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ORIGINAL RESEARCH ARTICLE

published: 02 January 2012

doi: 10.3389/fnins.2011.00143

Piéron’s law and optimal behavior in perceptual

decision-making

Leendert van Maanen*, Raoul P. P. P. Grasman, Birte U. Forstmann and Eric-Jan Wagenmakers

University of Amsterdam, Amsterdam, Netherlands

Edited by:

Michael Platt, Duke University, USA

Reviewed by:

Benjamin Hayden, Duke University

Medical Center, USA

Willem Huijbers, Harvard Medical

School, USA

*Correspondence:

Leendert van Maanen, Cognitive

Science Center Amsterdam, Plantage

Muidergracht 24, 1018 TV

Amsterdam, Netherlands.

e-mail: l.vanmaanen@uva.nl

Piéron’s Law is a psychophysical regularity in signal detection tasks that states that mean

response times decrease as a power function of stimulus intensity. In this article, we extend

Piéron’s Law to perceptual two-choice decision-making tasks, and demonstrate that the

law holds as the discriminability between two competing choices is manipulated, even

though the stimulus intensity remains constant. This result is consistent with predictions

from a Bayesian ideal observer model. The model assumes that in order to respond opti-

mally in a two-choice decision-making task, participants continually update the posterior

probability of each response alternative, until the probability of one alternative crosses a

criterion value. In addition to predictions for two-choice decision-making tasks, we extend

the ideal observer model to predict Piéron’s Law in signal detection tasks. We conclude

that Piéron’s Law is a general phenomenon that may be caused by optimality constraints.

Keywords: Piéron’s law, decision-making, random-dot motion paradigm, Bayesian ideal observer

INTRODUCTION

Cognitive science features several psychophysical laws. These laws

are not only inherently interesting, but often they also bring

together phenomena that seem unrelated at ﬁrst sight. One exam-

ple of a psychophysical law is Piéron’s Law. Piéron’s Law states

that mean response times (MRT) decrease as a power law with

increasing stimulus intensity I :

MRT = αI

−β

+ γ. (1)

Here α and β are scaling parameters that determine the slope

of the function and γ is an intercept (Luce, 1986). The speciﬁc

parameter values differ between application domains.

Originally, Piéron’s Law was formulated as an effect of stimu-

lus intensity (Piéron, 1914). For example, when participants were

instructed to press a button as soon as a light was switched on,

MRTs were found to follow a power law decrease with increas-

ing luminance of the light. Over the last century, the law has

been reported in many different domains, including brightness

detection (Piéron, 1914), tone detection (Chocholle, 1940), taste

detection of dissolved substances (Bonnet et al., 1999), odor detec-

tion (Overbosch et al., 1989), heat detection (Banks, 1973), and the

go/no-go task (Jaskowski and Sobieralska, 2004).

In recent years, Piéron’s Law has been found to hold in two-

alternative forced choice (2AFC) tasks as well (Pins and Bonnet,

1996; Palmer et al., 2005; Stafford et al., 2011). Similar to the initial

studies that reported Piéron’s Law, MRTs were found to decrease

as a power law with increasing stimulus intensity, even though the

task was not a signal detection task but instead a choice task. For

example, Stafford et al. (2011) demonstrate that reaction times in

the Stroop color naming task scale according to a power law with

the luminance of the color dimension.

These studies raise the question whether Piéron’s Law describes

a speciﬁc relation between stimulus intensity and MRT or whether

Piéron’s Law is related to the more general notion of discriminabil-

ity in (perceptual) decision-making. Thus, the typical power law

decrease in MRT is not only observed with increasing stimulus

intensity, but also when a decision becomes increasingly easy. One

reason for arguing in favor of this hypothesis is that the typi-

cal stimulus detection task that is associated with Piéron’s Law

can be thought of as a 2AFC task. The decision that is required

is that between providing a response or withholding a response.

The stimulus intensity can now be thought of as a factor on the

decision difﬁculty, because a low intensity stimulus discriminates

poorly between the two response alternatives (respond or withhold

a response).

In this paper we hypothesize that Piéron’s Law extends to the

more general notion of stimulus discriminability. We ﬁrst show

data that support that MRTs in a 2AFC task decrease as a power

law with increasing discriminability of the choice alternatives. In

particular, we show that the relation between MRT and stimu-

lus discriminability is better supported by a power function than

by an exponential function. This comparison quantiﬁes the like-

lihood that stimulus discriminability indeed leads to Piéron-like

behavior. The experimental result supports the idea that Piéron’s

Law is more broadly applicable than only in stimulus detection

paradigms. Next, we discuss how Piéron-like behavior may be a

consequence of optimal decision-making. That is, we present a

Bayesian ideal observer model (e.g., Brown et al., 2009) and show

that it predicts that MRT decreases as the power of the discrim-

inability of the choices increases. Finally, we extend the Bayesian

ideal observer model to include stimulus detection behavior, and

discuss under what assumptions optimal stimulus detection would

lead to Piéron’s Law.

In the process of studying Piéron’s Law in a 2AFC task, we also

make a methodological point: Although never explicitly formu-

lated in this way, Piéron’s Law relates stimulus intensity to mean

detection time, rather than mean response time. To account for the

www.frontiersin.org January 2012 | Volume 5 | Article 143 | 1

van Maanen et al. Piéron’s law in RDM

non-detection related components of the response time, an inter-

cept parameter is added (γ in Equation 1). To study the relation

between stimulus discriminability and decision time in the 2AFC

paradigm

1

, we separately estimate the non-decision time using a

more elaborate choice-response time model (the Linear Ballistic

Accumulator model, Brown and Heathcote, 2008). For compari-

son, we also include the analysis without this additional procedure,

and show how this leads to different – implausible – results.

