PLoS BIOLOGY
Monotonic Coding of Numerosity in Macaque
Lateral Intraparietal Area
Jamie D. Roitman1¤*, Elizabeth M. Brannon2,3, Michael L. Platt1,3
1 Department of Neurobiology, Duke University, Durham, North Carolina, United States of America, 2 Department of Psychology and Neuroscience, Duke University,
Durham, North Carolina, United States of America, 3 Center for Cognitive Neuroscience, Duke University, Durham, North Carolina, United States of America
As any child knows, the first step in counting is summing up individual elements, yet the brain mechanisms responsible
for this process remain obscure. Here we show, for the first time, that a population of neurons in the lateral
intraparietal area of monkeys encodes the total number of elements within their classical receptive fields in a graded
fashion, across a wide range of numerical values (2–32). Moreover, modulation of neuronal activity by visual quantity
developed rapidly, within 100 ms of stimulus onset, and was independent of attention, reward expectations, or
stimulus attributes such as size, density, or color. The responses of these neurons resemble the outputs of
‘‘accumulator neurons’’ postulated in computational models of number processing. Numerical accumulator neurons
may provide inputs to neurons encoding specific cardinal values, such as ‘‘4,’’ that have been described in previous
work. Our findings may explain the frequent association of visuospatial and numerical deficits following damage to
parietal cortex in humans.
Citation: Roitman JD, Brannon EM, Platt ML (2007) Monotonic coding of numerosity in macaque lateral intraparietal area. PLoS Biol 5(8): e208. doi:10.1371/journal.pbio.
0050208
(Figure 1). The mode-control model by Meck and Church [31]
proposes that number is represented by accumulating a fixed
number of pulses from a pacemaker for each event or object
enumerated (Figure 1A). Accumulation occurs as a serial
process, with larger numbers represented by a greater
accumulated magnitude over a longer interval. In a neural
network model by Dehaene and Changeux [32], objects are
represented on a retina as n areas of activation (Figure 1B). A
normalization stage maps each object according to size and
location, so that these attributes are not lost, but each object
is represented by an equivalent amount of activation.
‘‘Summation’’ units integrate the activity on the normalization map in a parallel manner to extract stimulus quantity.
‘‘Numerosity’’ units then encode a specific cardinal value,
such as 3 or 5, depending on the magnitude of the responses
in the summation units. Similarly, a neural network model
proposed by Verguts and Fias [33] utilizes hidden ‘‘summation’’ units that represent accumulated magnitude (Figure
1C). The outputs from these units are combined in an
additive or subtractive manner to estimate cardinal value in
‘‘number’’ units.
In all these models, accumulator neurons are predicted to
respond in a graded fashion with ordinal numerical quantity,
in contrast with ‘‘numerosity’’ or ‘‘number’’ units that
Introduction
Humans and animals perform similarly in tasks that
require representing the numerosity of a visual or auditory
array, or making a given number of responses, suggesting that
the underlying representation of quantity across species is
similar [1–4]. It has been speculated that quantitative judgments are based on an analog magnitude representation of
quantity akin to a ‘‘mental number line’’ that is either
logarithmically compressed or linearly scaled with variance
that increases proportionally with number [3,5]. Both lesion
and functional imaging studies suggest that numerical
processing is supported by neurons near the intraparietal
sulcus in humans [6–10]. Moreover, in monkeys, neurons
within the fundus of the intraparietal sulcus as well as in
prefrontal cortex (PFC) respond selectively to a specific
number of elements in a visual array [11–13]. Together, these
findings implicate parietal cortex as a critical component of
the neural circuitry responsible for encoding the mental
number line. The fact that brain regions sensitive to number
lie within the dorsal visual processing stream suggests that
numerical quantity may be derived from the visuospatial
representations [14–16] processed in this cortical pathway.
In monkeys, single neurons in posterior parietal cortex
(PPC) encode intended locations for eye movements [17–19]
and reaches [20,21], and reflect elapsed time when this
information guides saccades [22]. Neurons in the lateral
intraparietal area (LIP) reflect decisions based on randomdot motion stimuli [23], reward expectation [24], and the
probability of a initiating a behavioral response to a
particular location [24] or at a particular time [25]. All of
these tasks rely on the accumulation of spatial, temporal, and
reward information [26–28], thus it has been proposed that
LIP neurons function as neural integrators that accumulate
information relevant for guiding behavior [29,30].
Integration by neurons has also been proposed as a critical
stage in computational models of numerical representation
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Academic Editor: Stanislas Dehaene, Service Hospitalier Frederic Joliot, France
Received December 18, 2006; Accepted May 29, 2007; Published July 24, 2007
Copyright: Ó 2007 Roitman et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: ANOVA, analysis of variance; CI, confidence interval; LIP, lateral
intraparietal area; PFC, prefrontal cortex; PPC, posterior parietal cortex; RF,
receptive field; RT, response time; sp/s, spikes/second; VIP, ventral intraparietal area
* To whom correspondence should be addressed. E-mail: jroitman@uic.edu
¤ Current address: Department of Psychology, University of Illinois at Chicago,
Chicago, Illinois, United States of America
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Monotonic Coding of Numerosity in LIP
expectation by presenting the same numerical value at
different frequencies with different associated reward outcomes. On each trial, monkeys planned eye movement
responses to a target located distal to the numerical stimulus,
thus reducing any potential impact of motor planning on
neural responses. In addition, we used a large range of
numerosities (2–32), which facilitated discrimination of
neurons representing accumulated quantity from neurons
encoding cardinal numerical value. On each trial, the monkey
viewed an array of 2, 4, 8, 16, or 32 elements located in the
visual periphery for 400 ms. The numerosity of the array
predicted the amount of reward the monkey would receive
following a gaze shift to a response target in the opposite
hemifield. On 50% of trials, a standard number was
presented, predicting a standard reward size (0.10-ml fluid
delivery, Figure 2B). On each of the remaining trials, a
deviant number was shown, predicting a larger reward (0.15ml fluid delivery). Standard and deviant numerosities were
varied across blocks to dissociate reward-related salience and
presentation frequency from visual quantity. Because complementary stimulus attributes often co-vary with numerosity, we also implemented stimulus controls to balance total
number of pixels, element size, density, and, for completeness, color (Figure 2C).
