagram shown in Fig The theory of stochastic counters provides a rationale for Weber’s law, and predicts the Weber fraction in terms of counter accuracy It is straightforward to compute this result approximately (see the Appendix): The Weber fraction may be stated as a coefficient of variation, which is predicted to be FIG (A) Mean counts registered for probabilities of successful transmission of p ϭ 90 (bottom curve) and p ϭ 95–.99 counters (middle to top curves) (B) Weber functions for counters of differing accuracy (C) Residuals from a regression to the 99 counter SCALAR COUNTERS 69 σ/µ ϭ [p(1 Ϫ p)] 1/2 for this stochastic counter The coefficient of variation is constant (Weber’s law), at a value predicted from the probability of a successful transition A p ϭ 95 counter will give rise to a Weber fraction of 0.22 Some validation of these calculations is found in the careful work of Kristofferson (1977, 1984) who showed that the Weber function shifts from one like the top curve in Fig 4B to one like the bottom curve after scores of sessions of practice for each datum The conformity of the model for a p ϭ 99 counter to his asymptotic data is shown in Fig 5, unique to a scale transformation (viz., the period of the pacemaker, imputed here as 13 ms) The doubling of the steps in the standard deviations at times that are powers of indicate that this must be a binary counter Wrinkles in Time There are other significant phenomena that may help test and constrain the theory of stochastic counters This article focuses on the small systematic deviations in temporal productions and discriminations noted by Collyer, Church, and associates (e.g., Collyer, Broadbent, & Church, 1994; Collyer & Church, 1998) These are remarkable and well-replicated data An example is given in Fig 6, drawn from the research of Collyer et al (1992) In this experiment humans were asked to synchronize their finger tapping with a metronome After 50 synchronization taps, they were to continue tapping at the same rate However, their intertap interval drifted significantly from that of the metronome If the continuation interresponse intervals (IRIs) are plotted against those of the metronome, the deviations are barely visible; but if the residuals from a linear regression are plotted, the deviations are systematic and substantial The deviations are vaguely reminiscent of those shown FIG Standard deviations of temporal discriminations (circles) and of a 99 counter driven by an 80-Hz pacemaker (curve) The curve and data were brought into alignment by assigning the period of 13 ms to the period of the pacemaker 70 PETER R KILLEEN FIG Systematic deviations in the residuals from a time-production task in Fig 4, but are not the same They cannot be the same, because those in Fig are from a model for interval representation/discrimination, whereas the task of Collyer and associates was one of production For this the theory must birth a new model Production A discrimination paradigm asks an observer whether the (count) representation of a particular stimulus (e.g., the number of pulses from a pacemaker associated with 750 ms), is different from some other count (perhaps a criterion, or a prior stimulus) The production paradigm asks a participant to respond at some regular period or after some latency The underlying assumption of both SET and TSC is that individuals set a criterion number of counts, and respond when their counter reaches that value The resulting IRIs are called hitting times Figure pictures the properties of hitting times driven by a stochastic counter Figure shows that the mean number of counts registered increases approximately as a power function of the input with exponent of approximately 1/log (2p) Hitting times are rougher functions of input than registration times (Fig 4), because the abscissae are fixed, so that all variance is associated with the ordinates The internal criterion count is always either on the low side or on the high side of a critical point: If your task is to count to 30 many times, you may make mistakes in the responses (hitting times), but you will probably not change your criterion from 30 to, say, 35 There may be significant variance associated with the hitting times for any criterion (reflected in the ordinates) but the criterion is constant Conversely, in the case of discrimination, variance is associated with both axes: A 35-pulse stimulus may fall below the 32-count critical point (due to prior errors) or SCALAR COUNTERS FIG (Left) Mean hitting times for various counters (Middle) Weber functions for productions based on counters of differing accuracy (Right) Residuals (observed–regressed) from a linear regression of the mean