Perception & Psychophysics 1981,29 (5), 516·520

Notes and Comment The parallel-clock model: Replies to critics and criticisms

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HANNES EISLER University ofStockholm, S-113 85 Stockholm, Sweden

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The parallel-clock model was presented 5 years ago (Eisler, 1975). It deals with the behavior of observers required to judge some relationship between two sequentially presented durations, and accounts well for data obtained in a number of scaling experiments. Its generality has been demonstrated by application to scaling data from more than 100 studies, resulting in a collection of about 500 exponents of power functions for duration (Eisler, 1976). Although the 1975 study focused on scaling rather than on discrimination of duration, it showed that a number of peculiarities of duration discrimination data could be explained by the parallel-clock model. As demonstrated recently (Eisler, 1981), the model, in fact, accounts very well for discrimination data. This is particularly interesting, since it is unusual for the same model to be applicable to these two types of task. New models can expect criticism. The aim of the present paper is thus to refute the criticism leveled at the parallel-clock (PC) model since its introduction. Since most of it seems to derive from misunderstandings, I shall begin by restating the model itself. The Parallel-Clock Model When an observer is required to deal with some relationship between two successively presented durations, it is usually assumed that the first duration is stored in memory and that this memory is then compared with the second duration. In the PC model, on the other hand, two sensory registers are assumed. Typically, the first accumulates the total subjective duration, from the onset of the first duration to when the required response is given, and the other accumulates the duration from the start of the second, again until the response is given. I The subject deals with two quantities: the difference in content between the two registers and the content of the second register. These two quantities are compared continuously until the moment of the response, which depends on the task. If the observer is required to reproduce the first This investigation was supported by the Swedish Council for Research in the Humanities and Social Sciences. Requests for reprints should be directed to Hannes Eisler, Department of Psychology, University of Stockholm, Box 6706, S-11385 Stockholm, Sweden.

Copyright 1981 Psychonomic Society, Inc.

516

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Onset of first duration

Offset of first

and onset of second duration

- - - Physical

Offset of second durat ion

time - - -

Figure 1. Subjective duration as a fundion of physical duration according to tbe parallel-clock model. The upper curve indicates tbe accumulation of the total duration (first + second); note that it does not terminate at tbe offset of the first duration. Tbe lower curve indicates tbe accumulation of tbe second duration. The twopointed arrows sbow the compared quantities, tbe difference between total and second subjective duration, and tbe second subjective duration, respectively. For tbe middle pair of arrows, first and second durations are experienced as being equal; for the left pair, it is the first duration, and for the rigbt, the second tbat is experienced as longer.

duration, for example, she will'terminate the second duration (by pressing a button, say) when these two quantities are equal. Figure 1 will help to clarify this. The upper curve shows the total subjective duration, and the lower, the subjective duration of the second stimulus. Denoting the total subjective duration by \liT and the second subjective duration by \liz, the two quantities mentioned above are \liT - \liz, represented by the vertical distance between the two curves in Figure I, and 'Pz, the ordinate of the lower curve at the same abscissa. A "comparator" is fed with \IIT\liz from one side, and with \liz from the other. At the onset of the first duration, the comparator is strongly "deflected" to one side (cf. a galvanometer). The deflection starts to decrease at the onset of the second duration, passes through zero when \liT - \liz equals \liz (the middle vertical line in Figure 1), and subsequently increases on the other side. The passage through the zero point indicates (subjective) equality of the first and second durations to the observer. In this way, the comparator is supposedly used as a null instrument, conferring a quality on the task of duration reproduction that differs from, say, that of halving durations, for which task the amount of "deflection" has to be quantified. For quantitative treatment, this process model is 0031-5117/81/050516-05$00.75/0

NOTES AND COMMENT specified by introducing Stevens' power function 1jJ:. a(~-~o)f3, where ~ is physical time in, say, seconds and a, (3, and ~o are constants, viz., the scale unit, the exponent, and the subjective zero, respectively. For the case of reproduction of a time interval, we thus have (1a) that is, (1) and, accordingly, in physical terms, where ~1 is the to-be-reproduced duration (the standard) and ~2 the reproduced duration (the variable), with a set = 1,

