J Neurophysiol 109: 344 –362, 2013.
First published October 17, 2012; doi:10.1152/jn.01201.2011.
Dichotomy in perceptual learning of interval timing: calibration of mean
accuracy and precision differ in specificity and time course
Hansem Sohn1 and Sang-Hun Lee1,2
1
Interdisciplinary Program in Neuroscience, Seoul National University, Seoul, Republic of Korea; and 2Department of Brain
and Cognitive Sciences, Seoul National University, Seoul, Republic of Korea
Submitted 30 December 2011; accepted in final form 16 October 2012
time perception; perceptual learning; mean accuracy and precision;
Bayesian observer model; prior and likelihood
in an ever-changing and noisy environment,
our brain has to dynamically calibrate its spatiotemporal representations of the objects by interacting with the environment.
Among those representations in need of constant calibration,
interval timing (IT) is of particular interest because its perception is often subject to illusion by many endogenous and
exogenous factors (Eagleman 2008) such as attention and
arousal (Stetson et al. 2007; Wittmann 2009), emotional states
(Wittmann and Paulus 2008), eye movements (Burr et al.
2007), stimulus visibility (Terao et al. 2008), visual adaptation
(Johnston et al. 2006), and temporal context (Jazayeri and
Shadlen 2010). Therefore, it is not surprising to find that IT
undergoes not only incessant calibration for veridical sensation
and action but also perceptual learning (PL) for improved
temporal precision (Bartolo and Merchant 2009; Buonomano
et al. 2009; Karmarkar and Buonomano 2003; Meegan et al.
TO ACT ON OBJECTS
Address for reprint requests and other correspondence: S.-H. Lee, Dept. of
Brain and Cognitive Sciences, Seoul National Univ., 599 Gwanak-ro, Gwanakgu, Seoul 151-742, Republic of Korea (e-mail: [email protected]).
344
2000; Stetson et al. 2006; White et al. 2008). Despite this
strong evidence for PL of IT, how the perceived time interval
is adjusted by experience remains largely unknown.
Time encoding is distinguished from other types of sensory
encoding in that time is a ubiquitous feature dimension composing all sensory events, yet with no single sensory organ
directly dedicated to its encoding (Wittmann and van Wassenhove 2009). Because of this nature, previous attempts at
investigating mechanisms of time perception were prone to
confounding by other sensory, motor, or executive functions.
Hence it is crucial to avoid those confounds by precluding any
potential associations between IT and other nontiming variables. We measured IT based on natural reactions to an
invisible object in motion that was once visible and tracked by
observers while varying the time interval by manipulating the
speed and distance of motion (Fig. 1, A and B). The essential
element of our IT task is that perceived time interval (⌬t) is
revealed implicitly as a component of the equation that has to
be extrapolated from a speed (V) in an online fashion when the
spatial information (⌬X) is available (⌬T ⫽ ⌬X/V; Fig. 1C)
(Coull and Nobre 2008; Merchant and Georgopoulos 2006;
Rohenkohl et al. 2011). This feature allows us, unlike previous
timing procedures (e.g., reproduction or discrimination of time
intervals), to vary not only target durations but also other
indirect features such as speed, location, and motion direction
on a trial-by-trial basis (Buonomano et al. 2009; Livesey et al.
2007; Ryan and Fritz 2007; Staddon 2005), preventing IT from
being associatively conditioned to specific sensory events
(Janssen and Shadlen 2005; Leon and Shadlen 2003; Mita et al.
2009; Rakitin et al. 1998) or being affected by stimulus-evoked
sensory processes (Johnston et al. 2006).
Behavioral consequences of PL in IT can be probed mainly
by changes in two descriptive statistics (Fig. 1C): “mean
accuracy (constant error)”—the degree of match between physical (⌬T) and subjective [␮(⌬t)] time intervals on average—
and “precision (temporal variance)”—the reciprocal of variance [␴(⌬t)] of perceived time intervals across repeated trials
(McAuley and Miller 2007; Merchant et al. 2008b; Zarco et al.
2009). Despite the intimate relationship between the two measurements, which has been demonstrated in many visual tasks
(Ahissar and Hochstein 1997; Gold et al. 2010; Herzog et al.
2006; Wenger et al. 2008), simultaneous measurements or
conjunctive analyses of mean accuracy and precision data have
been rare in studies on IT. Only a few studies have reported
concurrent improvement (Meegan et al. 2000) or distinct effects of interstimulus interval or adaptation on discrimination
threshold and apparent duration (Buonomano et al. 2009;
0022-3077/13 Copyright © 2013 the American Physiological Society
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Sohn H, Lee SH. Dichotomy in perceptual learning of interval
timing: calibration of mean accuracy and precision differ in specificity
and time course. J Neurophysiol 109: 344 –362, 2013. First published
October 17, 2012; doi:10.1152/jn.01201.2011.—Our brain is inexorably confronted with a dynamic environment in which it has to
fine-tune spatiotemporal representations of incoming sensory stimuli
and commit to a decision accordingly. Among those representations
needing constant calibration is interval timing, which plays a pivotal
role in various cognitive and motor tasks. To investigate how perceived time interval is adjusted by experience, we conducted a human
psychophysical experiment using an implicit interval-timing task in
which observers responded to an invisible bar drifting at a constant
speed. We tracked daily changes in distributions of response times for
a range of physical time intervals over multiple days of training with
two major types of timing performance, mean accuracy and precision.
We found a decoupled dynamics of mean accuracy and precision in
terms of their time course and specificity of perceptual learning. Mean
accuracy showed feedback-driven instantaneous calibration evidenced
by a partial transfer around the time interval trained with feedback,
while timing precision exhibited a long-term slow improvement with
no evident specificity. We found that a Bayesian observer model, in
which a subjective time interval is determined jointly by a prior and
likelihood function for timing, captures the dissociative temporal
dynamics of the two types of timing measures simultaneously. Finally, the model suggested that the width of the prior, not the
likelihoods, gradually shrinks over sessions, substantiating the important role of prior knowledge in perceptual learning of interval timing.
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
345
Johnston et al. 2006; Stetson et al. 2006). Our paradigm
allowed us to track PL in IT over a long time period by
assessing ongoing changes in its mean accuracy and precision
simultaneously.
Given the hierarchical nature of brain anatomy and functions
supporting sensory perception, neural substrates underlying PL
have been widely addressed by assessing the specificity of
learning for particular stimuli or tasks during training (Fahle
2005; Fine and Jacobs 2002). A high specificity has been
interpreted as adaptive changes at a relatively early, low-tier
processing stage (Adini et al. 2002; Gilbert et al. 2001; Tsodyks and Gilbert 2004), whereas a nonspecific transfer of
learning was taken as evidence for changes in top-down signals
such as selective attention during PL (Ahissar and Hochstein
1997; Ahissar et al. 2009; Schäfer et al. 2009; Yu et al. 2004).
We examined learning specificity on a fine scale for a wide
range of ⌬Ts. In particular, we obtained “feedback transfer
curves” that quantify the degree of generalization along the
broad timescale around the interval trained with feedback
throughout 10 daily sessions (Fig. 1D). These extended followups of learning transfer curves on such a fine and wide
timescale allowed us to observe dynamic time courses of PL in
IT, distinct from previous studies, in which transfer curves
were compared only between before and after training (Bartolo
and Merchant 2009; Karmarkar and Buonomano 2003; Meegan et al. 2000; Nagarajan et al. 1998; Wright et al. 1997).
Our results revealed clear dissociations between mean accuracy and precision of IT in terms of both time course and
specificity of learning. Mean accuracy reacted immediately to
feedback by showing a partial transfer around the interval
trained with feedback, while precision did not exhibit any
specificity and improved slowly throughout the whole sessions
without dependence on feedback. A Bayesian observer model
described well the observed different dynamics in IT between
mean accuracy and precision and provided a parsimonious
explanation that PL of IT is promoted by the gradual reduction
of the width of the prior distribution, substantiating the important role of prior knowledge in PL of IT.
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Fig. 1. Stimuli and experimental procedure. A: visual stimuli were composed of a thin white bar (length 1° of visual angle), a black annulus (radius 5°, thickness
1°), and a pair of white nonius lines (length 0.5°) that denotes the occlusion arc. The small black-white ring was also shown around the fixation cross to help
define the boundary and location of the occlusion arc. B: on each trial of the interval timing (IT) task, the bar first appeared and remained stationary for a brief
time (“cue” period), then began to move with a constant speed (“visible motion” period), and then disappeared once it moved into the arc-shaped region of the
annulus defined as the occlusion zone (“invisible motion” period). The subject had to predict when the bar would reappear at the end of the occlusion arc by
pressing a button key. From the 5th to 10th “specific feedback” (SF) sessions, feedback was given by showing a snapshot of the physical location of the moving
bar at the time of button press only on trials where the physical time interval was 1.32 s. C: the structure of the implicit IT task is schematically depicted in a
time-space domain. The slopes of the linear lines are an example speed of the moving bar, which are constant during both the visible (solid lines) and invisible
(dashed gray lines) periods. Subjects have to extrapolate the bar’s arrival time at the end of the occlusion arc from the speed and distance (⌬T ⫽ ⌬X/V),
information portrayed during the visible portion of the trial. The Gaussian curve is drawn on the time axis to schematically illustrate how we estimate the mean
and standard deviation of response times in repeated trials at a certain physical time interval. The bracket on the space axis is drawn to illustrate that the feedback
for timing error is provided as a spatial offset and can be computed as multiplication of the bar’s speed V and the deviation of a perceived interval (⌬t) from
a physical interval (⌬T). D: the whole experiment consisted of 4 daily sessions of “no feedback” (NF) conditions (open bars) and an ensuing 6 sessions of “specific
feedback” (SF) conditions (hatched bars). A session started with the IT task, followed by the speed discrimination (SD) task (gray bars). There is no difference
between NF and SF conditions except that 35 additional training trials with feedback were inserted randomly between regular trials in the SF condition. The motor
control trials, which were also randomly inserted throughout the whole sessions, were identical to the IT task trials except that the moving bar did not disappear
when passing through the occlusion arc.
346
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
METHODS
Subjects and Apparatus
Four naive subjects (2 women, 2 men; age 21–25 yr) participated
in the experiment. All subjects had normal or corrected-to-normal
vision and gave written informed consent. This study was approved by
the Institutional Review Board of Seoul National University.
Stimuli were presented at 1,024 ⫻ 768-pixel resolution (40 ⫻ 30
cm) with a refresh rate of 60 Hz on a cathode ray tube monitor (LG
HiSync 291U) with 81-cm distance from eyes to the monitor. A chin
rest was used to minimize head movements. A button press was
recorded by a numeric keypad connected to an Apple Power Mac G5
computer through a USB port.
