J Neurophysiol 115: 39 –59, 2016.
First published October 14, 2015; doi:10.1152/jn.00225.2015.
CALL FOR PAPERS
Review
Decision Making: Neural Mechanisms
Dynamics of individual perceptual decisions
Daniel M. Merfeld,1,2 Torin K. Clark,1,2 Yue M. Lu,3 and Faisal Karmali1,2
1
Jenks Vestibular Physiology Lab, Massachusetts Eye and Ear Infirmary, Boston, Massachusetts; 2Department of Otology and
Laryngology, Harvard Medical School, Boston, Massachusetts; and 3Harvard School of Engineering and Applied Sciences,
Cambridge, Massachusetts
Submitted 6 March 2015; accepted in final form 13 October 2015
drift diffusion; signal detection; threshold
on perceptual decision-making
dynamics. For simplicity, we refer to “perceptual decision
making” as “decision making” throughout. Thorough definitions and descriptions follow later in this review; we begin by
briefly defining decision-making dynamics as the study of time
variations in the signals leading to individual decisions, including with the impact of stimulus frequency (i.e., studies of the
frequency response of the decision-making system). We provide an illustrative example in the following paragraph. To
keep the review focused on decision making, our primary focus
is on those processes that contribute to (and occur before) each
individual decision; we specifically choose not to cover all the
processes and/or processing that occur in conjunction with
decision making [e.g., confidence in a decision (Stankov et al.
2012), evidence accumulation after a decision boundary is
reached (Huk and Shadlen 2005), etc.] or interactions between
sequential decisions. To help raise an awareness of the importance of dynamics, our primary goal is to present and discuss
a variety of findings that pertain to decision-making dynamics.
To preview the endpoint, we provide our final thesis here:
perceptual decision-making data across multiple sensory do-
THIS REVIEW FOCUSES PRIMARILY
Address for reprint requests and other correspondence: D. M. Merfeld, Jenks
Vestibular Physiology Lab, Rm. 421, Mass. Eye and Ear Infirmary, Boston,
MA 02114 (e-mail: [email protected]).
www.jn.org
mains using various psychophysical tasks appear consistent
with the hypothesis that decision-making processes include
dynamic elements (i.e., phasic neural mechanisms). Furthermore, evidence also suggests that one or more midtrial decision
mechanisms (described later) may also be essential. More
specifically, we discuss the hypothesis that decision making
may reflect high-pass filtering of the relevant information
leading to an individual decision, but this review does not
hinge on the veracity of that hypothesis. Alternate hypotheses
(e.g., attentional influences, the increasing prevalence of driftdiffusion boundary crossings with time, shifting decision
boundaries, including costs of accumulating evidence) are
described elsewhere in the manuscript and assembled in DISCUSSION, Other Plausible Considerations.
We begin with an illustrative example. Recent studies (e.g.,
Grabherr et al. 2008) have shown that direction-recognition
thresholds for whole body rotations, primarily evoked by
signals from the vestibular system (Valko et al. 2012), vary
with frequency. Specifically, thresholds for lower frequency
(longer duration) rotation have been shown to have higher
thresholds than for higher frequency (shorter duration) rotation. Rotation perception assayed using psychophysical techniques that do not utilize the standard discrete decision-making
techniques does not show the same frequency dependence
0022-3077/16 Copyright © 2016 the American Physiological Society
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Merfeld DM, Clark TK, Lu YM, Karmali F. Dynamics of individual perceptual decisions. J Neurophysiol 115: 39 –59, 2016. First published October 14, 2015;
doi:10.1152/jn.00225.2015.—Perceptual decision making is fundamental to a
broad range of fields including neurophysiology, economics, medicine, advertising,
law, etc. Although recent findings have yielded major advances in our understanding of perceptual decision making, decision making as a function of time and
frequency (i.e., decision-making dynamics) is not well understood. To limit the
review length, we focus most of this review on human findings. Animal findings,
which are extensively reviewed elsewhere, are included when beneficial or necessary. We attempt to put these various findings and data sets, which can appear
to be unrelated in the absence of a formal dynamic analysis, into context using
published models. Specifically, by adding appropriate dynamic mechanisms
(e.g., high-pass filters) to existing models, it appears that a number of otherwise
seemingly disparate findings from the literature might be explained. One
hypothesis that arises through this dynamic analysis is that decision making
includes phasic (high pass) neural mechanisms, an evidence accumulator and/or
some sort of midtrial decision-making mechanism (e.g., peak detector and/or
decision boundary).
Review
40
PERCEPTUAL DECISION-MAKING DYNAMICS
Review “Roadmap”
To provide an overview and to help readers with different
interests and backgrounds navigate this review efficiently, we
provide an outline of the remaining sections. The first section
immediately following this roadmap (Defining Terms, Experimental Tasks, and Models) thoroughly defines terms, tasks,
and models. This section is primarily included for nonexperts
to introduce how terms, tasks, and models are commonly used.
It is not our intent to provide a comprehensive review of
advanced topics related to these definitions and descriptions,
which could be the topic of an entire review not focused on
dynamics. Rather, we focus on basic definitions and descriptions that capture the most common ways the terms are defined
and the tasks and models that are most commonly used.
Next, since the topic of dynamics may not be familiar to
many readers, we provide a short primer on dynamics (Filtering Dynamics) that provides a basic introduction to first-order,
linear analog, continuous-time high-pass and low-pass filters.
Some readers prefer that this primer be provided before
the data sections that immediately follow. Other readers prefer
the context provided by the data to help them focus on the
important aspects of this filtering section. To accommodate
both styles, we provide the section just before the data sections.
Individuals who prefer having context provided by experimental findings before reading about simple filtering might consider skimming this section when first reading this review and
then returning to read this section more thoroughly after
reading the data sections that follow.
As noted, several sections after the filtering dynamics primer
describe published findings. Some readers interested primarily
in empirical findings (and less in theory and modeling) may
consider these the key sections of this review. In the first data
section, we discuss the vestibular threshold literature that
demonstrates frequency responses consistent with high-pass
filtering; these data first suggested to us that dynamic elements
may be contributing to decision making (Vestibular Threshold
Frequency Responses). In the second data section, we discuss
data from several visual perception studies that appear to show
similar patterns of thresholds vs. frequency (Visual Perceptual
Responses). In the third data section, we present data showing
how speed and accuracy of decision-making co-vary depending on experimental constraints (Speed-Accuracy Trade-offs).
These three data sections appear to support the hypothesis that
perceptual decision-making processes include mechanisms that
have dynamic characteristics that mimic high-pass filter characteristics.
We then include two sections focused on drift-diffusion
models (and variants). Such models have long been used to
model time-to-respond data, so there is an immense literature
that we have reviewed and could draw upon. However, much
of this literature has been reviewed (Bogacz et al. 2006;
Busemeyer and Townsend 1993; Ratcliff 1978; Ratcliff and
Starns 2013; Usher and McClelland 2001), and most of this
literature does not consider dynamic elements beyond integration/accumulation. Furthermore, this literature does not appear
to clearly support or refute any specific dynamics theory,
high-pass filtering or other. Therefore, we provide an overview
of diffusion models with a focus on system dynamics (Dynamic Decision-Making Modeling). We then present several
recent studies (Drift-Diffusion Studies Investigating Dynamics)
that specifically investigate dynamic contributions to decision
making, with differing conclusions. The penultimate section
describes different ways that the brain might accumulate information for decision making (Relating High-Pass and LowPass Filtering), and the last section is a standard DISCUSSION
section wherein we consider several additional issues related to
decision-making dynamics.
Defining Terms, Experimental Tasks, and Models
Before proceeding, we need to define and introduce “dynamics” and then “decision-making” before we can precisely
define “decision-making dynamics.” We close this section with
a brief description of “magnitude estimation.”
Dynamics. System dynamics is the mathematical approach
that helps explain and understand the behavior of systems over
time. The genesis of dynamics can be traced to Newton’s
invention of differential equations to help explain planetary
motion using his models of gravity and motion. Dynamic
systems can be 1) electrical, mechanical, chemical, social,
economic, biologic, etc., 2) discrete time or continuous time (or
both), and/or 3) linear (e.g., Kailath 1980; Ogata 2004) or
nonlinear (e.g., Shelhamer 2006; Strogatz 1996; Thompson
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(Bertolini et al. 2011; Bronstein et al. 2008; Okada et al. 1999;
Sinha et al. 2008). As will be discussed in much greater detail
later, the variation of rotation thresholds with frequency have
led to a hypothesis that the brain might high-pass filter the
available perceptual signals as part of the decision-making
processes. The plausibility of this hypothesis is bolstered by
data showing that 1) visual motion thresholds show similar
variations with frequency, and 2) speed-accuracy trade-offs
provide evidence suggesting similar dynamic processes.
As defined in more detail later, perceptual decision making
is the process of making a choice or judgment resulting in
perceptual categorization (e.g., deciding if an object is hot or
cold). Decision making is challenging because the nervous
system does not perfectly represent information. For example,
various forms of noise can arise externally (i.e., static noise
heard and/or seen while “skyping”), during transduction,
and/or internally in the neural circuitry. Decision making
becomes especially challenging when stimuli are small in
comparison to noise, such as hearing a whisper at a loud party.
When decision making is studied in the laboratory, one of two
experimental paradigms is typically used. For one task, the
subject is provided stimuli and told that he/she must respond
when signaled by the operator after the stimulus is complete
(often called “forced choice”). For the second task, the subject
is instructed to respond as soon as a decision is reached (often
called “response time” or “time to respond”). Such experimental tasks are typically accompanied by one of two classes of
standard theoretic frameworks (Gold and Shadlen 2007): 1)
signal detection or 2) drift diffusion (formally referred to as
sequential analysis). Multiple reviews and books focused on
decision making have been published relatively recently (e.g.,
Bogacz 2007; Bogacz et al. 2006; Gold and Shadlen 2007;
Heekeren et al. 2008; Macmillan and Creelman 2005; Purcell
et al. 2010; Ratcliff and McKoon 2008; Ratcliff and Starns
2013; Sanfey et al. 2006; Shadlen and Kiani 2013; Sugrue et al.
2005). The focus of this review on decision-making dynamics
differs from the earlier reviews.
Review
PERCEPTUAL DECISION-MAKING DYNAMICS
standing of the system. In general, neither the time- or frequency-domain approach is “better,” nor does either provide a
more fundamental understanding.
Decision making. Decision making is studied using several
different approaches. Because these approaches are somewhat
different, we briefly describe several common approaches.
Decision making studied with the use of the signal detection
approach assumes that a forced-choice binary decision (e.g.,
left/right or yes/no) is made and requires an answer after each
trial is complete, often without penalty for deliberation time.
These studies almost always assume that there is a single
decision boundary (e.g., Fig. 1, A and B) that separates inputs
into one of two categories. Let us consider a standard motion
direction-recognition task (also sometimes called a directiondiscrimination task), where the subject’s task is to decide if
motion was leftward or rightward. If the decision variable is
“positive” relative to the decision boundary (i.e., is greater than
the decision boundary), the subject will decide “positive” (e.g.,
motion to the right), whereas if the decision variable is less
than the decision boundary, the subject will report “negative”
(e.g., motion to the left).