EXPERIMENT 1: STIMULUS DISCRIMINABILITY IN THE RDM

TASK

A 2AFC task that is perfectly suited to address the question at

hand is the random-dot motion task (RDM, Ball and Sekuler,

1982; Britten et al., 1992; Salzman and Newsome, 1994; Shadlen

and Newsome, 2001; Churchland et al., 2008; Forstmann et al.,

2008, 2010a,b; Niwa and Ditterich, 2008; Ho et al., 2009; Mulder

et al., 2010; Van Maanen et al., 2011). In this task participants are

required to indicate the apparent direction of motion of a cloud of

dots that is presented on a computer screen. Typically, a percentage

of the dots moves in a designated direction (the target direction),

while the remaining dots move randomly. The choice alternatives

are presented on a ﬁxed distance from the center of the dot cloud,

and their similarity can be manipulated by changing the angular

distance between the alternatives, as in Figure 1.

In Experiment 1, participants were asked to make a choice

between two alternative directions of motion. We manipulated

the angular distance between the response alternatives to opera-

tionalize choice difﬁculty. Response alternatives that were in close

spatial proximity were hypothesized to be hard to discriminate.

Response alternatives that were not in close spatial proximity were

hypothesized to be easy to discriminate (Figure 1). Note that in

this experiment we do not manipulate stimulus intensity, as the

dot cloud remains the same in all conditions. Therefore, evidence

for a power curve in this experiment would support the hypothesis

that Piéron’s Law is a general relation between MRT and stimulus

discriminability.

1

We use decision time to refer to the time required to make a choice in 2AFC tasks,

and detection time to refer to the time required to detect a stimulus in a stimulus

detection task.

FIGURE 1 | RDM stimulus display. A proportion of the dots move in a

target direction indicated by the arrows, while the remaining dots move

randomly. The discriminability of the t arget is manipulated by changing the

angular distance (d ) between the alternatives.

METHODS

Participants

Six students (three female, age range 18–20) from the University

of Amsterdam participated for course credit. All had normal or

corrected-to-normal vision.

Random-dot motion stimulus

To create the moving-dot kinematogram we used the Variable

Coherence Random-Dot Motion (VCRDM) library for Psych-

toolbox in Matlab (Brainard, 1997)

2

. The appearance of motion

in VCRDM is created by controlling the location of a subset of

dots for three frames in a row. That is, when the second frame

is drawn, the location of a subset of dots will be recomputed

to align with the target direction. The location of the remaining

dots is randomly assigned. The size of the subset –often referred

to as the coherence level – is under the experimenter’s control.

We set the coherence at 25%. Pilot studies with several levels of

coherence indicated that at a coherence of 25% participants per-

formed above chance for the hardest conditions, whereas they

performed below ceiling at the easiest conditions. Each dot con-

sisted of three by three pixels, and the initial locations of each dot

sequence were uniformly distributed in an aperture of 5 visual

degrees diameter.

Design and procedure

Participants were instructed to indicate the apparent direction

of motion of the random-dot kinematogram. The two-choice

alternatives were represented by two circles (one blue, one yel-

low) that were located at 5 visual degrees from the center of

the aperture. If the direction of motion was toward the blue

alternative, the participants were required to press “z”; if the

direction of motion was toward the yellow alternative they were

required to press “m.” The location of the alternatives was ran-

domized over the top half of the imaginary circle. However, the

blue alternative always remained to the left of the yellow alter-

native. Therefore, the “z” and “m” response mapping was always

congruent with the target positions on the screen. The angular

distances between the alternatives were exponentially distributed,

to maximize the probability of diverging model ﬁts. Consecu-

tively, the angular distances were 11.5˚, 16˚, 22.5˚, 32˚, 45˚, 64˚, or

90˚. The presentation order of the different angular distances was

pseudo-randomized in such a way that there were never more than

two consecutive trials with the same angular distance between the

alternatives.

After a short training session with 14 trials (2 for each angle) the

experiment was presented in seven blocks of 210 trials each. After

each block, the participant could take a short break. Participants

received feedback on the accuracy of their response: A screen stat-

ing “Correct!” or “Incorrect!” remained visible for 400 ms. At the

beginning of each trial, a red ﬁxation dot was presented together

with the alternatives. After 500 ms, the RDM stimulus was pre-

sented which remained on screen until the participant made a

response. If the response was faster than 200 ms a feedback screen

appeared that stated “Te snel!” (too fast). If the participant did not

respond within 2000 ms, the feedback “Incorrect” was given.

2

The library can be downloaded from http://www.shadlen.org/Code/VCRDM.

Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 2

van Maanen et al. Piéron’s law in RDM

Analyses

Based on the hypothesis that MRT and difﬁculty have a power

relationship, we ﬁtted a three-parameter power function to the

data:

MRT = αd

−β

+ γ. (2)

Note that Equation (2) is almost identical to Piéron’s Law

(Equation 1), except that the variable indicating stimulus intensity

(I) is replaced with a variable indicating angular distance (d). In

addition, we ﬁtted a three-parameter exponential function to the

data to study if another functional relationship could account for

the observed effects:

MRT = αe

−βd

+ γ. (3)

We ﬁtted both functions using standard simplex optimization

routines (Nelder and Mead, 1965). This was repeated 10,000 times

with randomized initial values to avoid local minima. To deter-

mine goodness of ﬁt, we computed the correlation between the

MRT for each angle and both models’ predictions for each angle.