Experimental Procedure and Stimuli for IT Task
Control for Sensorimotor Function and Speed Perception
In the SD task trials, subjects performed a two-interval forced-choice
task, in which they viewed two bars drifting on the visible arc in separate
intervals and judged which one moved faster. While setting the pedestal
speed to 8°/s, which roughly matched the mean speed (8.22°/s) in the IT
task, we used an evenly distributed set of six test speeds ([4, 5.6, 7.2, 8.8,
10.4, 12]°/s; method of constant stimuli), each shown in 30 trials. Similar
to the IT task, we prevented subjects from judging the relative speed of
the two bars based on only temporal or spatial information by varying the
arc length to be traveled by the two bars within the uniform distribution
ranging from 3° to 4°. From data in each session, we plotted a psychometric curve of proportion of “faster” responses as a function of test
speed and fit a cumulative Gaussian function to the curve with the
maximum likelihood estimation method (Wichmann and Hill 2001).
Then we estimated a point of subjective equality (PSE) and sensitivity by
finding its mean and standard deviation, respectively.
The motor execution (ME) task trials were intermittently inserted
between the IT task trials (20 ME trials per 270 IT trials of a daily
session). This was possible because the ME trials were exactly the
same as the IT trials in task structure except that the moving bar was
always visible. Without knowing ahead whether a given trial would be
a ME or IT trial until the moving bar entered the occlusion arc,
subjects were instructed to simply press the key at the point of time
when the bar arrived at the end of the occlusion arc. To minimize
unwanted potential influences of ME trials on timing performances, a
stimulus duration on each ME trial was randomly sampled from a
continuous uniform distribution, the range of which matched that of
the intervals in the main timing task. Because the distribution of errors
in timing data from the ME trials did not vary systematically depending on the ⌬Ts, we merged the data across all the ME trials with
different ⌬Ts within each session and simply computed the mean and
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At the beginning of each IT task trial, subjects viewed a black (0.81
cd/m2) annulus [radius ⫽ 5 degrees of visual angle (°); 1° thickness]
around the fixation cross at the center of a screen with gray (19.53
cd/m2) background (Fig. 1A). During the “cue” period (Fig. 1B, left),
a stationary thin (0.5° width; 1° height) white (74.58 cd/m2) bar was
shown in a randomly selected position on the annulus with a pair of
white nonius lines (0.3° width; 0.5° height) at the edges of the
annulus. These nonius lines demarcated an occlusion arc in which the
bar underwent invisible motion. In addition, a black-white annulus was
also shown near the fixation to help subjects recognize the size and
location of the occlusion arc (Fig. 1A). After the cue period of 250 ms, the
bar started to move in either a clockwise or a counterclockwise
direction along the annulus toward one of the nonius lines at a
constant speed, which was varied across trials. After this “visible
motion,” the duration and trajectory of which were varied pseudorandomly across trials (750 ⫾ 630 ms for duration; uniform distribution
of 3– 4° arc length; 0.6 – 0.8 rad), the moving bar became invisible
while passing through the “occlusion” arc (Fig. 1B, center). Subjects
judged when the bar would reappear at the end of the occlusion arc by
pressing a button key with the right index finger. The length of the
occlusion arc (⌬X) was varied across trials within a uniform distribution of arc length 3–15° (0.6 –3 rad). At the time of a key press, a thin
line was flashed briefly (a single monitor frame of 16.67 ms) inside the
nonius line as an indication that a response was made. A trial was
terminated when subjects did not press the key until after the time passed
more than twice a physical interval (⌬t ⬎ 2 ⫻ ⌬T). The proportion of
these “miss” trials was ⬍2% of the total number of trials. To examine the
specificity of learning, feedback was only given in trials with the median
(1.32 s) of nine test intervals sampled in a logarithmic scale (from 0.5 s
to 3.5 s). The feedback was provided at the time of a key press by
showing subjects a snapshot of the actual location of the moving bar for
750 ms (Fig. 1B, right). Hence an amount of error in time (⌬t ⫺ ⌬T) can
be derived by dividing the spatial gap between the feedback bar and the
nonius line by the speed of the moving bar.
The speed of the bar on a given trial was determined by dividing a
randomly selected length of the occlusion arc by a test time interval
(V ⫽ ⌬X/⌬T). The resulting distribution of speeds was positively
skewed (across-session mean, standard deviation, skewness ⫽
8.22°/s, 6.15°/s, 1.28°/s) and ranged from 0.95°/s to 29.29°/s. We
underscore that stimulus features other than the ⌬T were either
counterbalanced (direction of bar motion) or varied in a pseudorandom manner (V, ⌬X, arc length and starting position of the visible
motion). Thus, even when a given set of trials had an identical test ⌬T,
those individual trials had highly dissimilar nontemporal variables. Because trial-by-trial changes in those nontemporal variables may affect
timing performance, we inspected whether the time course or learning
specificity of timing accuracy and precision in each session varied in a
manner dependent on the across-trial differences in V, ⌬X, or duration of
visible motion. To do this, we split trials with the same ⌬T into a lower
half and a higher half according to each of the nontemporal variables and
compared the two halves of trials in timing accuracy or precision as a
function of ⌬T. We found no qualitative difference between the two split
halves in any of the nontiming variable dimensions, indicating that the
stimulus features other than ⌬T had hardly any impacts on timing
performance. We also emphasize that we varied the test ⌬T on a
trial-by-trial basis instead of presenting a single interval repeatedly within
a block of trials. This “roving” was introduced to prevent subjects from
employing a feedback-driven adaptive strategy to adjust a response
threshold (Whiteley and Sahani 2008), which may generate unwanted
sources of variance in timing measurements.
It is worth noting some of the descriptions offered by subjects
during postexperimental debriefing. First, none of the subjects reported being aware that feedback was given only to trials with a
specific test interval (1.32 s). Second, subjects described adopting
either a “dynamic imagery” strategy (Pearson et al. 2008), where they
responded to an imaginary bar in motion, or a “spatiotemporal
equation” strategy, where they estimated the interval by combining
spatial (e.g., ratio between the visible and invisible arc lengths) and
temporal (e.g., duration of the visible motion) information. Finally,
the debriefing confirmed that none of the subjects mentioned using a
verbal or nonverbal chronometric counting strategy, as explicitly
instructed before the experiment.
The whole experiment consisted of four daily sessions of “no
feedback (NF)” conditions and an ensuing six sessions of “specific
feedback (SF)” conditions (Fig. 1D). For a given subject, all the
sessions were carried out at a fixed time in a day to prevent potential
interactions between circadian rhythm and IT (Pashler and Medin
2004). The SF sessions were identical to the NF sessions except that
35 additional training trials with feedback were inserted and randomly
intermingled with the other 270 test trials. As a practice for the IT
task, each subject experienced 15 IT trials with no feedback on the
first day of the experiment. In each daily session, subjects first
performed 270 trials (30 trials for each of the 9 test intervals) of the
IT task and then 180 trials (30 trials for each of the 6 test speeds) of
the speed discrimination (SD) task, as described below.
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
347
standard deviation of motor errors (⌬t ⫺ ⌬T), which here represent
the accuracy and precision, respectively, of the sensorimotor function.
No feedback was provided either in the ME trials or in the SD task,
while there were 3 and 12 practice trials for the ME trials and the SD
task, respectively, before the whole experiment began.
Data Analysis
Dk ⫽
兺j ⱍd jkⱍ ⁄ N j where d jk ⫽
␮(⌬t) jk ⫺ ⌬T jk
⌬T jk
.
(1)
Here, the divisive normalization with ⌬Tjk was done as correction for
the scalar property of IT. Nj, the number of ⌬Ts, was always 9.
To index the overall precision of timing behaviors in a session, we
averaged the interval-specific coefficient of variation cvjk across
intervals for a session k:
CVk ⫽
␴(⌬t) jk
兺j cvjk ⁄ N j where cv jk ⫽ ␮(⌬t) jk .
(2)
Across-session analysis of accuracy and precision in IT. To test the
monotonicity of learning curves across sessions, we fit an exponential
function of the kth session (Dosher and Lu 2007), aebk ⫹ c, to the
observed time course of the two measures Dk and CVk, using a
method of nonlinear least squares and Levenberg-Marquardt algorithm in MATLAB (MathWorks). If a and b have different signs, the
learning curve is interpreted to decrease monotonically. In addition,
Spearman’s rank correlation test was also used to test whether the
long-term time course of timing performance increases or decreases
with an increasing order of daily sessions.
Analysis of interval-specific learning in IT. We evaluated feedback
effects on timing accuracy and precision by assessing the degree to
which PL in a specific time interval with feedback (1.32 s) is
transferred to other neighboring intervals without feedback in the
following steps. First, for each of the “NF” and “SF” sessions, we
SF
obtained 1) “deviation (DNF
j , Dj )” curves of absolute normalized
SF
deviations (|djk|) and 2) “variability (CVNF
j , CVj )” curves of coefficient of variation (cvjk) as a function of ⌬Tjk by averaging them across
sessions separately for NF and SF conditions (Eqs. 1 and 2 with
averaging across session index k, not across interval index j, and
replacing Nj with NNF
and NSF
k
k , 4 and 6). We then estimated learning
indexes for accuracy (LIA) by dividing the difference between the
pre- and postfeedback deviations by their sum:
LIA j ⫽
SF
DNF
j ⫺ Dj
DNF
j
⫹
.
DSF
j
(3)
We obtained a “feedback transfer curve” for timing accuracy by
plotting these LIAs as a function of ⌬T. Note that the feedback
transfer curve is different from the specificity measures used previ-
Fig. 2. Example IT responses in a single session from a representative subject.
A: IT responses are plotted against corresponding physical time intervals. The
dashed line is an identity line (⌬t ⫽ ⌬T). The dotted lines above and below the
identity line are ⌬t ⫽ 1.5 ⫻ ⌬T and ⌬t ⫽ 0.5 ⫻ ⌬T, respectively. The solid
line is a linear regression of the means of timing responses to physical time
intervals. The error bars represent SE of means at ⌬Ts. The raw data manifest
the scalar property (broader ⌬t distribution with increasing ⌬T) and Vierordt’s
law [under- and overestimated ␮(⌬t) for long and short ⌬T, respectively].
B: to illustrate the scalar property of the data shown in A, the standard
deviations of the data distributions [␴(⌬t)] are plotted against the corresponding means of the distributions [␮(⌬t)]. The solid black line is a linear function
with a slope indicating a constant coefficient of variation (CV), which was
fitted with linear regression.
ously in that the test ⌬Ts have not been monitored during training for
the pre- and posttraining comparison in previous studies. Similarly,
we also acquired a feedback transfer curve with learning indexes for
precision (LIP) based on the CV data:
LIP j ⫽
SF
CVNF
j ⫺ CV j
SF
CVNF
j ⫹ CV j
.
(4)
The perceptual learning indexes range from ⫺1 to ⫹1, positive values
indicating improvements by feedback and negative values indicating
deteriorations. Finally, we averaged the accuracy (|djk|) and precision
curves (cvjk) separately across subjects within each session to assess
dynamics of interval-specific feedback effects. The 95% confidence
intervals (C.I.) for those average accuracy and precision measure-
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For each session, we removed from analysis outlier trials in which
a ⌬t fell away from the mean response time ␮(⌬t) for a given ⌬T by
more than 3 standard deviations [⬎3 ⫻ ␴(⌬t)] because both of the two
major statistics in our study are sensitive to extreme outliers. The
fraction of the outliers did not exceed 2% of the total trials. No miss
or outlier trials were found in the control (ME and SD tasks) trials. In
addition, we excluded from analysis the trials that were trained with
feedback in the SF condition.