We recorded the activity of single LIP neurons using
standard techniques published previously [34] while monkeys
performed the implicit numerical discrimination task. For
each neuron, the monkey performed at least two blocks of
this task so that each standard number was also tested as a
deviant. This strategy permitted us to dissociate neural
responses associated with numerosity from those driven by
attention, which can be influenced by task variables like
reward expectation or the frequency of presentation of
particular numerical values. Notably, extensive training was
not required since monkeys were only required to shift gaze
to the response target; reward was delivered regardless of
whether or not monkeys attended to the numerical cue
(Figure 2B). In addition, we used up to 32 elements to study
each neuron, a greater than 6-fold expansion over the
numerosities used in prior electrophysiological studies of
numerical representation. We hypothesized that the activity
of LIP neurons would encode the quantity of visual elements
placed within their classical RF in a graded manner,
Author Summary
As any child knows, to answer the question ‘‘how many,’’ one must
start by adding up individual objects in a group. Extending beyond
humans, this cognitive ability is shared by animals as diverse as birds
and monkeys. Surprisingly, the exact brain mechanisms responsible
for this process remain unknown. Damage to a brain area known as
the parietal cortex disrupts basic mathematical skills, and functional
imaging studies show that this area is activated when people
perform basic computations. To understand how parietal cortex
contributes to numerical behavior, we studied the activity of
neurons in this area in monkeys while they looked at arrays of dots
on a computer screen. We found that parietal neurons responded
progressively as the total number of elements in the display was
varied across a wide range of values (2–32). These neurons resemble
‘‘accumulator neurons’’ that have been suggested to serve as the
first stage in counting. This information could be used by other
neurons that respond best for a particular cardinal number, such as
‘‘4,’’ as has been reported in prior studies. Our findings support
computer models that separate the processes of summing and
numerical identification, and may also explain the fact that parietal
cortex damage causes both numerical and spatial confusion.
respond maximally for a particular cardinal value. Although
prior studies have revealed evidence for a population of
neurons selective for the cardinal number of elements in an
array in both PFC and parietal cortex [11–13], evidence for
putative numerical accumulator neurons remains elusive. A
logical extension of the idea that neurons in LIP function as
neural integrators is the hypothesis that these neurons
accumulate numerical quantity. The inherent difficulties in
dissociating sensitivity to number from sensitivity to space,
salience, reward expectation, and motor preparation—all of
which could potentially modulate neural responses—make
this endeavor particularly challenging.
To address this idea, we studied the activity of single
neurons in parietal area LIP in monkeys performing an
implicit numerical discrimination task (Figure 2A). This task
permitted us to dissociate spatial sensitivity, attention,
reward expectation, and motor preparation from neural
coding of number. We controlled for spatial sensitivity by
optimizing the position of the numerical stimulus so that it
was placed in each neuron’s classical spatial receptive field
(RF). We dissociated salience from numerosity and reward
Figure 1. Models of Numerical Estimation Rely on an Accumulator Stage
(A) In the mode-control model [31], an ‘‘accumulator’’ receives pulses from a pacemaker and stores an analog magnitude proportional to the quantity
of stimuli enumerated. The accumulator value can represent elapsed time or quantity depending on the mode in which pulses are gated to it.
(B) In the Dehaene and Changeux neural network model [32], ‘‘summation units’’ accumulate in parallel to the quantity of stimuli. This information is
used by ‘‘numerosity’’ units, which represent the cardinal value of the set of stimuli.
(C) The neural network model by Verguts and Fias [33] utilizes a hidden ‘‘summation’’ layer in which numerosity is represented in a monotonic fashion.
Cardinal number is derived from summation units by ‘‘number’’ units.
doi:10.1371/journal.pbio.0050208.g001
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Monotonic Coding of Numerosity in LIP
Figure 3. LIP Neurons Respond to the Onset of the Numerical Cue,
Located within the RF, But Not to the Saccade Target, Located in the
Opposite Hemifield
For all 57 neurons studied, average firing rate was calculated in 20-ms
bins. Trials are first aligned to the time that the saccade target appears
(indicated by arrow). The break in the abscissa acknowledges the
variable length of the delay before number cue onset. Following the
appearance of the numerical cue in the RF (arrow), there is a large
increase in LIP activity.
doi:10.1371/journal.pbio.0050208.g003
reward size on saccade latency [35,36]. The subtle RT
difference observed here suggests the monkeys attended to
numerosity, despite no requirement to do so.
The monkeys’ sole task was to shift gaze to a visible target
following the disappearance of the fixation point (Figure 2A).
Because LIP neurons are spatially selective [17–19], each
neuron’s response field was first mapped while monkeys made
delayed saccades to visible targets (see Materials and
Methods). To measure neural responses to the number of
elements in a stimulus, we arranged the task geometry so that
the entire numerical cue was presented in the neuron’s RF
and the saccade target was located in the opposite hemifield.
For monkeys O and W, the average distances from the
fixation point to the center of the RF, and thus the center of
the numerical stimulus, were 15.28 and 17.58, respectively.