In addition we computed Bayesian Information Criterion (BIC,

Schwarz, 1978; Raftery, 1996) values for each model to obtain the

evidence ratio. The evidence ratio for the power function over the

exponential function is computed according to e

−

1

2

(BIC

pow

−BIC

exp

)

(Wagenmakers and Farrell, 2004). This expression quantiﬁes how

many times more likely the data are to have occurred under the

power function than under the exponential function, given their

respective BIC values.

In addition to these simple model ﬁts, we ﬁt a more complex

model in which the intercept parameter (γ) is ﬁxed for all condi-

tions. This method provides more constraint on the parameter that

we are least interested in (γ), allowing for a clearer interpretation

of the non-linear model ﬁts that describe the choice behavior. We

ﬁrst estimate the time required for non-decision related processes

using the full RT distribution and the error rate and then use these

estimates to ﬁx γ in the power and exponential curves.

Linear ballistic accumulator model

To estimate γ, we ﬁt the linear ballistic accumulator (LBA, Brown

and Heathcote,2008) model to the data (Figure 2). The LBA model

assumes that a decision is made by the accumulation of evidence

for a particular option until a decision threshold has been reached.

In the LBA model, the decision threshold b is a free parameter. The

starting point of the accumulation is drawn from a uniform dis-

tribution [0, A], with A as a free parameter. The speed of the

accumulation of each alternative i is controlled by a speciﬁc drift

parameter v

i

(and typically a common drift variance parameter

s). Because each response alternative is represented by a sepa-

rate accumulator, the accumulator that reaches the threshold the

quickest determines the response, and the time required to reach

the threshold is the decision time. Crucially, the LBA model also

has a parameter that quantiﬁes the amount of time required for

peripheral, non-decision processes (that is, the intercept γ).

In LBA and related response-time models, stimulus differences

are often modeled by allowing drift rate to vary (e.g., Brown and

Heathcote, 2008; Wagenmakers et al., 2008; Ho et al., 2009; Va n

Maanen et al., 2009; Van Maanen and Van Rijn, 2010). Therefore,

in the ﬁrst LBA model that we ﬁt to the data (i.e., Model 1), we

allowed the drift rate of the model to vary over the different angu-

lar distances, with all other parameters ﬁxed. In addition to drift

rate, Model 2 also allows the non-decision time to vary over the

angles. Although there is no theoretical reason for this assumption,

it is important to assess that in the data the non-decision time does

not vary over angles, so that we can estimate a single value for γ.

Models 3 and 4 are identical to Models 1 and 2 but in addition

allow the drift rate variance to vary over angles (Churchland et al.,

2011). Parameter values were optimized using quantile maximum

likelihood estimation (Heathcote et al., 2002), using the 0.1, 0.3,

0.5, 0.7, and 0.9 quantiles of the RT distributions for correct and

error responses.

RESULTS AND DISCUSSION

We excluded trials in which the participants failed to respond in

time (over 2000 ms, 3.3% of the trials) as well as trials in which the

participants responded too fast (faster than 200 ms, 0.9%). Table 1

presents the mean accuracy per condition; Figure 3 presents MRT

of the correct responses, for each participant separately. The results

show that MRT decreases with the angular distance, whereas accu-

racy increases. This is consistent with the idea that smaller angular

distances are more difﬁcult and would therefore lead to more

errors and slower correct responses. Participant 3 displays behav-

ior that does not clearly follow this result, although the smallest

angular distance yields slower responses than the widest angle. We

could not ﬁnd a reason why this participant displays behavior that

does not follow our hypotheses. Therefore, there is no reason to

FIGURE 2 | LBA model. The accumulator representing response alternative

1 is on average faster than the competing accumulator. The model therefore

predicts that response alternative 1 will be selected on the majority of

trials. See text for details.

Table 1 | Average accuracy scores per angular distance (and average

within-subjects standard error of the mean, Loftus and Masson, 1994).

Angle Mean accuracy (SEM)

11.5 0.67 (0.032)

16 0.71 (0.030)

22.5 0.77 (0.027)

32 0.82 (0.025)

45 0.85 (0.023)

64 0.89 (0.020)

90 0.88 (0.021)

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van Maanen et al. Piéron’s law in RDM

FIGURE 3 | Mean response times (MRT) as a function of angular distance. Each plot represents the data from one participant. (Error bars represent

within-subjects standard error of the mean, Loftus and Masson, 1994). The best ﬁtting power function and the best ﬁtting exponential function are overlaid.

Table 2 | Correlation coefﬁcient (ρ), BIC values, and evidence ratios

(Wagenmakers and Farrell, 2004) for the three-parameter power

function and the exponential function for each participant.

Participant ρ BIC Evidence ratio

Exp Pow Exp Pow

1 0.98 0.97 −64.22 −61.66 0.28

2 0.96 0.95 −54.56 −53.08 0.48

3 0.74 0.71 −45.52 −44.88 0.66

4 0.98 0.98 −58.47 −55.21 0.19

5 0.98 0.97 −56.40 −54.47 0.38

6 0.99 0.99 −64.90 −62.46 0.29

exclude this participant from the analyses. However, it should be

noted that the results do not hinge on this participant, and would

stand even if we would exclude this participant.

Table 2 presents the goodness of ﬁt of the power and expo-

nential functions presented in Equations (2) and (3), as well as

evidence ratios. The fact that the evidence ratios are between 0

and 1 indicates that there is slightly more evidence in favor of the

exponential function than the power function.