Within-session analysis of accuracy and precision in IT. A response
time (⌬tijk, where i, j, and k indicate a trial number, an index for ⌬T, and
a session number, respectively) was sorted into one of the nine sets
defined by the jth physical time interval in a given session k (⌬Tjk), as in
Fig. 2A. For each of these sets, the mean and standard deviation of
perceived time intervals were computed [␮(⌬t)jk and ␴(⌬t)jk].
To characterize the overall mean accuracy for a given session k, we
computed Dk, an across-⌬T mean of normalized deviations of ⌬tijk
from corresponding ⌬Tjk:
348
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
^
⌬d j,k⫽5⫽
兺 共ⱍd j,kⱍ⫺ⱍd j,k⫺1ⱍ兲 ⁄ Nk⫽5 for j ⫽ non-feedback ⌬Ts
k⫽5
⫽ 0 for j ⫽ feedback ⌬T.
(5)
Second, we then corrected the original feedback effects at the fifth
session for the central bias-induced changes in accuracy by parsing
out ⌬d̂j,k⫽5 from the original feedback effects:
BCFE j ⫽ 共ⱍd j,k⫽4ⱍ⫺ⱍd j,k⫽5ⱍ兲 ⫹ ⌬d j,k⫽5 ,
^
(6)
where BCFEj is a feedback effect (|dj,k⫽4| ⫺ |dj,k⫽5|) that is corrected
for bias (⌬d̂j,k⫽5) at the jth ⌬T.
To statistically test whether these BCFEjs are tuned around the
feedback ⌬T, we performed the Wilcoxon signed-rank test on the 16
pairs of the feedback ⌬T and ⌬Tfars (4 subjects ⫻ 4 ⌬Tfars), where
⌬Tfars were defined as the two shortest and two longest ⌬Ts. In
addition, to evaluate feedback-induced timing enhancements relative
to the variability of changes in timing accuracy in the remaining pairs
of consecutive sessions, we transformed BCFE values into z values by
dividing them by the standard deviation of changes in timing accuracy
between all possible pairs of the consecutive sessions except those
involved in the fourth and the fifth sessions. We found that the
normalized BCFE value (merged across subjects) was significantly
different from zero only around the feedback interval [z ⫽ 2.011 (P ⬍
0.05) for 1.32 s and z ⫽ 2.05 (P ⬍0.05) for 1.69 s] and decreased
gradually as a function of temporal distance from the feedback
interval, indicating that the introduction of feedback at the 1.32-s
interval in the fifth session generated highly substantial degrees of
feedback interval-specific accuracy enhancements, which are well
beyond the range of feedback-unrelated changes in timing accuracy.
Bayesian observer model for PL of IT. To predict ⌬t for a given ⌬T,
the model combines a likelihood function tuned for a ⌬T with a prior
probability distribution, both of which we assumed as Gaussian
distributions. We opted for the Gaussian prior and Gaussian likelihood
for the sake of computational and interpretive convenience by exploiting the fact that the Gaussian distribution is a conjugate prior of the
Gaussian likelihood (see DISCUSSION for the assumption). The Gaussian distribution is completely characterized by two statistics, the mean
and standard deviation, for which each of the prior and the likelihood
has its own value (␮prior, ␮likelihood, ␴prior, ␴likelihood) in the model. By
applying Bayes’ rule, the mean and standard deviation of the posterior
distribution can be obtained as follows:
␮ posterior ⫽
␴2prior
2
2
␴ prior ⫹ ␴likelihood
· ␮likelihood ⫹
2
␴likelihood
2
␴2prior ⫹ ␴likelihood
· ␮ prior , (7)
␴2posterior ⫽
␴2prior
2
␴2prior ⫹ ␴likelihood
2
· ␴likelihood
.
(8)
The posterior mean, which is equal to a mode or median for the
Gaussian distribution, is a weighted average or compromise of ␮prior
2
and ␮likelihood with their relative proportion of ␴2likelihood and ␴prior
as
2
weights. The posterior variance is a product of ␴prior
and ␴2likelihood
divided by their sum. We assumed that 1) ␮likelihood is unbiased and
equal to ⌬T, although the likelihood can be contaminated by sensory
noise, and 2) the relationship between ␮likelihood and ␴likelihood is
defined by the scalar property. The assumption of unbiased ␮likelihood
has been adopted in previous studies and is known to have negligible
impacts on modeling outcomes (Alais and Burr 2004; Ernst and
Banks 2002; Jazayeri and Shadlen 2010; Körding and Wolpert
2004; Stocker and Simoncelli 2006). The latter assumption enabled
us to greatly reduce the number of free parameters using a single
CV value instead of using nine (1 for each of the 9 ⌬Ts) ␴likelihoods
(Jazayeri and Shadlen 2010; Miyazaki et al. 2006). Our assumptions were validated by extensive model comparisons (see below
for comparisons of nested model variants). Accordingly, three free
parameters, including ␮prior, ␴prior, and CV, were fitted to a data set
from a daily session. The resulting posterior can be written as a full
conditional probability function:
ⱍ
P共⌬t ⌬T, ␴ prior, ␮ prior, CV兲
⬃N
冉
␴2prior
␴2prior ⫹ (CV · ⌬T)2
· ⌬T ⫹
(CV · ⌬T)
2
␴2prior ⫹ (CV · ⌬T)2
· ␮ prior,
冑
␴2prior · (CV · ⌬T)2
␴2prior ⫹ (CV · ⌬T)2
冊
(9)
For each individual session from each subject, we estimated the
best-fitting parameters that maximize the likelihood function of the
␴prior, ␮prior, and CV, assuming statistical independence among trials.
We also fitted the model to data from a pilot experiment, which is
identical to the main experiment except for the number of trials (⬃100
trials per session), sessions (6 daily sessions without feedback and 9
sessions with specific feedback to 2 s), session structure (the trials
with feedback are separately blocked and placed before the main IT
task in the SF condition), and sampled intervals (random sampling
from continuous uniform distribution from 0.5 s to 6.5 s). Although
the nondiscrete sampling hinders application of the above analysis for
accuracy and precision, we were able to fit the three-parameter
Bayesian model to the pilot data from six additional subjects. Similar
to the main data (see RESULTS), we found the ␴prior (rank correlation
for across-subject average, ␳ ⫽ ⫺0.95, P ⬍ 0.001), not that of the
likelihood (rank correlation for across-subject average CV, ␳ ⫽
⫺0.02, P ⫽ 0.94), gradually decreased along the sessions.
Comparison of nested model variants. We evaluated eight nested
model variants in terms of goodness of fit, using Bayesian information
criteria (BIC), which take into account sample size and number of
parameters for the model complexity to place competing models on
equal footing (Pitt and Myung 2002). The tested model variants,
which differ in assumptions for prior and likelihood distributions, can
be sorted out in decreasing order of number of free parameters: 1) an
“exaggerated” model with interval-specific priors and interval-specific
likelihoods as free parameters; 2) a “full” model, same as 1 but now
with a single prior common to all ⌬Ts set to be free; 3) a “fixed mean
of likelihood” variant of the reduced model, same as 2 but with
␮likelihood fixed to corresponding ⌬Ts; 4) and 5) “fixed mean of prior”
variants of the reduced model, same as 3 but now with the ␮prior fixed
to be equal either to the mean (4) or the median (5) of the entire set
of ⌬Ts, respectively; 6) a “single CV” variant of the reduced model,
same as 3 but now with a single CV parameter that generates the
␴likelihood based on the scalar property; and 7) and 8) minimal models,
same as 6 but now with the ␮prior fixed either to the mean (7) or the
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ments were obtained by resampling IT data for a ⌬Tjk in a daily
session with replacement and applying the same procedure to the
1,000 bootstrapped data sets to estimate our measures.
Dissociation of Vierordt’s law-associated and feedback-driven improvement in timing accuracy. To test a possibility that the Vierordt’s
law-associated effect (see RESULTS)— overestimated and underestimated ␮(⌬t) for relatively short and long ⌬Ts— confounds specificity
of the feedback effect, we linearly decomposed the improved accuracy
at the first SF session into contributions from those two effects and
compared the bias-corrected feedback effect (BCFE) across the feedback and nonfeedback ⌬Ts. First, for each ⌬T in each individual, a
“change in accuracy performance from the fourth to the fifth session
that can be attributed to Vierordt’s law,” ⌬d̂j,k⫽5, was estimated by
averaging the changes in accuracy performance between all possible
pairs of the neighboring sessions except that involved in the fifth
session (averaging across the pairs only within the NF condition also
leads to qualitatively similar and statistically significant results):
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
RESULTS
cient of variation (CV) values (mean across sessions and
subjects, 0.24; standard deviation, 0.048) were also comparable
to those in previous studies (Lewis and Miall 2009; Wearden
2008).
In addition, the ␮(⌬t)s departed from their corresponding
veridical ⌬Ts in an orderly way: ␮(⌬t) was overestimated for
short ⌬Ts and underestimated for long ⌬Ts, which has been
referred to as Vierordt’s law (Fig. 2A) (Fortin et al. 2009; Jones
and McAuley 2005; Kanai et al. 2006; Lewis and Miall 2009;
McAuley and Miller 2007; Penney et al. 2008; Wearden 2008;
Zarco et al. 2009). The statistical significance of Vierordt’s law
in our data can be tested by regressing ␮(⌬t) as a linear
function of ⌬T because the bias toward the center of sampled
⌬Ts is translated into a linear line with a slope smaller than 1
and an intercept larger than 0. The slopes (mean 0.77, standard
deviation 0.20) were significantly smaller than 1 [t(39) ⫽
⫺7.55, P ⬍ 10⫺8, 1-sample left-tailed t-test], and the intercepts
(mean 0.32, standard deviation 0.18) were significantly larger
than 0 [t(39) ⫽ 11.55, P ⬍ 10⫺13, 1-sample right-tailed t-test].
This pattern of bias that conforms to Vierordt’s law tended to
become gradually stronger in magnitude and more consistent
across subjects as an order of daily sessions (Fig. 3). The
intercept and the slope were found to increase, when averaged
across subjects (Spearman’s rank correlation, ␳ ⫽ 0.91, P ⬍
0.001), and to tend to decrease (rank correlation, ␳ ⫽ ⫺0.56,
P ⫽ 0.09), respectively. This gradual augmentation of bias in
mean accuracy will be addressed again below when our Bayesian model predicts the intimate relationship between biases in
mean accuracy and improvements in timing precision.
Scalar Property and Vierordt’s Law in Response Time
Distributions
Dissociation Between Timing Accuracy and Precision Across
Sessions
In our implicit IT task, unlike typical IT tasks, subjects were
not asked to judge or reproduce directly perceived time intervals (⌬t). Instead, after a drifting bar became invisible, subjects
simply reacted to the invisible bar by estimating when it would
arrive at a designated location on the annulus (Fig. 1B). By
measuring the time that elapsed between the disappearance of
the visible bar and subjects’ response, we obtained an estimate
for ⌬t as a component of the physical equation, ⌬T ⫽ ⌬X/V,
when the information about V and ⌬X is available to subjects
(Fig. 1C; see METHODS for the specific task strategies adopted
by subjects). Across 10 daily sessions (Fig. 1D), we collected
sets of ⌬t for a range of physical time intervals (⌬T) and
computed their means [␮(⌬t)] and standard deviations [␴(⌬t)]
(Fig. 2).
