SET and Human Timing

Abstract

SET was developed in several different forms. One of these was an information-processing account, shown in Fig. 3.1, which bears a striking resemblance to Treisman’s model from 1963. Stimuli and other events are timed by a pacemaker-accumulator internal clock similar to that proposed by Creelman (1962) and Treisman (1963), but the SET model involved more than just an internal clock. Two memory stores were proposed. One was a working memory, intended to retain time representations temporarily, and reflecting, more or less faithfully, the contents of the accumulator. In fact, some versions of SET combine the accumulator and working memory. There is, however, another memory store, one which retains references or standards which are used for a number of trials or a whole experimental session. SET’s treatment of data from animal experiments is discussed in greater detail in Chap. 9, but the difference between the memory stores proposed by SET can be nicely illustrated by a simple procedure with animals.

Simple Mathematics of Pacemaker-Accumulator Clocks

In this section, I provide some simple mathematics of a pacemaker-accumulator clock like that proposed by SET, and in particular, I want to illustrate what might be expected if experimental manipulations are claimed to change pacemaker speed or alter switch processes. The sole focus of interest here is the number of clock ticks accumulated in various conditions, which is considered to represent the “raw material” for time judgements.

Suppose that we have a pacemaker that ticks on average at r ticks per second. When an event to be timed begins, the switch closes with some non-zero latency l _c , so pulses begin to flow, and when the event to be timed ends, the switch opens again, with some non-zero latency l _o . This means that the effective duration of the event or stimulus lasting t seconds is

$$ t+{l}_c-{l}_o $$

(3.1)

such that the total number of ticks accumulated, P, is just

$$ P= r\left(t+ {l}_c\hbox{--} {l}_o\right) $$

(3.2)

To simplify, suppose that l _c  − l _o  = d, in that d represents the difference between the closing and opening latencies of the switch, and d can be negative, of course, if the opening latency is greater than the closing latency. This means that

$$ P=r\left(t+ d\right) $$

(3.3)

As t is varied between conditions, the number of accumulated ticks, P, is a linear function with slope r and intercept rd.

Suppose that a claim is made that the pacemaker rate has been changed by some manipulation. How should P change? Suppose that we have two pacemaker rates, r ₁ and r ₂, and we can arbitrarily define their relation as r ₁ > r ₂. Now, we use Eq. (3.3) to derive P ₁ and P ₂, the number of ticks accumulating with pacemaker rates r ₁ and r ₂.

From Eq. (3.3), P ₁ = r ₁(t + d) and P ₂ = r ₂(t + d), but we are probably interested in the difference between the number of ticks accumulated in the two cases, P ₁ − P ₂. This is

$$ {P}_1-{P}_2=\left(t+d\right)\left({r}_1\hbox{--} {r}_2\right) $$

(3.4)

Equation (3.4) shows that the difference between the number of ticks accumulated is dependent on the difference between the rates of the pacemakers (r ₁ − r ₂), but also on the quantity (t + d). If d is constant as the time, t, to be judged changes, then the quantity (t + d) is dependent on t. As t becomes larger, the difference P ₁ − P ₂ also increases, and so the effects of the changing pacemaker speed should be greater at longer times than those that are shorter.

For readers who do not like algebraic solutions, a numerical example may help. The numbers used here are completely imaginary, and employed simply for purposes of illustration. Suppose that it takes 20 ms to close the switch to start timing, but 40 ms to open it again to stop, and so d = 40 − 20 = 20 ms. This adds 20 ms to the effective duration of the stimuli. To illustrate, suppose we have event durations of 200, 400, 600, 800, and 1000 ms, and two pacemaker rates, 100 and 150 ticks per second. The number of ticks accumulated is shown in Fig. 3.9. It is clear that the pacemaker rate difference manifests itself in a difference in slope in the conditions which are compared. The difference in the number of ticks accumulated between the two conditions increases as the duration timed lengthens: for example, the difference is 11 ticks at 200 ms but 51 at 1000 ms.

