Time and memory: Towards a pacemaker-free theory of interval timing

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR
1999, 71, 215–251
NUMBER 2 (MARCH)
TIME AND MEMOR Y: TOWARDS A
PACEMAKER-FREE THEOR Y OF INTERVAL TIMING
J. E. R. STADDON AND J. J. HIGA
DUKE UNIVERSIT Y
A popular view of interval timing in animals is that it is driven by a discrete pacemaker-accumulator
mechanism that yields a linear scale for encoded time. But these mechanisms are fundamentally at
odds with the Weber law property of interval timing, and experiments that support linear encoded
time can be interpreted in other ways. We argue that the dominant pacemaker-accumulator theory,
scalar expectancy theory (SET), fails to explain some basic properties of operant behavior on inter-
val-timing procedures and can only accommodate a number of discrepancies by modifications and
elaborations that raise questions about the entire theory. We propose an alternative that is based on
principles of memory dynamics derived from the multiple-time-scale (MTS) model of habituation.
The MTS timing model can account for data from a wide variety of time-related experiments: pro-
portional and Weber law temporal discrimination, transient as well as persistent effects of reinforce-
ment omission and reinforcement magnitude, bisection, the discrimination of relative as well as
absolute duration, and the choose-short effect and its analogue in number-discrimination experi-
ments. Resemblances between timing and counting are an automatic consequence of the model. We
also argue that the transient and persistent effects of drugs on time estimates can be interpreted as
well within MTS theory as in SET. Recent real-time physiological data conform in surprising detail
to the assumptions of the MTS habituation model. Comparisons between the two views suggest a
number of novel experiments.
Key words: timing, clock, habituation, recall, Weber law, choose short, pacemaker
There is a developing consensus that inter-
variability (Weber law) problem, but BeT
val timing in animals is driven by a discrete
solves it in a less ad hoc (and more easily dis-
pacemaker-accumulator mechanism that
provable) way than SET (Bizo & White, 1995,
yields a linear scale for encoded time (e.g.,
1997). We focus on SET to limit the length
Gibbon, 1991). But these mechanisms are
of this paper, because some critiques of BeT
fundamentally at odds with the Weber law
already exist (e.g., Church, Meck, & Gibbon,
property of interval timing, and experiments
1994; Gibbon & Church, 1992) and because
that support linear time can be interpreted
in terms of citations and numbers of pub-
in other ways. In this article we first review
lished papers, SET is by far the most popular
the experimental and theoretical evidence
theory of interval timing. The review con-
for the dominant pacemaker-accumulator
cludes that SET fails to explain some basic
theory, scalar expectancy theory (SET; Gib-
properties of operant behavior on interval-
bon, 1977). This review does not deal with
timing procedures and can only accommo-
the major competitor to SET, the behavioral
date a number of quantitative discrepancies
theory of timing (BeT: Killeen & Fetterman,
by modifications and elaborations that raise
1988). Both theories confront the Poisson
questions about the entire theory. The sec-
ond part of the paper suggests an alternative
approach based on known principles of mem-
We thank Melissa Bateson, Alex Kacelnik, Warren
Meck, and their timing research group for several helpful
ory dynamics. This alternative lacks the for-
discussions of the issues raised in this paper. Juan Delius,
mal analytic base of SET, but is pacemaker
John Gibbon, and Peter Killeen also commented on ear-
free, is simpler in concept, and addresses a
lier versions of the present paper. A summary of the main
wider range of data.
argument was presented at the November 1997 meeting
of the Psychonomic Society in Philadelphia. We are grate-
ful to the National Science Foundation, the National In-
SCALAR EXPECTANCY THEORY
stitutes of Mental Health and Drug Abuse, and the Al-
exander von Humboldt Foundation for research support.
The scalar expectancy theory of timing
J. J. Higa is now at Texas Christian University.
(Gibbon, 1977; Gibbon & Church, 1984;
Address correspondence to J. E. R. Staddon, Depart-
Treisman, 1963) has provided a valuable
ment of Psychology: Experimental, Duke University, Box
90086, Durham, North Carolina 27708-0086 (E-mail:
framework for the study of interval timing in
staddon@psych.duke.edu).
animals. The theory has motivated and or-
215

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216
J. E. R. STADDON and J. J. HIGA
ganized the majority of recent behavioral ex-
total is compared with the current total, and
periments on interval timing (cf. Gibbon,
when the difference falls below a threshold
1991). It has also provided suggestive insights
(which may also vary), responding at a steady
into pharmacological effects (e.g., Meck,
rate begins. There is not yet consensus about
1996). We believe that two features of SET
the learning process: how many values are
may well be retained by future theories of in-
stored in reference memory, how these are
terval timing: the idea that the current time
selected for comparison, and so forth (Brun-
estimate and the memory for times rein-
ner, Fairhurst, Stolovitsky, & Gibbon, 1997).
forced in the past follow independent laws;
The credit-assignment problem—how does
and the notion that behavior is driven by
the system ‘‘know’’ what stimulus to use to
some kind of comparison between current
reset the accumulator (i.e., how does it iden-
and remembered time of reinforcement.
tify the time marker?)—is left open by SET.