Previous work has shown that at these eccentricities, RF
widths in LIP exceed the dimensions of the stimuli used here
[37,38]. Figure 3 shows that onset of the saccade target did not
evoke a neural response from the neurons studied. The
neural population response remained low throughout the
delay period (minimum, 500 ms) during which the target was
present before the onset of the numerical cue, thus
confirming the saccade response target was outside the
classical RF. Presentation of the numerical cue itself elicited
a large neural response, however, thus confirming its placement within the classical RF (Figure 3).
Neuronal responses to the cue showed graded modulation
by the numerosity of the stimulus. Figure 4 shows four
examples of single LIP neurons with activity that either
increased (Figure 4A and 4C) or decreased (Figure 4B and 4D)
with increasing number of elements during stimulus presentation. In each example, firing rate depended on number
(analysis of variance [ANOVA], p , 0.05), but not standard
value (indicated by line color, ANOVA, p . 0.05). In the
studied population, 35 neurons (61%) were sensitive to visual
quantity (ANOVA, p , 0.05), and for the vast majority, (34/35,
97%), activity did not depend on which number served as
standard or deviant (ANOVA, number 3 standard interaction, p . 0.05), thus effectively ruling out reward expectation
or stimulus salience as factors modulating response to the
numerical cue. For these examples, the modulation of firing
rate over the range of numerosities was quantified using
Figure 2. Implicit Numerical Discrimination Task
(A) Sequence of trial events. The RF of the LIP neuron is first mapped
using a delayed saccade task. Completion of each trial is followed by a
fluid reward.
(B) Sample trial sequence in which 8 is the standard numerosity and 2, 4,
16, and 32 are deviants. The standard is randomly presented on 50% of
trials, and predicts a standard reward (0.1-ml fluid, indicated by smaller
digit size). On the other trials, one of the deviant numerosities is shown,
predicting a larger reward (0.15-ml fluid, larger digit size). Arrow
identifies an example of numerosity that would be presented for an
‘‘8’’ trial.
(C) Examples of dot arrays for the numerosities tested: 2, 4, 8, 16, and 32.
Stimuli with the same average total number of pixels are in the first row;
same radius and overall extent in the second row; and same radius and
density in the third row.
doi:10.1371/journal.pbio.0050208.g002
independent of low-level visual features, attention, or reward
expectation.
Results
Although monkeys were not required to process the
numerical cue, we observed subtle but consistent effects of
numerical deviance, and thus, expected reward size, on
saccades. Response time (RT) between fixation offset and
saccade initiation decreased as the numerical distance
between standard and deviant increased (5.57 ms for
maximum difference of 30, confidence interval [CI]: 8.68
to 2.46, nested F ¼ 12.33, p , 0.0001, Materials and Methods,
Equation 1). The small magnitude of the difference in RT is
not surprising, given the task was merely to shift gaze to a
target that had been visible throughout the trial. Using
similar tasks, other studies have also found small effects of
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Monotonic Coding of Numerosity in LIP
Table 1. Preferred Numerosities for Neurons Found
to Be Significantly Modulated by Number
Number Cue
Number of Neurons
2
4
8
16
32
16
1
1
5a
12
Preference was determined by ANOVA (as in [11–13]; n ¼ 35/57). The majority of neurons
preferred either the minimum (2) or maximum (32) value.
a
Two neurons that preferred 16 were tested with 16 as the maximum number, i.e., range
of 2–16.
doi:10.1371/journal.pbio.0050208.t001
For neurons with an increasing relationship between firing
rate and number, the average correlation coefficients were
0.86 6 0.23 (mean 6 standard deviation) and 0.82 6 0.18 for
linear and logarithmic-scaled number, respectively. For
neurons with a decreasing relationship between firing rate
and number, the average correlation coefficients were 0.73
6 0.13 for a linearly scaled number and 0.82 6 0.11 for its
logarithmic transform. Across these two groups of neurons,
we compared how well the average population responses
were fit when scaled linearly or logarithmically. Neurons with
a negative relationship between firing rate and number were
slightly better fit using the logarithmic transform of number
(F ¼ 3.54, p , 0.05), whereas there was no difference between
firing rate and number for neurons with a positive relationship (F ¼ 1.69, p ¼ 0.19).
Thus, LIP neurons responded in a graded fashion to the
total number of elements within their response fields. Our
data also suggest that one subpopulation of LIP neurons was
better described as systematically responding to number on a
logarithmic scale. Therefore, we used the logarithmic transform of number in the remaining analyses; however, all
analyses were repeated with raw numbers, and the same
overall results were obtained.
The degree of response modulation by number, measured
as the change in firing rate over the range of numerosities
presented, for all numerosity-related neurons in the population, is shown in Figure 5 (Materials and Methods,
Equation 2, p , 0.05). There was a strong correspondence
between ANOVA and regression tests for numerical modulation: 28 neurons were found to have significant modulation
by numerosity when tested by both ANOVA and regression,
including the five cases in which the largest response was to
an intermediate value. The proportion of neurons that
preferred small or large values was not notably different in
the two monkeys tested. Of the 31 neurons recorded from
monkey O, 11 preferred small numerosities and eight
preferred large, whereas of the 26 neurons recorded from
monkey W, six preferred small numerosities and six preferred
large. Thus, the majority of numerosity-sensitive neurons in
our population were not tuned to cardinal numerical values,
but instead represented numerosity in a roughly monotonic
fashion—in stark contrast with neurons sensitive to cardinal
numerical value reported in the ventral intraparietal area
(VIP) and PFC.