Thresholds are often the measure resulting from the application of signal detection theory to human perceptual decision
making. A threshold is a representation of the ratio of signal to
noise; signal detection thresholds are always defined with
respect to noise. In fact, when the noise is zero mean (or
near-zero mean as is often assumed and/or found for directionrecognition tasks), one way to define a direction-recognition
threshold is as the signal level that equals the standard deviation of the noise (T ⫽ ␴). Using this definition of a direction-
Fig. 1. Decision-making models applied to perceptual motion direction-recognition tasks. We emphasize that the models shown all apply to tasks whereby
subjects are asked to classify the direction of motion (e.g., left vs. right), which is a type of motion direction discrimination sometimes referred to as direction
recognition. A: for untimed forced-choice tasks, the signal detection model assumes that a physical stimulus is contaminated with noise to yield a perceptual
decision variable distribution. One sample from this distribution for each trial yields the decision variable for that trial. If the sampled decision variable for a
given trial falls to the left of the decision boundary, “left” is reported. If the decision variable falls to the right of the decision boundary, “right” is reported. B:
to allow a direct comparison with C and D, we rotate the same decision variable and signal detection model from A by simply swapping the abscissa and ordinate.
C: for response-time tasks, a drift-diffusion model is often used to help model the time dependencies of the data. A time plot for a drift-diffusion model shows
a decision variable for a single trial plotted against time. When the decision variable crosses a decision boundary, a decision is made: “left” for top decision
boundary and “right” for bottom decision boundary. The time of the decision is indicated as tdecide. Before the decision variable crosses a decision boundary,
the subject is undecided. The distribution of the decision variables across many trials at time 0 (t0) is also shown. The average evidence accumulation corresponds
to the physical stimulus and in this example is assumed to be constant over time. D: a signal detection model that includes uncertainty as one of the options
provides a simple extension to the signal detection model shown in A and B. Here the same decision variable is sampled but with 2 decision boundaries. If the
sampled decision variable falls to the left of the most leftward decision boundary, “left” is reported. If the sampled decision variable falls to the right of the most
rightward decision boundary, “right” is reported. If the sampled decision variable falls between the 2 decision boundaries, “uncertain” is reported. Note the
intentional similarity of the probability distribution at t0 in C and the probability distribution in D.
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and Stewart 2002). Simple dynamic systems include high-pass
filters and low-pass filters (e.g., moving averages) that are
commonly employed during physiological data acquisition.
Dynamic analyses are commonly used to help describe timevarying changes in biological processes such as sensory transduction, sleep cycles, and muscle actuation. System dynamics
are typically investigated experimentally and analyzed using
one of two complementary approaches, time-domain or frequency-domain approaches (e.g., Bendat and Piersol 1971,
1980; Ljung 1987; Oppenheim and Willsky 1997). Both timeand frequency-domain approaches will yield the same resultant
model of the system dynamics when properly applied to a
linear system, but dynamic analyses are not limited to linear
systems.
To investigate system dynamics in the time domain, system
input signals are typically impulses, ramps, steps (or approximations thereof; e.g., trapezoids), random noise, or pseudorandom noise with the system output measured as a function of
time. The frequency-domain approach often utilizes finiteduration sinusoidal inputs having an integer number of cycles
at various frequencies, but frequency-domain approaches do
not necessarily require sinusoidal inputs (e.g., Soyka et al.
2011, 2012). When sinusoidal inputs are utilized, the amplitude
and/or phase of the steady-state sinusoidal output is often
quantified. If a system is (or can be approximated as) linear and
time invariant, both time- and frequency-domain analyses can
be used to deliver a transfer function that models (i.e., represents) the input-output relationship of that linear system. In the
event that significantly different models result from these two
analyses, it shows that we do not have a satisfactory under-
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PERCEPTUAL DECISION-MAKING DYNAMICS
Shadlen (2007), Ratcliff and McKoon (2008), and Smith and
Ratcliff (2004). Implicitly, this approach adds a third “uncertain” categorization between the two decision boundaries (Fig.
1C). Drift-diffusion modeling almost always focuses on timedomain analyses because the subject is instructed to register
his/her choice as soon as he/she makes a decision, which
means that the stimulus duration at the time of the decision,
and hence, the frequency content of the stimuli, cannot be
controlled by the experimenter.
For both signal detection and response-time approaches,
real-life decisions and laboratory tasks come in many varieties;
we list a few (but far from all) varieties below. We first
consider decisions where time to respond is not a critical
variable. Experimentally, such decisions are typically investigated using tools and approaches provided by signal detection
theory. One relatively simple binary decision is “am I moving?” Another binary decision is “am I moving to the right or
left?” Although both are binary decisions, these are fundamentally different from the perspective of signal detection theory.
Binary discrimination tasks that require yes/no answers are
classically referred to as detection tasks as suggested by Swets
(1996), who wrote that “The task of detection is to determine
whether a stimulus of a specified category (call it category A)
is present or not.” Binary recognition tasks that classify stimuli
into one of two categories (e.g., leftward or rightward motion,
faces vs. houses) are a second form of discrimination as
suggested by Swets (1996), who wrote that “The task of
recognition is to determine whether a stimulus known to be
present, as a signal, is a sample from signal category A or
signal category B.” For further details on the differences
between these different binary tasks, see Chaudhuri et al.
(2013), Macmillan and Creelman (2005), and Swets (1973).
Such tasks are often called forced-choice tasks, because subjects are instructed that they must provide an answer after each
trial. Note that for such binary forced-choice tasks, the subjective probability of category A (pA) by definition equals 1 minus
the subjective probability of the alternative category (pB), i.e.,
pA ⫽ 1 ⫺ pB. Hence, from a statistical viewpoint, a single
variable can encapsulate all probabilistic information regarding
such binary forced-choice decisions.
We next consider decisions where time to respond is
important. For example, time to respond is critical when
deciding whether “I am starting to fall to the right or left.”
Relevant time-to-respond tasks that have a binary classification goal have been analyzed (e.g., Laming 1968; Ratcliff
1978; Stone 1960). Such paradigms cannot strictly be considered “forced choice” because a third category of “I don’t
know” is implicit in the categorization before the subject
provides their response. Specifically, beginning with stimulus presentation, there is a period of time during which the
subject is uncertain (i.e., until he/she makes the decision).
Hence, a third subjective probability, that of being uncertain
(pU), is applicable prior to the subjective decision. With the
sum of the three probabilities equaling 1 (1 ⫽ pA ⫹ pB ⫹
pU), at least two variables are required to encapsulate all
probabilistic information regarding such decisions until the
decision is made. Drift-diffusion models often include two
independent variables that accumulate information pertinent
to the decision [or a single accumulation variable that can be
interpreted as the difference of these two accumulation
variables (Bogacz et al. 2006)]. Since the subject is
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recognition threshold, a subject should get 84% of binary
forced-choice trials correct when stimuli having this magnitude
are presented (Merfeld 2011). For example, see Merfeld
(2011), which is focused on vestibular system “self-motion”
sensing applications.
This signal detection approach assumes that signals are
contaminated by the presence of noise and lends itself directly
to a temporal frequency domain approach via the use of stimuli
that vary sinusoidally with time (e.g., Benson et al. 1986, 1989;
Grabherr et al. 2008; Lim and Merfeld 2012; Soyka et al.
2011). Analysis with signal detection theory yields a threshold
corresponding to the smallest stimuli the subject can categorize
with a defined level of reliability, and thus includes decisionmaking processes when subjects decide how to categorize their
perceptions of physical stimuli. Thresholds measured using the
signal detection approach are influenced by the characteristics
of the physical stimuli (i.e., frequency content and/or time
course) as well as the dynamics of the transduction processes,
sensory processing, and decision-making processes. Thus measuring thresholds as a function of frequency can elucidate the
dynamic properties of these components. Decision-making
dynamics can be isolated if transducer dynamics are known
(e.g., through neural recordings or behavioral studies) and if
behaviorally relevant stimulus frequencies overlap with frequencies influenced by decision-making dynamics. In the past,
thresholds have often been measured as a function of stimulus
frequency without necessarily improving our understanding of
decision-making dynamics. For example, hearing (e.g., Von
Békésy and Wever 1960) and tactile (e.g., Lofvenberg and
Johansson 1984; Von Bekesy 1959) thresholds change with the
frequency of the applied pressure variations, and visual thresholds vary with the temporal frequency of light (Cornsweet
1970). Given the dynamic ranges (i.e., frequency ranges)
investigated, these measured threshold variations as a function
of frequency typically reflect peripheral transduction processes and typically have not told us much about decisionmaking dynamics. With the exception of some vestibular
threshold studies (e.g., Benson et al. 1986, 1989; Grabherr
et al. 2008; Haburcakova et al. 2012; Karmali et al. 2014;
Lewis et al. 2011a, 2011b; Lim and Merfeld 2012; Mardirossian et al. 2014; Priesol et al. 2014; Soyka et al. 2011,
2012; Valko et al. 2012), decision-making studies using
signal detection methods have rarely focused on dynamics
(e.g., perceptual decisions as a function of frequency, where
the frequency is in a range relevant to decision-making as
opposed to sensory transduction). As discussed later in this
review, such vestibular threshold studies may help inform us
about decision-making dynamics because behaviorally relevant stimulus frequencies overlap with frequencies influenced by decision-making dynamics.
Decision making is also often studied using a response-time
task, in which subjects are asked to indicate their decision as
soon as they make it. Data from such tasks are often analyzed
using a drift-diffusion (i.e., sequential analysis) approach. In
this model, evidence from a noisy signal is accumulated over
time. It is generally assumed that the input (i.e., stimulus) as a
function of time is constant. This noisy accumulation process
leads to “drift” of a decision variable. When this decision
variable crosses one of two decision boundaries, a decision is
made. This task requires that the subject respond as soon as
possible. For more comprehensive reviews, see Gold and
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PERCEPTUAL DECISION-MAKING DYNAMICS
Filtering Dynamics
As mentioned in the Introduction (and detailed in subsequent sections), it has been suggested that a high-pass filter
(HPF) might help explain why thresholds increase as frequency decreases (Fig. 2). Here, as background for that
section, we discuss filter responses. A high-pass filter passes
high-frequency signals and noise (i.e., those frequency components above the cutoff frequency) while attenuating lowfrequency signals and noise (i.e., drift). For simplicity, we
refer to signal components below a cutoff frequency as low
frequency and signal components above the cutoff as high
Fig. 2. The high-pass filter (HPF) hypothesis predicts that thresholds increase
at low frequencies. A: a typical cosine bell physical stimulus with a frequency
of 0.2 Hz, corresponding to a period of 5 s, with the peak signal indicated by
the gray filled triangle. B: high-pass filtered version of the 0.2-Hz signal, which
yields the hypothesized decision variable, with the peak signal indicated by the
open triangle. Similarly, 0.5- and 1-Hz signals are shown before (C, E) and
after HPF (D, F). For illustration purposes, stimulus amplitudes have been
selected so that the decision variable always has a peak of 1. This requires
lower frequency stimuli to have larger amplitudes to yield the same peak
decision variable after filtering. G: stimuli peak at a range of frequencies, with
gray filled symbols indicating the peak for the examples shown by corresponding symbols in A, C, and E. H: the peak decision variable (DV) across a range
of frequencies, with the open symbols representing the peaks shown by
corresponding symbols in B, D, and F. High-frequency signals are relatively
unaffected by an HPF; low-frequency signals are attenuated. Thus larger
signals are required at low frequencies to yield the same signal amplitude after
a high-pass filter compared with high frequencies.
frequency. The steady-state output of such a filter always
includes some noise. For simplicity, we will assume that the
noise present on the filter output is zero-mean Gaussian
[N(0, ␴)] and stationary (i.e., not changing with time). Note
that the frequency characteristics of the noise are assumed to
be broad and independent of the signal, so the noise characteristics are the same regardless of the signal frequency.
In other words, the primary influence of this filter is on the
sensed stimuli/information (i.e., the signal) with little or no
impact on the noise.