Nevertheless, we will argue that despite its
We will see in a moment that a related prob-
formal structure and proven usefulness, SET
lem—how does the system recognize a tri-
is unnecessarily elaborate and redundant,
al?—is also not addressed by SET.
and it faces empirical shortcomings that are
The pacemaker concept has always been
sufficient to motivate the search for an alter-
troubling, because the properties of real tim-
native.
ing are fundamentally at odds with it. The
SET was devised to explain two things: (a)
problem is that timing with a pacemaker-ac-
that steady-state measures of time discrimi-
cumulator implies greater relative accuracy at
nation such as wait time (or, more precisely,
longer time intervals. If there is no error in
break point; Schneider, 1969) on fixed-interval
the accumulator, or if the error is indepen-
(FI) schedules or peak-rate time (on the peak
dent of accumulator value, and if there is
procedure) are proportional to the to-be-
pulse-by-pulse variability in the pacemaker
timed inter val (proportional timing; Dews,
rate, then by the law of large numbers, rela-
1970); and (b) that the standard deviations
tive error (standard deviation divided by
of such dependent measures are proportion-
mean, coefficient of variation) must be less at
al to their means (e.g., Catania, 1970; Stad-
longer time intervals. This relative improve-
don, 1965). The latter property is just We-
ment with absolute duration is independent
ber’s law applied to the dimension of time
of the type of variability in the pacemaker. In
(Weber timing). In the SET context it is termed
fact, coefficient of variation is approximately
scalar timing (Gibbon, 1977).
constant (Weber’s law, the scalar property)
In fact, there is some confusion between
over a limited range of times. At longer times
the terms scalar and proportional in the liter-
the coefficient of variation tends to increase
ature, which is why we reserve the term pro-
(rather than decrease) with the duration of
portional timing for any linear relation be-
the timed interval (e.g., Gibbon, Malapani,
tween an independent temporal variable
Dale, & Gallistel, 1997; Zeiler, 1991). Gibbon
(like interfood interval) and a dependent var-
(1977) was aware of this problem early on,
iable (like wait time or peak time), and scalar
but chose to deal with it in ways that preserve
(Weber law) timing for the constant ratio be-
the pacemaker-accumulator properties of
tween the mean and standard deviation of a
SET.
dependent temporal variable.
In an exploration of ways to reconcile the
The essence of scalar expectancy theory is
pacemaker-accumulator idea with Weber’s
straightfor ward: A Poisson-variable ‘‘pace-
law, Gibbon and Church (1984) showed that
maker’’ begins emitting pulses a short time
the simple pacemaker-accumulator model
after the onset of the time marker, and these
needs to be modified if it is to match the data.
are accumulated until a short time after re-
First, Gibbon and Church acknowledge that
inforcement, at which point the value of the
Poisson variance alone does not yield the sca-
accumulator is stored in a reference memory and
lar property: ‘‘In the Poisson system, variance
the accumulator is reset to zero. Parameters
increases directly with the mean, so that the
in the simple theory are the start and stop
system is more efficient, i.e., the ratio of stan-
delays and the rate of the pacemaker (which
dard deviation to mean is lower at long times
determines the variability of time estimates).
than at short times’’ (1984, p. 475); and later,
To generate behavior, the stored accumulator
‘‘These results, we feel, rule out Poisson var-

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TIME AND MEMORY
217
iance acting alone’’ (p. 477). Then, they go
SET thrived in spite of this difficulty? One
on to deal with the pacemaker-accumulator
reason is surely because the pacemaker no-
problem as follows:
tion is so intuitively plausible: Modern clocks
An alternative [to pulse-by-pulse variability]
(but not older devices such as the hourglass
source of pacemaker variance is a drifting rate.
and the clepsydra) all rely on the counting of
Imagine that the time between pulses . . . is
discrete pulses. Neurophysiology also pro-
fixed on any trial, but from trial to trial, the
vides ample evidence for pacemakers with pe-
pulse rate . . . varies normally around a mean.
riods in the SET range (1 to 50 per second)
. . . A more realistic version might allow rapid
(e.g., Spitzer & Sejnowski, 1997). But traces,
variation of local pulse rate both within and
integrators, and long-period oscillators are
between trials, but for our present purposes, it is
also widespread in neural and other tissue,
simplest to think of a locally constant rate which
even at the level of individual cells (Bu
¨ nning,
varies from trial to trial [italics added] . . . we
show that assuming a normal form for local
1973; Kondo et al., 1997). Physiology is rich
rate . . . we arrive at a system that is linear in
enough to support almost any hypothetical
real time with the scalar property: variance in
mechanism, so that physiological plausibility
the accumulator, and therefore the memory,
rarely distinguishes among behavioral theo-
increases approximately as the square of the
ries (Staddon & Zanutto, 1998). Many drug
mean. (p. 477)
effects find a natural interpretation within
So, rather than give up the pacemaker idea,
SET (e.g., Meck, 1996), but they can also be
a core assumption of the theory, Gibbon and
explained (we will argue) by competing the-
Church reconcile it with the scalar property
ories, so this evidence is also not decisive.