hitting time on the input of a 99 counter 71 72 PETER R KILLEEN above (risking catastrophe as the counter attempts 32) This additional source of variance in the abscissae smears the cliffs shown in Fig into the hills shown in Figs and The right panel of Fig shows some resemblance to the data in Fig The similarities are: (a) There is a spiky appearance; (b) The shortest values are overestimated The latter occurs because the mean hitting times are concave up functions of the criterion (i.e., the exponents are greater than 1) Any linear regression to such functions will have a negative intercept This is true for almost all of the empirical regressions to data reported from paradigms such as this (c) The slope down is gradual, and the slope up abrupt This is because of the potential for catastrophe at powers of 2, and the slowerthan-catastrophic growth of error between those critical points A linear regression thus shows a big error of underestimation at those points, and a gradual recovery up to the next such point These particulars suggest that we may be on the right track However, the match between model and data is not otherwise good Attention to two details may improve the coincidence of model and data (a) Whereas most pictures of these residual ‘‘oscillator signatures’’ by Collyer and associates show deviations from a linear regression, as those of Fig 7, the data in Fig show deviations from real time (b) Subjects were asked to set a new criterion, established by a new metronome beat, every few hundred responses Figure shows the results of a general model; but what is needed is a model for this particular experiment This was constructed by a tandem discrimination–production implementation of the theory N pulses from a regular pacemaker were fed to a stochastic counter, and the reading on the counter was noted This was done 50 times The average reading was imputed as the typical ‘‘representation’’ of those n pulses and was used as the criterion, C n An additional series of pulses were fed into this C n counter When they equaled C n , a response was scored (a ‘‘hitting time’’) Fifty of these hitting times were averaged to provide the critical data These average hitting times were subtracted from the metronome time, n, to give the comparable residuals Note that this paradigm has more verisimilitude than the counter statistics, as it permits the observer to calibrate his or her behavior to the task demands It does not matter what the nominal setting of the counter criterion is, as long as it appropriately compensates for stochastic error in the counter An error-prone counter that represented 100 as 70 on the average would yield a criterion of C n ϭ 70 when measuring 100; and the hitting times for a criterion of 70 on the same counter would in turn be close to 100 (Killeen & Taylor, 2000a, Equation 22) Thus discrimination and production distortions approximately compensate The results are shown in the left column of Fig Well, things have changed However, they have not necessarily improved The bottom left panel may come from the same family as Fig 6, but it is at most a kissing cousin, not an identical twin It is time to consider alternatives to the basic model SCALAR COUNTERS 73 FIG Simulation of the Collyer, Broadbent, and Church (1992) tap-timing experiment using stochastic counters with different accuracies The x-axis represents the number of seconds in the synchronization interval (divided by the period of the pacemaker) Thus, if the period is 25 ms, 16 pulses corresponds to an interval of 400 ms The residuals are output– input, and would be scaled by the period of the pacemaker to coincide with the data Each trial involved 50 synchronization intervals and 50 continuation intervals Each symbol derives from the continuation part of 100 such trials This process was replicated, giving two data for each condition RESET FAILURE The set-fail stochastic counter was a generalization of the binary counter to permit error in the setting of a bit This is not the only place where error may occur A bit may be set when its driving bit (its lower-order neighbor) starts to reset, but the reset of the driving bit may be incomplete (Haăusser, Major, & Stuart, 2001), or it may rebound to the on state Review the top of Fig For an accurate transition from to 2, the first stage must reset as it sets the second If it does not, both the second and first stages will be set, with the resulting count jumping to rather than Or, in the transition 74 PETER R KILLEEN from to 6, the first bit may fail to reset, leaving the count at Or, in the transition from to 8, all of the first three bits may activate their neighbors but fail to reset themselves, moving