It is misleading to accept these model results at face value,

because the parameter estimates cannot be interpreted in psy-

chological terms. A plausible interpretation of these functions

should be that each response consists of a decision and additional

processes such as stimulus encoding and response execution (cf.

Sternberg, 1969; Luce, 1986). The additional processes are cap-

tured by the intercept γ, whereas the actual decision process is

Table 3 | Best ﬁtting parameter values for the three-parameter power

and exponential functions.

Participant Exponential Power

αβγ(ms) αβγ(ms)

1 0.16 0.04 540 0.51 0.18 320

2 0.23 0.05 600 0.91 0.68 550

3 1038 0.82 810 3722 4.41 810

4 0.26 0.03 810 1.31 0.08 0

5 0.28 0.05 620 0.84 0.47 510

6 0.28 0.05 570 0.94 0.58 500

captured by the remaining time course. This interpretation entails

that γ should be in a reasonable range of ∼200 to ∼500 ms. How-

ever, γ ranges here from 0 to 810 ms (Tab l e 3 ). While a negative

value for this parameter is impossible because it would suggest

a negative duration

3

,avalueofγ =0 ms suggests that no time

was required for processes unrelated to the decision. On the other

hand, a value of γ =810 ms is higher than some of the observed

response times (and even higher than the MRT for large angles),

which also makes it impossible to interpret γ as non-decision time.

Thus, the under constraint of these non-linear models leads to

implausible parameter estimates, and therefore the results lack

psychological credibility. Because of the unexpected behavior of

3

We constrained the simplex routine to only allow positive values. Without this

constraint, the best ﬁtting models included values of γ < 0 ms.

Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 4

van Maanen et al. Piéron’s law in RDM

Participant 3, the best ﬁtting parameters take different values than

for the remaining participants.

For these reasons, we take a more elaborate approach in ﬁtting

the power and exponential functions by ﬁrst ﬁtting an LBA model

to the data and then estimating the parameters of the non-linear

functions on the decision time. The LBA model that best balanced

model ﬁt with the number of free parameters was Model 1, the

model in which only drift rate varied over angles. The average BIC

value for Model 1 over participants was 7092, which was consid-

erably less than any of the other models (7149, 8276, and 7307,

respectively)

4

. Thus, the BIC results indicate that only drift rate

variations were required to account for the data and no additional

parameters were needed.

For Model 1, the best ﬁtting parameter values are presented

in Table 4

5

. As expected, the LBA model captures the increase

in mean drift rate with increasing angular distance. The estimated

non-decision time parameters (γ) seem to be in a reasonable range

of 270–500 ms.

A power function ﬁts better

The LBA model allowed a more constrained estimation of the

non-decision time parameter γ. We can now estimate power and

exponential curves using this parameter estimate to assess whether

the data follows a Piéron-like pattern. We ﬁtted Equations (2) and

(3) to the data of Experiment 1, allowing two free parameters (α

and β), and one ﬁxed parameter γ. Again, we ﬁt both functions

using standard simplex optimization routines (Nelder and Mead,

1965). The best ﬁtting exponential and power curves are presented

in Figure 3 (the best ﬁtting parameters are presented in Ta ble 6 ).

The results show that the power function was the best ﬁt-

ting model for all participants. Tab le 5 presents the goodness of

ﬁt for the exponential and power functions presented in Equa-

tions (3) and (2). The power function is consistently the better

model, as indicated by the lower BIC values and higher correla-

tions. Although the data of Participants 3 and 4 do not clearly

distinguish between the models, the power function clearly is the

better model for the remaining participants. In terms of the evi-

dence ratio, the power function is 15 to 415 times more likely

than the exponential function for these participants. This suggests

that an increase in angular distance yields a power law decrease

in MRT, and not an exponential law. The model comparison of

the psychologically constrained models thus shows that behavior

4

These BIC values are higher than the BIC values obtained with the power and

exponential functions because here we ﬁt the quantiles instead of MRT.

5

The model ﬁt itself is presented in the Appendix.

in 2AFC tasks is Piéron-like when the discriminability of choice

alternatives is manipulated.

BAYESIAN IDEAL OBSERVER

In detection tasks, Piéron’s Law relates MRT to stimulus intensity

by means of a power function (Equation 1). Experiment 1 showed

that MRT and choice difﬁculty in a 2AFC task are also related by

means of a power function. This result suggests that Piéron’s Law

may generalize to choice behavior in which the discriminability of

the correct alternative relative to the incorrect alternative is manip-

ulated, instead of the intensity of the stimulus. To understand

better why participants in Experiment 1 behave in accordance to

Piéron’s Law, we developed a Bayesian ideal observer model. This

model assumes that participants make the optimal choice, given

the uncertainty of the task (for example, a noisy stimulus).

METHODS

The RDM stimulus consists of a set of dots each moving in a

particular direction. A proportion of dots move in the same direc-

tion whereas the remaining dots move in random directions. In

order to make a correct decision, the observer needs to decide

whether a certain amount of evidence for a particular response

alternative outweighs the evidence for other response alternatives.

If the observer performs this task on average in the minimum time

required for a particular error rate, the observer is said to be opti-

mal (Bogacz et al., 2006; Brown et al., 2009). Optimal behavior

is achieved if the observer computes for each response alternative

the posterior probability that it is the target based on the evidence

observed so far (Baum and Veeravalli, 1994):

P(H

i

|D) =

P(D|H

i

)P(H

i

)

j

P(D|H

j

)P(H

j

)

,

with H

i

the hypothesis that motion direction i generated the

RDM stimulus and x

1

, ..., x

t

∈D the observed motion directions

over time. Here we assume that the prior probabilities for each

alternative are equal and hence can be ignored:

P(H

i

|D) =

P(D|H

i

)

j

P(D|H

j

)

.