by means of two additional assumptions: that
We suspect that the main reason the Pois-
pacemaker rate varies (a) only from trial to
son pacemaker is accepted despite its theo-
trial (even though the system has no princi-
retical inconvenience is that in recent expo-
pled way to distinguish trial-onset stimuli
sitions of SET it receives only lip service: It is
from other stimuli) rather than simply pulse
assumed but not really used. For example, it
by pulse, and (b) normally rather than Pois-
is often suggested that the Poisson pacemaker
son (note that normal variation in pulse rate
has a ‘‘high rate’’ (Gibbon, 1991, p. 22) so
assumes a much more complex process than
that ‘‘Poisson [pacemaker] variance rapidly
random—Poisson—variation in time of occur-
becomes swamped by scalar variance [i.e., the
rence of each pulse).
noisy multiplier]’’ (Leak & Gibbon, 1995, p.
Given these constraints on the pacemaker,
18). Gibbon (1992) has shown how assump-
it is not clear why it is needed at all. Why not
tions about memory encoding and decoding
just assume a linear time code (no counts)
‘‘allow multiplicative random variables to in-
with a slope that varies normally from trial to
tervene . . . between the value of a count in
trial? In this bald form the arbitrariness of the
the accumulator at reinforcement time and
assumption of trial-by-trial variation would be
its value after retrieval when it is used for
more apparent. We argue below that current
comparison’’ (1992, p. 289). He concludes,
expositions of the theory in fact take some-
‘‘Scalar variance is induced by randomizing
thing close to this form, but because the
the Poisson mean with bias, or encoding the
pacemaker-accumulator framework is re-
retrieval variance in the memory system. The
tained, the redundancy of the pacemaker as-
components of variance multiply the repre-
sumption is not obvious.
sentation of criterion times and hence induce
It is worth noting that the accumulator as-
the scalar property’’ (p. 293). And, most re-
sumption is itself problematic, because it im-
cently and simply, ‘‘We have proposed that
plies a biological process that can increase
[Weber’s law in timing] reflects an underly-
without limit. SET assigns no upper bound to
ing random variation in a multiplicative noise
the duration of intervals that can be timed,
variable’’ (Gibbon et al., 1997, p. 170). Thus,
so if the time code is linear, there is no limit
the awkward property of any pacemaker-ac-
on the accumulator total.
cumulator system—increasing relative accu-
The fundamental contradiction between
racy at longer times—is sidestepped, because
the pacemaker-accumulator idea and the We-
variance due to pacemaker rate variation is a
ber law property of timing should be fatal to
trivial part of the whole. In both early and
any pacemaker-accumulator theory. Why has
current versions of SET, the Poisson pace-

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218
J. E. R. STADDON and J. J. HIGA
maker assumption is redundant: Either the
coded internal time
and real time t. Gibbon
Poisson property is replaced by a constant
concludes, ‘‘The variation induced by these
rate that varies from trial to trial, or, alterna-
parameters [k and ] scales with t* so that the
tively, residual Poisson variability is deemed to
distributions on memory are (nearly) scale
make a negligible contribution to total vari-
transforms of each other’’ (1991, p. 23). As
ance.
we have seen, the variation in pacemaker
The core assumptions of SET as it is actu-
rate,
, is assumed to be negligible; hence,
ally used seem to be something like the fol-
the major contribution to these distributions
lowing. SET, and perhaps any theory of op-
is variation in parameter k. Because k is a mul-
erant
timing,
seems
to
require
three
tiplier, any variation in
will automatically be
time-related variables: real elapsed time, the
proportional to t (i.e., scalar, in SET termi-
encoded value of current time, and the re-
nology). Thus, SET explains the scalar prop-
membered value for times encoded in the
erty by assuming (a) that there is a multipli-
past. We denote real time as t , where
cative transformation between encoded time
i
i indi-
cates the relevant time marker. The encoded
and remembered time; (b) that temporal
value for t is
. The remembered value for a
judgments represent a comparison between
i
i
past
we indicate by
. An asterisk denotes
long-term remembered time and short-term-
i
i
the value of each variable that is associated
encoded current time; and (c) that most of
with reinforcement. The internal variable for
the variability in remembered time is due to
current time,
, is always referred to in SET
the multiplicative relation (
i
k) between encod-
as ‘‘number of pulses,’’ but because the pace-
ed and remembered time. The Poisson pace-
maker-accumulator assumption is in fact un-
maker-accumulator system seems to be com-
necessary, it could simply be any internal var-
pletely redundant in this most recent version
iable proportional to real time.
of SET. The effect of independent variables
SET assumes that the relation between re-
such as drugs on a previously learned perfor-
membered time and real time is linear (cf.
mance is attributed either to a change in k
Figure 3, below; Leak & Gibbon, 1995, Figure
(the translation between working and refer-
1, and many such figures in earlier papers).
ence memory), a change in the slope of
Formally,
kt (subscripts neglected for
Equation 1b (i.e., in the more or less constant
simplicity); that is, remembered time,
, is
pacemaker rate), or a change in
*, the re-
proportional to real time, t, and k is a con-
membered time of reinforcement (in refer-
stant of proportionality. But this cannot be a
ence memory).