We confirmed the validity of our data as a necessary condition of IT by examining whether the ⌬t distribution conforms
to the scalar property, a hallmark signature of IT (Buhusi and
Meck 2005; Wearden 2008). The linear relation between ␮(⌬t)
and ␴(⌬t) was quite evident from the ⌬t distributions that were
sorted by physical ⌬T: spread of the distributions increased
monotonically with an increasing ⌬T (Fig. 2A for data from a
representative subject). To quantitatively evaluate the scalar
property in our data, we tested whether the correlation coefficient between ␮(⌬t) and ␴(⌬t) is significant within each
session (Fig. 2B for representative data). The correlation between ␮(⌬t) and ␴(⌬t) was significant in all of the sessions for
all subjects (r ⫽ 0.89 – 0.99; P ⫽ 10⫺3-10⫺7). When ␴(⌬t) was
normalized by the corresponding ␮(⌬t), the resulting coeffi-
To monitor changes in mean accuracy of IT across sessions,
we first normalized offsets of mean response times at corresponding physical intervals by dividing them by physical
intervals {d ⫽[␮(⌬t) ⫺ ⌬T]/⌬T; Eq. 1}. Then, by averaging
the absolute values of normalized deviations or constant errors
(|d|) across ⌬Ts in a given session, we indexed the overall
degree of accuracy (D) in IT for that session (Fig. 4A; Eq. 1).
To examine whether the averaged accuracy increased or decreased monotonically as a function of session order, we fitted
an exponential function to the time course of mean accuracy
and inspected the signs of the multiplication and exponent
terms (see METHODS). If the multiplication term a and the
constant b in the exponent have different signs, the learning
curve can be interpreted as decreasing monotonically. The
multiplication of those two terms was not consistent across
subjects, being negative for two subjects [S1, a ⫻ b ⫽ ⫺0.050
(C.I. ⫽ [⫺0.060, ⫺0.036]); S2, a ⫻ b ⫽⫺8.3 ⫻ 104 (C.I.⫽
[⫺2.6 ⫻104, ⫺1.3 ⫻104])] and positive for the other two
subjects [S3, a ⫻ b ⫽ 0.009 (C.I. ⫽ [0.006, 0.014]); S4, a ⫻
b ⫽ 0.36 (C.I. ⫽ [0.23, 0.45])]. For goodness of fit, the
percentage of variance explained ranged from 19.2% to 62.4%,
indicating that the overall mean accuracy D neither decreased
nor increased monotonically across sessions (rank correlation
for D averaged across subjects, ␳ ⫽ ⫺0.16, P ⫽ 0.66).
Note that the feedback was not provided until the fifth daily
session and was given only to one particular interval (⌬T ⫽
1.32 s) from the fifth session (Fig. 1D). In the “specific
feedback (SF)” sessions, feedback was provided by showing
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median (8) of the entire set of ⌬Ts, respectively. For details of these
models, see Table 1, which compares the models in terms of assumptions for prior and likelihood distributions, number of free parameters,
and BIC values from best-fitting parameters.
We fitted models separately to data from individual sessions in
each subject, using the maximum likelihood estimation method with
the “fmincon” function in MATLAB (MathWorks). We found that the
“single CV” variant of the reduced model (the 6th model, with free
␮prior, ␴prior, CV and the veridical ␮likelihood) produced the minimum
BIC values in the majority of sessions (mean rank ⫾ SE ⫽ 1.78 ⫾
0.17 among 8 tested models). This model also turned out to be the best
for all the subjects when BIC values were pooled over the entire
sessions, indicating that the “single CV” model variant was the best
among all the model variants when both predictive power and model
complexity were considered. The exaggerated model and the full
model did not differ significantly in predictive power, indicating that
the multiple excess priors assumed in the exaggerated model are
unnecessary. Even when the models were compared only in log
likelihood, not in BIC, the “single CV” variant with 3 free parameters
performed only marginally worse than the “fixed mean of likelihood”
variant (the 3rd model), which has 11 free parameters (mean rank ⫾
SE ⫽ 2.95 ⫾ 0.25, the second minimum among the 8 models), despite
the considerable difference in degrees of freedom (3 vs. 11).
We confirmed the stability of results by systematically varying
initial guesses for free parameters. We also examined effects of lower
bounds for parameters on fitting results. Obviously, ␴likelihood and
␴prior have a constraint of positivity for their sign. However, it is less
obvious whether we should impose the positivity constraint on ␮prior
and ␮likelihood as well. We found that the lower bounds of zero for
␮prior and ␮likelihood show little effect on the BIC values.
349
A common prior to
⌬Ts (1)
Mean(⌬T) as a
constant ␮prior
across sessions (0)
Median(⌬T) as a
constant ␮prior
across sessions (0)
A common prior to
⌬Ts (1)
Mean(⌬T) as a
constant ␮prior
across sessions (0)
Median(⌬T) as a
constant ␮prior
across sessions (0)
Reduced model with
veridical ␮likelihood
Reduced model with
veridical ␮likelihood and
mean(⌬T) as ␮prior
Reduced model with
veridical ␮likelihood and
median(⌬T) as ␮prior
Reduced model with
veridical ␮likelihood and
free CV, ␴prior, ␮prior
Minimal model with
mean(⌬T) as ␮prior
Minimal model with
median(⌬T) as ␮prior
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
A common prior
to ⌬Ts (1)
Multiple priors for
each ⌬T (9)
␴prior
Constant and equal
to ⌬Ts (0)
Constant and equal
to ⌬Ts (0)
Constant and equal
to ⌬Ts (0)
Constant and equal
to ⌬Ts (0)
Constant and equal
to ⌬Ts (0)
Constant and equal
to ⌬Ts (0)
Free for ⌬Ts (9)
Free for ⌬Ts (9)
␮likelihood
110 ⫽ (1 ␮prior ⫹ 1 ␴prior ⫹ 9 ␴likelihood) ⫻
10 sessions
100 ⫽ (1 ␴prior ⫹ 9 ␴likelihood) ⫻ 10
sessions
100 ⫽ (1 ␴prior ⫹ 9 ␴likelihood) ⫻ 10
sessions
30 ⫽ (1 ␮prior ⫹ 1 ␴prior ⫹ 1 CV) ⫻ 10
sessions
20 ⫽ (1 ␴prior ⫹ 1 CV) ⫻ 10 sessions
20 ⫽ (1 ␴prior ⫹ 1 CV) ⫻ 10 sessions
Free for ⌬Ts (9)
Free for ⌬Ts (9)
Free for ⌬Ts (9)
Veridical ␮likelihood
(⫽⌬T) ⫻ CV (1)
Veridical ␮likelihood
(⫽⌬T) ⫻ CV (1)
Veridical ␮likelihood
(⫽⌬T) ⫻ CV (1)
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
S1
S2
S3
S4
360 ⫽ (9 ␮prior ⫹ 9 ␴prior ⫹ 9 ␮likelihood ⫹
9 ␴likelihood) ⫻ 10 sessions
200 ⫽ (1 ␮prior ⫹ 1 ␴prior ⫹ 9 ␮likelihood ⫹
9 ␴likelihood) ⫻ 10 sessions
Subject
2,686.7
4,498.2
5,767.1
5,034.5
1,424.4
3,124.6
4,329.1
3,550.7
959.14
2,581.1
3,696.1
2,909.7
1,965.9
3,045.9
4,253.6
3,279.8
1,695.5
3,018.6
4,339.7
3,468.6
631.62
2,108.5
3,127
2,620.8
1,540
2,598.1
3,884.4
3,286
1,201.8
2,563.7
4,030.8
3,596.9
BIC value
BIC of Best Fits for
Subjects
No. of Total Free Parameters
Free for ⌬Ts (9)
Free for ⌬Ts (9)
␴likelihood
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Note that number of the whole data points for each subject is approximately 2,700 after excluding outliers [9 physical time intervals (⌬Ts) ⫻ 30 trials ⫻ 10 sessions]. CV, coefficient of variation; BIC,
Bayesian information criteria.
A common prior to
⌬Ts (1)
Multiple priors for
each ⌬T (9)
␮prior
Full model
Exaggerated model
Model Name
Model Assumptions (no. of free parameters for a daily session)
Table 1. Comparison of several nested model variants
350
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
351
subjects a snapshot of the true location of the moving bar at the
time of a key press (Fig. 1B, right). In other words, the
feedback was a spatial offset of the bar from the nonius line at
the moment of a key press, which in principle can be translated
into a temporal offset (⌬t ⫺ ⌬T) when divided by the speed (V)
(Fig. 1C). We examined whether abrupt changes in timing
accuracy occurred after the feedback was provided. A significant change was observed between the last “no feedback
(NF)” session and the first SF session [t(3) ⫽ 3.61, P ⫽ 0.04;
a planned paired t-test], indicating that the feedback improved
the overall timing accuracy immediately even though feedback
was given only to one particular interval. To further examine
the abrupt enhancement of timing accuracy, we tracked the
time course of accuracy over individual trials within the first
SF session. The deviations of ⌬ts from corresponding ⌬Ts
became smaller only after several feedback trials and then
quickly reached an asymptote, remaining relatively stable.
These asymptotic and stable deviations are in great contrast
with those observed in the preceding NF sessions, which
fluctuated unpredictably (data not shown). Across the sessions
both before and after the instantaneous feedback effect, the
mean deviation tended to increase gradually (Fig. 4A), in line
with the slowly growing biases toward the center ⌬T. Intervalspecific impacts of feedback on timing accuracy are addressed again in another section below, where we inspect
the specificity of perceptual learning and its relationship
with the gradual buildup of central biases in ␮(⌬t) toward
the center ⌬T.
Session-by-session changes in timing performance were also
examined in terms of overall precision by averaging CV values
across intervals within each session (Eq. 2). By contrast with
the time course of timing accuracy, the mean CV gradually
decreased over sessions (Fig. 4B; rank correlation for CV
averaged across subjects, ␳ ⫽ ⫺0.90, P ⬍ 0.001). The monotonic decrease in timing precision over sessions was evident in
all of the subjects, which was confirmed by significant negative
values for multiplication of the constant and exponent terms in
the fitted exponential function [S1, a ⫻ b ⫽⫺0.25 (C.I. ⫽
[⫺1.10, ⫺0.06]); S2, a ⫻ b ⫽ ⫺0.008 (C.I. ⫽ [⫺0.01,
⫺0.006]); S3, a ⫻ b ⫽ ⫺0.07 (C.I. ⫽ [⫺541, ⫺0.007]); S4,
a ⫻ b ⫽ ⫺0.03 (C.I. ⫽ [⫺0.05, ⫺0.02])]. The goodness of fits
was better than those for the accuracy data: the percentage of
variance accounted for ranged from 55.8% to 88.5%. Again in
contrast with the accuracy data, there hardly existed any
impacts of feedback on time courses of timing precision. The
amount of difference in the mean CV between the last NF and
the first SF sessions was not significant [t(3) ⫽ ⫺0.20, P ⫽
0.86; a planned paired t-test]. Instead, the monotonic decrease
in precision seemed to be present even in the NF sessions alone
(rank correlation for CV averaged across subjects, ␳ ⫽ ⫺1.00,
P ⫽ 0.08).