the count up, not to 8, but to 15 This reset-fail architecture is a natural complement to set-failure Call the probability of a reset r The counter will increment from to with probability r, and from to with probability Ϫ r Figure is the transition diagram Again, catastrophic error occurs at critical points— kϪ1 —where there is a small possibility of the count almost doubling, and an increasing (with count) probability of it incrementing more than it should This counter has the same type of self-similarity as the set-failure counter It has one additional property of interest: It is possible to get from one number to a higher one without traversing all the numbers in-between If r ϭ 90, then one time in 10 the counter will go from to 5, skipping 4; one time in 100 it will go from to skipping 4, 5, and It is as though when swimming laps in a pool, a swimmer anticipated the tens increment and forgot the unit’s decrement, so at turnaround her count went from 29 to 39 The statistical properties of reset-fail counters are shown in Fig 10 Consider first discriminations, which are mediated by an individual deciding whether a stimulus has given rise to a number of counts that exceed his or her threshold for saying long The top left panel shows that a given number of input pulses will often give rise to more counts than were input The mean number registered is an increasing function of input, well captured by an appropriate power function The deviations from a linear regression are shown in the panel below Because the registrations are concave-up, deviations from it are U-shaped functions Spikes occur at powers of 2, where there is a possibility of the counts doubling The standard deviation is an approximately linear function of input In the case of productions, the mean hitting times (top right panel) are concave-down functions of the criterion for responding This is because there FIG The transition diagram of a reset-fail stochastic counter After activating a downstream bit, the probability of a successful reset is r Note for example the transition from to 4, which requires that the first two bits reset, which happens with probability r If both fail to reset, the resulting count is (2 ϩ ϩ 2) SCALAR COUNTERS 75 FIG 10 Characteristics of counters in which bits reset with probability r ϭ 9, 94, and 98, after setting a downstream bit The left column shows the statistics for discriminations and the right for productions The residuals are deviations from linear regressions is some possibility of achieving a high count by a reset failure rather than by an honest input The residuals are inverted-U shaped, as expected They are not so extreme as is the case for set-failure counters This is because in the production mode a set failure counter is Sisyphean, with the possibility that the count will fail and fall to zero, whereupon the whole task, with the 76 PETER R KILLEEN same possibilities for failure, must be revisited all over again And possibly again And again There is a finite, if minuscule, probability that the task will never be completed, home never reached Reset-fail counters are Ozzian; the next click of a counter, as of someone’s heels, may carry the counter over the top, a top that had seemed so remote just a moment before The reset-failure model cannot be used directly with Collyer and associates’ (1992) data, but must be embedded in the same experimental protocol as the set-failure model When this is done, the middle column of Fig results The residuals have a very different character, both from the set-fail counters and—alas—from the data shown in Fig SET–RESET FAILURE In any real system it is unlikely that only errors of omission (set fail) or commission (reset fail) will happen; the causes for one will probably affect the other Therefore a model with more face validity would suffer both set failures and reset failures The resulting process will be dynamic, with chaotic properties (Killeen & Taylor, 2000b) Counting will sometimes yield larger numbers, and sometimes smaller numbers, than the things counted A set–reset failure counter with equal probabilities of set and reset failure was simulated in the paradigm of Collyer and associates’ (1992) experiment The results are shown in the right column of Fig Again the residuals have character, but not that of Fig Although with a squint the middle panel just might pass THIRD INNING The next move is to expand the database, to ensure that the data of Fig are not idiosyncratic, and after that, to investigate yet another failure mode for stochastic counters Additional Data Collyer et al (1992) reported the percentage residual bias (PRB) for several conditions of tap-timing for three