As discussed earlier, one common representation of a perceptual threshold is when the decision-making signal equals
the standard deviation of the noise (i.e., ␴). Therefore, if a
high-pass filter is processing a given cognitive or sensory
signal as part of a decision-making process, the resultant
high-pass-filtered output signal amplitude should equal ␴ at
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(strongly) encouraged to choose one of two options, we will
call this a binary “response-time” (or time to respond) task.
A signal detection model might be considered as a snapshot
of an evolving process that results from a drift-diffusion model.
In fact, a signal detection task, where time to respond is not a
factor, that includes these same three categorizations (signal
category A, signal category B, or uncertain) has recently been
reintroduced (e.g., Fechner 1966; García-Pérez and AlcaláQuintana 2011, 2012). This might be considered a snapshot at
time t̃ from Fig. 1C. This too cannot be considered a binary
forced-choice task, because the uncertain option means that
subjects are not forced to make binary decisions (Fig. 1D). As
for the time-to-respond task described in the previous paragraph, at least two variables are required to encapsulate all
probabilistic information regarding such decisions. Figure 1, B
and D, depicts probability density on the x-axis and the perceptual decision variable on the y-axis. Although not the focus
of this review, this graphically shows the similarity of the
signal detection approach with an uncertainty option (Fig. 1D)
to the time-to-respond approach prior to crossing a decision
boundary (Fig. 1C).
Decision-making dynamics. Having introduced and defined
both “dynamics” and “decision making,” we now combine
these definitions to define the focus of our review, which is
“decision-making dynamics.” We succinctly define the study
of decision-making dynamics as the study of how decision
making systemically varies with frequency and/or time. We
emphasize that while we investigate decision-making dynamics by varying the frequency content and/or the time course of
stimuli, the dynamic characteristics of decision making are
fundamental characteristics of the neural pathways that contribute to decision making.
Magnitude estimation. Magnitude estimation is a standard
psychophysical method in which subjects provide values
according to perceived stimulus magnitude (e.g., Stevens
1974; Wolfe et al. 2014). Such responses can involve tasks
that match one modality with another. For example, subjects
can be asked to vary the brightness of a light to match their
perception of sound loudness. Magnitude estimation tasks
do not utilize the forced-choice or time-to-respond techniques described before that are typically utilized for studies
of decision making. Hence, perceptual responses assayed
using these methods presumably avoid the neural mechanisms associated with decision making. If, as hypothesized
above, high-pass filtering impacts decision making, this
impact should not be observed when perception is assayed
using magnitude estimation techniques that do not utilize
these decision-making mechanisms.
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PERCEPTUAL DECISION-MAKING DYNAMICS
frequency are directly related to one another using the same
relationship shown above.
If we combine a linear, first-order, continuous-time highpass filter and an integrator, a linear, first-order, continuoustime low-pass filter results (Fig. 3B). Because each of the
elements is linear and time invariant, the relative order of the
filtering does not matter; high-pass filtering followed by integration is the same as integration followed by such high-pass
filtering.
As just one example, it is straightforward to show that that
the integral of the output of our linear, first-order high-pass
filter for a step input [兰oHPF(t=)dt=] equals the output of our
linear, first-order low-pass filter output (oLPF) shown above,
i.e.,
t
兰 Ke
⫺t'⁄␶
dt' ⫽ K␶共1 ⫺ e⫺t⁄␶兲 ⫽ K'共1 ⫺ e⫺t⁄␶兲 .
0
In summary, a low-pass filter can be achieved by integration
followed by high-pass filtering (or high-pass filtering followed
by integration), and all yield leaky integration. We also note
that this section introduces stationary linear, first-order, continuous-time filters. We do so for simplicity but explicitly note
that the filtering performed by the brain may not be stationary,
linear, first order, and continuous time.
But perfect integration is difficult to achieve; what if integration were imperfect? Specifically, what if a high-pass filter
were combined with a leaky integrator instead of a perfect
integrator? Since there are a number of neural leaky integrators
(Bogacz et al. 2006; Raphan et al. 1979), such as the oculomotor integrator (e.g., Kamath and Keller 1976; Robinson
1975) and velocity storage integrator (e.g., Raphan et al. 1979),
with time constants of 20 s or more, here we consider a “leaky
integration” time constant of 30 s. Figure 3C shows the
response of a high-pass filter combined with a leaky integrator,
demonstrating that it differs little from the response of a simple
low-pass filter. This is generally true when the time constant of
the leaky integrator is substantially larger than the time constant of the high-pass filter.
Vestibular Threshold Frequency Responses
Fig. 3. Relationship between low- and high-pass filters in the time domain. A:
in response to a step input (gray line), an HPF responds (thick black line) with
an initial step, followed by an exponential decay with a time constant of 1 s.
B: since an integrator followed by an HPF is equivalent to a low-pass filter
(LPF), we show that an LPF responds (thick black line) to a step input (gray)
with an exponential rise with a time constant of 1 s. C: when an HPF is
combined with a leaky integrator with a time constant of 30 s, a leaky LPF
(thick dashed black line) results. It has a response that differs little from the
LPF (thick black solid line) for the first second or two.
The studies described in this section use a signal detection
model/approach to analyze data obtained using what we sometimes refer to as a direction-recognition task (e.g., Chaudhuri et
al. 2013), which means that the subject experienced a single
motion and then needed to decide which of two directions (e.g.,
left vs. right) he/she moved and must indicate that decision
after being informed that the stimulus was complete. Typically,
subjects were in complete darkness during motion, to eliminate
visual cues (Chaudhuri et al. 2013). Sound contributions were
typically minimized via sound-reducing earpieces or auditory
masking noise (or both). Tactile contributions were often
minimized via the use of padding to distribute any such cues as
much as possible. That the vestibular system is the predominant contributor for at least some motions has been demonstrated via elevated thresholds in patients having total bilateral
vestibular loss (Valko et al. 2012).
As noted earlier, measuring how the response of a system
varies with frequency (i.e., the “frequency response”) provides
one way to study system dynamics. Benson and colleagues first
explicitly measured human vestibular perceptual thresholds as
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threshold, independent of the frequency content and amplitude
of the original input signal. (For simplicity, we assume that
downstream noise is small relative to the noise on the perceptual input signal.) To keep the resultant output signal constant
as a function of frequency, the input signal amplitude will need
to vary with frequency. Since a high-pass filter attenuates
low-frequency signal components, a high-pass filter will require a larger input signal at low frequencies (Fig. 2G) to
deliver the same output amplitude (Fig. 2H). Hence, under this
simple model, thresholds would be predicted to increase at
frequencies below the cutoff frequency of a high-pass filter
(Fig. 2G).
For a linear, first-order, continuous-time system, the cutoff
frequency (f) of a high-pass filter is inversely related to its time
constant (␶; i.e., ␶ ⫽ 1/2␲f). Hence, a 1-s time constant
corresponds to a cutoff frequency of 0.16 Hz for a first-order
linear filter. Given an input step signal at time 0, a high-pass
filter responds with an output (oHPF) that includes a step at time
0 in the same direction as the input, followed by a decaying
exponential with time constant ␶, oHPF(t) ⫽ Ke⫺t/␶ for t ⱖ 0
(Fig. 3A). This equation defines what is called the step response
of this high-pass filter.
In contrast to our earlier description for high-pass filters,
low-pass filters (LPF) attenuate frequency components above
the cutoff frequency and pass input frequency components
below the cutoff frequency. Given a step signal as an input, a
linear, first-order, continuous-time low-pass filter responds
with an output (oLPF) that demonstrates a gradual rise with a
time constant ␶, oLPF(t) ⫽ K=(1 ⫺ e⫺t/␶) for t ⱖ 0. (This gradual
exponential response of a low-pass filter is considered later
when we discuss speed-accuracy trade-offs.) This equation
defines what is called the step response of this low-pass filter.
The time constant of a first-order low-pass filter and its cutoff
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PERCEPTUAL DECISION-MAKING DYNAMICS
Fig. 4. Comparison of monkey vestibulo-ocular reflex (VOR) thresholds and
human perceptual thresholds as a function of frequency. Open symbols show
the average human perceptual thresholds from Grabherr et al. (2008) converted
to “one-sigma” thresholds. Note the plateau in human perceptual thresholds for
frequencies above 1 Hz and substantive threshold increase as frequency
decreases below 1 Hz. The solid curve shows the least-square fit to the human
threshold data, with the shaded area indicating 1 SD from the mean. Filled
symbols show the VOR thresholds for each of 3 monkeys. [Modified from
Haburcakova et al. (2012), where details can be found.]
tibulo-ocular reflex (VOR)1 thresholds were measured in rhesus monkeys (Fig. 4). The monkey VOR thresholds at 1
through 3 Hz were roughly the same as human perceptual
thresholds at these frequencies (Haburcakova et al. 2012),
which is suggestive of a common noise source determining
threshold performance. However, unlike human perceptual
thresholds, the measured VOR thresholds did not show the
same pattern of increasing thresholds with decreasing frequency. Specifically, the measured pattern of monkey VOR
thresholds as a function of frequency (Haburcakova et al.
2012) does not mimic the human perceptual threshold pattern
(Grabherr et al. 2008), which shows a substantial threshold
increase as frequency decreased from 0.2 to 0.05 Hz (Fig. 4).
The study concluded that one explanation is that perceptual
thresholds are influenced by high-pass filter contributions to
the perceptual decision-making process. However, other effects, including a difference between humans and nonhuman
primates, could not be ruled out (Haburcakova et al. 2012).
The perceptual threshold data of Grabherr et al. (2008) were
later quantitatively fit (Lim and Merfeld 2012) using several
modeling approaches. This modeling study reported that Grabherr et al.’s threshold data were best fit via a first-order
high-pass filter. The average threshold plateau across subjects
was 0.5°/s, and the fitted high-pass filter cutoff frequency was
0.26 Hz, which corresponds to a time constant of 0.6 s. The
cutoff frequency was specifically noted as being well above the
cutoff frequency of the semicircular canals (around 0.03 Hz;
Fernandez and Goldberg 1971) and also well above the cutoff
frequency associated with velocity storage (around 0.01 Hz;
Raphan et al. 1977; Robinson 1977a). Lim and Merfeld specifically wrote that “threshold dynamics do not match the
dynamics of human yaw rotation perception measured using
magnitude estimation (Bertolini et al. 2011; Bronstein et al.
2008; Okada et al. 1999; Sinha et al. 2008), which show decay
time constants between 10 and 25 s, corresponding to cutoff
frequencies between 0.006 and 0.016 Hz.” Since the perceptual
dynamics assayed using magnitude estimation techniques
show a passband that is much wider than that measured via
thresholds, Lim and Merfeld suggested that a high-pass filter
could contribute to decision making but did not rule out other
contributions (e.g., attention or other cognitive or decisionmaking elements).
The basic frequency effect first reported by Grabherr and
colleagues for yaw rotation has since been confirmed experimentally for yaw rotation by two independent studies (Soyka et
al. 2012; Valko et al. 2012) that each also extended these
findings in different ways. Valko et al. (2012) showed that
qualitatively similar frequency characteristics applied for other
transient stimuli, specifically, inferior-superior (z-axis) translation as well as interaural (y-axis) translation, and Soyka et al.
(2012) were the first to show that the dynamic high-pass
filtering effect generalized to other stimulus shapes (e.g., triangles and trapezoids). For yaw rotation, the cutoff frequency
that best fit Soyka’s data was 0.23 Hz, which corresponds to a
time constant of 0.68 s, similar to the time constant fit by Lim
and Merfeld to a different data set. In fact, Soyka et al. (2011)
had previously applied these fitting methods to interaural
1
Angular vestibulo-ocular reflexes are reflexive eye movements evoked by
the vestibular system’s semicircular canals that rotate the eyes to help compensate for head rotation.