Of course, the use of " + " and" - " signs should not be taken literally, as real digital operations. They simply serve to map a process in the observer using mathematical symbolism. The + sign between ~1 and ~2 means that the register is not interrupted at the offset of the first duration (more about that below), and the term -~o denotes that the registers do not start accumulating before the real-time moment, ~o. The process described by Equation la is thought to be achieved by a kind of analog-coupling. It would not be difficult to devise, say, an electrical circuit achieving just this, but I do not think that such a wiring diagram would promote understanding. It follows that the observer has access only to (her) subjective duration, although of course she does also have access to temporal point-events for which there is no difference between the physical and the subjective (apart from a possible small displacement on the time axis). The point-events can be external, such as on- and offset of the time intervals, or internal, such as zero deflection of the comparator. One problem that needs discussing is the behavior of the register during the interstimulus interval (lSI, the time interval between the offset of the first duration and the onset of the second). Assuming that the register works as a counter, accumulating subjective duration units, I hypothesize that the counter is in operation during the lSI too, but that it accumulates "zeros" rather than duration units during this realtime period, which is bounded by point events (offand onset of first and second durations, respectively). The accumulation of zeros is assumed to take place only when there is more subjective time to accumulate, that is, during an interruption. The problem of the subjective zero, ~o, can be dealt with on different levels. On a purely formal level, Stevens' power law, including ~o, was derived from reproduction data (Eisler, 1974, 1975). On the neurophysiological level, it can be hypothesized that an ongoing inhibition is to be counteracted, requiring

517

a certain strength of stimulation. On a level more directly connected with psychophysical experiments, ~o seems to be related to the absolute threshold (Ekman, Eisler,& Kunnapas, 1960). For the continuum of time, in particular, ~o might be a "deterministic internal delay" (Kristofferson, 1976), perhaps simply the afferent latency, that is, the time interval between the onset of the stimulus and the onset of the percept (Kristofferson, 1976, 1977). However, I do not conceive of the mechanism producing ~o as a "clock," since it, rather than "accumulating" duration, results in a fixed time interval. I would prefer to compare the mechanism functionally with the adjustment of the sensitivity of the annulospiral receptors of muscle spindles by efferent gammainnervation (see, e.g., Matthews, 1972).

Parallel Processing The name "parallel-clock model" derives from the assumption of two sensory registers ("clocks") running simultaneously. Curtis and Rule (1977) carried out two experiments in which observers were required to estimate the total duration of pairs of simultaneously presented durations and the average duration of pairs of successively presented durations. Their results show that the "composition rules" defining the way in which duration information is combined are different for the two tasks. The PC model, they argue, implies parallel processing even with successive presentation of the durations. Were this correct, simultaneous presentation, which necessarily requires parallel processing, should not entail different composition rules. Disregarding differences of task and response format between Curtis and Rule's experiments and those on which the PC model was based, their "simultaneous presentation" experiment is, nevertheless, inapplicable to the PC model. It is true that the PC model does require parallel processing, but not all parallel processing is pertinent to the PC model. The point of the PC model is that, at any point in time, the quantities IjJT -1jJ2 and 1jJ2 are available. But Curtis and Rule started the two durations simultaneously, and accordingly, until the moment of the response, only the longer duration is in progress, and thus available. All that remains of the shorter duration is presumably the memory. Had the onset of the shorter duration been chosen so that both durations terminated simultaneously instead, the experiment might have had some bearing upon the PC model. The Validity of Ratio Setting Experiments In a recent comment, Allan (1978) contested the finding (Eisler, 1975, 1976) that time perception, rather than being veridical, follows a power function with an exponent of about .9. She bases her conclusion on diverse findings from earlier durationdiscrimination studies, on Kristofferson's (1977) realtime criterion model, and on lack of invariance of