Modulation of neuronal activity by numerosity did not
Figure 4. Single LIP Neurons Respond in a Graded Fashion to the Total
Number of Elements in the Response Field
Firing rate for each number (mean 6 standard error [SE]) in the 400-ms
epoch beginning 50 ms after stimulus onset, plotted for each standard
block tested (line color). The change in firing rate (D) over the range of
numerosities tested was estimated by linear regression with the
logarithm of stimulus number (all p , 0.01; Materials and Methods,
Equation 2). Numerical sensitivity did not depend on standard block
(ANOVA, number 3 standard: [A] p ¼ 0.59, [B] p ¼ 0.99, [C] p ¼ 0.10, and
[D] p ¼ 0.17). Neurons in (A) and (C) increased firing with larger
numerosities, whereas those in (B) and (D) had larger responses for
smaller numerosities.
doi:10.1371/journal.pbio.0050208.g004
linear regression (Materials and Methods, Equation 2). We
observed neurons with both increasing and decreasing
numerical response profiles in both monkeys.
The majority of neurons modulated by number preferred
either the smallest or largest numerosity tested. In previous
studies [11–13], ANOVA was used to determine whether or
not neurons were modulated by numerosity, with each
neuron’s ‘‘preferred’’ numerosity assigned as that which
elicited the largest response. ANOVA thus provides a
model-free means for assessing preferred numerosity. In
our study, the majority of the neurons (30/35) found to be
modulated by numerosity using ANOVA preferred either the
largest or smallest value tested (Table 1).
Because we hypothesized that neural activity in LIP would
encode the total number of elements within the RF in a
roughly monotonic fashion, rather than encode a specific
numerical value, we next used linear regression to quantify
the relationship between firing rate and numerosity for all 57
neurons (Materials and Methods, Equation 2). Fourteen
neurons had a significant positive relationship (as in Figure
4A and 4C), whereas 17 showed a significant negative
relationship between numerosity and neural response (as in
Figure 4B and 4D). An important question is whether the
relationship between neuronal response and numerosity is
scaled linearly or logarithmically. For 28 of these 31 neurons,
firing rate varied significantly with both raw number and its
logarithmic transform (Materials and Methods, Equation 2).
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Monotonic Coding of Numerosity in LIP
Figure 5. Sensitivity of the LIP Neuronal Population to the Total Number
of Elements within the Response Field
For each neuron, the D firing rate across range of numerosities was
estimated by linear regression with the logarithmic transform of number
(Materials and Methods, Equation 2). Red: neurons with significant
negative slope (prefer small numerosity); purple: neurons with significant
positive slope (prefer large numerosity).
doi:10.1371/journal.pbio.0050208.g005
Figure 6. Numerical Sensitivity in LIP Does Not Reflect Attention or
Reward Expectations
For 31 neurons significantly modulated by number, the average
response was compared for sets of trials in which a particular number
was presented as both standard and deviant in the same recording
session. The line shows the best fit to the data with an intercept of 5.5
sp/s and a slope of 0.92.
doi:10.1371/journal.pbio.0050208.g006
depend on whether the numerical cue was presented as the
standard or as one of the deviant values. For the neurons with
significant modulation by numerosity (Figure 5), we compared responses on trials in which the same numerical value
was presented as both standard and deviant. Because neurons
were typically tested in approximately three standard blocks,
the comparison could not be made for all numerosities for
every neuron. Figure 6 shows the average firing rate for all
cases in which the same numerosity was tested as standard
and deviant for a given neuron. The average firing rate did
not systematically differ between standard and deviant trials.
Nested regression was used to test whether there was
modulation of firing rate due to the presentation of a given
numerosity as standard or deviant in addition to the effect of
numerosity. Across all neurons that preferred small numerosities, neural responses were modulated by numerosity, but
not by assignment as standard or deviant (number: 20.31
spikes/s [sp/s], CI: 23.10 to 17.53, F ¼ 102.9, p , 0.0001;
standard: 0.06 sp/s, CI: 1.91 to 1.80, nested F ¼ 0.003, p ¼
0.95; Materials and Methods, Equation 3). Similarly, neurons
that preferred large numerosities showed no effect of
presentation as standard or deviant on firing rate in addition
to the effect on numerosity (number: 23.07 sp/s, CI: 18.0 to
28.2, F ¼ 30, p , 0.0001; standard: 0.45 sp/s, CI: 3.76 to 2.87,
nested F ¼ 0.70, p ¼ 0.79; Materials and Methods, Equation 3).
Because there were no differences between neural responses
on standard and deviant trials, all trials are included in
subsequent analyses.
The time course of the neural response to visual arrays
revealed two phases of discrimination. In the 200 ms before
stimulus onset, baseline activity was low, at 14.2 6 0.2 sp/s
(Figure 3). Following a transient response that began
approximately 40 ms after stimulus onset, we observed
persistent modulations in activity related to the number of
elements in the visual arrays (Figure 7). Figure 7A shows the
average firing rate over the period of stimulus presentation
for the 14 neurons that preferred larger numerosities. The
transient increase in firing rate was higher for larger
numerosities, and was followed by activity that decreased
more quickly and to a lower extent with smaller numerosities.
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The level of activity that persisted throughout the rest of the
interval remained above baseline and increased 22.4 sp/s over
the range of numerical values presented (Figure 7A, inset; CI:
17.5 to 27.3, F ¼ 80.22, p , 0.0001; Materials and Methods,
Equation 2). Neurons that preferred small numerosities had a
similar, but weaker, transient increase in activity that was
larger for large numerosity arrays (Figure 7B). However,
following the initial, brief positive modulation by numerosity,
activity decreased more quickly and to lower levels with
greater numbers of elements. Sustained activity in these
neurons decreased 20.3 sp/s over the range of numerosities
shown (Figure 7B, inset; CI: 23.1 to 17.5, F ¼ 205.9, p ,
0.0001; Materials and Methods, Equation 2), but overall,
remained greater than baseline activity.