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a function of frequency nearly 30 years ago (Benson et al.
1986, 1989) using single cycles of sinusoidal acceleration.
These single-cycle stimuli allow accelerations to be focused
near a specific frequency by simply varying the period (i.e.,
duration) of the sinusoidal stimuli. For example, a single cycle
of sinusoidal acceleration having a 2-s period has most of its
energy focused near 0.5 Hz. (See Appendix for theoretic
analysis of frequency content in single-cycle sinusoids.) Benson et al. (1986) first measured interaural (y-axis) translation
thresholds across a range of frequencies and reported that the
thresholds monotonically decreased as frequency increased
(i.e., duration decreased) from 0.14 to 1.02 Hz. A few years
later, Benson et al. (1989) reported a similar threshold response
pattern for yaw rotation thresholds for frequencies between
0.05 and 1.11 Hz.
Recently, Benson’s study of yaw rotation thresholds vs.
frequency was repeated utilizing a broader range of frequencies
(Grabherr et al. 2008) covering two full decades between 0.05
and 5 Hz. By extending the frequency range above 1 Hz, this
study reported a plateau in thresholds at higher frequencies but
noted the same pattern previously reported by Benson and
colleagues for frequencies below 1 Hz (Fig. 4). Grabherr and
colleagues suggested that these data appeared consistent with
high-pass filtering of a decision-making variable. Specifically,
Grabherr and colleagues suggested noise should be relatively
unaffected by the stimulus characteristics and that the plateau
could be rationalized as the passband (i.e., above the cutoff
frequency) of a high-pass filter with the rise at lower frequencies caused by signal attenuation of the signal at lower frequencies by the high-pass filter (Fig. 2G).
Comparing these perceptual threshold decision-making dynamics with motor responses originating from the same sensory organs can help separate transduction effects from the
effects of decision making. For example, yaw rotation ves-
45
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46
PERCEPTUAL DECISION-MAKING DYNAMICS
find that the model is able to predict dynamics of a human
decision-making process. Specifically, the predicted thresholds
closely match experimental (⫻ symbols in Fig. 5) results
(Grabherr et al. 2008). For example, at frequencies above 1 Hz,
the particle filter predicts a plateau in threshold, which becomes larger as frequency decreases. To confirm that the
observed dynamics originate in decision making and not transduction or other central processes, we also show the dynamics
of two other high-pass filters that impact rotation perception.
First, the frequency response of a common first-order model of
the semicircular canal (SCC; the peripheral organ that senses
Fig. 5. A relationship between noise, signal detection theory, and dynamics is
demonstrated by a parallel, dynamic particle filter model (Karmali and Merfeld
2012). A: a block diagram showing the relationship between the particle filter
and the decision-making filter. The only addition made to the published model
was the addition of linear, first-order high-pass filtering of the estimated
angular velocity. The input to the particle filter is physical angular velocity ␻,
and an output is estimated angular velocity ␻
ˆ. This model includes both
sensory and central processes, including ␻
˜, the afferent signals carried from the
periphery. The decision-making process high-pass filters ␻
ˆ to determine the
decision variable, d. The cutoff frequency (fco) of the filter was 0.25 Hz, which
corresponds to a time constant of 0.6 s. SSC, semicircular canal. B: simulated
gain of the system as a function of frequency. C: the same data plotted as
threshold as a function of frequency. The predicted threshold for each frequency (circles) is determined by determining the signal-to-noise ratio for an
input at that frequency. “Signal” is the peak estimated angular velocity when
the filter is simulated with no noise and an input of a single cycle of a sinusoid
in acceleration at the indicated frequency, equivalent to a cosine bell in
velocity; this ensured that transient effects of filtering a single-cycle signal
were taken into account (Lim and Merfeld 2012). “Noise” is the standard
deviation of the estimated angular velocity when the filter is simulated with
noise but no signal. As detailed in the text, experimental data (crosses;
Grabherr et al. 2008) are inconsistent with the frequency response of the
semicircular canals and perceived angular velocity, suggesting that another
process, such as a high-pass filter decision-making mechanism, may be a better
explanation for the results.
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(y-axis) translation thresholds before investigating yaw rotation
thresholds. They reported frequency characteristics for y-translation that mimicked those for yaw rotation. The fitted cutoff
frequency they reported was 0.26 Hz, corresponding to a time
constant of 0.62 s. This model fit appears consistent with
Valko’s interaural translation thresholds described above. The
dynamics of translation perception are not as well established
as for yaw rotation. When studied using magnitude estimation
tasks, translation perception appears transient in nature (Seidman 2008), so it is more difficult to separate translation
perception dynamics as assayed via magnitude estimation techniques from decision-making dynamics quantified via threshold studies for y-translation than for yaw rotation. Nonetheless,
the similarity of these various fitted high-pass filter parameters
for both yaw rotation and y-translation across two laboratories
certainly are not inconsistent with the hypothesis that vestibular perceptual threshold decision making can be modeled by
high-pass filtering velocity with a time constant of around
0.5– 0.7 s.
Consistent with these findings, vestibular response-time
tasks have suggested that when the acceleration rate is small,
subjects often do not make a decision regarding their motion
direction (Arrott et al. 1990; Melvill Jones and Young 1978).
For example, for an acceleration of 0.005 G, humans were
reported to either be incorrect or not respond at all on over half
of the trials (Melvill Jones and Young 1978). This would be
consistent with the influence of high-pass filter that would
inhibit the decision variable from reaching the decision boundary for small acceleration rates, even when the acceleration is
maintained constant for around 5 s, but also may be consistent
with drift-diffusion analyses that predict boundary crossings
due to integration of a noisy signal for extended periods of
time.
Finally, we note that several recent vestibular threshold
studies (Coniglio and Crane 2014; Crane 2012a, 2012b; Roditi
and Crane 2012) reported aftereffects on threshold processes
when a threshold motion was presented shortly after another
motion. Specifically, these studies reported a bias in a direction-recognition task; subjects were more likely to report that a
second motion was in the direction opposite a prior motion.
The papers reported that this effect was largest just after a
stimulus was provided and that the effect decreased with a time
constant of about 1 s that was explicitly noted to be independent of velocity storage. We note that this is exactly the effect
that one would predict if a high-pass filter contributed to
decision making, since a high-pass filter would yield an aftereffect in the opposite direction immediately after a unidirectional stimulus. We further note that the estimated aftereffect
time constant (⬃1 s) is near that reported to explain frequencydependent threshold variations.
To demonstrate the relationship between noise, signal detection theory, and dynamics, we developed a computational
model using a technique called particle filtering (Karmali and
Merfeld 2012). This study showed that Bayesian optimal
processing of rotational cues from the semicircular canals
could model rotation perception and that processing could be
implemented using a parallel computational structure that resembles the underlying physiology. When the rotation perception predicted by this model is processed by a high-pass filter
decision-making element that was added to the previously
published model to represent decision-making dynamics, we
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PERCEPTUAL DECISION-MAKING DYNAMICS
rotation) is shown (dashed line) with a time constant of 5.7 s
(Fernandez and Goldberg 1971; Melvill Jones and Milsum
1971). Also, the frequency response of “velocity storage”
(Raphan et al. 1977; Robinson 1977b) (solid line) is shown,
which is the frequency response measured when humans are
asked to indicate how much they have moved via a magnitude
estimation task (e.g., Bertolini et al. 2011; Okada et al. 1999;
Sinha et al. 2008) rather than which direction they have moved.
Visual Perceptual Responses
visual motion perception assayed via techniques that do not
require binary decision making (e.g., magnitude estimation)
might be unaffected by decision-making dynamics, so a large
difference in dynamics assayed via visual motion decision
making from those assayed using other psychophysical techniques that do not invoke decision making might provide one
way to isolate decision-making dynamics from other visual
processing dynamics.
Studies examining how the ability to correctly determine
motion direction of a moving random dot scene improves with
viewing duration have found that there are neuronal correlates
of this improvement (Britten et al. 1992; Gold and Shadlen
2000), which are well-described by models that incorporate
dynamics (Grossberg and Pilly 2008). An examination of these
results suggests an increase in performance with time that
reaches a plateau in a little less than 0.5 s, which is consistent
with the behavior of a dynamic decision-making element but
likely reflects visual transduction and visual processing dynamics, as well.
Speed-Accuracy Trade-Offs
A number of studies (e.g., Corbett 1977; Corbett and Wickelgren 1978; Dosher 1976; Hintzman and Curran 1997; Ratcliff
1981, 2006; Reed 1973, 1976; Schouten and Bekker 1967;
Wickelgren 1977; Wickelgren et al. 1980) have utilized speedaccuracy trade-off methods to study cognitive and perceptual
dynamics. Such studies typically measure response accuracy as
a function of stimulus duration, where the operator controls
stimulus duration. Subjects almost always become more accurate with increasing stimulation duration. Often, an exponential
growth function that asymptotes over time fits the data. In fact,
a modeling summary (Ratcliff 2006) reported that “the exponential function generally provides good fits to response signal
data” and also reported that an exponential function with a time
constant around 300 – 400 ms fit their data, as well: in this case,
data representing binary categorization of the number of visible
dots as either a “small” or “large.” The article further noted that
“The major problem with the exponential function . . . is that it
is not theoretically based. There is no model . . . that gives rise
to the exponential function. . . . ” As described earlier (and
below), our recent analysis suggests a model that gives rise to
these predicted exponential functions. Specifically, such exponential functions are exactly what a linear, first-order low-pass
filter (i.e., integration followed by high-pass filtering) predicts.
Several exemplar findings are briefly described in the following paragraphs.
Reed (1976) studied the accuracy with which subjects could
recognize whether a probe consonant was on a previously
presented list of consonants and reported that accuracy increased with probe presentation duration. d-prime (d=), a standard signal detection measure representing the ability to distinguish stimuli, increased from zero (or near zero) at 0.2 s
toward a plateau for stimulus duration ⬎1 s. The time course
appeared exponential. This time course is consistent with the
behavior of a linear, first-order low-pass filter having a time
constant ⬍1 s.
Corbett and Wickelgren (1978) used a semantic memory
retrieval task to test how rapidly associations between examples and a predefined category develop. Using the same d=
measure as Reed, this study showed a similar exponential
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A few studies have examined dynamics for visual motion
perception using signal detection theory approaches. Nakayama and Tyler (1981) determined the threshold as a function of the temporal frequency of motion. In their study,
subjects detect horizontal linear motion of dots in the simultaneous presence of random dot motion. The display specifically
removed displacement cues and provided only motion cues.
They found that threshold decreased with a slope consistent
with a first-order high-pass filter (e.g., Figs. 2 and 3) between
0.1 and 1 Hz, and then flattened before climbing again above
2 Hz.
The Nakayama and Tyler study differed somewhat from
other studies mentioned in this review, since they used the
method of adjustment in which the subjects turned a dial to
adjust the motion amplitude until they perceived the continuous sinusoidal motion. A second study (de la Malla and
López-Moliner 2010), using forced-choice procedures, found
similar results between 0.3 and 3.3 Hz. A third study (LagacéNadon et al. 2009) tested roll motion (i.e., about the line of
sight) using a forced-choice recognition approach for continuous sinusoids between 0.25 and 2 Hz and also reported similar
results, with a slope consistent with a high-pass filter up to 1
Hz, with a possible flattening at around 2 Hz.