participants The PRB equals 100 ϫ (IRI Ϫ Regress)/Regress, where IRI is the measured Inter-Response Interval, and Regress is the value predicted from a linear regression Figure 11 shows the median PRBs over conditions for each of these three subjects, along with smooth curves representing the average loci of the data The top panel is reminiscent of Fig Notice that in the top panel the low points occur at 350 and 700 ms, and in the bottom panel the high points occur at 400 and 800 ms This suggests a binary counter at work Counting Cascades Consider each stage in a counter to be an idealized neuron A single spike to it raises its generator potential partway to the threshold for firing, and a second spike raises it over threshold, causing it to fire and set the next neuron SCALAR COUNTERS 77 FIG 11 The percentage deviation between the measured intertap interval and those predicted by a linear regression 78 PETER R KILLEEN in the chain Other inhibitory input to the neuron may cause a set-failure, a situation treated in the first section of this paper Other excitatory input may cause the first spike to carry the neuron over the threshold for firing, causing a premature setting of the next stage This is new If the counter is at zero, a single pulse may set it to 1; or, in the case of a ‘‘preignition’’ may raise it over its threshold, causing it to fire and reset, in the process setting the second bit This is a double set error In the case where the counter was at, say, 1, a double set can no more than carry it farther over threshold No immediate harm done However, in setting the second bit a double set error has another opportunity to occur, carrying the counter to If the first bit were at 0, a double set will skip and carry the counter to In general, whenever a signal is sent to the next stage an opportunity for error arises, as the next bit may prematurely discharge Whenever a most significant bit double sets, it doubles the number of counts When this happens, there is a chance that the bit it sets also goes over threshold, doubling the count once again In the case of an input to a cleared counter, preignition of the first bit could carry it to 2; if the second bit preignites also, the count will double to Whenever the counter is incremented, there is a chance for a double set, and this possibility propagates through the downstream stages with decreasing probability Call the probability of a double set q An input to a counter registering will go to and stay there with probability Ϫ q (the second bit does not double set, with probability Ϫ q) It will go from to and stay there with probability q(1 Ϫ q) (the first bit double-sets but the second does not) It will go from to and stay there with a probability of (1 Ϫ q)q It will stop at a count of k with probability (1 Ϫ q)q k , with k limited only by the number of stages in the counter The term q k gives the probability of k successive errors; the term (1 Ϫ q) gives the probability that the string of failure stops at the bit in question The expected number of counts registered on a counter starting at zero and receiving a single input is the sum of the series (1 Ϫ q)(2q) i, from i ϭ to k, where k is the size of the counter in stages, or bits For an indefinitely large counter, the expectation is (1 Ϫ q)/(1 Ϫ 2q), q Ͻ This is not bad for small to moderate values of q: For q ϭ 10 the expected count is just 9/8 The expectation becomes extreme, however, as q approaches 5, where the count approaches infinity This is truly a counting cascade The above calculations started with a cleared counter and a single input Double set errors can occur from any starting position Figure 12 gives a flow diagram of the process To follow the action, it helps to refer to a binary representation such as the one shown in Fig The probability of an accurate increment to the next count is always Ϫ q This is because in a binary counter an accurate increment entails the setting of only a single bit (in this analysis resets are assumed error-free) With probability (1 Ϫ q)q j the count erroneously increments instead to k The numbers on the elbows of the flow chart give the values for k and j The exponent k gives the power of that the counter may mistakenly reach, and the exponent j gives the number of SCALAR COUNTERS 79 FIG 12 The flow diagram for double-set counters Normal transition from left to right occurs with probability Ϫ q at each stage, where q is the probability of a pulse prematurely carrying the next bit over its threshold for firing Thus upon a single input, the counter will move from to and stay there with probability Ϫ q; from to with the same probability, and from n to nϩ1 with that same probability With a