direct relation, because remembered time,
,
Response rule. Predictions of response pat-
is not stored directly—what is stored is en-
tern versus time (and sources of variability)
coded time,
—and indeed Gibbon (1991)
are obtained through a threshold assumption
writes ‘‘When, on a given trial, reinforcement
(note that the assumption that response rate
is obtained at a time t*, the value for accu-
stops and starts around the time of reinforce-
mulated pulses [i.e., , encoded current time,
ment is tied very much to a particular exper-
in our terminology] stored in working mem-
imental procedure, the peak procedure, dis-
ory on that trial is translated to reference
cussed below, because response rate is not so
memory [i.e., converted from * to
*] via
simply related to t* on other interval-timing
the proportionality constant, k [our sym-
schedules):
bols]’’ (p. 23). So the correct relations must
be
if
*
,
response rate
x,
(2a)
k
(1a)
otherwise,
response rate
0,
and
where x is a constant and
is a threshold.
Because both
and
are linear with respect
t,
(1b)
to real time (Equation 1), t may be substitut-
so that
ed so that Equation 2a is thus shorthand for
( cancels)
k t,
(1c)
where
denotes the pacemaker rate or, in
if kt *
t
,
response rate
x,
(2b)
our terms, the scale relation between linearly
otherwise,
response rate
0.

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TIME AND MEMORY
219
Although Equation 2a is the simplest form of
where a and k are constants. Exponential de-
i
threshold assumption, in most expositions of
cay is not consistent with Weber’s law but, as
SET (cf. Church & Gibbon, 1982, Equation
Fechner showed many years ago, a logarith-
1; Gibbon, 1991, Equation 1) a ratio version
mic function like Equation 4 is. If internal
is preferred:
noise, , is independent of
[i.e.,
k1
k2ln(t)
], then Weber’s law (the scalar
if (kt *
t)/kt *
,
response rate
x,
property) can be obtained directly, just from
otherwise,
response rate
0.
the form of , because the slope of Equation
(2c)
4 is inversely related to t: d /dt
k2/t. [Be-
cause it does not matter whether f(t) is in-
Equation 2c, linear time encoding with a ra-
creasing or decreasing, we use the decreasing
tio response rule, is equivalent to logarithmic
form of log function for comparability with
time with a difference response rule:
the other decreasing functions, and because
(kt *
t)/kt *
we will later introduce the idea of a decreas-
ing memory trace as a basis for the timing
t/kt *
1
ln t
ln t *
,
(2d)
function.] Given a constant variation in re-
where
is a threshold value. Nevertheless,
sponse threshold, therefore, the variation in
the ratio-rule-with-linear-time version is pre-
the time of onset of responding (Equation 2)
ferred in SET because of the commitment to
will be proportional to the slope of f(t),
a pacemaker.
hence (for the log function) proportional to
The essential features of SET as it has been
t. This is less ad hoc than the early version of
used in recent papers are thus relatively sim-
SET, which assumes linear f(t) plus trial-
ple: linear encoded and remembered time,
locked variation in pacemaker rate to achieve
related multiplicatively, and all-or-none be-
the same result. It is more complicated than
havior generated via a thresholded compari-
the later SET, which uses a multiplicative
son between them. We will argue in a moment
translation from working to reference mem-
that time is in fact encoded approximately
ory (k), but that version runs into difficulties
logarithmically, as Equation 2d implies.
with bisection data.
The log-time assumption is consistent with
Alternatives to Linear Encoded Time
temporal bisection data, which show that an-
It is important to recognize that the only
imals judge an event of duration x to be
necessary requirement for interval-time dis-
equally like two comparison durations y and
crimination is some internal variable that
z if x
yz, that is, at the geometric mean
changes in a reliable monotonic way with
(e.g., Church & Deluty, 1977; Stubbs, 1968).
time elapsed since a time marker. Moreover,
In a typical bisection experiment the organ-
as long as there is a unique value of the var-
ism has two choices and is presented on each
iable for each time, it makes no difference
trial with one of two stimulus durations, TS
whether the variable increases or decreases
and TL. Reinforcement is delivered for Re-
with time. Given such a monotonic function,
sponse A following the short duration, TS,
time can be told by associating specific values
and Response B following the long, TL. In oc-
of the variable with reinforcement or its ab-
casional probe trials, intermediate durations
sence and responding accordingly. Given that
are presented. The typical finding is that sub-
SET-type theories of timing are essentially a
jects are indifferent between A and B when
comparison between some internal time-re-
the probe duration, TP, is equal to the geo-
lated variable,
f(t), and the remembered
metric mean of TS and TL, TP
(TS·TL)½. This
value of that variable at the time of reinforce-
is what would be expected given symmetrical
ment, there is no up-front reason to restrict
variation around TS and TL on a logarithmic
such models to a linear function. A couple of
psychological time scale: Responses A and B
nonlinear possibilities are exponential,
should have equal strength (point of subjec-
tive equality) at the geometric mean of the
e at,
(3)
short and long training times. This result is
different from what would be predicted by
and logarithmic,
SET (cf. Gibbon, 1981; Leak & Gibbon, 1995,
k1
k2ln(t),
(4)
Figure 1). For example, given TS
1 and TL

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220
J. E. R. STADDON and J. J. HIGA
Argument Against Log Time
Given the apparent equivalence of models
with a linear time scale and a ratio-difference
response rule versus a logarithmic scale and
a constant-difference response rule, it is ob-
viously important for SET to rule out the hy-
pothesis that time is encoded logarithmically.