Karmali et al. (2014) recently tested visual motion perception using a direction-recognition rather than motion-detection
task and single-cycle acceleration motions rather than multiple
sinusoidal cycles, over a broad frequency range of 0.05 to 5
Hz. The study also used a “natural” visual scene that included
typical objects seen in daily life rather than random dot
patterns. Karmali et al. (2014) found that, again, consistent
with a high-pass filter, thresholds decreased as frequency
increased up to 1 Hz and then flattened between 2 and 5 Hz.
Between 0.1 and 5 Hz these responses are consistent with a
high-pass filter and broadly similar to the results described
above for vestibular thresholds (Benson et al. 1989; Grabherr
et al. 2008; Lim and Merfeld 2012; Valko et al. 2012). It is
important to note that the transduction and processing dynamics are much more complex for vision than for the vestibular
system, which, as noted earlier, are known (Bertolini et al.
2011; Bronstein et al. 2008; Fernandez and Goldberg 1971;
Okada et al. 1999; Raphan et al. 1977; Robinson 1977a; Sinha
et al. 2008) to have a relatively constant response over a broad
frequency range. Thus it is harder to attribute visual motion
perception dynamics to a decision-making filter. Nonetheless,
the similarity in dynamic responses is notable. Although separating visual processing dynamics from decision-making dynamics is not trivial, such analyses could provide tests for the
specific decision-making filtering hypothesis discussed herein.
While speculative, one plausible approach is suggested by
analogy to the vestibular studies described above. Specifically,
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PERCEPTUAL DECISION-MAKING DYNAMICS
Dynamic Decision-Making Modeling
This section provides a brief overview of several decisionmaking models. All of these models accumulate noisy evidence over time and make a decision once sufficient confidence
has been established. They differ primarily in the specific ways
that the evidence is accumulated.
For example, the so-called “race model” separately integrates evidence favoring each of the two alternatives and
makes a decision as soon as one of the integrators reaches a
given threshold (e.g., Boucher et al. 2007). In contrast, the
drift-diffusion models (DDM) only maintain a single abstract
integrator, accumulating the difference between the evidence
supporting either of the alternatives. A decision is then made
once the integrated difference crosses a positive or a negative
threshold. Such DDMs have long been used to model accuracy
and reaction time datasets (e.g., Laming 1968; Ratcliff 1978;
Stone 1960). In statistical decision theory, DDM may be
regarded as an instance of a standard sequential probability
ratio test, which is optimal in the sense that it is the test that
provides the quickest decision under a given accuracy requirement. (The trade-off between accuracy and decision delay can
be achieved by varying the decision thresholds.) In a recent
article (Bitzer et al. 2014), the authors have also established an
interesting connection between the DDM and a Bayesian
model for decision making.
The DDM and its various extensions (e.g., extended DDM)
provide good fits to available data, albeit at the cost of requiring several parameters that have yet to be directly verified.
Specifically, because decision making is not simple, the extended DDM2 includes free parameters that represent mean
drift rate, drift rate variability, the initial value of the driftdiffusion variable, and physiological noise variance. Additional assumptions include distribution of the physiological
noise and distribution of the initial conditions.
Bogacz et al. (2006) have provided a comprehensive analysis of seven models of dynamic decision making. Models
analyzed there include those that have a single accumulation
variable (i.e., DDM, extended DDM, and Ornstein-Uhlenbeck
models) and those that have more than one parallel variable to
accumulate evidence in support of each hypothesis (i.e., race
models, mutual inhibition models, feedforward inhibition models, and pooled inhibition models). In their analysis, Bogacz et
al. showed that, with the exception of the race model, all the
models mentioned above can be reduced to DDM under suitable parameter choices. In the following paragraphs, we briefly
summarize aspects of their analysis that directly relate to
decision-making dynamics. (For those interested in details, we
recommend the article.)
From the perspective of decision-making dynamics, Bogacz
et al. (2006) note that neither the DDM (e.g., Bogacz et al.
2006; Ratcliff 1978) nor the extended DDM (e.g., Bogacz et al.
2006; Ratcliff 1978; Ratcliff et al. 1999) include dynamic
elements beyond integration/accumulation of evidence. Similarly, race models and feedforward inhibition models include
simple integration as the primary dynamic element (Bogacz et
al. 2006). Therefore, the following paragraphs focus on the
three models that include dynamic elements beyond integration
and accumulation. Diffusion models lend themselves to timedomain rather than frequency-domain analyses because, as
noted earlier, the investigator cannot control frequency content
of the stimuli at the time that the subject responds during a
typical time-to-respond task.
The simplest of these models (called the Ornstein-Uhlenbeck model) includes a feedback term controlled by an additional rate parameter (Busemeyer and Townsend 1993; Buse2
Because the distinction between the standard drift-diffusion model and the
extended drift-diffusion model is not critical to our review of dynamics, we do
not expound on the differences herein. We simply note that the extended
drift-diffusion model includes two added assumptions required to better fit
data: 1) the drift rate for a given stimuli included variability across trials
(instead of always having the same drift rate for the same stimuli), and 2) the
initial value of the drift-diffusion variable was selected from a uniform
distribution (instead of always starting at zero). We refer the interested reader
to Bogacz et al. (2006; p. 704-705) for more details.
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approach to an asymptotic accuracy. These speed-accuracy
trade-off data were consistent with an exponential growth
function having a time constant of around 250 ms. Dosher
(1976) used the speed-accuracy trade-off method to study the
retrieval dynamics for recognition memory for parts of sentences. Recognition accuracy increased with time to an asymptotic accuracy at about 3 s, with the rate of accuracy increase
similar across conditions.
Using a visual dot motion discrimination task, Ratcliff and
McKoon (2008) studied the speed-accuracy trade-off in a
time-to-respond task. On separate blocks of trials, human
subjects were instructed to respond “as quickly as possible” or
“as accurately as possible.” The effect of the instructions was
fit by altering only the separation of the decision boundaries in
the extended drift-diffusion model (details of this model are
discussed in Dynamic Drift-Diffusion Modeling). This suggests
that the accumulation process (e.g., drift rate and variability)
and non-decision components are unaffected by the speedaccuracy trade-off. The precise shape of an exponential growth
function could not be demonstrated with only two conditions
(“quickly as possible” or “accurately as possible”), but the data
are qualitatively consistent with the exponential growth reported above since the instructions emphasizing accuracy led
to longer response times that yielded improved accuracy. A
very recent study (Hanks et al. 2014) in monkeys yielded
results more or less consistent with these human findings,
except that neural recordings showed little evidence of a shift
in the decision boundary, but instead suggested an increase in
the neural firing rate prior to evidence accumulation at the
beginning of each trial. Another recent speed-accuracy tradeoff study has shown that an urgency signal might contribute to
both “deciding” and “acting” (Thura et al. 2014).
Swensson (1972) studied the speed-accuracy trade-off in a
visual discrimination task of shape orientation by manipulating
subject rewards (i.e., encouraging faster or more accurate
responses by providing payouts that reward each of these
strategies). As for studies discussed above (e.g., Corbett and
Wickelgren 1978; Ratcliff 2006; Reed 1976), accuracy improved with increasing time to respond, but accuracy did not
approach perfect performance (i.e., a performance asymptote
was evident), and data (Swensson 1972) were again consistent
with an exponential growth function that asymptotes over time.
Also, as shown in Filtering Dynamics, exponential growth in
evidence accumulation is the predicted output of a low-pass
filter (i.e., leaky integrator), which can be modeled as an
integrator followed by a high-pass filter (or vice versa).
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PERCEPTUAL DECISION-MAKING DYNAMICS
meyer and Townsend 1992). The equation for the standard
DDM can be written as simple diffusion equation:
dx
dt
⫽A⫹c
dW
dt
where x represents the accumulated evidence, A is the evidence
being accumulated, and the last term represents Gaussian white
noise with variance c2.
For the Ornstein-Uhlenbeck model, the above equation is
modified as
dx
dt
⫹ ␭x ⫽ A ⫹ c
dW
dt
Drugowitsch et al. 2012; Ratcliff and Frank 2012). This can be
used to model the “urgency” effect in decision making. Also,
whereas the models we describe in this section all involve
some kind of random walk or diffusion processing during the
decision-making process at each trial, there are other decisionmaking models proposed in the literature that are not based on
this random-walk formulation. One prominent example is the
LATER model (Carpenter et al. 2009; Carpenter and Williams
1995; Reddi et al. 2003), which accumulates evidence at a
fixed rate during a trial, with stochastic variations appearing
across different trials.
Drift-Diffusion Studies Investigating Dynamics
Recent studies have investigated whether drift-diffusion
models that combine an integrator with an information accumulation bound better explain data than the same model with
leaky integration and no accumulation bound. As summarized
below, some found that data were best explained by the
presence of “leaky integration” alone (i.e., high-pass filtering
combined with integration), whereas others found more support for the presence of an information accumulation bound
alone.
Consistent with reports from the earlier Speed-Accuracy
Trade-offs section, an article using data acquired from two
monkeys confirmed that monkey visual motion direction-discrimination thresholds measured with different levels of random dot motion coherence demonstrate a roughly exponential
time course when display duration is varied between 80 and
1,000 ms (Fig. 3 in Kiani et al. 2008). This study (Kiani et al.
2008) then further investigated whether the leveling off exhibited by the exponential function described in detail in the
previous section might be due to the presence of leaky integration alone (i.e., high-pass filtering combined with integration) or to the presence of an information accumulation bound
alone (i.e., without high-pass filtering). (This study did not
investigate leaky integration combined with an accumulation
bound.) These modeling analyses show that information provided earlier in the trial is more important for the bounded
information accumulation mechanism than for the leaky integration mechanism. On the other hand, modeling also showed
that later information was more pertinent for a leaky integration mechanism without bounded information accumulation
than for bounded information accumulation alone.
The study yielded two experimental findings, both of which
supported bounded information accumulation. One result derived from a reverse correlation analysis that was performed
for those trials in which the stimulus had no available motion
information (0% motion coherence). When this analysis was
performed for the model simulations, both the experimental
data and the bounded accumulation model showed that early
information correlated best with the final decision, whereas the
leaky integration model diverged, showing that late information correlated best with the final decision. The second finding
was that pulses of information provided early in the viewing
period had a greater influence on the final decision than pulses
of information provided late in the viewing period. As noted
above, this finding best corresponds to the predictions from the
bounded accumulation model and is opposite to predictions
from the leaky integration mechanism. On the surface, these
results appear to contradict the high-pass filter hypothesis.
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where ␭ represents a feedback parameter. The introduction of
the factor ␭x allows one to introduce several new effects. For
example, as described by Busemeyer and Townsend (1993),
when the feedback parameter is between 0 and 1, this parameter produces “recency effects so that the recent samples have
greater impact.” When negative, this parameter produces “primacy effects so that earlier samples have greater influence.”
These statements seem to require the assumption that the
information being accumulated is constant following an abrupt
on-transition, assumptions that were met by the experiments
being modeled, but such assumptions are not generally true,
even in laboratory conditions. From a dynamic systems perspective, the sign of the feedback parameter (␭) affects the
stability of the model. Positive feedback yields an unstable
system; negative feedback yields leaky integration, which, as
noted earlier (see Filtering Dynamics), is the same as low-pass
filtering (and both are the same as integration combined with
high-pass filtering). Since there is little evidence that perceptual decision making is unstable, from this point forward we
consider only systems with negative feedback that are stable
and do not consider unstable variants.