Both groups of neurons showed a larger initial neural
response for larger numerical arrays before quickly diverging.
For each group, modulation of activity by the number of
stimulus elements was computed in 40-ms windows, calculated every 20 ms (Figure 7C). Initially, all neurons showed a
reliable positive response modulation, indicating a larger
response for larger numerosities. After approximately 40–60
ms, the responses for the two groups of neurons began to
separate. Neurons that preferred large numerosities continued to show a positive modulation of activity (purple line)
throughout stimulus presentation. However, the modulation
of neurons that preferred small numerosities reversed such
that larger numerosities evoked smaller responses (red).
These two response profiles persisted throughout the entire
stimulus viewing period.
Differential neuronal responses to numerosity emerged at
a similar time across all values. Figure 8 shows the time at
which the neural response to each numerosity began to differ
from that of the value(s) closest to it (i.e., 2 versus 4, 4 versus 8,
8 versus 16, and 16 versus 32 [one-tailed, two-sample t-tests]).
Before stimulus presentation (at time ¼ 0), the neural
responses for each pair of numerosities did not differ
statistically from each other. For neurons that preferred
large numerosities, the difference in firing rate for each pair
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Monotonic Coding of Numerosity in LIP
Figure 7. Numerical Selectivity Evolves Rapidly in LIP
(A) For 14 neurons preferring large numerosity, averages were calculated in 20-ms bins for trials grouped by stimulus numerosity. Responses are aligned
to stimulus onset (dashed vertical line at zero) and shown for the full stimulus period (400 ms). Numerosity is indicated by line color. Inset: average
response (mean 6 SE) during the epoch 50–450 ms following stimulus onset.
(B) Average response of 17 neurons preferring small numerosity. Same conventions as (A).
(C) Modulation of response for the two groups of neurons shown in (A) and (B). Linear regression was used to estimate the change in firing rate over the
range of numerosities presented in a 40-ms window centered at the time point, sampled every 20 ms. Positive modulation indicates responses that
increase with number; negative modulation indicates responses that decrease with increasing number. Arrow: time at which modulation first
significantly differs from zero; subsequent time points are also significantly modulated by number (p , 0.05).
doi:10.1371/journal.pbio.0050208.g007
of numerosities, except 8 versus 16, became significant
approximately 40 ms after stimulus onset. The differences
in firing rate emerged simultaneously for neurons that
preferred small numerosities as well, although slightly
delayed from the neurons that preferred large numerosities.
For these neurons, the time to first discrimination did not
occur at exactly the same time, but there were no systematic
differences to suggest that discrimination of larger values
occurred later. These results support parallel encoding of
numerosity, with additional time required to generate
negative modulations in response to increasing numerosity.
The modulations in firing rate observed were not due to
differences in other stimulus attributes. Because total number
of pixels in the stimulus, size of the individual elements, and
stimulus density typically covary with numerosity, several sets
of stimuli were used that balanced these attributes while
varying numerosity. In 17 of the 31 neurons significantly
modulated by numerosity, these controls were instituted in a
blockwise fashion. In the remaining 14 neurons, stimuli from
all of the different types of controls were interleaved during
recording. Neural responses on trials grouped by total pixels,
element size, color, density, or numerosity are shown in
Figure 9 for the 14 neurons tested with interleaved stimulus
controls. We estimated the amount of modulation due to
different stimulus attributes (Materials and Methods, Equation 4a). Seven of the neurons tested with interleaved
stimulus controls preferred small numerosities (Figure 9,
top row; modulation ¼12.1 sp/s, CI: 16.4 to 7.8, F ¼ 30.7, p
, 0.0001, Equation 4a). The responses of these neurons
during stimulus viewing were not affected by cumulative
number of pixels (modulation ¼ 0.003 sp/s, CI: 0.006 to 0.011,
F ¼ 0.32 p ¼ 0.57, Equation 4a), element size (modulation ¼
0.002 sp/s, CI: 0.0002 to 0.005, F ¼ 3.20, p ¼ 0.08, Equation 4a),
or density (modulation ¼ 4.08, CI: 3.76 to 11.9, F ¼ 1.05, p ¼
0.31, Equation 4a). For these neurons, there was modulation
due to color, with responses 6.12 sp/s and 8.96 sp/s higher
across numerosities for the red and green arrays, respectively,
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compared with blue arrays (CIred: 2.56 to 9.68, CIgreen: 5.39 to
12.53, F ¼ 34.61, p , 0.0001, Equation 4a). When neural
response was modeled with color and numerosity as factors,
we found that significant modulation by numerosity remained (modulation by number ¼ 9.70, CI: 13.2 to 6.2,
nested F ¼ 14.83, p , 0.0001; Materials and Methods, Equation
4b). The remaining seven of the neurons with interleaved
stimulus controls showed greater activity for large numerosities (Figure 9, bottom: modulation ¼ 24.0 sp/s, CI ¼ 16.3 to
31.7, F ¼ 37.4, p , 0.0001). In these neurons, we found no
significant effect of total number of pixels (modulation ¼
0.002 sp/s, CI: 0.006 to 0.01, F ¼ 0.17, p ¼ 0.68, Equation 4a),
element size (modulation ¼ 1.34 sp/s, CI: 1.10 to 3.78, F ¼
1.16, p ¼ 0.28, Equation 4a), or density (modulation ¼ 2.88 sp/s,
CI: 2.97 to 8.73, F ¼ 0.93, p ¼ 0.33, Equation 4a) on neural
responses during stimulus viewing. There was a 15.1 sp/s
larger response for red stimuli than green or blue (CI: 9.08 to
21.28, F ¼ 13.25, p , 0.0001, Equation 4a). Again, modeling
the neural response with both color and numerosity as factors
showed that a significant modulation by numerosity persisted
(modulation by number ¼ 19.81, CI: 13.46 to 26.16, nested F ¼
18.7, p , 0.0001, Equation 4b). We performed an additional
regression to test whether the effect of numerosity on neural
response persisted when all other stimulus attributes were
included. We found that including numerosity as a regression
factor with cumulative number of pixels, element size,
density, and color resulted in a significant improvement in
the fit to the data (prefer small: modulation by number ¼
11.57 sp/s, CI: 20.60 to 2.54, nested F ¼ 6.31, p ¼ 0.01;
prefer large: modulation by number ¼ 17.93 sp/s, CI: 4.07 to
31.8, nested F ¼ 6.43, p ¼ 0.01; Materials and Methods,
Equation 5). Just as neuronal responses were typically not
affected by low-level stimulus attributes, neuronal activity
also did not depend on whether a particular numerosity
served as standard or deviant (prefer large: modulation ¼
2.47 sp/s, CI: 7.48 to 2.54, nested F ¼ 0.94, p ¼ 0.33; prefer
small: modulation ¼ 1.57 sp/s, CI: 4.51 to 1.37, nested F ¼
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Monotonic Coding of Numerosity in LIP
Figure 8. LIP Neurons Differentiate Numerosities Simultaneously
For the two groups of neurons, ‘‘prefer large’’ and ‘‘prefer small,’’ trials were sorted by number, and the average response was calculated in 40-ms bins.