The main argument against the assumption
that time is encoded in a log-like way is based
on ‘‘time-left’’ choice experiments (Gibbon &
Church, 1981). In the time-left procedure,
rats or pigeons must choose between a short
fixed delay and a longer fixed delay that has
already partly elapsed (time left). In their Ex-
Fig. 1.
Some memory-decay functions. Parameters
periment 1, for example, rats chose between
(Equation 16): a
.7, .9, and .994, for three cascaded
i
two levers, one fixed and the other retract-
integrators; b
.2. k
.94 (exp). k
.2 (log). m
.5
(power). The MTS function is derived from the recovery
able, representing different delays until food
portion of a habituation series. Scale parameters are ig-
reinforcement. The fixed lever signaled a
nored. Note that the power and MTS functions are very
fixed delay, CTO (timed from trial onset), un-
similar.
til food. The delay on the retractable lever,
SLP (timed from lever presentation), present-
ed TTO s into the trial, was also fixed, but
2, logarithmic time predicts bisection at TP
shorter than CTO. The experimental question
1.414 and scalar timing at TP
1.33, the
is: How will preference for fixed delay SLP ver-
harmonic mean (linear time would place the
sus time-left delay CTO
TTO change as a
point of subjective equality at 1.5).
function of TTO? In particular, will the ani-
There are several other functions that have
mals prefer the time-left lever when CTO
very similar properties to the logarithmic:
TTO
SLP? In fact, rats are indifferent when
power (e.g., Staddon, 1984), the sum of ex-
delays to food are equal on both levers, which
ponentials (multiple time scale: MTS1; Stad-
Gibbon and Church take as evidence against
don, 1997; Staddon & Higa, 1996), and oth-
logarithmic coding of time: ‘‘A logarithmic or
ers. Four functions are shown in Figure 1.
indeed any curvilinear subjective time scale
The logarithmic, power, and MTS (but not
ought to result in a greater preference for the
the single-exponential) functions can approx-
time-left side of the choice when the real time
imate both Weber’s law and necessary mem-
left to food is equal on both alternatives’’ (p.
ory properties such as Jost’s law. The power
92).
function is the candidate offered by Wixted
This argument is equivalent to saying that
and Ebbesen (1991, 1997) for the forgetting
because the log form for
changes less dur-
function in a range of species. The log and
ing the second half of a timed interval than
power functions were the best fits to the large
during the first half, the second half is more
human-forgetting-curve data set reviewed by
valuable to the animal, because it is objective-
Rubin and Wenzel (1996; they did not look
ly smaller than a half-length interval just be-
at MTS-type functions). The resemblance be-
ginning (cf. Gibbon & Church’s, 1981, dis-
tween the MTS and power functions will be-
cussion of Figure 2). Thus, if a 30-s lever (S
come relevant when we present a memory-
LP
30) is introduced half way through the in-
based approach to time discrimination in the
terval on a 60-s lever (C
second half of the paper.
TO
60), the animal
is supposed to compare the small upcoming
difference,
, between 30 and 60 on the 60-
1 But note that because of the nonlinear dynamics of
s lever with the larger upcoming difference
the MTS model, the form of the function is not invariant,
between 0 and 30 on the inserted lever and
but depends on the system history. For example, see how
choose the 60-s lever, rather than be indiffer-
the MTS trace declines more slowly after a history of low-
rate stimuli versus a history of high-rate stimuli in Figure
ent between the two as linear timing would
5, below.
require.

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TIME AND MEMORY
221
This analysis is flawed, because it takes an
ought to result in a greater preference for the
assumption about the form of an internal
time-left side of the choice when the real time
‘‘time-coding’’ variable and then assumes a
to food is equal on both alternatives. (p. 92)
direct relation between that variable and re-
They do acknowledge a potential problem
inforcement value: the smaller
, the greater
with this interpretation: ‘‘Suppose that each
the value.2 The problem is that a theory
of the three entry points [i.e., 15 s, 30 s, and
about how time is encoded [e.g., the assump-
45 s] is learned separately as a paired-associate
tion that
f(t)] says nothing at all about
subtask [italics added]. Were this the case, the
how an animal will act with respect to a given
performances should be comparable with
change in . Nor does it say whether a given
ones in which a 45-sec FI is pitted against a
change always has the same value, indepen-
30-sec FI, two 30-sec FIs are pitted against
dent of the starting value for . For example,
it is possible that the animal will evaluate the
each other, etc.’’ On this basis they conclude
30-s alternative in a conditional way, depend-
that ‘‘The mechanism for producing an ap-
ing on when it occurs during the 60-s inter-
propriate performance in each of the three
val, or that it will begin timing both outcomes
subproblems then might be of almost any
from the beginning of the 30-s interval once
sort, independently of the character of the
the 30-s lever is introduced. Moreover, Gib-
subjective time scale’’ (p. 93). Nevertheless,
bon and Church’s (1981) assumption that
they reject this line of reasoning because pref-
value is inversely related to
, independently
erence for the 60-s lever when the 30-s choice
of the starting value, leads to obvious coun-
is inserted at 15 s (45 vs. 30 s to food) is the
terfactuals. After a very long time, even a
same as the preference for the 30-s lever
large time-left delay (very small
at that
when it is inserted at 45 s (15 s vs. 30 s to
long time) should be preferred to a short
food). But this argument is also not decisive,
comparison delay (large
), for example. Ei-
because the supposedly symmetrical data
ther the log-time assumption is wrong, or the
(their Figure 3) are not very convincing
assumption that value is inversely related to
(there is much individual variability among
encoded (as opposed to real) time is wrong.