probability p given by the box, the count instead will increase to k Where this leaves the count depends on the count with which it entered the box The numbers at the elbows in the flow diagram are the parameters k, j, where k is the power of to which it increases, and j is the power of q, which gives the probability Thus, will increase to (viz., 3) instead of if the second bit double-sets This happens with probability q It will stop at if the third bit does not misfire, which happens with probability (1 Ϫ q) There is a possibility that whenever a leading bit is set, either correctly or due to a double-set error, it also will misfire, setting the next power of 2, and so on ad infinitum This is represented by the recurrent flow diagram double-sets necessary to get there Once there, however, there is a possibility that the errors may propagate This is indicated in the flow diagram by incrementing both indices, and iterating the calculation This continues until k exceeds the size of the counter The probability of large errors decreases exponentially with their size (viz., as q j ) Once again the count achieved by double-set prone counters is approximately a power function of input Figure 13 shows other relevant statistics of double-set prone counters in the production mode The top panel shows the residuals from a power-function for a q ϭ 06 counter Compare it with Fig Residuals from q ϭ 02 and q ϭ 10 counters are similar The bottom panel shows that the standard deviations from this failure mode are approximately linear functions of the input value Once again, the counters are scalar The real test of this model is to embed it in the relevant experimental design and inspect the resulting residuals This was done for a sampling of parameters that yielded Weber fractions consistent with the data of Collyer and associates (1992) Fifty synchronizing inputs established a criterion, and then the machine produced 50 intervals, tapping when the count equaled or exceeded that criterion and then resetting and restarting the counter In this 80 PETER R KILLEEN FIG 13 (Top) The residuals from power-function regression to a double-set counter (Bottom) The standard deviations of the counts achieved on such counters as a function of input case, both set errors and double-set errors were possible Figure 14 shows the results The scale of the x-axes depends on the speed of the pacemaker, and here is a free variable The question is, can counter signatures, unique up to dilation along the x-axis, be found that coincide with the signatures shown in Fig 11? Finding a good match to the data shown in Fig 11 still proves difficult One successful example is given in Fig 15 The data from other participants were more recalcitrant This may be due to the lack of an efficient search algorithm, as there is a large search space—rates of each kind of failure, and period of the pacemaker—and the character of the residuals changes as a function of the range of data included in the regressions It is therefore difficult to automate the generation and testing of simulation templates, such as the one shown in Fig 15 In an analogous experiment, Crystal, Church, and Broadbent (1997) trained 10 rats to respond on a ramped interval schedule that started between 20 and 150 s, and changed by s after each reinforcement until they had SCALAR COUNTERS 81 FIG 14 The residuals from a linear regression to the productions of a set/double-set counter for a sample of parameter values This counter was trained and tested in a close simulation of the procedure of Collyer and associates (1992) The curve is a smoothed characterization of the average loci of the data 82 PETER R KILLEEN FIG 15 Data from observer HB, with the signature of a set/double-set counter superimposed The probability of each failure mode was 6%, and it was driven by a pacemaker with a period of 39 ms reached the limit, when the progression reversed direction and continued to change in 2-s steps They found similar patterns of residuals, both from a linear regression of average pause against interval length, and also from a regression of the standard deviations of pause duration against interval length This shows that Weber functions are subject to microdeviations similar to those found for the mean Examples of these were shown in Fig for the discrimination data of Kristofferson Crystal and associates found a significant negative correlation between mean residuals and standard deviation residuals The simulation used for Fig 15 also produced systematic deviations from Weber’s law, as expected These were also negatively correlated with the deviations around the mean performance (r ϭ Ϫ.80) Thus, the map to the statistics of