At each time point, responses were compared for pairs of numerosities closest to each other using a one-tailed, two-sample t-test to determine the time
at which the responses began to differ. The resulting p-value is plotted as a function of time. The time to first discrimination did not occur later for
larger numerosities, suggesting that numerosity is encoded via a parallel rather than serial process.
doi:10.1371/journal.pbio.0050208.g008
1.10, p ¼ 0.29; Materials and Methods, Equation 3), thus
effectively ruling out salience, attention, or reward expectations as factors that could account for the effect of
numerosity on neuronal responses.
integration of visual motion information [28,30]. Our data
suggest that, in addition to integrating with respect to time,
LIP neurons also integrate the number of objects within their
response fields.
The decrease in activity related to element number for
neurons preferring small numerosities is also consistent with
the hypothesis that some LIP neurons may signal oculomotor
intentions. Multiple-element arrays could be interpreted as
multiple saccade targets, so that the intention to look at any
one would be divided among the n possibilities. Consistent
with this idea, single neurons in superior colliculus show
reduced responses with increasing number of potential
saccade targets throughout the visual field [42] and suppression by multiple stimuli located within the same RF [43]. In
light of these findings, it is worth noting that the monkeys in
our study never made erroneous saccades to the numerical
array.
Although lesion and neuroimaging studies in humans
suggest that parietal cortex is critical for estimating numerical quantity, previous neurophysiological studies in monkeys
have not shown an extensive representation of number in
PPC. Nieder and Miller [12] found that approximately 20% of
PPC neurons, but only 13% (10/77) of neurons in LIP, were
modulated by the number of visual elements shown during
the sample period in a numerical match-to-sample task.
Further, those neurons responding to number did not
resemble accumulators, but rather were tuned for particular
cardinal values. Neurons in VIP were also found to have
cardinal tuning for numerical stimuli presented sequentially
(22%) as well as simultaneously (12%) [13]. Even when stimuli
were presented in succession, number-related activity was
tuned for a preferred position within the sequence, rather
than changing systematically with number (see also [40]). VIP
neurons typically have multimodal response fields aligned to
the face and central visual field [44–46]; consequently, in the
match-to-sample task used by Nieder and colleagues [12],
numerical stimuli, ranging from one to five elements, were
uniformly placed near central fixation. In contrast with
neurons in VIP, LIP neurons typically have peripheral
Discussion
We found that a population of neurons in LIP encodes the
number of elements in a visual array in a roughly monotonic
manner. Parietal cortex has been implicated as a principal
node of the neural circuit necessary for numerical processing,
and previous evidence in monkeys has revealed neurons in
VIP that are tuned to the cardinal value of elements within an
array. Moreover, recent neuroimaging studies of responses to
dot arrays like those used here have reported activation in
human parietal cortex modulated by the number of elements
in the array [10]. Accumulation of quantity has been
proposed as a critical initial stage of enumeration, a process
that could be served by ‘‘integrator’’ neurons in LIP [30,39].
Endorsing this idea, we found that a majority (.50%) of LIP
neurons had activity that either increased or decreased with
number. This response pattern differs both quantitatively
and qualitatively from previous physiological studies [11–
13,40].
All of the neurons that were modulated by numerosity
showed an initial, transient response that increased with
number (Figure 7). The transient increase in activity for
larger arrays suggests an initial salience signal modulated by
the number of possible objects to attend [41]. This transient
response was followed by a longer-lasting signal correlated
with accumulated numerical value. Rapid modulation by
number, occurring in the first 40–60 ms, supports the
hypothesis that numerical accumulation occurs simultaneously, in a parallel manner [12]. The initial, positive
modulation is followed by persistent number-related activity
that is analogous to temporal integration previously shown in
LIP. Responses of LIP neurons while monkeys perform
random-dot motion discrimination have been shown to
increase or decrease proportionally to stimulus strength as
a function of time [23,29], a process likened to temporal
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Monotonic Coding of Numerosity in LIP
Figure 9. Numerical Sensitivity in LIP Does Not Reflect Low-Level Stimulus Attributes
For 14 of the 31 number-sensitive neurons, stimuli that balanced total number of pixels, individual element size, color, or array density were all
interleaved throughout each experiment. The top row shows responses for seven neurons that preferred small numerosities, and the bottom row
shows responses for seven neurons that preferred large numerosities. In each panel, trials are divided according to one stimulus attribute only. Firing
rate was calculated and then plotted in 20-ms bins.
doi:10.1371/journal.pbio.0050208.g009
response fields located in the contralateral visual field [17–
19]. Therefore, we tailored task geometry to the spatial
sensitivity of the neural population tested by mapping the
area of the RF that extended beyond the borders of the
numerical cue. This may account for the higher proportion of
number-sensitive neurons found here (.50%) than in
previous studies. Most importantly, neurons in LIP represented numerical magnitude in a roughly monotonic manner,
consistent with neural accumulation.