The mutual inhibition model (e.g., Bogacz et al. 2006; Usher
and McClelland 2001) can be considered as two parallel
Ornstein-Uhlenbeck models with additional mutual inhibition
terms that link these two models. More specifically, the positive strengthening of evidence in favor of one alternative will
inhibit (i.e., reduce) the accumulated evidence for the other
alternative. Given this, it seems understandable that detailed
analyses (Bogacz et al. 2006; Usher and McClelland 2001)
show that the difference between the competing accumulators
mimics the Ornstein-Uhlenbeck model dynamics. Similarly,
the pooled inhibition model can be shown to be equivalent to
an Ornstein-Uhlenbeck model (Bogacz et al. 2006). Hence, all
three models reviewed by Bogacz and colleagues that include
dynamic elements can be considered, from a dynamics perspective, nearly equivalent. Each essentially low-pass filters
the available information, a concept also previously proposed
and investigated by Smith (1995, 2000). To summarize, we
emphasize that when the rate parameter is negative, this leaky
integration model low-pass filters the available information. As
noted earlier, low-pass filtering is nothing more than integration combined with high-pass filtering.
Finally, we note a few variations on these models that
impact dynamic analysis. For most models in the DDM family,
the decision thresholds are kept fixed. However, one can
consider the use of time-varying thresholds that, for example,
“collapse” over time (Churchland et al. 2008; Ditterich 2006;
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variable peaks. Since we are unaware of drift-diffusion
analyses that have combined leaky integration with any such
midtrial decision mechanism, this may remain an open issue
that requires new analyses and explicit experimental investigations.
Before proceeding, it is important to note that the DDM (and
variants) include a substantial number of free parameters that
provide the freedom to match data from a wide variety of
studies quite well. This is good as this provides the freedom to
match the complex behavior reported/observed. However, this
freedom may come at the cost of not always providing direct
hypothesis testing. As explicitly shown in the previous paragraph, the frequency-domain approaches provide an alternative
approach that can provide useful information that might sometimes help resolve discrepancies.
Relating High-Pass and Low-Pass Filtering
We have noted that the dynamics of decision making
found for vestibular direction-recognition tasks were consistent with a filtering element having a time constant on the
order of one-half second (see Vestibular Threshold Frequency Responses), as are the reported frequency responses
for visual perception (see Visual Perceptual Responses).
Furthermore, literature suggests that speed-accuracy tradeoffs for semantic decision-making yielded similar time constants (see Speed-Accuracy Trade-offs). We did not highlight earlier that the vestibular threshold frequency responses were fitted using a high-pass filter whereas the
speed-accuracy trade-off data were consistent with a lowpass filter. We now describe one explanation for how these
results could arise from a single mechanism. We specifically
note that the goal of this review is to present relevant data
and to provide plausible hypotheses, not to provide definitive proof for any single hypothesis or viewpoint. More
data, alongside more thorough dynamic analyses of decision
making, are needed to reach definitive conclusions.
The information for speed-accuracy assays is obtained by
turning the stimulus on for each trial. This initiates the start of
evidence accumulation, which proceeds until the stimulus is
turned off or the subject provides a response. As described
earlier, it is commonly assumed that the information being
accumulated is constant as a function of time, and such accumulation is often modeled using an integrator (Fig. 6A), which
can be shown to be a statistically optimal process (Bitzer et al.
2014; Bogacz et al. 2006). If decision making includes the
additional contributions of a first-order high-pass filter, this
would be represented by adding a first-order high-pass filter to
the integration process (Fig. 6B). As noted earlier, since both
integration and high-pass filtering can be represented as linear
time-invariant processes, the order of these processes is irrelevant, so we could reverse the order of these processes to
indicate high-pass filtering of information followed by evidence accumulation (Fig. 6C). Either way, from a functional
input/output perspective, this integration and high-pass filtering can be combined into a single first-order low-pass filter
(Fig. 6D) that can (as discussed earlier) explain the exponential
relationship commonly reported for such speed-accuracy tradeoffs.
In contrast, we note that “vestibular” stimuli cannot be
turned on or off. Even in the absence of motion, the vestibular
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On the other hand, a previous human study (Experiment 3
presented in Usher and McClelland 2001) did not come to the
same conclusion. In that study, humans viewed an interleaved
stream of S’s and H’s and reported at the end of the stream
which letter had appeared the most. When the number of S’s
and H’s were the same, one of the letter’s might appear more
often early in the trial with the other appearing later. Of the six
subjects tested, two showed an effect indicating a larger influence of the early information, two showed an effect indicating
a larger influence of the late information, and two showed
about the same impact of early and late information. At a
minimum, this suggests that any such primacy/recency influence might vary in humans (e.g., depend on subject and/or
perceived task constraints).
Given these apparent discrepancies, a recent study (Tsetsos
et al. 2012) mimicked the methods used by Kiani et al. (2008)
except that human subjects were tested. Specifically, a moving
dot display with varying motion coherence, like that used by
Kiani et al., was employed to test highly trained subjects.
Furthermore, the time-pressure to respond was manipulated
across conditions. The article reported a replication of the
Kiani findings when the participants were pressured to respond
quickly (i.e., data best mimicked a bounded accumulation
model without high-pass filtering), but when the task explicitly
provided less time pressure, influence of early information was
much weaker. For 2 of 10 subjects, late information was more
influential than early information when the time pressure to
respond quickly was lessened. (A full summary of the 4
experiments exceeds the space available here; see the original
paper for details.) In summary, Tsetsos et al. (2012) wrote that
their studies showed that the influence of earlier information is
equally consistent with the bounded accumulation model and
the leaky competing accumulator model.
We are not in a position to adjudicate. We observe that
these comparisons reflect a view that the explanation is
either bounded accumulation or leaky integration. Given
this observation, we return to the vestibular threshold frequency responses described above. The reported increase in
thresholds at lower frequencies cannot be explained by
bounded accumulation alone. Bounded accumulation in the
total absence of any dynamic mechanisms would predict
that the thresholds would be constant as a function of
frequency (or perhaps even increase with frequency under
some assumptions) with the bound crossed at the same
velocity independently of frequency. However, the reported
thresholds are also inconsistent with a leaky integration
mechanism in isolation (i.e., without boundary crossings or
some other similar mechanism). Specifically, since the motion trajectories used continuous velocity waveforms that
first increased and then decreased back to zero just before
the response was requested, it would be meaningless in this
context to ignore boundary crossings and/or the peaks in
the decision variable expected to occur transiently near the
middle of the motion trajectory. In fact, analysis of the
vestibular thresholds implicitly assumed both 1) a filtering
mechanism and 2) a decision mechanism that would reflect
the decision variable found near the middle of the trial when
the peak velocity occurred. Such a midtrial decision mechanism could be a boundary crossing, a latch that records/
maintains peaks in the decision variable, or a mechanism
that provides a post hoc assessment of recent decision-
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PERCEPTUAL DECISION-MAKING DYNAMICS
system encodes that the motion is “zero” (e.g., Fernandez and
Goldberg 1971, 1976; Goldberg and Fernandez 1971). This is
different from the absence of incoming information as occurs
when visual (or other) stimuli are turned off. In fact, the brain
continuously processes vestibular information (e.g., Angelaki
and Cullen 2008; Goldberg et al. 2011; Merfeld 2014; Raphan
et al. 1977; Robinson 1977b), yielding underlying motion and
orientation states (e.g., Angelaki et al. 2004; Clark et al. 2014;
Glasauer 1992; Karmali and Merfeld 2012; Laurens and
Droulez 2007; Macneilage et al. 2008; Merfeld et al. 1999;
Zupan et al. 2002) that contribute to perception and reflexive
responses. Furthermore, a number of studies suggest that the
brain optimally processes this information (Borah et al. 1988;
Butler et al. 2010; de Winkel et al. 2010; Fetsch et al. 2009,
2010, 2011; Gu et al. 2008; Jurgens and Becker 2006; Karmali
et al. 2014; Karmali and Merfeld 2012; Laurens and Droulez
2007; Paulin et al. 1989; Selva and Oman 2012; Young 2011).
If the information is, in fact, being optimally processed to yield
optimal motion and orientation estimates, additional integration as part of the decision-making process might not make
sense. (If integration yielded “better” information, then integration might have been included as part of the original optimal
processing.) When information is continually processed optimally, integration would be suboptimal under at least some
conditions, because integration of optimally processed information would yield suboptimal information.
Instead, we hypothesize that it would make sense for decision making to utilize the optimal information (î in Fig. 6E)
already available rather than to integrate the incoming sensory
information (i) or to integrate the optimally processed information (î). Schematically, we represent such neural processing
using a block labeled “neural processing” in Fig. 6E. For
vestibular responses, this could represent the optimal processing of vestibular information. In a general sense, the neural
processing shown in Fig. 6E could also represent the integration explicitly included in Fig. 6, A–C (and implicitly included
in D), since integration is a form of processing that seems
particularly pertinent and may even be near-optimal when
constant transient information is provided (Bitzer et al. 2014;
Bogacz et al. 2006) for short periods of time.
More generally, such neural processing could represent
other processing dynamics. For example, auditory (Burbeck
and Luce 1982) or visual (Smith 1995) response times may be
better modeled by including transient processes in addition to
constant sustained accumulation. Furthermore, the drift rate
may be time-varying when the stimulus is continuous but
time-varying (e.g., Drugowitsch et al. 2014) or when the
stimulus is discrete (e.g., tone bursts in an auditory discrimination task; Brunton et al. 2013). Thus we emphasize that
many sensory information sources just may not be turned on
and/or may not have constant, sustained accumulation; future
models should aim to account for these variations. In particular, these cases may help illuminate dynamic mechanisms that
contribute to the decision-making process.
DISCUSSION
In this article, we have reviewed literature that seems relevant to the development of an understanding of decisionmaking dynamics. As presented, both data and models suggest
that a dynamic decision-making element having a time constant ⬍1 s appears consistent with available data. At the same
time, it is important to note that definitive evidence for such a
dynamic element is lacking. Although neural recordings have
shown phasic decision-making response components consistent with a high-pass filter (Aston-Jones and Cohen 2005a,
2005b), this is far from a proof that the phasic component
represents decision-making dynamics. What other factors
could explain these reported results that appear consistent with
dynamic filtering?
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Fig. 6. Evidence accumulation models. A: standard evidence accumulation is
represented by a simple integration of noisy information, represented in the
diagram as the sum of noise (n) and information (i). The output is the decision
variable (d). As noted in the text, such integration might provide optimal
processing for information sources that begin abruptly. B: if an HPF contributes to decision making, it would filter the accumulated evidence. C: alternatively, information could be accumulated following high-pass filtering. D: an
LPF can represent the processes shown in B and C, because, as presented
earlier, integration combined with high-pass filtering, in either order, is
equivalent to low-pass filtering. Stated explicitly, from an input-output perspective, B, C, and D are identical. E: when information is continually
processed optimally, integration of such information is suboptimal under many
conditions, because integration of optimally processed information yields
suboptimal processing. More generally, the continual processing shown in E
could also represent the integration included in A–D, since integration is a form
of continual processing that is particularly pertinent when a transient information source is turned on.
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Other Plausible Considerations
3
If the noise is assumed to be zero-mean, white Gaussian noise, the integral
of this noise yields a Brownian noise random walk whose standard deviation
increases unbounded with the square root of time. If the information being
accumulated is constant, the integral yields a decision-variable that increases
linearly with time. The ratio of these (i.e., the signal divided by the noise)
determines a threshold, so under these conditions, thresholds would be expected to decrease as the square root of time in an unbounded manner, which
is not what is typically found experimentally. See Speed-Accuracy Trade-Offs
for references and descriptions of some relevant data.
