19

Effect  of  Numeric  Order  on  Subjective  Duration  of  Following  Stimulus  Tomomitsu  Herai1,2,3,  Ken  Mogi3      Tokyo  Institute  of  Technology,  2JSPS  Research  Fellow,  3Sony  Computer  Science  Laboratories,  herai@brn.dis.titech.ac.jp  kenmogi@csl.sony.co.jp       

1Abstract.   A   perceived   duration   is   affected   by   the   magnitude   of   a   stimulus.   In   most   studies,   the   magnitudes   are  attributed   to   the   target   stimulus   itself.   We   focused   on   a   representation   of   magnitude   organized   in   the   brain,   using  ascending  and  descending  numbers.  We  investigated  whether  these  representations  affected  the  duration  judgment.  A  subject   watched   the   ascending,   descending,   or   randomly   ordered   numbers   and   judged   the   duration   of   a   square  presented   just   after   the   last   digit.   In   all   conditions,   subjects   overestimated   the   duration   of   the   target   stimulus.   The  overestimation   was   larger   in   the   ascending   than   in   the   descending   condition.   The   temporal   judgments   were   not  significantly  different  when  only  a  single  digit  was  presented  before  the  target.  Thus,  the  order  of  presentation  has  a  significant  effect  in  the  temporal  processing  of  following  visual  stimulus.  We  conclude  that  the  priming  by  the  order  of  numbers  affected  to  the  temporal  processing  of  subsequent  stimulus.     Keywords:  time  perception,  duration  estimation,  temporal  illusion  1         Introduction  Temporal processing is ubiquitous in our life. The neural basis of temporal processing and its relation to the subjective
sense of time are still unclear. Many psychological studies have shown that our subjective sense of time is easily
manipulated in laboratory experiments [1].
An Oddball stimulus appeared in a sequence of an identical stimulus is perceived longer than its actual interval [2].
This oddball effect happened even when the subjects knew the position when the oddball stimuli appeared in the train of
stimulus [3]. The authors argued that oddball stimuli captured the SHUFHLYHU¶V DWWHQWLRQ, resulting in an expansion ofsubjective time. In fact, the attended object was perceived longer than its actual presented duration [4].
When the presented stimulus could be predicted, the subject underestimated the presentation interval [@6XEMHFW¶V
expectation of the stimuli might "shrink" the presented duration of the stimulus. This result indicated that higher
cognitive process affected the time perception. In contrast, the underestimation of repeated presentation of the identicalstimulus occurs on briefly presented stimuli, each of which were presented too short to reach the awareness of subject[6]. This indicated that a lower mechanism was also involved in constructing the psychological time.
The property of stimulus itself affects our perceived duration. Non temporal factors such as spatial size andmagnitude of stimulus also affect its perceived subjective duration. The presented duration of larger stimulus wasestimated to be longer than that of the smaller one [7]. This effect also occurred with more conceptual stimuli such as
numbers [8]. A theory of magnitude [9] hypothesizes various dimensions such as space, time, quantity are represented
by common neural mechanism.
In those studies, however, the magnitude which affected the temporal estimation have been attributed to the stimulus
itself. It is unclear whether a low perceptual or a high cognitive mechanism is engaged to the temporal effect ofmagnitude. In this study, we investigated whether higher mechanism relating the magnitude could induce a temporal
effect.
Human being has an ability to extract abstract information from external stimulus and categorize them in various
dimension and magnitude, subsequently using this information to predict a following stimulus. Therefore, wehypothesized that these magnitudes and predictions which were synthesized internally in the brain would also affect thetemporal estimation.
Even if we can predict the next stimulus when the digit appears successively in order, our introspection is different
EHWZHHQDQDVFHQGLQJDQGDGHFOLQLQJQXPEHUV$VFHQGLQJQXPEHUVPD\EHFDWHJRUL]HGWRDUHSUHVHQWDWLRQOLNHDV³XS´
DQG ³H[SDQVLRQ´ ZKLOH GHFOLQLQJ QXPEHUV PD\ EH FDWHJRUL]HG WR D UHSUHVHQWDWLRQ OLNH DV ³GRZQ´ DQG ³FRQWUDFWLRQ´
Subjects also may predict that the next stimulus will be bigger/smaller in ascending/declining numbers. Our hypothesis
was that these insights would affect our time perception. We expected that ascending numbers would induce temporal
overestimation of the stimulus presented just after numbers.