the 4 rats), and because such a result is in
Gibbon and Church (1981) have a second
any case not quite what we would expect from
argument for the linear time assumption:
other choice data. In other experiments with
delayed reward (e.g., Chung & Herrnstein,
The increase in preference for time left over
1967; Shull, Mellon, & Sharp, 1990) prefer-
indifference when the standard lever entered
ence is proportional to relative immediacy.
at 45 sec was the same as the decrease in pref-
Thus, given two delays on left and right d
erence for time left when the standard lever
L
entered at 15 sec. Thus, preference was sym-
and dR, the ratio of responding, BL/BR
(1/
metrical around indifference. These results
dL)/(1/dR)
dR/dL. Hence, in the time-left
strongly suggest that time is appreciated in a
experiment, we might expect B45/B30
30/
linear fashion in both intervals. A logarithmic
45
.67, whereas B30/B15
15/30
.5, that
or indeed any curvilinear subjective time scale
is, the two ratios should not be the same, as
Gibbon and Church contend. They did not
2 Gibbon and Church’s argument is deceptively similar
test between these two possibilities in this ex-
to the optimality argument that derives risk aversion from
periment, but in a later ones (Brunner, Gib-
decreasing marginal utility. The crucial difference is that
bon, & Fairhurst, 1994; Gibbon, 1986) they
the risk-aversion argument derives from a plot of subjec-
confirmed that reinforcement delays con-
tive value versus objective quantity, whereas their argu-
ment derives from a plot of objective time code versus
form to the idea that the animals value a
objective real time. The problem in their analysis is that
choice according to relative immediacy. Thus,
subjective is not the same as encoded. Time may well be
in the Gibbon (1986) experiment, pigeons
(and is, we contend) encoded nonlinearly, in the sense
were roughly indifferent between two equi-
that it is mapped onto some internal variable that in-
creases with elapsed time in a negatively accelerated way.
probable delays of 15 and 240 s (harmonic
Nevertheless, subjective time, like subjective weight and
mean: 28 s) and a fixed delay of 30 s.
other examples, is roughly proportional to real time. We
There are at least two other ways to inter-
argue that encoding determines experimental results
pret the time-left experiment that can rec-
that depend on discriminability, but subjective value de-
termines results that depend on value (e.g., choice ex-
oncile these results with nonlinear encoded
periments).
time:

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222
J. E. R. STADDON and J. J. HIGA
First, consider Gibbon and Church (1981),
There is a second way to look at the time-
Experiment 2 (Figure 6). There are two
left experiment. First, we must acknowledge
choices, each available at a variable time after
that an organism may be able to assess (i.e.,
trial onset (TO): call that time TTO. Choice
respond selectively to) the rate of occurrence
W gives food after a time, CTO, that is variable
of an event like food reinforcement without
from the time of choice presentation, but is
being directly sensitive to the time at which
fixed from the onset of the trial. Thus the
food occurs. For example, a simple organism
time marker for Choice W is trial onset. For
whose behavior is guided by a leaky integra-
this choice, therefore, we have a real time,
tor can assess reinforcement rate via the state
tTO, and an encoded time, TO, both measured
of the integrator but may nevertheless be un-
from time marker TO. On a given trial, at
able to anticipate a periodic event (cf. many
time T the value of Choice W is given by the
stochastic learning models, beginning with
immediacy of food on that choice (according
Bush & Mosteller, 1955). Second, we assume
to numerous delay-of-reinforcement experi-
that the organism can learn the association
ments, e.g., Chung & Herrnstein, 1967),
between particular stimuli and particular
which is given by the difference between the
rates of reinforcement.