Drift-Diffusion Considerations
It has been noted that the drift-diffusion model suffers a
shortcoming (e.g., Usher and McClelland 2001) in that it
predicts that accuracy can improve without limit, whereas data
(and our own personal experience) show imperfect performance even when participants are given unlimited time to
respond. To address this concern, in his extended drift-diffusion model, Ratcliff (1978) assumed that the average drift rate
is not the same on each trial, even when identical stimulation
is provided (i.e., drift rate variations exist across trials) and
showed that when such drift rate variations exist, accuracy
does not improve without limit.
As an alternative, Usher and McClelland (2001) described a
leaky accumulator model, which, as noted earlier, is the same
as a low-pass filter model and the same as a high-pass filter
combined with an integrator. Usher and McClelland specifically noted that accumulated information might decay. In
response to Ratcliff’s (1978) use of an additional free parameter, variability across trials in the information accumulation
rate (i.e., drift rate), they wrote, “While not denying such a
factor [i.e., accumulation variability] is likely to play a
role . . . , limitations on discriminability may also arise from
characteristics of the information accrual process itself. Specifically, we suggest that information that has been accumulated may be subject to leakage or decay.” Usher and McClelland also specifically noted that leakage imposes limits on the
temporal window over which information can be accumulated.
Usher and McClelland (2001) provide a number of direct
comparisons for how well their leaky accumulator model
compares to the extended drift diffusion model. For one
forced-choice experiment in which subjects judged which side
of a nearly square rectangle was longer, they reported “small
but systematic advantages” of the fit of their model relative to
the drift-diffusion model for fitting the experimental timeaccuracy curves. For a difficult time-to-respond experiment,
Usher and McClelland reported that their leaky integration
model could fit experimental skewed reaction time distributions as well as the drift-diffusion model. Also, Ratcliff and
Smith (2004) reported that performance of the diffusion model
with leaky integration is about the same as that of the extended
drift-diffusion model across three experiments. Given these
findings, it would seem reasonable to think that such dynamic “leakage” effects would typically be considered, but
that does not seem widely true. One explanation may be that
the predicted differences between the models were small
relative to variability in the data. In fact, Ratcliff and Smith
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We briefly discuss a few explanations that might be considered instead of or in addition to high-pass-filtering of the
decision variable.
1) It is certainly possible that attentional mechanisms contribute to decision making. As one example, threshold decision-making tasks require attentional resources to monitor the
stimuli to avoid missed stimuli. It is possible that these mechanisms have limits that manifest with characteristics that mimic
filtering dynamics when stimuli lasting more than a second (or
so) are provided. We are unaware of studies that specifically
study attentional contributions to threshold decision making,
except to minimize lapses wherein a subject misses a stimulus
(or several stimuli). To avoid excessive speculation in the
absence of data, we simply note this as an area deserving study.
2) If the drift-diffusion model (i.e., without high-pass filtering) is an accurate representation of response-time decision
making, it predicts that diffusion will lead to decisions even
when no stimuli are presented. Similarly, such diffusion would
contribute to incorrect decisions for subthreshold stimuli. Such
diffusion contributions presumably increase with time. If true,
longer duration (loosely equivalent to “lower frequency”)
small stimuli would lead to more incorrect decisions due to
diffusion, which could lead to higher thresholds for longduration than short-duration (loosely equivalent to “higher
frequency”) stimuli. This would mimic the general characteristics of a high-pass filter illustrated in Fig. 2, but would not
explain (at least without further development and/or information) the similar time constants reported for a) vestibular
thresholds as a function of frequency and b) vestibular aftereffects as well as c) visual thresholds as a function of frequency
and d) speed-accuracy trade-offs (each presented above). Furthermore, for constant input stimuli, the drift-diffusion model
predicts that accumulated information would increase with
time3 in an unbounded manner that is not consistent with the
performance plateaus reported earlier after about 1 s or so for
speed-accuracy trade-offs.
3) We have posited that high-pass filtering of a decision
variable accompanied by a constant decision boundary could
explain some of the findings reported earlier. Are there alternatives that yield the same functional effect without high-pass
filtering the decision variable? One alternative would be that
the decision variable is unfiltered but that the decision boundary increases with a time course that yields the same relative
spacing between the decision variable and decision boundary.
As mentioned earlier, recent neural recordings (Hanks et al.
2014) provide little evidence of an increased decision boundary, despite looking explicitly for decision boundary changes.
We cannot identify another mechanism equivalent to high-pass
filtering of the decision variable at this time, but that does not
mean that another such mechanism may not exist.
4) The cost of accumulating evidence could also impact
perceptual decision making (Drugowitsch et al. 2012). This
modeling work shows that such costs could help explain
speed-accuracy trade-offs, but explicit prediction of frequency
effects, like those reported earlier for both vestibular and visual
stimuli, have not yet been published. For time-to-respond tasks
like those investigated, it is certainly plausible that time could
be seen as a cost, but it is not clear that forced-choice signal
detection tasks should include such a cost, since a subject’s
responses have no impact on stimuli presentation in such tests.
Adding a phasic dynamic element, like a high-pass decisionmaking filter, to such a model could also help improve these
model predictions.
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PERCEPTUAL DECISION-MAKING DYNAMICS
frequency stimuli). (This relates to material presented earlier in
Drift-Diffusion Studies Investigating Dynamics.)
As discussed in an earlier paper (Haburcakova et al.
2012), there is another potential rational role for such a
high-pass filter in decision making. Almost every signal that
contributes to decision making will include noise. The
accumulation of such noise via integration leads to a random
walk (Fig. 7A) having variance that grows with time (Fig.
7B). For example, if we assume that the noise is Gaussian
white noise, the integration of this noise leads to Brownian
noise. One characteristic of Brownian noise is that the
standard deviation grows with the square root of time in an
unbounded manner. Such unbounded signal variance with
time would not form an optimal decision variable. One way
to create bounded variance is to high-pass filter the decision
signal. For example, high-pass filtering Brownian noise
leads to a constant value of the standard deviation after just
a couple of filter time constants (Fig. 7C).
Potential Behavioral Benefits of High-Pass Filtering
Since a high-pass filter as part of decision making has been
suggested by previous studies, we next consider some potential
functional benefits that such a high-pass filter could contribute
to decision making. Without such a high-pass filter, the influence of a previous stimulus could linger for many seconds or
longer. For example, we consider a relatively dramatic vestibular example. “Velocity storage” has a time constant of 10 –20
s that that typically yields an illusory rotation aftereffect lasting
up to a minute following unidirectional rotation. High-pass
filtering perceptual signals as part of the decision-making
process could help reduce the impact of such lingering perceptual aftereffects on perceptual decision making.
We do not suggest that this potential behavioral benefit does
not come without costs. Specifically, high-pass filtering may
impair our ability to make accurate decisions in response to
low-frequency stimuli. In the case of vestibular processing, this
could impair our ability to recognize motion direction at low
frequencies in the dark. Thus the utilization of a high-pass filter
in the decision-making process may help de-emphasize older
information that may no longer be relevant, at the expense of
suboptimal decisions for longer duration stimuli (i.e., low-
Fig. 7. Effect of a linear, first-order HPF on drift-diffusion trajectories. All
simulations have a drift rate of zero and include no decision boundaries to
show the effect of accumulation of noise over time. A: an example diffusion
trajectory with (gray line) and without (black line) the HPF. The HPF has little
effect early but later draws the trajectory back toward the starting level. B:
mean ⫾ 2SD of 100 diffusion simulations without filtering over time. These
could represent either ensemble parallel population coding or the expected
outcome of sequential trials. Whereas the mean remains near the starting level,
the SD of the trajectories grows without bound. C: the same trajectories as in
B, but with the addition of the HPF with a time constant (␶) of 0.5 s. The HPF
causes the SD of the trajectories to remain relatively constant after roughly 2
time constants and to remain strictly bounded in time even for long periods of
time.
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(2004) showed that the extended drift-diffusion model and
one with leaky integration have substantial mimicry in many
simulations. Another explanation is that adding the complexity of dynamic “leakage” to the already complex driftdiffusion model does not endear itself to experimental
scientists (but this is probably no less true of the extended
drift-diffusion model).
A third explanation could be that people are content to
study forms of decision making where the existing driftdiffusion model is adequate. For example, Ratcliff and
McKoon (2008) have correctly noted that the drift-diffusion
model is generally appropriate for periods of up to only 1 or
2 s. If one is content to accept such limitations, then the
influence of the dynamic effects, like those reported by
Usher and McClelland (2001), are small. It is only when the
time period of interest is large relative to the decisionmaking time constant that these dynamic influences become
evident. As just one example, our self-motion threshold
studies (e.g., Grabherr et al. 2008) sometimes used motions
having periods on the order of 5–10 s. Such longer duration
stimuli may demonstrate the dynamics of decision making
more readily than stimuli lasting only a second or so, but
this is far from proven and may only hold true for some
forms of decision making. Finally, the presence of dynamic
“leaky” integration vs. the trial-to-trial variability of drift
rate in the extended drift-diffusion model may depend on the
specifics of the experimental task. Usher and McClelland
(2001) suggest that the concept of “drift variance is a
plausible one, particularly when it is applied to experiments
involving highly inhomogeneous items, such as items from
a memory experiment. In such cases, item differences could
easily arise from a number of sources, includes adequacy of
study or relative similarity of a test item with other studied
items, and these would likely play an important role.” This
further suggests that the experimental design and task may
be critical for cleanly observing the presence (or lack of
presence) of dynamic elements such as “leaky” integration.
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Other Dynamic Considerations
Dynamic Decision Making is Not the Same as the Dynamics
of Decision Making
To avoid confusion, we clearly distinguish the dynamics of
decision-making (also called decision-making dynamics),
which is our focus herein, from dynamic decision-making,
which has been defined (Fox et al. 2013) as the process of 1)
Moving Forward
We want to emphasize that this review is written with an
incomplete understanding of decision-making dynamics, because our understanding beyond simple integration remains
rudimentary. As such, our goal in writing this review is to
accelerate progress by bringing together different perspectives
that, to our knowledge, have never before been gathered
together. Given the state of the field, we expect some specific
topics included in this review may eventually be found to be
only indirectly pertinent. We hope that such inconsistencies
will not detract from our central thesis that advances in our
understanding of decision making will likely accrue with
advances in our understanding of the dynamics of decision
making.
Summary
The various signal detection and sequential analysis decision-making models (e.g., the DDM family) have contributed
to our understanding of decision making. It seems likely that
future progress will be enhanced by additional theoretical and
experimental approaches that augment the measurements and
approaches previously utilized. For example, neural recordings
in animals will continue to expand our understanding of decision making; simultaneous recordings of multiple neurons
relevant to a decision-making task in different brain areas
could provide a large step toward our understanding of nonhuman decision-making dynamics. Advances in our understanding of human decision-making dynamics also seem likely
to arise from new techniques and/or new measurements that
augment traditional measures like those focused on herein. For
example, human evoked response potential recordings that
simultaneously include response components representing sensory information, decision making, and motor responses (e.g.,
Kelly and O’Connell 2013; O’Connell et al. 2012) provide
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We note that different decision-making processes could
share common decision-making dynamics, for example, a
high-pass filter. When true, such common dynamics would be
reflected in the data, which is why we compare visual, vestibular, and speed-accuracy dynamics herein. At the same time, it
is important to remember that other dynamic elements, beyond
just the decision-making dynamic element focused on herein,
likely contribute to decisions. For example, sensory transduction dynamics contribute to decision making, as do central
processing mechanisms. (For example, see Fig. 8 of Haburcakova et al. 2012). Such dynamic elements might need to be
modeled to represent the pertinent dynamics of other transduction and/or physiological/cognitive processes performed as part
of the decision-making response being studied.