Neural network models of numerical representation [32,33]
derive cardinal number from elements with just the type of
graded coding shown by LIP neurons in this study. Although
each model of numerical processing (Figure 1) includes an
accumulator stage, they make different predictions about the
manner in which numerical magnitude is represented. The
accumulator in the mode-control model of Meck and Church
[31] utilizes a serial mechanism for counting that produces an
accumulated value linearly proportional to the number of
events enumerated. The summation units in the neural
network model of Dehaene and Changeux [32] use a parallel
process to encode number as proportional to the total
amount of activity following a normalization stage. Our
findings are consistent with the summation units in the
Dehaene and Changeux model. We observed a differential
response to number that emerged simultaneously (Figure 8),
regardless of whether the stimulus number was small or large,
consistent with parallel encoding of numerosity. The Dehaene and Changeux model predicts that number is
ultimately represented on a logarithmic scale by ‘‘numerosity’’ units, which are driven by the activation of summation
units that exceed a particular threshold. Such numerosity
units are consistent with neurons in VIP and PFC that were
previously found to encode cardinal numerical value on a
logarithmic scale. Our data do not strongly support either
linear or logarithmic scaling of neuronal sensitivity to
number in LIP. The graded responses we observed are also
consistent with the neural network model proposed by
Verguts and Fias [33], which predicts numerical summation
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represented by activity that increases or decreases monotonically with increasing quantity.
Our task was designed not only to consider spatial
selectivity of LIP neurons, but also to reduce the influence
of task demands such as reward expectation and training.
Expectations about impending rewards have been shown to
influence neural activity in such areas as LIP, PFC, and
posterior cingulate cortex [24,35,36,47]. Such reward-modulation has been observed when the target of the upcoming
reward was located in the neuron’s RF (but see [35]). Here, we
have positioned the target in the opposite direction of the RF.
It is conceivable that the cue response would be influenced by
reward magnitude or the frequency of a particular value’s
presentation, both of which differed according to a number’s
status as standard or deviant [9,24]. However, we did not
observe this type of modulation. Training has also been
shown to modulate neural responses. Monkeys performing
the match-to-sample task based on numerosity had extensive
training to explicitly categorize stimuli according to number
[11]. This type of training has been shown to influence neural
responses in PFC [48] and LIP [49]. Although our monkeys
had ongoing experience with the task and their RTs suggests
they at least implicitly processed the numerical array, there
was no explicit requirement to discriminate number. This
suggests the possibility that quantity may be encoded
spontaneously in LIP, a hypothesis warranting further study
in naive monkeys.
The neuronal sensitivity to number we observed is
consistent with a spatial representation of accumulated
magnitude. Coding of quantity within spatially selective
response fields may offer a mechanism for the process of
enumeration [33]. That there is a relationship between the
representations of space and number offers potential
explanation for the overlap of spatial and numerical deficits
that characterize Gerstmann syndrome [50] which follows
parietal damage, as well as the psychophysical observations of
interactions between spatial and numerical judgments [14]
(see also [15]). By taking the spatial selectivity of LIP neurons
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Monotonic Coding of Numerosity in LIP
into account, we have found encoding of numerical magnitude in the activity of single neurons. The neurons sampled
by Nieder and colleagues in other regions of PPC and PFC
[11–13] may compute cardinal numerical representations
based on inputs from neurons such as those reported here. In
other words, these two classes of number-selective neurons
may be the physiological instantiation of the summation units
(ordinal) and numerosity units (cardinal) proposed in neural
network models of numerical representation [32,33]. Quantity may thus be another aspect of visual information that is
processed hierarchically within the dorsal visual stream, in
which information such as color and orientation, typically
associated with the ventral visual stream, is not lost at higher
levels of processing, but is carried forward as more complex
RFs are built [51,52].
maintained fixation on a central point while a saccade target was
placed in a random location either to the right (monkey O) or left
(monkey W) of fixation (Figure 1A). Following a variable prestimulus
delay, an array of 2, 4, 8, 16, or 32 circles was presented on the
opposite side of the screen for 400 ms. When the fixation point was
extinguished, the monkey shifted his gaze to the saccade target for a
fluid reward. The likelihood that a particular numerosity was
presented on each trial varied in a blockwise fashion. In each block,
one number was selected as the standard, presented on 50% of trials.
On each of the remaining trials, a deviant number was shown (Figure
1B). We encouraged monkeys to attend to the numerical stimulus by
administering a standard reward (fluid delivered for 100-ms open
solenoid time, for 0.10 ml) on successful standard trials and a larger
reward (150-ms open solenoid time of fluid, for 0.15 ml) on all
successful deviant trials. Once the RF was mapped, monkeys
performed an average of 372 (range: 143–688) numerical discrimination trials. Each block consisted of approximately 100 trials, with
two to five blocks per neuron.