time at which the choice must be made, TTO,
Now consider the decaying ‘‘memory’’ of a
and the time when food is predictably deliv-
time marker. At different delay times, 1, 2, 3,
ered, CTO. Because
f(t), this value func-
and so on, this memory takes on different
tion is just
values, 1, 2, 3, and so on. On the time-left
choice, each of these ‘‘memories’’ is a dis-
1
1
V(T )
V( C
T)
1/[ f
( C)
f
( T)]
criminative stimulus for an outcome that has
1/(C
a certain rate of reinforcement (the recipro-
TO
TTO),
(5)
cal of the time left). (This is a ‘‘paired-asso-
where f 1 denotes the inverse function. The
ciate subtask,’’ in the words of Gibbon and
situation for the other choice in this experi-
Church, 1981.) The reinforcement rate sig-
ment, Choice G, is simple: Food arrives after
naled by each stimulus value can therefore be
a fixed delay time, S, from choice presenta-
compared with the rate signaled by the fixed-
tion (CP, not trial onset) which is indepen-
delay ‘‘standard’’ stimulus. Clearly, the ani-
dent of T. Thus,
mal should be able to choose either the time-
V(T)
V(
left stimulus,
, or the standard stimulus,
i
S)
1/f 1( S)
1/SCP. (6)
depending on which is associated with the
The choice then is trivial: Choose the shorter
higher rate of reinforcement. This interpre-
real-time delay, which is what the animals do.
tation is independent of the particular form
In other words, although time is coded ac-
of memory-decay function. Perhaps this is
cording to some function f(t) that determines
what Gibbon and Church meant when they
the accuracy of time discrimination (this is a
write (correctly, in our view) that ‘‘The mech-
confusion scale in the traditional psychophysi-
anism for producing an appropriate perfor-
cal terminology), behavioral actions are de-
mance in each of the three subproblems then
termined by the inverse function, which is the
might be of almost any sort, independently of
animal’s estimate of real time—the ecologi-
the character of the subjective time scale’’
cally relevant variable. The claim is that no
(1981, p. 93).
matter what the animal’s internal code for
The essential feature of our second argu-
elapsed time, it will also have some kind of
ment against Gibbon and Church’s interpre-
compensatory perceptual constancy mecha-
tation of the time-left experiment is separa-
nism (we suggest a specific possibility below)
tion between the animal’s capacity to assess
that allows it to behave appropriately with re-
reinforcement rates and its capacity to use a
spect to the real world (i.e., real time). This
decaying memory trace as a stimulus. There
way of doing it avoids the non sequitur of as-
is in fact no necessary relation between an
suming that just because long times cannot
organism’s ability to learn to identify a partic-
be estimated as accurately as short times, time
ular point in time and its sensitivity to rates
intervals long delayed from a time marker are
of reinforcement The theoretical proof is the
somehow more valuable than time intervals
existence of reinforcement-rate-sensitive
little delayed from a time marker.
learning models that lack any timing capabil-

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TIME AND MEMORY
223
ity (e.g., Dragoi & Staddon, in press). The
measure of perceptual constancy (cf. the first
empirical proof is that even in situations
interpretation, above). Thus, we may be less
where time discrimination is possible, such as
accurate at large values (weights, times) than
concurrent variable-interval variable-interval
short, but we know perfectly well that a
choice, time discrimination (momentar y
weight or a time has approximately doubled
maximizing), and choice performance often
in value, even though our internal code has
operate independently (cf. Hinson & Stad-
changed by less than a factor of two.3
don, 1983; Williams, 1988); and some organ-
isms—some fish, for example—are able to
Argument for Log Time
choose on the basis of reinforcement rate but
The time-left experiments are not a con-
seem to be rather poor at estimating time in-
vincing argument against logarithmic encod-
tervals (Rozin, 1965). Thus, Gibbon and
ing of time. Is there any evidence in favor of
Church’s interpretation of the time-left pro-
log-like encoding, beyond the Weber law stan-
cedure is by no means forced. Hence their
dard-deviation property and geometric-mean
data do not constitute evidence against the
bisection data? There is some indirect sup-
idea that encoded time is nonlinear.
porting evidence that derives from the wide-
These examples are just illustrative. The
spread finding of power-law relations in in-
fundamental flaw in the time-left argument is
terval-timing experiments. Platt (1979) has
in fact conceptual—namely the assumption
reviewed numerous studies showing a power-
that an organism has access to (and its be-
law relation between temporal dependent
havior is directly determined by) the objec-
and independent variables in temporal dif-
tive properties of its own internal represen-
ferentiation and discrimination experiments:
tation, or, to put the same thing in more
‘‘psychological’’ terms, that subjective (how
b
qts,
(7)
long a time appears to be to the animal)
where b is the observed behavior (e.g., re-
equals objective (how much its internal time
sponse duration, waiting time), t is the re-
code changes). This is an old error in psy-
quired duration, q is a constant, and s is an
chology, and there are numerous illustra-
exponent (usually close to one). Power-func-
tions. For example, in the neural homuncu-
tion relations with exponents different from
lus in the human brain, the representation of
unity cannot easily be reconciled with SET,
the hands is much larger than the represen-
but there is a theoretical argument that ties
tation of the back, and this is reflected in the
them to logarithmic internal coding. The ar-
smaller two-point threshold on the hands.
gument is as follows.