On the basis of the inspection of the overall pooled-across-⌬T
error metrics in timing performance over sessions, we conclude
that accuracy improved only immediately after the introduction of
feedback, whereas precision improved steadily throughout the
whole sessions regardless of the feedback.
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Fig. 3. Across-session changes of ␮(⌬t) ⫺ ⌬T with linear fits as a function of ⌬T. Mean perceived intervals [␮(⌬t)] were regressed as a linear function of their corresponding
veridical ⌬T for each session (solid gray lines). For illustration, ␮(⌬t) ⫺ ⌬T is shown with ⌬T as the x-axis in logarithmic scale. The ␮(⌬t) became increasingly overestimated
for short ⌬T and underestimated for long ⌬Ts over sessions compared with the horizontal line [␮(⌬t) ⫽ ⌬T; dashed black lines].
352
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
Across-Session Dynamics of Interval-Specific Perceptual
Learning
We first assessed impacts of feedback on timing performance by comparing the mean accuracy and precision before
(NF sessions) and after (SF sessions) the feedback. For timing
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Fig. 4. Time courses of timing accuracy and precision averaged across time
intervals. A: to estimate the overall mean accuracy of IT, we first computed the
deviations of ␮(⌬t)s from their corresponding physical time intervals with
correction for across-interval differences in error magnitude due to the scalar
property (see METHODS; Eq. 1). The across-interval means of these normalized
deviations are plotted against the order of learning sessions. Different symbols
in gray indicate different subjects. Open and filled symbols represent sessions
without feedback (NF) and with specific feedback (SF), respectively. Black
circles are the means across subjects. Error bars are 95% confidence intervals
that were estimated from 1,000 data sets resampled from the bootstrap
procedure. Despite no noticeable monotonic trend across sessions, the mean
deviation decreased significantly between the last NF session and the first SF
session [P ⫽ 0.04, t(3) ⫽ 3.61], indicated by the asterisk. B: CV estimates, an
example of which is shown in Fig. 2B, are plotted against the order of learning
sessions (Eq. 2). Unlike in the mean accuracy data, the CV values exhibited a
persistent reduction on a long-term scale without showing a significant change
between the last NF session and the first SF session (n.s., not significant).
Symbols are identical to those in A.
accuracy, we averaged absolute normalized deviations (|d|)
across the NF and the SF sessions separately, resulting in the
“prefeedback” and “postfeedback” deviation curves, respecSF
tively, as a function of ⌬T (DNF
j , Dj ; Fig. 5A). Then, for each
⌬T, we indexed perceptual learning by dividing the difference
between the pre- and postfeedback deviations by their sum (Eq.
3). We obtained a “feedback transfer curve” for timing accuracy by plotting these LIAs as a function of ⌬T relative to the
feedback ⌬T (Fig. 5B). Similarly, we also acquired a feedback
transfer curve for timing precision based on the CV data (CVs
for Fig. 5D; LIPs for Fig. 5E and Eq. 4).
There were interval-specific improvements in timing accuracy after the feedback: the improvement in timing accuracy
was maximal around the feedback ⌬T and gradually diminished as the difference between a test and the feedback intervals increased (Fig. 5A), yielding smooth transfer curves for all
subjects (Fig. 5B). Statistically significant improvements in
accuracy were indicated by the positive LIA values whose 95%
confidence intervals estimated by a nonparametric bootstrapping were above zero in Fig. 5B. Conversely, postfeedback
improvements in timing precision were not confined to the
feedback ⌬T (Fig. 5D). Although the overall degree of improvements in precision (estimated by LIP) varied among
subjects, the feedback transfer curve did not show any systematic patterns as in the timing accuracy (Fig. 5E), indicating that
the improvements by feedback in timing precision did not vary
depending on test intervals relative to the feedback ⌬T.
Given that the feedback was given to the median ⌬T and there
were the slowly growing biases toward the central ⌬T (Fig. 2A),
the observed “feedback interval-tuned” improvements in timing
accuracy could also be attributed to the growing central biases,
which were likely to lead to deterioration in timing accuracy at the
short and long ⌬Ts predicted by Vierordt’s law (Fig. 3, Fig. 4A).
To check this possibility, we estimated the contribution of the
central biases to the feedback interval-tuned calibration of mean
accuracy at the fifth session with ⌬d̂j,k⫽5, that is, the averaged
changes in the deviations between all possible pairs of the neighboring sessions except the fourth and fifth (jth ⌬T, k for session;
Eq. 5), and examined whether the feedback specificity in accuracy
enhancement remains to exist in the feedback effects that were
corrected for the contribution of the central biases. We found that
the bias-corrected feedback effects (Eq. 6) were maximal around
the feedback ⌬T and then decreased as it moved away from the
feedback ⌬T toward either the shortest or longest ⌬Ts (P ⫽
0.0052, z ⫽ ⫺2.792, Wilcoxon signed-rank test). The same results were obtained when we reinspected the bias-corrected feedback effects after normalizing them for the interval-specific variability of changes in timing accuracy in all possible pairs of the
neighboring sessions except the fourth and fifth (see METHODS for
details). On the basis of these results, we conclude that the
feedback-specific improvements in timing accuracy are still present even after the potential contribution of the growing central
biases to feedback effects was parsed out.
To reveal further the dissociative nature of timing error
measures, we also tracked dynamic changes over daily sessions
in the deviation (d) and variability (CV) curves. We averaged
them across subjects and estimated confidence intervals for
those sample mean statistics as the same bootstrap procedure
adopted above (Fig. 5, C and F). Before feedback, the deviation curves did not exhibit a constant shape and fluctuated
unstably in an overall level with a tendency to increase in the
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
353
order of sessions. As soon as feedback was provided on trials
with the median ⌬T (1.32 s), the deviation curve changed
dramatically by conforming to a “V” shape. The reduction in
deviation peaked around the feedback ⌬T and decreased with
an increasing difference between the test and the feedback
intervals. This abrupt promotion of interval-specific learning in
accuracy by feedback can be readily appreciated by a comparison between the deviation curve from the last NF session
(lightest blue line in Fig. 5C) and that from the first SF session
(orange line in Fig. 5C). The across-session dynamics of the
variability curves was completely different from that of the
deviation curves in two respects. First, the variability curves
did not exhibit any interval-specific effects of feedback in any
daily sessions (Fig. 5F). Second, overall interval-nonspecific
improvements in timing precision developed slowly and
steadily throughout the entire daily sessions, including the NF
sessions (open symbols in Fig. 5F). This is in line with the
monotonic decrease of the across-⌬T mean CV (Fig. 4B).
Putting together the across-session dynamics of timing errors and the interval-specific effects of feedback, we conclude
that feedback interval-dependent learning occurs immediately
after the feedback for timing accuracy whereas the interval-
nonspecific learning gradually accumulates over sessions regardless of the feedback for timing precision.
Control for Sensorimotor Function and Speed Perception
We have been attributing the ⌬t distributions to the variations of IT. Because of the nature of our task, however, there
are two other sources that potentially contribute to the observed
⌬t distributions. First, the variation in speed perception could
have affected the ⌬t distributions because errors in perceived
speed of the bar during the “visible motion” would cause errors
in timing response (Fig. 1C). To check this possibility, we
asked subjects to perform a speed discrimination (SD) task
shortly after blocks of IT task trials in each daily session (Fig.
1D). On each trial, subjects viewed two moving bars sequentially and judged which of the two bars moved faster. While
setting the reference speed to 8°/s, which roughly matched the
mean speed (8.22°/s) in the IT task, we used an evenly
distributed set of six test speeds around the reference speed. By
computing the mean and the standard deviation of a cumulative
Gaussian function fitted to each session’s psychometric curve
(Fig. 6A), we assessed how the accuracy and precision in speed
perception changed over sessions. If the observed changes in
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Fig. 5. Interval-specific learning in timing accuracy and precision. A and D: timing accuracy error estimates (|djk| in Eq. 1; A) and timing precision error estimates
(cvjk in Eq. 2; D) were averaged across the NF sessions (open symbols) and the SF sessions (filled symbols) at each physical time interval (⌬T). Error bars are
SE. Data were from a representative subject, S4. B and E: changes in timing error between before and after the feedback were summarized by 2 types of learning
indexes, one for accuracy (LIA in Eq. 3; B) and the other for precision (LIP in Eq. 4; E). The learning index was computed by dividing the difference between
the prefeedback and postfeedback deviations (A) and cv values (D) by their sum (see METHODS for details). In B, the “feedback transfer” curves of LIA show
interval-dependent improvements in timing accuracy with maxima at around the feedback interval (note that peak of the feedback transfer curve seems slightly
shifted to long intervals in 2 subjects, which may be related to the issue of whether internal scale of time interval is linear or logarithmic). In contrast, the feedback
transfer curves of LIP show overall improvements in an interval-nonspecific manner. Error bars are 95% confidence intervals that were estimated from 1,000
data sets resampled from the bootstrap procedure. C and F: deviation curves (|djk| in Eq. 1; C) and variability curves (cvjk in Eq. 2; F) are averaged across subjects
and plotted for individual sessions to illustrate the development of interval specificity of learning over daily sessions. While the deviation curves exhibited a “V”
shape immediately after the feedback was provided to 1.32 s (C), the variability curves showed slow reduction throughout the whole daily sessions without any
interval-specific effect of feedback (F). Open and filled symbols represent data from the NF and SF sessions, respectively. Different colors indicate different
sessions, with temperature representing the order of sessions. Error bars are SE of the means across subjects.
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PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
timing accuracy and precision across sessions (Fig. 4) were
caused by changes in speed perception, the across-session
dynamics of speed perception should be similar to that of IT.
However, neither of the two measurements exhibited any
noticeable changes across sessions [Fig. 6, B and C; paired
t-test between PSEs of SD task in 4th and 5th sessions, t(3) ⫽
⫺0.44, P ⫽ 0.69; rank correlation for 1/sensitivity of SD task
averaged across subjects, ␳ ⫽ 0.15, P ⫽ 0.68], invalidating
speed perception as a source for systematic changes in IT that
we observed in the IT task across sessions.