these data is very good Other treatments of these residuals (e.g., Crystal, 1999) invokes machinery that embodies some of the key properties of stochastic counters (see the Appendix) SAVING THE BABY The set/double-set error model of counters is the best contender for describing the linear effects and the residuals of means (proportional timing), standard deviations (Weber’s law), and third moments (the skew of the counting and hitting time distributions) It is interpretable in terms of tonic levels of excitation or inhibition in the nervous system, and their behavioral concomitants Arousal manipulations affect time perception (Morgan, Killeen, & Fetterman, 1993; Penton-Voak, Edwards, Percival, & Wearden, 1996) Traditionally, this has been interpreted to be due to variations in the speed of the pacemaker However, the double-set error mode provides an alternative mechanism The distinction is testable: Shifts due to excitation or inhibition of the pacemaker should cause smooth, scalar shifts in the be- SCALAR COUNTERS 83 havioral measures, whereas shifts due to excitation of the pacemaker should cause jumps in the estimates of elapsed time (Fig 12) But, reset and double-set error modes sacrifice one of the signal accomplishments of the set-fail model, the fine accommodation of the data of Kristofferson shown in Fig 5? In theory they should not; the critical points, at which variance can explode, are still found at powers of the base Figure 16 shows Kristofferson’s data replotted, along with the trace of a set/doubleset counter with p ϭ q ϭ 0.001, driven by a 13-ms pacemaker (100 pulses correspond to 1300 ms) The map is as good as that shown in Fig Values of p ϭ q ϭ 01 provide an equally good fit to the data; however, they impute too large a standard deviation, requiring a substantial rescaling of the ordinates If the effects of pacemaker variance is small—as should be the case when a large number of pulses are summed (Gibbon, 1992; Killeen & Weiss, 1987)—we should be able to predict the left ordinate by multiplying the right ordinate by the period of the pacemaker In the present case, for a 13ms pacemaker, a standard deviation of four counts should correspond to a standard deviation of time discrimination of 13 ms/count ϫ counts ϭ 52 ms We see that this is approximately the case It is also possible that the period of the pacemaker is 26 ms, halving the count required, and leaving the correspondence between axes intact In this case the smallest ripple in the curve, near the ordinate and here not visible, would be magnified into the one above and to its right, and so on SCALAR COUNTERS Despite the name of his theory, perhaps the weakest part of Gibbon’s Scalar Expectancy Theory was his mechanism for achieving scalar timing FIG 16 Data from Kristofferson (1977) shown with the trace of a set/double-set fallible counter with a probability of both failure modes of 0.1% The curve is a running average of the simulations 84 PETER R KILLEEN Gibbon proposed three possible sources of scalar error (Gibbon, 1991; Gibbon & Church, 1984) One was a fluctuation of the speed of the pacemaker between timing epochs but not within timing epochs If the period varies randomly between intervals being timed, long counts would multiply this variance more than small counts, achieving precisely scalar timing However, for this scheme to be successful, it is important that the period not similarly vary during the epoch being timed, or the scalar effect will be washed out In standard timing studies such as the peak procedure, however, the time between intervals is negligible; yet it was here that all drift and variation in the pacemaker was to occur The timing interval itself, which might involve a large congeries of adjunctive and frustrative responses, allows much more time for the period to drift; but there it must hold constant This scenario predicts that Weber fractions will be an increasing function of the time between trials: If the drift is random, the Weber fraction will be proportional to the square root of the intertrial interval This source of scalar variability is clearly implausible A second source of variance is in the comparator The comparator plays the same role as the criterion in Fig If, instead of an absolute criterion there were proportional variability in the criterion, Weber’s law and scalar timing would obtain Gibbon’s use of ratio comparators and similarity comparators is tantamount to vesting the criterion with scalar error, as the denominator essentially multiplies the criterion, and with it any variance that it carries (But why should the criterion show scalar variance? Isn’t imputing it with ratio comparators begging the question?) If