On the basis of new incoming evidence, the model continually

updates the posterior probability of each response alternative,until

the probability of one of the alternatives crosses a preset response

criterion θ:

P(D|H

i

)

j

P(D|H

j

)

θ.

Table 4 | Best ﬁtting LBA model parameters for the data of Experiment 1.

Participant γ (ms) Abs v

11.5

v

16

v

22.5

v

32

v

45

v

64

v

90

1 416 0.28 0.28 0.38 0.61 0.65 0.80 0.86 0.97 1.04 1.01

2 389 0.50 0.50 0.54 0.92 1.03 1.12 1.19 1.32 1.41 1.30

3 306 0.90 0.90 0.59 0.66 0.84 0.71 0.93 0.79 1.05 1.03

4 504 0.53 0.57 0.32 0.57 0.59 0.64 0.69 0.81 0.89 0.96

5 271 0.60 0.60 0.31 0.61 0.56 0.59 0.73 0.76 0.86 0.86

6 402 0.43 0.43 0.41 0.74 0.84 1.06 1.09 1.18 1.29 1.33

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van Maanen et al. Piéron’s law in RDM

Table 5 | Correlation coefﬁcient (ρ) and BIC values for the

two-parameter power function and the exponential function for each

participant.

Participant ρ BIC Evidence ratio

Exp Pow Exp Pow

1 0.94 0.97 −58.73 −64.28 15.97

2 0.85 0.94 −48.30 −53.78 15.49

3 0.59 0.63 −45.47 −46.16 1.41

4 0.96 0.97 −55.69 −55.71 1.01

5 0.90 0.97 −47.97 −56.37 66.82

6 0.92 0.99 −50.97 −63.03 415.14

Table 6 | Best ﬁtting parameter values for the two-parameter power

and exponential functions.

Participant Exponential Power

αβ αβ

1 0.24 0.0076 0.45 0.27

2 0.33 0.0061 0.59 0.23

3 0.56 0.0018 0.67 0.072

4 0.52 0.0054 0.81 0.19

5 0.51 0.0049 0.81 0.19

6 0.33 0.0086 0.70 0.31

Note that the intercept is obtained through LBA model ﬁtting (see Table 4).

The time needed to reach that particular criterion θ determines

the decision time (DT). Because in the current context we only

discuss two-choice decisions, this equation reduces to a sequential

probability ratio test (SPRT, Wald, 1947)

P

(

D|H

1

)

P

(

D|H

2

)

= θ

∗

, (4)

in which alternative 1 is chosen if θ

∗

≥θ/(1 −θ) and alternative 2

is chosen if θ

∗

≤(1 −θ)/θ (Baum and Veeravalli, 1994).

RESULTS AND DISCUSSION

Piéron’s law in the ideal observer model for 2AFC tasks

We assume here that the evidence for the correct response alter-

native is represented by a Gaussian distribution with a mean μ

i

representing the direction of motion. The variance of the distrib-

ution σ

2

may represent individual differences in perceptual ability,

but will remain constant in our simulations. Thus, at each time

step the model samples from the Gaussian distribution:

f

(

x|μ, σ

)

=

1

√

2πσ

2

e

−(x−μ

i

)

2

/2σ

2

(5)

until the decision threshold has been reached

6

.

6

Because of the circular arrangement of response alternatives in the RDM task a

slightly more accurate sampling distribution would be the von Mises distribution,

Without loss of generality we assume that the correct response is

always alternative 1. The evidence for alternative 1 at any time step

is computed from the Gaussian distribution function (Equation

5). Substituting Equation (5) in Equation (4) and taking the log

shows that an evidence sample S

t

for any alternative is proportional

to the distance between the alternatives:

S

t

= log

1

√

2πσ

2

exp

−(x − μ

1

)

2

/2σ

2

1

√

2πσ

2

exp

−(x − μ

2

)

2

/2σ

2

∝

(

μ

1

− μ

2

)

2

(6)

Following this computation of the evidence at each time step

the ideal observer model predicts that decision times decrease as a

power curve with the angular distance between the alternatives, as

shown by Stafford and Gurney, 2004): Suppose that at each time

step, the posterior probability of either alternative is increased

with the average evidence sample E{S

t

}. Then a decision is made

as soon as Σ

DT

t=1

E{ S

t

} log θ

∗

, with a decision time of approx-

imately DT. Therefore DT ·E{S

t

} ≈log θ

∗

, which means that the

mean decision time (MDT) is

MDT = C · E

{

S

t

}

−1

(7)

with C =log θ

∗

. Because S

t

∝(μ

1

−μ

2

)

2

, Equation (7) entails

Piéron’s Law (Equation 1) with the intercept γ =0.The value of

the scaling parameters α and β depends on the threshold and the

variance in the assumed Gaussian distributions.

To corroborate this result, we explored the model’s behavior

as a function of angular distance by running a 2AFC simulation.

That is, the model had two response options and we manipulated

the angular distance from 11.5˚ to 90˚. In addition, we modeled a

range of response criteria (from 0.6 to 0.9). The variance of the

sampling distribution was kept constant at σ

2

=4. The result of

these simulations are presented in Figure 4. Each data point in

Figure 4 was estimated using 10,000 Monte Carlo samples. The

model’s behavior exhibits Piéron’s Law: when transformed to a

log-log-scale the relation between angular distance and MDT is

approximately linear. This pattern is observed independent of the

response criterion (θ

∗

) that is being adopted.