Second, the variation in IT may have also occurred at the
stage of motor execution (ME). To test this possibility, we
inserted ME task trials randomly between the IT task trials
(Fig. 1D). The ME task was exactly same as the IT task, except
that the moving bar did not disappear and continued to drift
even when passing through the occlusion arc. Because subjects
simply responded to the arrival of the bar at the nonius line by
pressing a button, there was no need for subjects to estimate
⌬Ts to perform the ME task. The ME data did not exhibit a
scalar property, indicated by a constant level of standard
deviation across different ⌬Ts (Fig. 6D). To know whether the
ME trial data exhibit patterns conforming to Vierordt’s law that
were observed in the IT data, we conducted linear regression
analysis separately on the ME trials of each session in all
subjects. The slope (0.98 ⫾ 0.02) and intercept (0.05 ⫾ 0.04)
were significantly smaller than 1 [t(39) ⫽ ⫺5.93, P ⬍ 10⫺6,
1-sample left-tailed t-test] and larger than 0 [t(39) ⫽ 7.36, P ⬍
10⫺10, 1-sample right-tailed t-test], respectively. Despite these
statistical significances of Vierordt’s law in ME data, these
results are unlikely to suggest that the error patterns in ME
trials determined the observed across-session changes in timing
accuracy in the main IT task (the promotion of central biases)
for the following two reasons. First, unlike the IT data, the
slope and intercept for the ME trials did not show gradual
changes across sessions (rank correlation for slope, ␳ ⫽ 0.21,
P ⫽ 0.56; rank correlation for intercept, ␳ ⫽ 0.39, P ⫽ 0.26).
Second, the effect sizes of the central biases in the ME trials
were marginal (average slope, 0.98; average intercept, 0.05)
and substantially smaller than those in the main IT trials [slope:
0.98 ⫾ 0.02 (ME) vs. 0.77 ⫾ 0.20 (IT), P ⬍ 10⫺5; intercept:
0.05 ⫾ 0.04 (ME) vs. 0.32 ⫾ 0.18 (IT), P ⬍ 10⫺7; paired
Wilcoxon signed-rank tests between ME and IT data]. Without
the correction for scalar property, the time course of mean
accuracy in the IT task, which may be primarily determined by
data with long ⌬T, differed from that of the motor control tasks
(Fig. 6E). In addition, the significant feedback effect was
absent in ␮(⌬t ⫺ ⌬T) for motor control [Fig. 6E; paired t-test
between 4th and 5th sessions, t(3) ⫽ ⫺0.50, P ⫽ 0.65].
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Fig. 6. Accuracy and precision for speed perception and motor execution. A: in the speed discrimination task, subjects indicated in which interval the bar moved
faster. The proportion of responses that subjects judged faster than the reference speed (8°/s) is plotted as a function of the speed of the test bar. Open and filled
symbols represent the NF and SF sessions, respectively. Dashed and solid lines are cumulative Gaussian functions fitted to the psychometric curves of the NF
and SF sessions, respectively. The data that are shown here were from a single representative subject, S2. B and C: by estimating the mean (point of subjective
equality, PSE) and standard deviation (slope) of the fitted cumulative Gaussian functions in A, we acquired the error measurements for speed discrimination
accuracy by computing deviations of PSEs from the reference speed (B) and those for speed discrimination precision from the inverse of sensitivity (C). D: in
the motor control task, subjects performed a task similar to the main timing task. The only difference was that the drifting bar did not undergo the “invisible”
period. Here, the motor response latency is plotted against the physical time interval ⌬T. Open and filled symbols represent the NF and SF sessions, respectively.
The data shown here were from the same subject as in A. E and F: means (E) and standard deviations (F) of motor errors are plotted as a function of the order
of daily sessions. The color and symbol scheme is same as in B and C. For comparison, the same procedure of estimating timing errors was applied to the IT
task data without correction for the scalar property. The across-subject means of these error estimates are plotted with star symbols.
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
Bayesian Modeling of Perceptual Learning in Timing
Accuracy and Precision
We found that the dynamics of timing accuracy was clearly
dissociated from that of timing precision in overall changes
across sessions (Fig. 4) and interval-specific effects of feedback (Fig. 5). Intriguingly, the framework of Bayesian probabilistic inference (Chater et al. 2006; Doya 2007; Friston 2010;
Griffiths et al. 2008; Kersten et al. 2004; Körding and Wolpert
2006; Maloney and Mamassian 2009) appears to offer a parsimonious way of explaining these seemingly complicated
dynamics of the timing accuracy and precision. Bayesian
models have been broadly applied to visual perception (Barthelmé and Mamassian 2009; Knill and Saunders 2003; Welchman et al. 2008; Zhaoping and Jingling 2008), multisensory
integration (Alais and Burr 2004; Burge et al. 2010; Ernst and
Banks 2002), and sensory-motor control (Dokka et al. 2010;
Hudson et al. 2008; Körding and Wolpert 2004; Stevenson
et al. 2009; Trommershauser et al. 2008), while providing
normative predictions about how to integrate a variety of
available information in a statistically optimal manner. When
the framework of Bayesian inference is applied to our IT task
situation, there are two types of information contributing to the
distribution of ⌬t. The first type is a probabilistic knowledge or
belief about physical time intervals [p(⌬T)], called a “prior
distribution,” which can be formed by previous experience or
naive expectation. Without incoming sensory events, an observer may rely on the prior to make a guess about how long
a ⌬T is. When confronting a sensory input such as ⌬T in our
case, the initial guess based on the prior can be updated with
additional information obtained from that sensory input
[p(⌬t|⌬T)], called a “likelihood function.” Final decisions
about time intervals across trials can be predicted from a
posterior function, which is jointly determined by the prior and the
likelihood [p(⌬T|⌬t) ⫽ p(⌬t|⌬T)p(⌬T)/p(⌬t)]. This leads to interesting predictions by the Bayesian model regarding two
timing error metrics of interest in our study. If we adopt
Gaussian distributions to model the prior and likelihood function, it can be analytically shown that the mean or maximum of
a posterior (MAP; ␮posterior) is the weighted average of the
means of the prior and the likelihood (␮prior, ␮likelihood) with
the relative proportion of their variance (␴2prior, ␴2likelihood) as a
weight to the counterpart mean (Eq. 7). As a result, the mean
of the posterior is predicted to fall midway in between the
␮prior and ␮likelihood. On the other hand, the variance of the
2
posterior (␴posterior
) is a product of the variances of the prior
and the likelihood divided by their sum (Eq. 8), consequently
becoming always smaller than the variance of the likelihood.
Of particular relevance, the Bayesian model provides a
parsimonious and analytical explanation for two systematic
across-session trends in our data: 1) the interval-nonspecific
improvement in timing precision across sessions (Fig. 4B, Fig.
5F, Fig. 7C) and 2) the increasing bias of ␮(⌬t) toward the
central ⌬T across sessions (Fig. 2A, Fig. 3, Fig. 4A, Fig. 7B).
Theoretically, Eq. 8 indicates that the interval-nonspecific
improvement in timing precision (␴posterior) could be caused by
reduction of either ␴prior or CV, which is analogous to
␴likelihood divided by ␮likelihood. However, the decrease in CV
alone is contradictory to the progressively increasing bias of
␮(⌬t) toward the center ⌬T with an increasing order of session
because it predicts a shift of the ␮posterior in the direction of the
␮likelihood (black vs. gray arrows in Fig. 7A, right), which is
opposite to what we observed (Fig. 3, Fig. 7B). By contrast, the
progressive reduction of ␴prior predicts both the nonspecific
reduction of ␴(⌬t) and the biased pattern of ␮(⌬t) during the
time course of PL (black vs. gray arrows in Fig. 7A, middle).
To confirm empirically the analytical explanation laid out
above, we developed several variants of nested Bayesian models, which differed in the number of free parameters and model
assumptions, and fitted them to a data set of ⌬t and ⌬T from
each daily session of each subject (Table 1; see METHODS for
details). Here, we assumed that the ␮likelihood is the same as ⌬T
in a given trial, as in previous studies on the Bayesian modeling of human psychophysical data (Miyazaki et al. 2006;
Stocker and Simoncelli 2006; Whiteley and Sahani 2008). The
scalar property was also assumed, leading to the linear association between ␮likelihood and ␴likelihood. Figure 7, B and C,
respectively, show the observed ␮(⌬t) and ␴(⌬t) over sessions,
along with the model fits, ␮posterior and ␴posterior, for a representative subject (S4). We found that, in all of the subjects, the
Bayesian model was successful in capturing the dynamics of
the observed ␮(⌬t) and ␴(⌬t), including the two major trends
of interest, the increasing bias of ␮(⌬t) toward the central ⌬T
(Fig. 7B) and the interval-nonspecific improvement in timing
precision (Fig. 7C). As analytically demonstrated in Fig. 7A,
the fitted model parameters (␴prior and CV) revealed that it is
the width of the prior (rank correlation for across-subject
averaged ␴prior, ␳ ⫽ ⫺0.94, P ⬍ 0.001), not that of the
likelihood (rank correlation for across-subject averaged CV,
␳ ⫽ 0.15, P ⫽ 0.68), that gradually decreased along the
sessions (Fig. 7D).
DISCUSSION
Using an implicit IT task, in which human subjects responded to an invisible bar drifting at a constant speed, we
tracked daily changes in distributions of timing responses for a
broad range of physical time intervals both before and after
feedback was given. Then we estimated two major types of
timing error based on the mean and standard deviation of a
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Regarding motor variability, however, the standard deviations
of motor latency decreased slowly across sessions [rank correlation for across-subject averaged ␴(⌬t ⫺ ⌬T), ␳ ⫽ ⫺0.77,
P ⫽ 0.01], indicating some degree of improvement in ME
precision, although there was no explicitly intended feedback
for the motor control task. This improvement occurred probably because subjects might calibrate their motor latencies by
comparing positions of the visible moving bar and another bar
presented briefly in between the nonius line at the moment of
button press (see METHODS). However, the reduction in motor
variability is unlikely to account for the observed changes in
timing precision. The size of motor variability reduction was
too small to explain the size of variability reduction in the IT
data, particularly at long ⌬T (Fig. 6F), probably due to the
absence of scalar property in motor variability. Moreover, the
time course of the motor variability did not correlate significantly with that of the timing precision except for only one
subject (S1; r ⫽ 0.71, P ⫽ 0.02). Thus we conclude that
across-session changes in the accuracy and precision of IT data
were largely independent of those of speed perception and
motor execution.
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PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
given distribution, the former associated with timing accuracy
and the latter with timing precision. The accuracy and precision estimates were dissociated from each other both in overall
time courses (Fig. 4) and in interval-specific effects of feedback (Fig. 5). The timing accuracy did not improve without
feedback but began to improve specifically at around the
interval with feedback instantaneously after the feedback was
introduced. By contrast, effects of the feedback hardly existed
in the timing precision: the timing precision improved for most
of the tested intervals slowly but steadily throughout the entire
sessions regardless of the feedback. The analysis of subjects’
performances in the SD and ME tasks indicated that neither of
them could explain the dynamics of timing behaviors in our
study. We found that a Bayesian observer model, in which a
subjective time interval is determined jointly by a prior and a
likelihood function, captures the dynamics of the two types of
timing measures simultaneously. The model suggests that the
width of the prior, not the likelihoods, gradually narrows over
sessions, demonstrating the critical role of prior knowledge in
PL of IT.