Volume 11, No. 2

Australian Journal of Intelligent Information Processing Systems

202         Experiment  1  2.1         Methods  Subjects. Nine subjects (8 naïve (3 female) and author T.H, mean s.d age = 28 2.4) participated in the experiment.
They have normal or corrected-to-normal vision. Written informed consent was obtained from all participants.
A pparatus and Stimulus. Stimulus presentation and collecting of VXEMHFW¶V UHVSRQVH were controlled by a PC
(Panasonic CF-W4). Visual stimuli were presented on a 17-inch monitor (Eizo FlexScan S1911/S1922) with a refresh
rate of 60 Hz. Subjects sat in front of the monitor with the distance of 60 cm with a chin rest. One of nine white digits(from µ¶ WR µ¶ LQ ZLGWK DQG  LQ KHLJKW  DQG D ZKLWH VTXDUH   were presented at the center of themonitor with a black background.

F ig. 1. Experimental procedures. (a) sequential numbers, (b) single number.

Procedure. The experiment consisted of a sequential numbers condition and a single number condition (Figure.1). In
the sequential numbers condition, each of numeric stimuli was presented successively with a standard duration (500ms).
Subjects were instructed beforehand that the presented duration of each digit were equal. There were three types (UP,DOWN and RAND) of orders in digits presentation. In the UP condition, each number appeared in an ascending order,
while number goes down in order in the DOWN condition. In the RAND condition, the presented orders of digits wererandomized. TherefoUH LQ WKLV FRQGLWLRQ VXEMHFWV FRXOGQ¶W SUHGLFW WKH QH[W QXPEHU 5$1' FRQGLWLRQ ZDV LQFOXGHG
for enhancing the subject's attention to the order of digits, by making the ascending and descending order salient.
RAND condition was also designed to avoid the subjects suspecting the aim of experiment.
After the last digit was presented, the white square was presented with one of nine durations (367, 400, 433, 467, 500,
533, 567, 600 and 633 ms). No delay was inserted within successive digits and between the last digit and the target.
After the stimulus presentation, participants judged whether the duration of the last square was longer or shorter thanthat of each of numeric stimulus and responded by pressing the button of mouse with their right hand.
Three types of number presentation were mixed randomly in one block. Each of test duration was presented once foreach of conditions in a block. Therefore, each block contained 27 trials (9 durations
3 conditions). Subjects
conducted a total of 10 blocks.
Furthermore, to investigate whether there was any effect of the last digit to the duration estimation, we conducted thesingle number condition. Experimental setting was the same as the sequential numbers condition except that only one of
WKUHHGLJLWV ³9´³1´RU³5´NINE, ONE or FIVE condition, respectively) was presented as a reference stimulus for
500 ms before the target appeared with valuable delays. The '9' and '1' were the last digits in ascending order and indescending order, respectively. 10 blocks were conducted in each condition. The sequential and single number
conditions were changed by five blocks and the orders of them were counterbalanced among the subjects.
Data A nalysis. 6XEMHFW¶V UHVSRQVHV ZHUH SORWWHG DV D IXQFWLRQ RI WKH WHVW GXUDWLRQV 8VLQJ PD[LPXP OLNHOLKRRG
estimation, the data for each subject for each condition were fitted to the Weibull function:­ x E½° § · °F ( x; D , E ) 1  exp ® ¨ ¸ ¾ .
°̄ © D ¹ °¿where F represents a proportion of µlong¶ responses. x is the test duration. Į and ȕ are fitting parameters. Fitting
parameters for each subject were used to estimate their PSE (Point of Subjective Equality: 0.5 point of fitted responding
curve) and JND (Just Noticeable Difference: a difference of estimated duration between 0.75 and 0.5 point). A PSE
indicated a target interval where the subjects perceived the target as the same duration with the interval of each digit.
JND indicated sensitivity of temporal discrimination.

Volume 11, No. 2

Australian Journal of Intelligent Information Processing Systems

21

2.2         Results  Figure VKRZV WKH PHDQ SURSRUWLRQ RI µORQJ¶ UHVSRQVHV SORWWHG DV D IXQFWLRQ RI WKH WHVW GXUDWLRQV LQ HDFK FRQGLWLRQ
Figure.3 shows the mean PSE for each condition. In the sequential numbers condition, PSEs of all conditions weresignificantly different from 500ms (UP: 433 36 ms (mean s.d), t(8)=-5.22, p<0.001, DOWN: 455 22 ms, t(8)=5.49, p<0.001, RAND: 455 48 ms ,t(8)=-2.56, p=0.033). In the single number condition, only ONE condition was
significantly different from 500ms (NINE: 449 69 ms, t(8)=-2.08, p=0.070, ONE: 452 57 ms, t(8)=-2.35, p=0.046,
FIVE: 461 52 ms, t(8)=-2.07, p=0.071). The difference of PSE between UP and DOWN condition was significant(paired t-test, t(8)=-2.37 p=0.044) but between ONE and NINE was not(paired t-test, t(8)= 0.34, p=0.73). Therefore, thesubjects overestimated the duration of target stimulus more in UP condition than in DOWN condition and thisdifference was not caused by the digit just before the target. The JNDs were not different significantly in both pairs(paired t-test, UP vs DOWN: t (8)=0.12, p=0.90, ONE vs NINE: t(8)=-0.68, p=0.51) (Fig.4).