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EISLER

the exponent of the power function of duration obtained from ratio-setting data. The first objection should be obviated by the results obtained when her own duration discrimination data (Allan, 1977) were treated in accordance with the PC model (Eisler, 1981). The model yielded an excellent fit, all but one of the 17 exponents computed falling below unity, with a mean of .74. I shall return to Kristofferson's real-time criterion model in the next section. That leaves only the main point in Allan's argument, her ratio-setting data. (In a ratio-setting task, the observer is required to set a variable stimulus to a prescribed ratio of the standard stimulus.) I agree that equality between the ratio prescribed by the experimenter and the subjective ratio corresponding to it is a necessary condition for the computation of a valid exponent. I likewise concede that this condition is not always met-although with the exception of duration reproduction, which formally corresponds to a ratio of unity. As I pointed out in the original paper (Eisler, 1975) and attempted to make clear in the section "The parallel-clock model" above, in a duration-reproduction task, the observer functions as a null instrument with no opportunity to deviate from the prescribed ratio. Allan's data (1978, Table 3) also show excellent agreement between the exponents obtained from the two reproduction experiments for three of the four observers, and fair agreement for the fourth. Accordingly, exponents obtained from reproduction should be correct, whereas data based on other ratios mayor may not agree with the former. The doubling data from Allan's (1978) observer S. K. constitute an interesting case. Whereas this observer's exponent from reproduction is about .78, doubling yielded an exponent of only .47. However, Figure 2 in Allan (1978) clearly demonstrates that S. K.'s data for the two reproduction tasks and the doubling task coincide within random errors. In this situation, one has the choice of considering the PC model invalid-or of assuming that the observer, rather than following a difficult instruction, relapses into an easier task, reproduction. Assuming the latter to be the case, an exponent of .80 is obtained, in excellent agreement with the two exponents deriving from reproduction (.81 and .74). The same relapse phenomenon seems to have occurred with one or two of my observers in their halving task (Eisler, 1975, Table 5, Observer 3 and possibly Observer 10), thereby explaining the low correlation of .14 between the exponents obtained from reproduction and halving over 12 observers, a result mentioned by Allan (1978, 1979). Since observers when relapsing typically do so into easier, rather than more difficult, tasks,' I judge this finding as highly supportive of the PC model in general and of the null-instrument analogy in particular.

The Real-Time Criterion Model

The experimental paradigm with which Kristofferson (1977) investigated his real-time criterion model can be characterized as a many-to-few single-stimuluscomparison design with four stimulus durations and two response alternatives (long, short). The model successfully predicted a number of features of the data, and my aim here is to demonstrate that the most important of these likewise follow from the PC model. As Kristofferson correctly points out, his results do not require interval-measures of duration. This holds for the PC model, too, if the observer takes into account only the direction of the "deflection of the comparator" at the moment of response. In applying the PC model to an apparently singlestimulus design, we have to make the basic assumption that an observer is unable to carry out the required task, but, instead, performs according to forced choice. As described in Eisler (1975), at the nth trial, one sensory register contains the subjective duration corresponding to the sum of the durations presented at trials n - 1 and n, and the other register concerns the subjective duration corresponding to the duration presented at trial n. After the response is given, the first register is cleared and, during trial n + 1, accumulates the duration presented then. At the same time, this duration is accumulated on top of the previous one in the second register. In this way, the two registers contain, alternately, the "total" duration (comprising the previous and the present trial) and the second (present) duration. This presupposes that the intertrial interval does not enter into the registers, as suggested in Eisler (1975), described above for the lSI, and supported by the findings by Michon (1977). In order to adapt Equation 2, which describes a reproduction task, to a discrimination task, we take the step from a deterministic to a stochastic model using Thurstone's law of pair comparisons (cf. Eisler, 1981). We assume that the probability p of the response "short" is given by the relation

where
NOTES AND COMMENT able to me.) The case for which this outcome is most unexpected for the PC model is a sequence when the two critical stimuli are the same, that is, the sequences +I,n-I - +I,n and +4,n-1 - +4,n. Let us first look at +1. Unlike Kristofferson, who assumes that the observer's training affects an internal standard equal to the mean of the four durations, I propose that the observer adjusts her subjective zero, +0, appropriately, which here means making it equal to the shortest stimulus, We thus obtain

+,.3

(+I,n-I

+ +I,n -

+01 -

2(+I,n - +01

= (+, ++.-+,t-2(+,-+.i =

+/J_ O = zo,

,/3

(4)

z=-. o

+/J,

We see that (I) z is positive and (2) when 0 ~ as is typically the case (see, e.g., Eisler, 1981), z is large. Thus, the response "short(er)" is practically ensured. A rough estimate from Kristofferson's (Note 2) data yields z = 10.4. For we obtain the following:

+.,

(+4,n-1

+'4,n-+ol - 2(+4,n-+o)l1 = (2+. -

+0)11 - 2(+. -

+01 =

zoo

To obtain the response "long," z has to be negative, yielding

which can be rearranged into 2+. - +0

< 21/ fJ •

+. - +0

Solving this inequality for {J yields the following condition (5)

which is satisfied for small {J or small +0 {and (J <1). The small value of 0 again yields a high absolute value of z, thereby practically ensuring the "long" response. An estimation from Kristofferson's data for the +4,n-1 - +4,n sequence gives z = -5.4. (2) The "short" response is time-locked to the offset of the duration. Look at Figure I. The left vertical line illustrates this case, where 4JT - 4Jn exceeds 4Jn . This means that at the moment of the offset of the duration, the "hand of the comparator" is still deflected. As long as the zero point is nOI yet reached,

519

which implies that the first (previous) subjective duration is longer, the response "short" (for the actual duration) can be given immediately at the offset. This is also in accordance with the reaction-time latency distribution demonstrated empirically by Kristofferson (1977).