Difference Between Previous Tasks and Our Task for
Estimating Interval Timing
Owing to the intangible but ubiquitous nature of the time
dimension in diverse perceptual or cognitive events (Ahrens
and Sahani 2008), judgments about time can be readily complicated by the other sensory, motor, and executive functions
(Buonomano et al. 2009). For example, classic timing tasks, in
which subjects are asked to reproduce a sample ⌬T or distinguish between two consecutively presented ⌬Ts, require active
retention of the reference ⌬T in working memory or motor
planning and execution (Bartolo and Merchant 2009; Buhusi
and Meck 2005; Buonomano et al. 2009; Gibbon 1977; Gibbon
et al. 1997; Karmarkar and Buonomano 2003; Meegan et al.
2000; Miyazaki et al. 2005). In a temporal bisection (or
generalization) task, subjects have to refer to long-term memory of the reference ⌬T (Rakitin et al. 1998; Wearden 2008).
Subjects often rely on associative memory among sensory or
response events in tasks where ⌬Ts are simply conditioned to
cues in other sensory domains (Janssen and Shadlen 2005;
Leon and Shadlen 2003; Mita et al. 2009). Verbal estimation of
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Fig. 7. Bayesian observer model for interval timing. A: in the Bayesian observer model, a posterior distribution [p(⌬T|⌬t), solid lines in color] is determined by
a compromise (Eqs. 7 and 8) between a sensory likelihood function [p(⌬t|⌬T), dotted lines in color] and a prior distribution [p(⌬T), dashed lines in black]. Left:
model fit to the actual data at the shortest (blue) and longest (brown) ⌬Ts in a single session from a representative subject. Black arrow indicates the direction
and amount of shift from the mean of the likelihood toward that of the posterior, illustrating the impact of the relative widths of prior and likelihood distributions
on the mean of a posterior distribution. Middle and right: 2 possible scenarios for how the timing inference is changed in a following session because of perceptual
learning: a reduction either in prior width or in likelihood width. Middle: “prior width reduction” scenario, predicting that the standard deviation and the mean
of the posterior are reduced and further shifted toward the mean of the prior (indicated by enlarged black arrow), respectively. Right: “likelihood width reduction”
scenario also predicts a reduction in posterior width similarly to the former scenario but predicts the opposite about the mean of the posterior distribution: the
amount of a shift toward the prior mean is reduced, resulting in a regression backward to the likelihood mean, as indicated by the shrunken length of the black
arrow. Gray arrows, middle and right, are copied from the black arrow, left (the posterior mean in the previous session) to help appreciate the critical difference
in relative shift direction of posterior mean between the predictions by the 2 scenarios. B and C: model estimates (solid lines) of means (␮posterior, B) and standard
deviations (␴posterior, C) of the posterior distributions are plotted against the order of sessions, in parallel with the observed (symbols) means [␮(⌬t), B] and
standard deviation [␴(⌬t), C] of timing responses. Data were from a single representative subject, S4. Different colors indicate different physical time intervals,
with temperature representing the increasing order of interval length. D: model estimates of standard deviations of the prior distributions (␴prior, solid lines) and
those of CVs of the likelihood distributions (CVlikelihood, dashed lines) are plotted against the order of sessions. Symbols in gray are the estimates from individual
subjects, whereas black symbols are the estimates averaged across subjects. Error bars are SE of the means across subjects.
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
Dissociations Between Timing Accuracy and Precision
Behavioral consequences of PL have been characterized by
two descriptive statistics, a mean and a variance, the former
indexing how close the overall mean of a response distribution
(point of subjective equality, PSE) is to a reference stimulus
(accuracy, also referred to as “constant error”) and the latter
indexing how reliable responses are across trials (precision,
sensitivity, discrimination threshold, or “temporal variance”)
(McAuley and Miller 2007; Merchant et al. 2008b; Zarco et al.
2009). A few studies on visual perception reported feedbackinduced rapid adjustments of decision criteria and relative
inertia of sensitivity changes (Ahissar and Hochstein 1997;
Berniker et al. 2010; Chalk et al. 2010; Herzog et al. 2006). To
the best of our knowledge, however, this is the first study to
report systematic dissociations between mean accuracy and
precision of IT in the two major aspects of PL, time course and
specificity. A few earlier studies on IT have reported simultaneous improvements in both accuracy and precision (Meegan
et al. 2000) as well as differences between the two measures
by changes in interstimulus interval (Buonomano et al. 2009), by
adaptation effects of visual flickers (Johnston et al. 2006), or by
pharmacological effects in relation to the internal clock and
reference memory (Buhusi and Meck 2005; Coull et al. 2010).
However, none of those studies identified systematic and clear
dissociations in overall temporal dynamics and specificity of
PL in the two timing errors.
The decoupled dynamics of the two timing metrics demonstrate that it can be misleading to characterize PL effects with
only one of those metrics. For example, if interpreted alone, the
steady reduction in the timing variability in our data (Fig. 4B)
suggests that prolonged practice induced fundamental enhancement in perceptual sensitivity of IT over sessions. When
considered together with the fact that the ␮(⌬t) at the short or
long ⌬Ts substantially deviated from the physical ⌬Ts (Fig. 2A,
Fig. 5B), however, the apparent enhancement in timing variability at those ⌬Ts is actually quite detrimental in a real-world
situation because veridical targets are missed in a highly
reliable manner for the short or long ⌬Ts. Our findings underscore the importance of assessing effects of PL by measuring
the two error metrics simultaneously and interpreting them
jointly in a coherent manner (Buonomano et al. 2009; Gold
et al. 2010; Johnston et al. 2006; Meegan et al. 2000; Stetson
et al. 2006).
In contrast with the slow and steady changes in timing
precision, we found that feedback calibrated mean accuracy
quickly within approximately dozens of trials. One potential
source contributing to the immediate impact of feedback on
accuracy is the parametric nature of the feedback in our task.
Unlike other studies, in which feedback was typically categorical, the feedback was delivered quantitatively on a fine scale
on a trial-by-trial basis. It is interesting to consider the different
temporal dynamics between accuracy and precision in the
framework of the reverse hierarchy theory for PL (Ahissar and
Hochstein 1997). According to this view, in hierarchical perceptual systems in which the initial feedforward sweep is
followed by the feedback processing with fine scrutiny in the
reverse order (Ahissar et al. 2009; Hegde 2008), PL occurs in
a top-down manner, first at a higher level relatively easily and
then at a lower level for a more difficult task. Our data are
consistent with this view in that the calibration in timing
accuracy requires a ⬎1-h session whereas the calibration in
precision occurs slowly over multiple daily sessions (Berniker
et al. 2010). One intriguing interpretation is that neural mechanisms involved in timing accuracy and precision are likely to
differ—the former located at a relatively high level in the
hierarchy with a high degree of malleability to learning and the
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⌬T is highly subject to cognitive bias with respect to our
nomenclature for time intervals. Finally, if sensory processes
carrying temporal information undergo dynamic changes, subjects might exploit those changes for estimation of ⌬Ts (Ahrens and Sahani 2011; Johnston et al. 2006; Kanai et al. 2006).
We devised the IT task to reduce these potential confounds
because improvements may arise in any of these nontiming
domains while subjects are repeatedly performing the same
task over multiple days. An implicit IT task like ours, which
mimics a natural situation where a predator must keep track of
its moving prey temporarily hidden in a cluttered visual scene
to make a timed response afterward (Hulme and Zeki 2007;
Shuwairi et al. 2007; Yi et al. 2008), obviates the need to
explicitly encode the ⌬T and to retrieve it later as in the
previous studies. Our task allowed us to vary unpredictably
nontiming variables such as the direction, speed, and location
of a moving bar and the length of occlusion distance on a
trial-by-trial basis while preserving independent control of the
main variable, the ⌬T. In addition, the nine test ⌬Ts were
shuffled randomly or roved within a block of trials throughout
the entire sessions, so that subjects could not know explicitly
which ⌬T was tested in a given trial. All these manipulations
made it unlikely that simple forms of associative conditioning
contribute to the observed variations of timing data. Moreover,
because there was no change in physical stimuli during our IT
task, timing performance in the task is hardly contaminated by
any information associated with transients in sensory input. In
summary, our task is designed to minimize the potential
confounding factors for PL in previous studies, while enabling
us to examine timing on a fine and broad timescale. On the
other hand, a potential downside of our task is that any
enhancement in speed estimation, motor planning, and compensation of delay in motor execution can affect the timing
performance in the IT task (Cardoso-Leite et al. 2009; Faisal
and Wolpert 2009; Harris and Wolpert 1998; Law and Gold
2008). For this reason, it has been necessary to directly measure performances in motor execution during IT tasks, particularly when the motor system is likely to contribute to IT
performance, as in the interval reproduction task (Cicchini
et al. 2012; Jazayeri and Shadlen 2010). However, our analysis
of the data from the control task trials (Fig. 6) showed that the
contributions of speed perception and motor execution were
too small to account for our timing data, although their roles in
IT task cannot be completely ruled out. It is left for future
studies to examine whether our findings can be generalized to
other types of IT tasks with explicit timing and different
timescales (Coull and Nobre 2008; Merchant et al. 2008a;
Rohenkohl et al. 2011; Zelaznik et al. 2002). We also note that
tasks or stimuli in the present study were similarly used in
several previous studies, albeit framed in different contexts
such as motion extrapolation, selective attention, coincidence
timing, and manual interception (Eskandar and Assad 1999;
Linares et al. 2009; Merchant and Georgopoulos 2006; Miyazaki et al. 2005; Rohenkohl et al. 2011; White et al. 2008).
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latter located at a relatively low level with rigidity to learning
(Summerfield and Koechlin 2008). It will be interesting to use
electrophysiological or brain imaging techniques to learn
whether neural signatures for calibrations of timing accuracy
and precision reside at different levels along the hierarchy of
cortical processing.
Bayesian Inferences in Interval Timing
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Different from the interpretation based on two heterogeneous mechanisms, our Bayesian model provides a parsimonious and coherent way of interpreting the seemingly dissociative dynamics of timing accuracy and precision in our data, at
both conceptual and empirical levels. The collectively fluctuating patterns of ␮(⌬t) along daily sessions are well predicted
by the Bayesian model fitted to the individual subjects’ data
(Fig. 7). The impact of repeated performance in our timing task
is unlikely to manifest itself as a change in the precision of
sensory measurements of ⌬T—the reliability of likelihood
functions— because this leads to the opposite direction of
accuracy bias to the observed (Fig. 7A). Instead, the internal
representation of prior distribution was calibrated during multiple days of extensive training, as the gradual reduction of the
␴prior succinctly captures the across-session dynamics of timing accuracy and precision simultaneously (Zhaoping and Jingling 2008).
The pattern of perceived time intervals being attracted to the
center of the interval range has also been reported in previous
studies on IT (Fortin et al. 2009; Jones and McAuley 2005;
Kanai et al. 2006; Lewis and Miall 2009; McAuley and Miller
2007; Penney et al. 2008; Wearden 2008; Zarco et al. 2009).