F ig.2. Response curves of the sequential numbers (left) and single number (right) conditions.

F ig.3. Mean of Point of Equality (PSE) of the sequential numbers (left) and single number (right) conditions.

3         Experiment  2  In experiment1, the mental representation induced by the number presentation influenced the temporal processing offollowing stimulus. A repetition of the same digit might also make a stable and invariant representation. This
representation was obviously different from representations made by ascending and descending numbers. We
hypothesized that time perception would be varied among these number presentations.
3.1         Methods  Subjects. Eight subjects (7 naïve (3 female) and author T.H, mean s.d age = 28 2.4) participated in the experiment.
They had normal or corrected-to-normal vision. Written informed consent was obtained from all participants.
A pparatus and Stimulus. The experiment was controlled on a desktop PC (EPSON Endeavor). Visual stimuli were
presented at the center of a 21-inch CRT monitor (Sony, Trinitron, CPD-G520) with a refresh rate of 100 Hz. Subjects
sat in front of the monitor at the distance of 60 cm with a chin rest. One of nine white digits µ¶WRµ¶ LQZLGWK
and 0.08 in height)) and a white rectangle (0.04° in width and 0.08 in height) were presented in a black background.

Volume 11, No. 2

Australian Journal of Intelligent Information Processing Systems

22

F ig.4. Experimental procedure

Procedure. Figure.4 shows the experimental procedures. Five digits were presented in order before the presentation of
the target stimulus. There were three conditions in number presentations. The digits IURPµ¶WRµ¶DQGfrom µ¶WRµ¶were presented sequentially in the UP and DOWN conditions, respectively. In the SAME condition, all of presented
digits were '5'. There was the blank interval between each digit. The blank interval was randomly chosen from 300, 350,
400, 450, 500 ms. We used the random interval to avoid the subjects using rhythmic information for temporalestimation. Each digit was presented 500 ms. The range of presentation interval of target stimulus was from 380 to 560
ms by 20 ms bin. The subjects judged whether the target interval was longer or shorter than the presented interval ofeach digit. Three conditions were mixed and randomly presented in a block. The subject responded by pressing the keyof mouse. The target interval for each condition was presented one time in one block. One block contained 27 trials.
The subjects conducted 10 blocks all.
3.2         Results  The data were analyzed in the same way as in experiment 1. The results are shown in figure.5. The PSEs in allconditions were significantly smaller than 500ms (one sample t-test, UP: 456 23 ms t(7)=-5.00, p=0.0015, DOWN:
459 29 ms, t(7)=-3.71, p=0.0074, SAME: 435 25 ms, t(7)=-6.53, p=0.00032). We conducted one way repeatedmeasures ANOVA for PSEs and JNDs. The ANOVA detected the significant effect of presented numerical order for
temporal estimation of the target interval ( F (2,14)=5.998, p=0.013). On the contrary, JNDs were not different
significantly among the conditions ( F (2,14)=1.348, p=0.29). We next conducted paired t-test for each pair (UP vs
DOWN: t(7)=-0.49, p=0.63, UP vs SAME: t(7)=2.77, p=0.027, DOWN vs SAME: t(7)=2.72, p=0.029). The PSE in the
SAME condition was significantly smaller than that in the UP and DOWN conditions. Results indicated that theperceived duration of target interval was overestimated more in the SAME condition than in the UP and DOWN
conditions. We estimated that the transition was lower in the SAME condition than in the UP and DOWN conditions.
Therefore, the saliency of target was higher in SAME condition than in UP and DOWN condition. So, the target caught
the subject's attention more in SAME condition than in UP and DOWN condition. Since attention affects the perceived
interval, the subjects overestimated the target interval more in the SAME condition than in the UP and DOWN
conditions.

F ig.5. Response curve and Point of Equality (PSE)

4         Discussion  In this study, we examined whether temporal estimation was affected by the context just before the target. In
experiment 1, when the digits were presented in ascending order (UP), subject perceived the following square as longer
in duration than in the descending order (DOWN). The non-significant difference of JNDs between the UP and DOWN
conditions indicated that the VXEMHFWV¶sensitivities of time estimation were not changed between these conditions. These
results, in all, indicate that the preceding context affects the subsequent temporal processing in the brain.
Volume 11, No. 2