But we do not feel that our hands are larger
First assume that temporal discrimination
than our back. Peripheral visual resolution is
is a comparison process in which an internal,
much worse than foveal, but the periphery
logarithmic temporal variable (reference
does not appear smaller than the area ob-
memory) is compared with an output vari-
served by the fovea. In the discrimination of
able (working memor y: encoded elapsed
weight (Weber’s original experiment, and the
time) that is also logarithmic: ‘‘Investigators
basis for his law), few doubt that the internal
. . . have suggested that performance in scal-
coding is logarithmic. Yet there is also no
ing experiments results from the subject
doubt that although the just-noticeable-differ-
ence (jnd) for a 1 lb weight is about 0.2 lb
3 John Gibbon (personal communication) has pointed
and the jnd for a 10 lb weight is about 2 lb,
out an apparent contradiction between the geometric-
the subject knows perfectly well that the sec-
mean bisection data, which are consistent with a log
ond increment is larger than the first. As Ste-
scale, and the time-left and other similar choice data,
vens wrote many years ago, ‘‘the jnd’s for
which are consistent with linear time. The difference is
that in the bisection case, the animal is judging the du-
loudness are unequal in subjective value. The
ration of a single past event (the sample time) and there-
same appears to be true of other intensive
fore chooses based on the log-scale point of subjective
attributes like subjective weight, brightness
equality, which is the geometric mean. But in a choice
and taste’’ (1951, p. 36). The general point
experiment (e.g., preference for a fixed delay, x, vs. two
is that discriminability does not determine
equiprobable delays, y and z) the animal is choosing be-
tween different expected rates of reinforcement, so that
perceived value. Every sensory dimension is
choice is at the value point of subjective equality, which
processed in a way that usually preserves a
is the harmonic mean: x
½(y
z)/yz.

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224
J. E. R. STADDON and J. J. HIGA
matching an internal representation of the
s
w / .
b
wt
(13)
stimulus to an internal representation of the
response. If both of these representations are
The two constants, K1 and K2, are scale factors
logarithmically related, the power law results’’
(Staddon, 1978), assumed to be constant
(Platt, 1979, p. 19; see Ekman, 1964; MacKay,
across different experimental procedures.
1963). In other words, if memory for a time
Notice that if the sensitivities (Weber frac-
interval is encoded logarithmically and if cur-
tions) of remembered time and elapsed time
rent elapsed time is also encoded logarith-
are the same, the exponent, s, is unity, and
mically, and if behavior involves comparison
behavior (waiting time), b, is linearly related
between the two, then the empirical relation
to elapsed time, t. This is a common, but not
between temporal independent and depen-
universal, result in temporal experiments.
dent variables will take the power form.
The exponent for the function relating wait-
This interpretation of the psychophysical
ing time to FI duration in steady-state para-
power law was extended by Staddon (1978)
metric FI experiments is usually close to one.
and the argument can be applied to time dis-
But the exponent in steady-state tracking ex-
crimination. We assume that the internal ef-
periments, in which the animal is repeatedly
fects, dz, of both remembered time (repre-
subjected to cyclically varying interfood inter-
sented in Equation 8) and elapsed time
vals, is typically less than one. This is just what
(represented by Equation 9) show Weber law
we would expect, given that the exponent s
sensitivity, according to sensitivity coefficients
w /
and that it is harder for the animal
b
wt
(Weber fractions), w and
:
to remember the upcoming interfood inter-
t
wb
val when several are intermixed in each ses-
dt
w
sion than when all the intervals are the same
t dzt
(8)
t
from session to session. If w , the Weber frac-
t
and
tion for remembered time, increases (i.e.,
poorer discrimination), then the exponent s
db
should decrease. As this argument suggests,
wbdzb.
(9)
b
Innis and Staddon (1971) found a less-than-
one power-function exponent of .824 in an
The first equation simply states that a small
early interval-tracking experiment in which
change in real time, dt, has a psychological
pigeons were repeatedly exposed to a cycle of
effect, dz, that is inversely related to t and the
seven ascending and seven descending inter-
Weber fraction w :
t
dz
dt/wtt; and similarly
food intervals. They also found that the ex-
for the second equation (Staddon, 1978).
ponent increased to .894 when different dis-
Integrating both sides of Equations 8 and
criminative stimuli signaled the ascending
9 yields
and descending parts of the cycle and pre-
sumably reduced memor y interference
ln t
K1
wtzt
(10)
among remembered intervals (cf. Staddon,
1974b).
and
If different experimental arrangements af-
ln b
K
fect sensitivities and the two sensitivities are
2
wbzb,
(11)
affected differentially, then the power-func-
a logarithmic relation between both remem-
tion exponent will be different in different
bered time, t, and elapsed time, b, and their
experiments. It follows from Equations 12
internal effects, z and
.
t
zb K1 and K2 are con-
and 13 that the slopes and intercepts of a set
stants of integration. In temporal discrimi-
of such functions will be linearly related:
nation experiments, the internal effects of re-
membered and elapsed time are equated, zt
ln q
sK1
K2,
(14)
z , which allows us to eliminate
b
z from
Equations 10 and 11. Rearranging yields the
which is a testable empirical prediction.
power relation (Equation 7), with
DeCasper (1974, cited in Platt, 1979) plotted
the slopes and intercepts of power functions
q
exp(wbK1/wt
K2)
(12)
obtained in four different temporal differ-
and
entiation experiments, with the results shown

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