Despite its frequency, this “regression to the mean” has been
overlooked (Bartolo and Merchant 2009; Zarco et al. 2009) or
diverse explanations have been offered based on, for example,
“central tendency” (Lewis and Miall 2009), Vierordt’s law
(Fortin et al. 2009; Tse et al. 2004), context-induced biases
(Jones and McAuley 2005; Kanai et al. 2006; McAuley and
Miller 2007), or abnormal timing in dopamine-deficient patients (Malapani 2002; Malapani et al. 1998). The Bayesian
model argues that the “migration” effect is a consequence of
strong contribution of prior distributions to IT, as suggested by
recent modeling works in time perception (Cicchini et al. 2012;
Jazayeri and Shadlen 2010). Specifically, the model predicts that
the degree of bias is determined by the width of the prior
distribution relative to that of the likelihood (Eq. 7). When the
scalar property is counted in, the model also offers quantitative
predictions about how both the size of migration and the variability vary systematically as a function of the ⌬Ts. The scalar
property dictates that the longer the time interval, the larger
variability of IT (␴likelihood) is. Accordingly, the longer the time
interval, the greater the relative contribution of the prior is, which
leads to larger migration toward the ␮prior as the weight to the
␮likelihood is inversely proportional to the ␴likelihood (Eq. 7; Fig.
7A). By the same token, the amount of variability reduction will
be greater for longer ⌬Ts because the weight to the ␴likelihood is
inversely proportional to itself (Eq. 8; Fig. 7C). All these detailed
model predictions about migration and variability reduction match
the observed data (Fig. 7).
Previous studies interpreted the reduction in timing variability as evidence for enhanced sensory representation of ⌬T via
noise reduction (Bartolo and Merchant 2009; Buonomano et al.
2009; Karmarkar and Buonomano 2003). However, our Bayesian modeling results provide an alternative explanation by
suggesting that the decreased timing variability in the previous
studies could arise from the sharpened prior distribution around
the test ⌬Ts that was overexposed to subjects through massive
training or procedural learning. In the same vein, an alternative
view can also be offered for the origin of the transfer in PL of
IT across different sensory channels (Buonomano 2003; Westheimer 1999; Wright et al. 1997) or modalities (Meegan et al.
2000; Nagarajan et al. 1998). The major sources of generalization of PL effects could be the update of prior distributions,
rather than an enhancement in a sensory mechanism for IT
common to different modalities, because the impact of updated
priors can be manifested in time intervals carried in different
sensory channels or modalities. The previous studies on IT
might not have been able to observe the contribution of priors
probably because the limited number of test intervals and the
wide separation between those intervals could make it difficult
to appreciate differences between the prior distribution and
likelihood.
What could have caused the narrowing prior during PL? In
general, reducing overall variability of perception is advantageous for achieving and maintaining accurate perceptual representation of sensory events in an uncertain environment
because it helps the adaptive perceptual system to be sensitive
to subtle changes in stimulus statistics and accordingly to
recalibrate sensory representations by minimizing the variance
of errors (Burge et al. 2010). Thus it is conceivable that
observers sequentially update their belief about ⌬T by using
the current posterior distributions, which already become
sharpened, as a prior for the future. We tested this “sequential
updating of prior” hypothesis by developing a simple generative model that uses the ⌬t distribution in the current session as
an empirical prior for the next session. However, this generative model failed to predict the complex and noisy time courses
of the observed timing accuracy and precision. One likely
reason for this failure is overfitting: including only the previous
daily session could be insufficient to estimate the prior. More
sophisticated variants of the generative model, such as those
that update priors by merging many previous daily sessions
with different weights or those that update priors based on
internal dynamics and feedback through reinforcement learning mechanisms (Gold et al. 2008; Nassar et al. 2010; Shibata
et al. 2009), may be needed to capture the intriguing dynamics
found in our study.
The results of our Bayesian modeling indicate that, despite
individual differences in our data that were reflected in highly
flexible and idiosyncratic priors across individual subjects, the
contribution of prior distributions provides a simple yet powerful explanation for the multiple aspects of our results simultaneously. It is possible that the high degree of uncertainty in
our IT task augmented the influence of prior knowledge. Note
that subjects had to judge a wide set of ⌬Ts ranging from
hundreds of milliseconds to several seconds, the order of which
was completely randomized on a trial-to-trial basis, without
any predictive cue for certain ⌬Ts. In this circumstance with
high uncertainty, the brain may need to rely heavily on prior
knowledge (Hudson et al. 2008; Körding and Wolpert 2004;
Lewis and Miall 2009; Zhaoping and Jingling 2008). From a
functional viewpoint, insurmountable demands for timing are
also quite evident in virtually all sensory, motor, and cognitive
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
tasks, not to mention various direct timing tasks such as IT,
temporal order judgment, rhythmic responses of temporal sequences, and causality inference (Ahrens and Sahani 2008;
Coull and Nobre 2008; Karmarkar and Buonomano 2007).
Thus the adaptive construction of the prior would be advantageous as well as efficient given the necessity of the mental
clock to be calibrated incessantly to the dynamic environment
and our limited capacity to deal with the virtually infinite range
of ⌬Ts.
Assumption of Gaussian Prior Distributions
An Interpretation of Feedback Effects
In our data, the impact of feedback was observed significantly only in mean timing accuracy, especially right after it
was first introduced. Meanwhile, the reduction in timing variability seemed to exist even before the introduction of feedback
and was hardly influenced by the feedback thereafter (Fig. 4B,
Fig. 5F). In our Bayesian modeling framework, this dissociation between the feedback effect and timing variability data
implies that the width of the Bayesian prior is to some extent
independent of the effect of feedback (Fig. 7D). On the other
hand, although the immediate effect of the feedback was not
observed in the ␮prior, the introduction of feedback made the
␮prior estimated from different subjects converge to the central ⌬T
particularly from the first SF session (data not shown). We
speculate that the feedback might have played a role of “anchoring” the prior near to the veridical center ⌬T, as hinted by the
convergence of ␮prior to the central ⌬T over the SF sessions.
As argued above, our modeling results suggest that the
contraction of the Bayesian prior distribution is responsible for
both the nonspecific reduction of timing variability and the
migration of ␮(⌬t) toward the center. How is this process
distinguished from the “anchoring” effect of the feedback?
Note that it is difficult to disentangle the feedback effect and
the growing biases toward the central ⌬T because we provided
the feedback to the center of sampled ⌬T. Nevertheless, one
possible interpretation is that the contraction of the prior width
builds up steadily as an unsupervised learning process via
intensive training, whereas the location of the prior center can
be quickly corrected by the feedback as a top-down process.
This interpretation is consistent with a line of studies that
reported immediate impacts of a variety of different feedbacks
(Lustig and Meck 2005), including even “fake or incorrect
feedback,” in visual PL (Shibata et al. 2009) and suggested an
important role of feedback as top-down processing in PL
(Sasaki et al. 2010; Zhang et al. 2008).
The Bayesian model does not explicitly specify how the error
feedback updates the prior distribution and how the selective and
immediate effect of the feedback on the mean accuracy occurs at
a mechanistic level. Without doubt, the elucidation of the exact
role of feedback in PL of IT and its relation to the evolving prior
requires further studies, especially in the context of Bayesian
modeling (Kersten et al. 2004; Shibata et al. 2009; Stevenson et al.
2009). For instance, by giving feedback for the shortest or longest
⌬T or by using fake feedback as in previous studies (Herzog 1997;
Shibata et al. 2009), we may learn more about specific contributions
of the feedback of PL in IT.
Implications for Neural Mechanisms for Interval Timing
The neural basis for IT became of great interest recently
(Ahrens and Sahani 2008; Coull et al. 2004; Wittmann and van
Wassenhove 2009). The present study can provide information
that constrains neural models of IT by evaluating which neural
models or theories are more compatible with our data. First, the
dissociation of timing accuracy and precision is not expected from
the view that the neural machinery for IT is composed of a single
centralized clock (Gibbon et al. 1997; Rakitin et al. 1998). If there
were one centralized clock or a tight coupling between individual
timers along different timescales, the trend of shifting toward the
central ⌬t (coexistence of under- and overestimation) would not
have been observed. Second, although it has been presumed that
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Despite its capability to describe our data parsimoniously
and to offer interesting explanations, our Bayesian observer
model of IT has a couple of limitations that need to be
addressed carefully. The most crucial of these is the assumption about prior distributions. In previous studies, the prior was
either reconstructed empirically (Brenner et al. 2008; Körding
and Wolpert 2004; Stocker and Simoncelli 2006) or often
presumed identical to physical distributions of stimuli (Jazayeri
and Shadlen 2010; Körding and Wolpert 2004; Miyazaki et al.
2006; Whiteley and Sahani 2008). Although test intervals in
our study were actually drawn from the uniform distribution in
a log scale, we opted to model the prior distribution with a
Gaussian function for several reasons. The adoption of the
physical prior function as a Bayesian prior may be inappropriate before subjects are exposed to the entire range of stimuli
over an extended period of training. In the present study, the
purpose of which was to track how effects of learning on IT
evolve over multiple days, we began to collect data after
providing subjects with minimal training (15 trials only at the
beginning of the first-day session). Another reason for preferring the Gaussian prior to the physical prior was that, in the
process of being internalized in the brain, the physical prior is
likely to be contaminated by internal noises (Cicchini et al.
2012). This would result in a prior whose distribution is
smoother than the physical prior, particularly around its boundaries. The smoothness of the internal prior would be even more
pronounced in our situation because IT responses are known to
have relatively large variability and to be prone to perceptual
illusions and recalibration (Burr et al. 2007; Eagleman 2008;
Johnston et al. 2006; Terao et al. 2008). Another problem with
the physical prior is that it does not allow posterior estimates to
fall outside the prior distribution because the prior probability
outside the range of ⌬Ts is eventually zero. This apparently
conflicts with the data points beyond the range of the stimulus
distribution, which have also been easily observed in previous
studies as well (Ahrens and Sahani 2008; Jazayeri and Shadlen
2010; Merchant et al. 2008b; Zarco et al. 2009). A recent study
(Jazayeri and Shadlen 2010), in which the physical distribution
of test stimuli was used as a Bayesian prior, had to create
another noise distribution at a motor execution stage with a
scalar property to get around this problem. Finally, it should be
stressed that our major conclusion about the relative contributions of the prior and likelihood over the time course of PL
does not rely critically on the detailed shape of the prior. The
steady and monotonic reduction of the “width” of the prior, not
the likelihood, is the crucial feature that is required for the
Bayesian model to capture the dynamics of both timing accuracy and precision simultaneously.
359
360
PERCEPTUAL LEARNING OF INTERVAL TIMING AND BAYESIAN MODELS
ACKNOWLEDGMENTS
The authors thank Randolph Blake, Daeyeol Lee, Marcus Kaiser, and three
anonymous reviewers for constructive comments on the earlier versions of this
manuscript and useful discussions.
GRANTS
This research was supported by the WCU program through the National
Research Foundation of Korea funded by the Ministry of Education, Science,
and Technology (R31-10089) and by the Seoul Science Fellowship to H. Sohn.
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: H.S. and S.-H.L. conception and design of research;
H.S. performed experiments; H.S. analyzed data; H.S. and S.-H.L. interpreted
results of experiments; H.S. and S.-H.L. prepared figures; H.S. and S.-H.L.
drafted manuscript; H.S. and S.-H.L. edited and revised manuscript; S.-H.L.
approved final version of manuscript.
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