These simulations are in line with the results from Experiment

1, and together the model and experiment support the view that

Piéron’s Law is a consequence of optimal behavior, at least in

perceptual 2AFC tasks. To extend these results to the traditional

ﬁeld of application of stimulus detection, we demonstrate how the

ideal observer model predicts Piéron’s Law in stimulus detection

experiments.

Piéron’s law in the ideal observer model for stimulus detection

In stimulus detection experiments, the intensity of a stimulus is

manipulated and participants are required to indicate the pres-

ence or absence of the stimulus. An ideal observer (ideal stimulus

detector) weighs the likelihood of the presence of the stimulus (i.e.,

which can be thought of as a Gaussian distribution wrapped around a circle. Because

this distribution leads to similar results we chose to use the more generally applicable

Gaussian.

Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 6

van Maanen et al. Piéron’s law in RDM

FIGURE 4 | Ideal observer model behavior as a function of the angular

distance between alternatives, for four different response criterion

values. Left: Mean decision time (MDT) vs angular distance. Middle: log MDT

vs log Angle. Right: Proportion correct vs Angle. θ

∗

: response criterion value.

P(D|H

1

) in Equation 4) against the likelihood of the absence of

the stimulus (i.e., P(D|H

2

)). Because any evidence supporting the

presence (or absence) of the stimulus contributes equal evidence

against the absence (or presence) of the stimulus, Equation (4) can

be written as

P

(

D|H

1

)

1 − P

(

D|H

1

)

= θ

∗

. (8)

Piéron’s Law emerges from an ideal observer model for stimulus

detection if the likelihood of the presence (or absence) of the stim-

ulus at each time step scales linearly with the stimulus intensity.

In that case, MDT will decrease with stimulus intensity accord-

ing to a power function (Equation 7) which will result in Piéron’s

Law. Figure 5 presents the results of a simulation of this process.

On each time step a value is drawn from a Gaussian distribution

with stimulus intensity as the mean and σ =0.005. The stimulus

intensities ranged from 0.01 to 0.05. Equation (8) was computed

until the posterior probability exceeded threshold (θ

∗

), and the

number of time steps required was recorded. Each data point in

Figure 5 was estimated using 10,000 Monte Carlo samples. Again,

the linearity in log-log space illustrates that the non-linear relation

in the MRT data follows a power curve.

GENERAL DISCUSSION AND CONCLUSION

Piéron’s Law has previously only been studied in experiments in

which stimulus intensity was manipulated. Here, we hypothesized

that Piéron’s Law may be a special case of a more general rela-

tionship between choice difﬁculty and mean response time. To

support this conjecture, we performed an experiment in which the

difﬁculty of choice in a random-dot motion paradigm was manip-

ulated by adjusting the angular distance between two response

alternatives. We found a power law relation between angular

distance (discriminability) and mean response time, supporting

our hypothesis of a general mechanism behind Piéron’s Law. A

Bayesian ideal observer model showed that participants perform-

ing a 2AFC task may respond in a Piéron-like manner because

it is the optimal way of minimizing overall response times (for a

ﬁxed error rate). The ideal observer model can also be extended

to stimulus detection behavior, showing that Piéron’s Law reﬂects

optimal detection behavior under varying stimulus intensities.

FIGURE 5 | Ideal observer model behavior as a function of stimulus

intensity, for ﬁve different response criterion values. Left: Mean decision

time (MDT) vs stimulus intensity. Right: log MDT vs log Intensity. θ

∗

:

response criterion value.

One particular aspect of our study deserves some additional

consideration. We conclude that the decline of MRT with choice

difﬁculty in Experiment 1 can be best described by a power func-

tion. An important step toward this conclusion was to determine

which part of the observed response time we wanted to explain.

As we have argued, directly ﬁtting the models to the observed

data does not provide enough constraint to make an appropri-

ate model selection inference. Therefore, we ﬁrst determined the

decision time by subtracting an estimate for the non-decision

time from the RTs. The non-decision time was estimated using

the LBA model of choice-response time (Brown and Heathcote,

2008).

The validity of this approach rests on the extent to which it

is reasonable to exclude non-decision time and focus on decision

time. Because the exponential and power functions also implicitly

focus on decision time, we believe that in this study, our approach

is plausible. After subtracting non-decision time the evidence in

favor of the power function was overwhelming, which was not

the case for the models that were directly ﬁtted to MRT. In some

instances, the traditional approach may lead to an accurate esti-

mate of the non-decision time parameter. In those cases, a more

elaborate two-step ﬁtting approach may not appear to be neces-

sary. However, even in such cases we recommend the two-step

ﬁtting approach.

www.frontiersin.org January 2012 | Volume 5 | Article 143 | 7

van Maanen et al. Piéron’s law in RDM

It should be noted that although we used a speciﬁc processing

model to obtain appropriate non-decision time estimates (the LBA

model), the same results could have been obtained with any other

model that takes the variance in the RT distribution into account

(e.g., Donkin et al., 2011). Where the LBA model is a full process

model that attempts to describe many aspects of decision-making,

we could have also used simpler process models (e.g., the EZ dif-

fusion model, Wagenmakers et al., 2007), or even any descriptive

model of the RT distribution (Matzke and Wagenmakers, 2009).