(3) The "long" response is time-locked to the onset of the duration and can be given before its termination. The response "long" is given as soon as the hand of the comparator passes through the zero point, that is, when 4JT - 4J n = 'lin. This not only means that the response can be given before the termination of the duration, in agreement with Kristofferson's (1977) finding, but also that the response, being dependent on the previous duration, is not perfectly time-locked. An estimation based on Kristofferson's (Note 2) data shows, however, that the variation is negligible. Furthermore, a time estimation latency distribution is expected for this case, in agreement with Kristofferson's (1977) findings. To conclude this section, I would like to point out that I consider the case of the real-time criterion model vs. the PC model far from closed. There are a number of quantitative predictions made by Kristofferson which I did not investigate with the PC model, mainly because of a scarcity of data. The main features are, however, well explained by the PC model. Conclusion The parallel-clock model accounts for both scaling and discrimination data. This is an appealing property, since it unites two different theoretical approaches and experimental paradigms. The four different formulations of the model in the original paper (Eisler, 1975) obviously failed to convey my ideas fully, since the objections raised against the model seem mostly to be based on misunderstandings. The present paper is an attempt to remedy these shortcomings. REFERENCE NOTES I. Hellstrom, A. Personal communication, July 1980. 2. Kristofferson, A. B. Unpublished data, May 1977.

REFERENCES L. G. The time-order error in judgments of duration. Canadian Journal of Psychology, 1977, 31, 24-31. ALLAN, L. G. Comments on current ratio-setting models for time perception. Perception & Psychophysics, 1978, 24, 444-450. ALLAN, L. G. The perception of time. Perception & Psychophysics, 1979,16,340-354. CURTIS, D. W., & RULE, S. J. Judgment of duration relations: Simultaneous and sequential presentation. Perception & Psychophysics, 1977, 22, 578-584. EISLER, H. The derivation of Stevens' psychophysical power law. In H. R. Moskowitz, B. Scharf, & J. C. Stevens (Eds.), Sensation and measurement, Dordrecht, Holland: D. Reidel, 1974. ALLAN,

520 EISLER,

EISLER H. Subjective duration and psychophysics. Psychological

Review, 1975. 82,429-450. EISLER,

H. Experiments on subjective duration 1868-1975: A

J. A. Holes in the fabric of subjective time: Figureground relations in event sequences. Acta Psychologica, 1977, 41, 191-203. .

MICHON,

collection of power function exponents. Psychological Bulletin, 1976,83,1154-1171. H. Applicability of the parallel-clock model to duration discrimination. Perception & Psychophysics, 1981,29,225-233. EISLER, H.. & ROSKAM, E. E. Multidimensional similarity: An experimental and theoretical comparison of vector, distance, and set theoretical models. I. Models and internal consistency of data . .4cta Psychologica, 197"7,41,1-46. EKMAJI:, G.. EISLER. H .. & KCSNAPAS, T. Brightness scales for monochromatic light. Scandinavian Journal of Psychology, 1960,1,41-48. KRISTOfFERSOS, A. B. Low-variance stimulus-response latencies: Deterrnimsuc Internal delays? Perception & Psychophysics, 1976,20,89·100. KRISTlIFFERSO:-;. A. B . A real-time criterion theory of duration discrimination Perception & Psychophysics, 1977,21,105-117.

EISLER,

P. B. C. Mammalian muscle receptors and their central actions. London: Edward Arnold, 1972.

MATTHEWS.

NOTES I. From a more everyday point of view, one could describe this process as the observer's dealing with two point events, the onsets of the two durations. In considering the two time intervals, what she has at hand at any moment is "how long ago" the first and the second point event, respectively, took place. This way of looking at the PC model was suggested by Ake Hellstrom (Note I). 2. Another instance of observer's performing a simpler task rather than the more demanding one they were Instructed 10 perform is that described in Eisler and Roskam (1977, p. 7) in which the observers estimated similarities instead of commonality ratios. 3. As a matter of Iact , this assumption is an outcome from my analysis of Krisrof'Iersons (Note 2) data.

(Received for publication August 14. 1980; accepted March 17, 1981.1

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