Australian Journal of Intelligent Information Processing Systems

23

Both the oddball [2] and the prediction [5] effects might contribute to the VXEMHFWV¶altered temporal estimation in thepresent study. In the sequential numbers condition, the oddball effect always occurred because the visual property oftarget was largely different from that of the digits. This effect expanded the perceived subjective duration of the target
stimuli. At the same time, in the UP and DOWN conditions, but not in the RAND condition, the subjects could predict
the next digit, resulting in the perceived duration of digit being contracted than actual. Therefore, it is possible that theoverestimation in the UP and DOWN conditions were induced by the oddball and the prediction effects, while that in
the RAND condition was induced by only the oddball effect. This suggestion is confirmed by the statistical analysis
where the shifts of PSE from the standard duration in the UP and DOWN conditions were more robust than that in the
RAND condition.
In addition to these effects, we propose based on the results that the presented order of numbers before the target
stimulus DIIHFWHG VXEMHFWV¶ WHPSRUDO HVWLPDWLRQ 7KH VXEMHFWV WHQGHG WR RYHUHVWLPDWH the target duration in the UP
condition to a significantly larger extent than in the DOWN condition. This contextual effect was not so strong as theoddball effect, possibly due to the representation generated in the brain being weakly linked to the target. We give
below a few possible explanations why the order of numbers DIIHFWHGWKHVXEMHFW¶VWHPSRUDOHVWLPDWLRQThe first possibility LVWKDWWKHVXEMHFWV¶H[SHFWDWLRQWRWKHQH[WVWimulus, which was made by the pattern of stimulus
presentation, affected the temporal estimation. In this case, the subjects expected that the next stimulus would be bigger(or smaller) than the previous one. So, they had larger (smaller) prediction when they watched the target stimuli in the
UP (or the DOWN) condition. In the same manner as larger stimuli was perceived longer, this expectation also affected
the processing of target stimuli and led to a more significant overestimation in the UP condition than in the DOWN
condition.
Another possible account is to deal with the train of digit more comprehensively. The ascending (declining) number
induced the internal representation of expansion (contraction) within the subject's brain, resulting in a differential
processing of the target stimulus subsequently entered. The processing of the target stimuli might be effected by thisbrain state, changing the perceived subjective duration. When the target stimuli interacted with the brain state which
represent either expansion or contraction, the target was consequently perceived as longer or shorter.
This internal brain state generated by the digits train may be also related to attentional state. Ascending numbers candraw up arousal level more than the declining numbers, resulting in the subjects attending to the stimuli more in the UP
condition than in the DOWN condition. Since the attention expanded perceived duration [2] [4], the subject could
overestimate the target stimuli more in the UP condition as a result.
In summary, we showed that the order of numbers presented before the target stimuli affected the perceived
subjective duration. When the integer presentation before the target was upward, the subjects overestimated the target
duration larger than when the order was downward. The digits train before the target might induce different brain states,
resulting in affected temporal processing of the target stimulus presented right after the numbers. We conclude that thecontext of stimulus presentation is important in the construction of our time perception.

References  1.  Eagleman,  D.:  Human  time  perception  and  its  illusions.  Curr  Opin  Neurobiol  18,  131-­136  (2008)  2.  Tse,  P.,  Intriligator,  J.,  Rivest,  J.,  Cavanagh,  P.:  Attention  and  the  subjective  expansion  of  time.  Percept  Psychophys  66,  1171-­1189  (2004)  3.  van  Wassenhove,  V.,  Buonomano,  D.,  Shimojo,  S.,  Shams,  L.:  Distortions  of  subjective  time  perception  within  and  across  senses.  
PLoS  One  3,  e1437  (2008)  4.  Mattes,  S.,  Ulrich,  R.:  Directed  attention  prolongs  the  perceived  duration  of  a  brief  stimulus.  Percept  Psychophys  60,  1305-­1317  (1998)  5.  Pariyadath,  V.,  Eagleman,  D.:  The  effect  of  predictability  on  subjective  duration.  PLoS  ONE  2,  e1264  (2007)  6.  Pariyadath,  V.,  Eagleman,  D.:  Brief  subjective  durations  contract  with  repetition.  J  Vis  8,  11.11-­16  (2008)  7.  Xuan,  B.,  Zhang,  D.,  He,  S.,  Chen,  X.C.:  Larger  stimuli  are  judged  to  last  longer.  Journal  of  Vision  7,  (2007)  8.   Oliveri,   M.,   Vicario,   C.,   Salerno,   S.,   Koch,   G.,   Turriziani,   P.,   Mangano,   R.,   Chillemi,   G.,   Caltagirone,   C.:   Perceiving   numbers  alters  time  perception.  Neurosci  Lett  438,  308-­311  (2008)  9.  
Walsh,  V.:  A  theory  of  magnitude:  common  cortical  metrics  of  time,  space  andquantity.  Trends  Cogn  Sci  7,  483-­488  (2003)  Volume 11, No. 2

Australian Journal of Intelligent Information Processing Systems