In summary, we have shown that MRT in a 2AFC task decreases

with stimulus discriminability according to a power function. In

addition, we provided a Bayesian ideal observer analysis of both

2AFC tasks and stimulus detection tasks. This analysis showed that

optimal behavior in these tasks follows a power function when

stimulus discriminability is manipulated. These results support

the view that Piéron’s Law originates from optimal information

processing.

AUTHOR NOTE

Leendert van Maanen, Raoul Grasman, Birte Forstmann, and

Eric-Jan Wagenmakers, Department of Psychology, University of

Amsterdam. Correspondence concerning this article should be

addressed to Leendert van Maanen,Cognitive Science CenterAms-

terdam,University of Amsterdam, Plantage Muidergracht 24, 1018

TV Amsterdam, the Netherlands, e-mail: l.vanmaanen@uva.nl.

We thank Helen Steingröver for data collection. This research

was supported by VENI and VIDI grants from the Netherlands

Organization for Scientiﬁc Research.

REFERENCES

Ball, K., and Sekuler, R. (1982). A spe-

ciﬁc and enduring improvement in

visual motion discrimination. Sci-

ence 218, 697–698.

Banks, W. (1973). Reaction time as a

measure of summation of warmth.

Percept. Psychophys. 13, 321–327.

Baum, C., and Veeravalli, V. (1994).

A sequential procedure for multi-

hypothesis testing. IEEE Trans. Inf.

Theory 40, 1994–2007.

Bogacz, R., Brown, E., Moehlis, J.,

Holmes, P., and Cohen, J. D. (2006).

The physics of optimal decision

making: a formal analysis of models

of performance in two-alternative

forced-choice tasks. Psychol. Rev.

113, 700–765.

Bonnet, C., Zamora, M. C., Buratti, F.,

and Guirao, M. (1999). Group and

individual gustatory reaction times

and Piéron’s law. Physiol. Behav. 66,

549–558.

Brainard, D. H. (1997). The psy-

chophysics toolbox. Spat. Vis. 10,

433–436.

Britten, K. H., Shadlen, M. N., New-

some, W. T., and Movshon, J.

A. (1992). The analysis of visual

motion: a comparison of neuronal

and psychophysical performance. J.

Neurosci. 12, 4745–4765.

Brown, S., and Heathcote, A. (2008).

The simplest complete model of

choice response time: linear ballis-

tic accumulation. Cogn. Psychol. 57,

153–178.

Brown, S., Steyvers, M., and Wagen-

makers, E.-J. (2009). Observing evi-

dence accumulation during multi-

alternative decisions. J. Math. Psy-

chol. 53, 453–462.

Chocholle, R. (1940). Variation des

temps de réaction auditifs en

fonction de l’intensité à diverses

fréquences. Année Psychol. 41,

65–124.

Churchland, A. K., Kiani, R., Chaud-

huri, R., Wang, X.-J., Pouget, A., and

Shadlen, M. N. (2011). Variance as

a signature of neural computations

during decision making. Neuron 69,

818–831.

Churchland, A. K., Kiani, R., and

Shadlen, M. N. (2008). Decision-

making with multiple alternatives.

Nat. Neurosci. 11, 693–702.

Donkin, C., Brown, S., Heathcote, A.,

and Wagenmakers, E. (2011). Dif-

fusion versus linear ballistic accu-

mulation: different models but the

same conclusions about psychologi-

cal processes? Psychon. Bull. Rev. 18,

61–69.

Forstmann,B. U.,Anwander,A., Schäfer,

A., Neumann, J., Brown, S., Wagen-

makers, E.-J., Bogacz, R., and Turner,

R. (2010a). Cortico-striatal connec-

tions predict control over speed

and accuracy in perceptual decision

making. Proc. Natl. Acad. Sci. U.S.A.

107, 15916–15920.

Forstmann, B. U., Brown, S. D., Dutilh,

G., Neumann, J., and Wagenmakers,

E. J. (2010b). The neural substrate

of prior information in percep-

tual decision making: a model-based

analysis. Front. Hum. Neurosci. 4:40.

doi:10.3389/fnhum.2010.00040

Forstmann, B. U., Dutilh, G., Brown,

S., Neumann, J., von Cramon,

D. Y., Ridderinkhof, K. R., and

Wagenmakers, E. J. (2008). Striatum

and pre-SMA facilitate decision-

making under time pressure.

Proc. Natl. Acad. Sci. U.S.A. 105,

17538–17542.

Heathcote, A., Brown, S., and Mewhort,

D. J. K. (2002). Quantile maximum

likelihood estimation of response

time distributions. Psychon. Bull.

Rev . 9, 394–401.

Ho, T. C., Brown, S., and Ser-

ences, J. T. (2009). Domain general

mechanisms of perceptual decision

making in human cortex. J. Neurosci.

29, 8675–8687.

Jaskowski,P., and Sobieralska, K. (2004).

Effect of stimulus intensity on man-

ual and saccadic reaction time. Per-

cept. Psychophys. 66, 535–544.

Loftus, G. R., and Masson, M.

(1994). Using conﬁdence intervals

in within-subject designs. Psychon.

Bull. Rev. 1, 476–490.

Luce, R. D. (1986). Response Times

.New

York: Oxford University Press.

Matzke, D., and Wagenmakers, E.-J.

(2009). Psychological interpretation

of the ex-Gaussian and shifted Wald

parameters: a diffusion model analy-

sis. Psychon. Bull. Rev. 16, 798–817.

Mulder, M., Bos, D., Weusten, J. M.,

van Belle, J., Dijk, S. C., Simen, P.,

van Engeland, H., and Durston, S.