this is the only source of scalar variance, the psychometric functions will be true distributions rather than pseudo-distributions, symmetric on a linear axis, not approximately symmetric on a logarithmic axis (Killeen et al., 1997) A thorough comparison of this difference has yet to be made, although early results favor the former (Allan, this issue) A third source of scalar variability occurs if the error in the counter is proportional to the number of counts (i.e., as in Gibbon’s memory translation constant, k*) However, what is the mechanism for such translation error? It will not for, say, a count to be dropped or added with a probability of p This does not generate a scalar count: It is a Bernoulli process that generates Poisson timing (Killeen & Fetterman, 1988, Eq A7) SET provided no mechanism for k* Scalar counters, such as those constructed here, provide a mechanism for proportional counter error, both in setting the criterion and in estimating individual intervals If criteria are based on a weighted average of what has been recently reinforced, criterial variance will be smaller than, but proportional to, the variance of their input In particular, assume that the criterion is adjusted by the following linear update rule: Move the criterion from where it is 10% of the distance to the most recently reinforced interval (as represented by the counts registered) This is also called an exponentially weighted moving average (Killeen, 1981); its standard deviation σ C ϭ 0.23σ S In general, if β is the SCALAR COUNTERS 85 proportion by which an organism updates its criterion, the standard deviation of the criterion will be σ C ϭ [β/(2 Ϫ β)] 1/2σ S In turn, the variance of the stimuli, σ S —the number of counts registered for a given number of pacemaker pulses input—has been shown here to be scalar for stochastic counters The combination of perceptual variance due to the representation of the stimulus as a stochastic count and a diminished criterial variance as the criterion tracks the reinforced percepts is likely to provide the best description of temporal discriminations Stochastic counters thus provide a mechanism for scalar timing They reconcile the occasional finding of less-than-scalar growth of variance (Allan & Kristofferson, 1974): For accurate counters, there will be long plateaus of less than proportional increases in variability, punctuated by jumps which, overall, cause the Weber function to approximate proportionality with the interval timed They provide microdeviations from smooth functions consistent with those discovered by Collyer and his colleagues (e.g., Collyer et al., 1992) The presentation of probe stimuli will give rise to pseudo-distribution psychometric functions, such as those recorded by (Allan, 2001b) If the criterion is updated substantially, the result will be scalar timing It is this last result, the forceful and elegant presentation of it, and the scientific progress on understanding the perception and production of time it has fostered, for which we owe so much to John Gibbon APPENDIX: WEBER’S LAW Assume that the only source of failure is the catastrophic one to that can occur when moving up to powers of the base; this ignores all the lesser failures The probability of a successful transition to k is p k; the probability of a failure to is (1 Ϫ p)p kϪ1 (see Fig 2) The sum of these is p kϪ1 The relative probability of successful transition is then p k /p kϪ1 ϭ p; and of failure to is (1 Ϫ p)p kϪ1 /pkϪ1 ϭ (1 Ϫ p) Then the mean count expected at this juncture is the probability of success ( p) times the new count (2 k) plus that of failure (1 Ϫ p) times the new count in that case (0): p2 k ϩ (1 Ϫ p)0 ϭ p2 k The variance σ is the expectation of the squares ( p2 2k) minus the square of the expectations ( p 222k): p2 2k Ϫ p 222k ϭ p(1 Ϫ p)2 2k Therefore the standard deviation is σ ϭ k[ p(1 Ϫ p)] 1/2 Now calculate the coefficient of variation, the standard deviation divided by the mean number of counts registered, which is one form of the Weber fraction In this case it is σ/µ ϭ k[ p(1 Ϫ p)] 1/2 /2 k, or σ/µ ϭ [ p(1 Ϫ p)] 1/2 APPENDIX: CONNECTIONIST MODELS Crystal et al (1997) generated a systematic pattern of residuals using the connectionist model of Church and Broadbent (1990, 1991; for a different approach to the residuals, see Rosenbaum, 1998) The important part of this model is a series of oscillators with periods increasing as powers of 2, and with correlated variability in those periods It is transparent that each stage in a binary stochastic counter is a flip-flop, or oscillator, with the period of each stage approximately twice that of the one upstream The main difference in the models