If we have a range of t values, then the manner in which the difference in pacemaker speed is manifested in behaviour depends on the task required. If a verbal estimation procedure is used, and we assume that the number of ticks accumulated is directly scaled into estimates, then the pacemaker speed difference should manifest itself as a difference in slope when the mean estimates from different t values with putatively different pacemaker rates are plotted against duration, with the higher pacemaker speed giving longer estimates, like the results shown in Fig. 3.12. Some experimental data support this contention (e.g. Penton-Voak et al., 1996; see Fig. 3.7). In contrast, if a method like bisection is used, then the difference between bisection points between the conditions reflects pacemaker rate, so this difference should increase as the stimuli in the bisection tasks increases in length. For example, if we have a 200/800-ms and a 400/1600-ms bisection pair, the difference between the bisection points derived from the conditions with different pacemaker speeds should then be greater in the 400/1600-ms than the 200/800-ms case. Again, there are data suggesting that this is true (Droit-Volet & Wearden, 2002). For another example using a production method, see Burle and Casini (2001; data in their Fig. 3.7, p. 202).

Fig. 3.12

Number of clock ticks accumulated for two different pacemaker rates (100 and 150 ticks per second) for event durations from 200 to 1000 ms. See text for other details

In contrast to changes in pacemaker speed, suppose we have some manipulation which changes the latencies to start and stop timing, d in our equations above. This change in start/stop latency is not dependent on the interval timed, and so should have a constant effect regardless of the duration to be judged. In terms of plots such as that in Fig. 3.9, this would manifest itself in a change in intercept rather than slope, or a constant difference in bisection point, irrespective of the range of intervals used.

This is easy to demonstrate numerically. Suppose that d, the difference between latencies to stop and start timing, changes from 20 ms, as in our example above, to 50 ms. If the pacemaker rate is 100 ticks per second, the effective duration of the stimuli timed then increases by 50 ms, or 5 ticks. Therefore, 25 rather than 22 ticks represents 200 ms, and 105 rather than 102 ticks represents 1000 ms. When the number of ticks accumulated is plotted against duration, the lines from the two conditions compared are parallel. Some studies report changes in condition, usually involving “attentional” manipulations, which show such constant effects. See, for example, Droit-Volet (2003).

In general, then, changing the pacemaker speed produces multiplicative effects on time judgements, such that effects are larger at longer times than shorter times, whereas changing the balance of start/stop latencies by changing switch processes produces constant effects, independent of the interval timed. Given that the idea of an arousal-sensitive pacemaker has been a popular one, multiplicative effects are sometimes referred to as being based on “arousal”, whereas constant effects are based on “attention” (see Burle & Casini, 2001, for discussion). However, some theories of attention in timing permit multiplicative effects, and so are difficult or impossible to distinguish from arousal effects, as discussed in Chap. 5.

The pacemaker of the internal clock is usually assumed to be a Poisson timer, in that ticks are emitted at random with a constant average rate. With this type of timer, the number of ticks grows linearly with real time, but it is the variance which increases linearly with real time rather than the standard deviation. This means that a relative measure of time such as the coefficient of variation (standard deviation/mean) will decrease as the time intervals judged lengthen, violating scalar timing. However, an underlying Poisson timer can be reconciled with scalar timing in various ways. Perhaps the easiest is to assume that the pacemaker is a Poisson timer, but that the pacemaker rate varies from trial to trial, and on each trial takes a value sampled from a Gaussian distribution, with a mean and some coefficient of variation. Such a process will generate average pacemaker output with scalar properties. Other ways of reconciling scalar timing with an underlying Poisson process are discussed in Chap. 9.

Poisson timers also have the property in which the standard deviation of the number of ticks produced grows as the square root of the pacemaker rate. This means that faster pacemakers produce relatively lower variability: the variability is a smaller fraction of the mean as the pacemaker rate increases. The implications of this property for measures of behaviour are discussed in Wearden and Jones (2013).