(2010). Basic impairments in reg-

ulating the speed-accuracy tradeoff

predict symptoms of ADHD. Biol.

Psychiatry 68, 1114–1119.

Nelder, J., and Mead, R. (1965). A sim-

plex method for function minimiza-

tion. Comput. J. 7, 308–313.

Niwa, M., and Ditterich, J. (2008). Per-

ceptual decisions between multiple

directions of visual motion. J. Neu-

rosci. 28, 4435–4445.

Overbosch, P., de Wijk, R., de Jonge,

T. J., and Köster, E. P. (1989). Tem-

poral integration and reaction times

in human smell. Physiol. Behav. 45,

615–626.

Palmer, J., Huk, A. C., and Shadlen, M.

N. (2005). The effect of stimulus

strength on the speed and accuracy

of a perceptual decision. J. Vis. 5,

376–404.

Piéron, H. (1914). Recherches sur les

lois de variation des temps de latence

sensorielle en fonction des inten-

sités excitatrices. Année Psychol. 22,

17–96.

Pins, D., and Bonnet, C. (1996). On the

relation between stimulus intensity

and processing time: Piéron’s law

and choice reaction time. Percept.

Psychophys. 58, 390–400.

Raftery, A. (1996). “Hypothesis test-

ing and model selection,” in Markov

Chain Monte Carlo in Practice,edsW.

Gilks, S. Richardson, and D. Spiegel-

halter (Boca Raton, FL: Chapman &

Hall/CRC), 163–187.

Salzman, C. D., and Newsome, W.

T. (1994). Neural mechanisms for

forming a perceptual decision. Sci-

ence 264, 231–237.

Schwarz, G. (1978). Estimating the

dimension of a model. Ann. Stat. 6,

461–464.

Shadlen, M. N., and Newsome, W. T.

(2001). Neural basis of a percep-

tual decision in the parietal cortex

(area LIP) of the rhesus monkey. J.

Neurophysiol. 86, 1916–1936.

Stafford, T., and Gurney, K. N. (2004).

The role of response mechanisms

in determining reaction time perfor-

mance: Piéron’s law revisited. Psy-

chon. Bull. Re v. 11, 975–987.

Stafford, T., Ingram, L., and Gurney,

K. N. (2011). Piéron’s law holds

during Stroop conﬂict: insights into

the architecture of decision making.

Cogn. Sci. 35, 1553–1566.

Sternberg, S. (1969). The discovery

of processing stages: extensions of

Donders’ method. Acta Psychol.

(Amst.) 30, 276–315.

Van Maanen, L., Brown, S. D.,

Eichele, T., Wagenmakers, E. J.,

Ho, T. C., Serences, J. T., and

Forstmann,B. U. (2011). Neural cor-

relates of trial-to-trial ﬂuctuations

in response caution. J. Neurosci. 31,

17488–17495.

Van Maanen, L., and Van Rijn, H.

(2010). The locus of the Gratton

effect in picture-word interference.

Top. Cogn. Sci. 2, 168–180.

Van Maanen, L., Van Rijn, H., and Borst,

J. P. (2009). Stroop and picture-word

Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 8

van Maanen et al. Piéron’s law in RDM

interferenceare two sides of the same

coin. Psychon. Bull. Rev. 16, 987–999.

Wagenmakers, E.-J., and Farrell, S.

(2004). AIC model selection using

Akaike weights. Psychon. Bull. Rev.

11, 192–196.

Wagenmakers, E.-J., Ratcliff, R., Gomez,

P., and McKoon, G. (2008). A

diffusion model account of cri-

terion shifts in the lexical deci-

sion task. J. Mem. Lang. 58,

140–159.

Wagenmakers, E.-J., van der Maas, H. L.

J., and Grasman, R. P. P. P. (2007).

An EZ-diffusion model for response

time and accuracy. Psychon. Bull.

Rev . 14, 3–22.

Wald, A. (1947). Sequential Analysis.

New York: Wiley.

Conﬂict of Interest Statement: The

authors declare that the research was

conducted in the absence of any com-

mercial or ﬁnancial relationships that

could be construed as a potential con-

ﬂict of interest.

Received: 20 September 2011; accepted:

12 December 2011; published online: 02

January 2012.

Citation: van Maanen L, Grasman

RPPP, Forstmann BU and Wagenmak-

ers E-J (2012) Piéron’s law and opti-

mal behavior in perceptual decision-

making. Front. Neurosci. 5:143. doi:

10.3389/fnins.2011.00143

This article was submitted to Frontiers

in Decision Neuroscience, a specialty of

Frontiers in Neuroscience.

Copyright © 2012 van Maanen, Gras-

man, Forstmann and Wagenmakers. This

is an open-access article distributed under

the terms of the Creative Commons Attri-

bution Non Commercial License, which

permits non-commercial us e, distribu-

tion, and reproduction in other forums,

provided the original authors and source

are credited.

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van Maanen et al. Piéron’s law in RDM

APPENDIX

Figures A1–A6 present the ﬁt of the Linear Ballistic Accumulator (LBA) model to the data of Experiment 1. Each ﬁgure

represents the data (black circles) and model ﬁt (red lines) for one participant. The top line represents the cumulative RT dis-

tribution of correct responses, the bottom line represents the cumulative RT distribution of incorrect responses. Both lines are

weighted to the probability of an incorrect response. Each graph represents the different angular distances (d) used in Experiment 1.

FIGURE A1 | LBA model ﬁt for Participant 1.

Frontiers in Neuroscience | Decision Neuroscience January 2012 | Volume 5 | Article 143 | 10