is the decision criteria In the connectionist model, the weights are conditioned, and responding begins when the average correlation among the weighted settings with time-through-trial exceeds a threshold In the stochastic counters presented here, it is assumed that the weights on each stage are wired to increase as powers of 2, and their weighted sum constitutes a count 86 PETER R KILLEEN The decision process is represented in Fig If the weights are conditioned, a single-purpose timer results, one that shows periodicities after omitted reinforcers (Killeen & Taylor, 2000a) Stochastic counters constitute a module that may be inserted in the connectionist theory to provide a testable mechanism for some of its features (Church, 1997) The conditioning histories that carry an organism from a single-purpose timer with weights tailored to a particular interval, to a timer of general utility, will form an interesting topic of study It has already been demonstrated that in the transition from one tailored timer to another steps occur, suggesting reconditioning of stages of a binary counter (Fetterman & Killeen, 1992; Meck, KomeilyZadeh, & Church, 1984) The details of such experiments may constrain possible structures and rules for conditioning binary counters REFERENCES Allan, L G (1999) Understanding the bisection psychometric function In P R Killeen & W R Uttal (Eds.), Fechner Day ’99 (vol 15, pp 204–209) Tempe, AZ: The International Society for Psychophysics Allan, L G (2002a) The location and interpretation of the bisection point The Quarterly Journal of Experimental Psychology, B, in press Allan, L G (2002b) Are the referents remembered in temporal bisection? Learning and Motivation, 33, 10–31 Allan, L G., & Gerhardt, K (2001) Temporal bisection with trial referents Perception & Psychophysics, 63, 524–540 Allan, L G., & Kristofferson, A B (1974) Judgements about the duration of brief stimuli Perception & Psychophysics, 15, 434–440 Church, R M (1997) Quantitative models of animal learning and cognition Journal of Experimental Psychology: Animal Behavior Processes, 23, 379–389 Church, R M., & Broadbent, H M (1990) Alternative representations of time, number and rate Cognition, 37, 55–81 Church, R M., & Broadbent, H A (1991) A connectionist model of timing In M L Commons, S Grossberg, & J E R Staddon (Eds.), Neural network models of conditioning and action (pp 225–240) Hillsdale, NJ: Erlbaum Church, R M., Broadbent, H A., & Gibbon, J (1992) Biological and psychological description of an internal clock In I Gormezano & E A Wasserman (Eds.), Learning and memory: The behavioral and biological substrates (pp 105–128) Hillsdale, NJ: Erlbaum 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455–468 Kristofferson, A B (1977) A real-time criterion theory of duration discrimination Perception & Psychophysics, 21, 105–117 Kristofferson, A B (1984) Quantal and deterministic timing in human duration discrimination In J Gibbon & L Allan (Eds.), Timing and time perception (Vol 423, pp 3–15) New York: New York Academy of Sciences Meck, W H., Komeily-Zadeh, F N., & Church, R M (1984) Two-step acquisition: Modification of an internal clock’s criterion Journal of Experimental Psychology: Animal Behavior Processes, 10, 297–306 Morgan, L., Killeen, P R., & Fetterman, J G (1993) Changing rates of reinforcement perturbs the flow of time Behavioural Processes, 30, 259–272 Penton-Voak, I S., Edwards, H., Percival, A., & Wearden, J H (1996) Speeding-up an internal clock in humans—Effects of click trains on subjective duration Journal of Experimental Psychology: Animal Behavior Processes, 22, 307–320 Rodriguez, P., Wiles, J., & Elman, J L (1999) A recurrent neural network that learns to count Connection Science, 11, 5–40 Rosenbaum, D A (1998) Broadcast theory of timing In D A Rosenbaum & C E Collyer (Eds.), Timing of behavior: Neural, psychological, and computational perspectives Cambridge, MA: MIT Press Rosenbaum, D A., & Collyer, C E (Eds.) (1998) Timing and behavior: Neural, psychological, and computational perspectives Cambridge, MA: MIT Press Received April 1, 2001 ... (bottom curve) and p ϭ 95–.99 counters (middle to top curves) (B) Weber functions for counters of differing accuracy (C) Residuals from a regression to the 99 counter SCALAR COUNTERS 69 σ/µ ϭ [p(1... critical point (due to prior errors) or SCALAR COUNTERS FIG (Left) Mean hitting times for various counters (Middle) Weber functions for productions based on counters of differing accuracy (Right)... to its right, and so on SCALAR COUNTERS Despite the name of his theory, perhaps the weakest part of Gibbon’s Scalar Expectancy Theory was his mechanism for achieving scalar timing FIG 16 Data