One reason that decision-making dynamics could be implicated to help explain yaw rotation thresholds measured as a
function of frequency (Benson et al. 1989; Grabherr et al.
2008; Lim and Merfeld 2012; Valko et al. 2012) is that the
transduction processes (Fernandez and Goldberg 1971) and
central processing elements (e.g., Karmali and Merfeld 2012;
Merfeld et al. 1993; Raphan et al. 1977; Robinson 1977b) are
known (or at least believed) to demonstrate dynamics that are
relatively constant over the range of frequencies where thresholds show relatively large changes with frequency (i.e., between 0.1 and 1 Hz in Fig. 4). Certainly, as noted earlier,
rotation perception measured using magnitude estimation has
been shown on several occasions using a variety of methods
(Bertolini et al. 2011; Bronstein et al. 2008; Okada et al. 1999;
Sinha et al. 2008) to have dynamics that demonstrate a broad
frequency range wherein the perceptual response is constant
(i.e., “velocity storage”).
This minimal impact of transduction and processing dynamics is certainly not universally true for all sensory modalities,
since other dynamic elements (e.g., transduction dynamics
and/or neural processing dynamics) impact the overall dynamics measured when studying decision making. Hence, one does
not necessarily predict identical overall decision-making dynamics for different decision-making processes even if the
decision-making element in isolation is identical across systems. As a simple extreme example, imagine that sensory
transduction could be modeled as a low-pass filter with a time
constant of 1 s for system A and 0.01 s for system B. When true,
such transduction dynamics would impact the information
signal sent to the decision-making mechanisms. Even if the
two systems had identical dynamics in their decision-making
element (e.g., a high-pass filter with time constant ⬃0.5 s), this
would lead to not-so-subtle differences in the overall measured
decision-making dynamics. Specifically, the overall decisionmaking dynamics for sensory system A that included the
relatively slow dynamic transduction processes would be predicted to be slower than for system B unless the transduction
dynamics were quantitatively factored into the analysis.
recognizing when a decision is needed, 2) determining the
options, 3) establishing the relevant criteria, 4) resolving conflicts, etc. Standard examples of dynamic decision-making
include firefighting, emergency medical care, sports, military
command, and air traffic control. As early as 1962 (Edwards
1962), dynamic decision-making had been characterized as
achieving a goal in real-time via a series of decisions that are
not independent where the problem changes with time. In other
words, dynamic decision-making includes the entire process of
framing what is being decided as well as how to make the
decision. In contrast, we have defined decision-making dynamics as the time course (or frequency response) by which each
specific individual well-defined decision is made.
A collection of papers on subtopics related to dynamic
decision-making has recently been published, including integrative approaches (Fox et al. 2013; Trueblood and Busemeyer
2012), preference-based decisions (Fiedler and Glöckner 2012;
Krajbich et al. 2012; Wollschläger and Diederich 2012), and
adaptive decision making (Dutt and Gonzalez 2012; Knox et
al. 2012; Lange et al. 2012; Osman and Speekenbrink 2012; Yu
and Lagnado 2012). The editorial (Usher et al. 2013) that
accompanies these articles as part of this decision making
collection summarizes these contributions in much greater
detail than is appropriate for this review of decision-making
dynamics.
Review
PERCEPTUAL DECISION-MAKING DYNAMICS
55
APPENDIX: FREQUENCY CHARACTERISTICS OF TRUNCATED
SINUSOIDS
One way to understand the dynamics of physiological systems is to
characterize the frequency response. Experimentally, frequency characterization often consists of measuring responses to finite-duration
sinusoidal stimuli over a range of frequencies and reporting gains or
thresholds as a function of frequency. For simplicity, such frequency
analyses often assume infinite-duration sinusoids. In this Appendix we
explore the quantitative effects of finite-duration sinusoids and the
theoretic impact on measurements.
Finite duration sinusoids can be represented as the product (multiplication) of an infinite-duration sinusoid (Fig. A1A) with a square
window having unity amplitude and a finite duration. The magnitude
of the frequency-domain representation (determined using Fourier
analysis) of the infinite-duration sinusoid is zero everywhere except
for a positive impulse at frequency ⫹f0 and a negative impulse at
frequency ⫺f0 (Fig. A1F; only positive frequencies shown). These
impulses are usually represented using Dirac’s delta functions, each
having an area of one-half: ␦(f ⫺ f0)/2 and ␦(f ⫹ f0)/2. The frequencydomain representation of the square window of width T is the sinc
function T: sinc (Tf) ⫽ sin (␲fT)/(␲f) (Fig. A1G, gray line). For our
single-cycle stimuli (Fig. A1B), the window length (gray line) is one
period of our acceleration stimuli (T ⫽ 1/f0), so the sinc function for
single-cycle stimuli is 1/f0·sinc (f/f0) ⫽ sin (␲f/f0)/(␲f). Since multiplication in the time domain corresponds to convolution in the
frequency domain, the frequency-domain representation of the truncated sinusoid (Fig. A1G, thick black line) is the convolution of the
sinc function and the impulses. Whereas the peak frequency is near
that of the infinite-duration sinusoid, the square-wave windowing
causes some spreading of frequency content around the fundamental
frequency. For more cycles, the window becomes larger. For example,
for three cycles, the window is T ⫽ 3/f0, which corresponds to a
frequency-domain sinc function 3/f0·sinc (3f/f0) ⫽ 3 sin (3␲f/f0)/(␲f).
Thus spreading of frequency content occurs even with three, five, nine
(Fig. A1, C–E, and H–J) or more cycles; no finite-duration sinusoid is
immune to this effect.
The ultimate goal of measuring physiological frequency responses
is often to characterize a filter in the brain, such as the velocity storage
filter (Raphan et al. 1977; Robinson 1977) or the perceptual decision-
Fig. A1. Frequency characteristics of truncated sinusoids. A: temporal representation of an infinite-duration sinusoid with frequency f ⫽ 1 Hz and period
T ⫽ 1 s. Acceleration is shown, but the analysis is general for any sinusoid.
B–E: finite-duration sinusoids ranging between 1 and 9 cycles; these are
mathematically realized by multiplying an infinite-duration sinusoid (thin
black line) with a square window (gray line). F: the frequency -domain
representation for an infinite-duration sinusoid has all its energy represented by
impulses at ⫹1 and ⫺1 Hz. G–J: the square window in time corresponds to the
sinc function (gray line) in frequency. Multiplication in time corresponds to
convolution in frequency; convolving the impulse (thin black line) with the
sinc function (gray line) yields the frequency spectrum of the finite-duration
sinusoid (thick black line). These finite-duration sinusoids exhibit small but
significant frequency distortions.
making filter (Grabherr et al. 2008; Lim and Merfeld 2012; Soyka et
al. 2011). We now describe differences in how filtering impacts
truncated sinusoids in comparison to infinite-duration sinusoids. In
this example, we use a first-order high-pass filter with a cutoff of 0.25
Hz (corresponding to a time constant of 0.6 s), the cutoff frequency
estimated for yaw rotation threshold decision making (Lim and
Merfeld, 2012). Figure A2A shows a portion of an infinite-duration
0.2-Hz sinusoid (thin gray line) and the result of high-pass filtering
(thick black line). Because the sinusoid frequency is below the cutoff
of the filter, the output is an attenuated sinusoid with the same
frequency as the input motion and with a phase lead present. We
emphasize that this is the steady-state response, which is reached after
a few time constants of the filter. Less attenuation occurs at higher
frequencies such as 0.5 and 1 Hz (Fig. A2, B and C), resulting in
larger peak amplitudes (open circles). When the corresponding 0.2-Hz
single-cycle sinusoid is filtered (Fig. A2D), the output (thick black
line) is not a pure sinusoid, but its largest magnitude (asterisk) is
similar to that of the filtered infinite-duration sinusoid, albeit in the
negative direction. The same is true for higher frequencies (Fig. A2,
E and F), although with less attenuation. In terms of a directionrecognition task, a stimulus that incorporates two equal half cycles in
opposite directions is usually nonsensical for subjects to interpret.
This is especially true after filtering, since the peak filtered signal
occurs in the direction opposite of the initial stimulus direction, and
the peak signal is the most likely cue for decision making (Lim and
Merfeld 2012; Soyka et al. 2011). Velocity, rather than acceleration,
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great promise for future advances in our understanding of
decision-making dynamics, as do recordings of confidence
(e.g., Kepecs et al. 2008; Kiani and Shadlen 2009; Kunimoto et
al. 2001; Ratcliff and Starns 2013; Rolls et al. 2010) with, of
course, appropriate considerations (Galvin et al. 2003) depending on task specifics.
Future psychophysical experiments are likely also required.
These studies should focus on perceptual decisions involving
sensory systems for which we have at least an approximate
understanding of peripheral/sensory dynamics such that central
decision-making dynamics can adequately be distinguished on
the basis of psychophysical responses (such as we suggest can
be done in the vestibular system). In general, an emphasis
should be placed on designing studies in which the “frequency”
of the stimuli is well controlled but varies across trials or
sessions. The role of phasic mechanisms (i.e., high-pass filtering) is difficult, if not impossible, to identify if all of the
presented stimuli have similar frequency content or if that
content is not well controlled by the experimenter. With these
experimental considerations, then the appropriate application
of a formal dynamics analysis could help identify the role of
phasic mechanisms in the decision-making process that exist
across sensory systems.
Review
56
PERCEPTUAL DECISION-MAKING DYNAMICS
GRANTS
This work was supported by National Institute on Deafness and Other
Communication Disorders Grants R01 DC04158 (to D. M. Merfeld) and R03
DC013635 (to F. Karmali) as well as National Space Biomedical Research
Institute Grant NCC9-58 (to T. K. Clark).
Fig. A2. A: a portion of a 0.2-Hz infinite-duration sinusoid (thin gray line) and
the steady-state response after application of an HPF (thick black line) with a
cutoff of 0.25 Hz, corresponding to a time constant of 0.6 s. Peak output
magnitude is indicated with an open circle. B and C: the same as A for 0.5 and
1 Hz, respectively. D–F: single-cycle sinusoids (thin gray line) and the result
of high-pass filtering (thick black line), with peak output magnitude indicated
with an asterisk. G–I: cosine bells (thin gray line), which are the integral of
single-cycle sinusoids, and the result of high-pass filtering (thick black line),
with peak output magnitude indicated with a filled diamond. J: gain, measured
as the ratio of peak output magnitude to peak input magnitude, for infiniteduration sinusoids, single-cycle sinusoids, and cosine bells. K: the corresponding threshold for the gains shown in J.
has been suggested as the salient cue in many facets of vestibular (e.g.,
Melvill Jones and Young 1978) and visual sensation (e.g., Nakayama
and Tyler 1981). Thus a stimulus in which acceleration is a singlecycle sinusoid, which after integration means that velocity is a cosine
bell, is often more appropriate for direction-recognition tasks, because
the cosine bell never reverses polarity. Frequency for a cosine bell is
often defined as the inverse of the period of the underlying singlecycle acceleration. Filtering a cosine bell with a frequency of 0.2 Hz
(Fig. A2G, thin gray line) using the high-pass filter described before
(time constant of 0.6 s corresponding to a cutoff frequency of 0.25 Hz)
yields a highly attenuated output (thick black line) that is almost
symmetric, but with the largest magnitude (filled diamond) always in
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the authors.
AUTHOR CONTRIBUTIONS
D.M.M., T.K.C., and F.K. prepared figures; D.M.M., T.K.C., Y.M.L., and
F.K. drafted manuscript; D.M.M., T.K.C., Y.M.L., and F.K. edited and revised
manuscript; D.M.M., T.K.C., Y.M.L., and F.K. approved final version of
manuscript.
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