Stimulus controls. Stimuli were generated using Matlab (The
Mathworks, http://www.mathworks.com). Several sets of stimuli were
constructed. All stimuli consisted of a set of 2, 4, 8, 16, or 32 circles
plotted in random locations in a 7.08 3 7.08 square (135 3 135 pixels,
except where noted). Circles were red, blue, or green on a black
background. Two hundred stimuli were created for each number and
each color, so that there were 3,000 bitmap images in each set of
stimuli. Several sets of stimuli were used: (1) random size with
balanced element density. Each element’s radius was randomly varied
between three to six pixels. Density was controlled by using a random
walk to place circles in contiguous locations (each possible location a
15 3 15 pixel box); (2) balanced total pixels. The total number of
pixels in each stimulus was chosen randomly from a uniform
distribution of 402–2,513 pixels; (3) balanced element size. The
radius for all the elements in a given stimulus was chosen from a
uniform distribution of four to ten pixels; and (4) balanced element
size and density. Element size was selected as in (2), but elements were
placed within backgrounds of different sizes, ranging from 2.58 to 8.38
square.
Analysis. First, we tested whether saccade RT depended on the
numerical distance between the standard and deviant value presented
(Matlab; The Mathworks). For each trial, RT was measured as the time
between fixation point offset and initiation of the eye movement. RT
was modeled as:
Materials and Methods
Subjects. Two adult rhesus macaque monkeys (Macaca mulatta)
weighing 7.5–8.5 kg served as subjects. All procedures were approved
by the Duke University Institutional Animal Care and Use Committee
and were designed and conducted in compliance with the Public
Health Service’s Guide for the Care and Use of Animals.
Surgical and training procedures. A head restraint prosthesis and
scleral search coil [53] were implanted during an initial, sterile
surgical procedure performed under isoflurane inhalant anesthesia
using standard techniques described in detail elsewhere [34]. Animals
received postoperative analgesics for 3 d and antibiotic prophylaxis
for 10 d after all surgeries. Following a 6-wk recovery period, animals
were habituated to head restraint and trained to perform oculomotor
tasks for fluid reward using custom software (http://www.
ryklinsoftware.com). Horizontal and vertical eye positions were
sampled at 500 Hz (Riverbend Instruments, http://www.
riverbendinst.com) and recorded by computer. Visual stimuli were
presented on a computer monitor (21 in, 1,024 by 768 pixels, 60 Hz
refresh) 46 cm in front of the monkey. Once monkeys could perform
the behavioral tasks, a second sterile surgical procedure was
performed to place a stainless steel chamber (Crist Instruments,
http://www.cristinstrument.com) over a 15-mm craniotomy over LIP
(5 mm posterior and 12 mm lateral of stereotaxic 0,0; left hemisphere
for monkey O and right for monkey W). Microelectrode recording
began 1 wk following the recovery period.
Microelectrode recording procedures. Before each recording
session, the cylinder was opened under aseptic conditions and
repeatedly flushed with sterile saline. A Teflon grid (Crist Instruments) was secured in the cylinder and an X-Y micropositioner (Crist
Instruments) and hydraulic microdrive (David Kopf Instruments,
http://www.kopfinstruments.com) were mounted onto the cylinder. A
tungsten steel (0.8–1.2 MX) electrode (FHC, http://www.fh-co.com) was
drawn into a 23-gauge hypodermic tube, which was used to puncture
the dura. Electrophysiological signals were amplified and filtered to
exclude power-line noise and signals of the magnetic fields used to
monitor eye position (band-pass, ;0.2 to 5 kHz). Individual action
potentials were identified in hardware by time and amplitude
criteria, and the times of spike occurrences were recorded by
computer.
Neuron selection and spatial mapping of response fields. A
hallmark of LIP neurons is their spatial selectivity [17–19,54,55], so
monkeys first performed a block of standard delayed-saccade trials to
map each neuron’s spatial RF. On each trial, a target was displayed in
the periphery while the monkey fixated on a central point. Following
a delay (700 to 1,500 ms), the fixation point disappeared and the
monkey was rewarded with fluid for shifting gaze to the target. Spatial
selectivity of each neuron was assessed online by initially presenting
targets widely dispersed throughout the entire visual field. When
elevated activity in one region was observed, saccade targets were
placed at multiple locations within an approximately 128 3 128 area
on approximately 80% of trials to test for an expanded area of
elevated neural response. The target was located in the opposite
hemifield on the remaining 20% of trials for comparison. Typically,
50–100 trials were used to map the RF in order to maximize data
collection in the numerical task. Recording sites were localized to LIP
using digital ultrasound imaging (monkey O, [34]) and histology
(monkey W).
Implicit numerical discrimination task. On each trial, monkeys
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Y ¼ b1 þ b2 M þ b3 T þ b4 D þ e
ð1Þ
where Y is RT, M is monkey, T is the time delay between number cue
offset and fixation point offset (0, 200, or 400 ms), and D is the
absolute value of the difference between standard and delay for each
trial (max ¼ 30) and e is random error. The coefficients, b1 through b4,
were estimated using weighted least-squares regression. An F statistic
was calculated using the principle of extra sum of squares [56]. This F
statistic is referred to in the text as ‘‘nested F,’’ and its corresponding
p-value is given, indicating the likelihood that the null hypothesis is
true. Here, the null hypothesis, that b4 ¼ 0, signifies that D had no
effect on RT.
Neural data were processed offline to extract the time at which
each trial event occurred as well as the time of each action potential.
Neurons were first analyzed using ANOVA (Statistica; Statsoft, http://
www.statsoft.com) to quantify whether there were any differences in
firing rate related to quantity, without assuming a linear relationship
(Figure 4 and Table 1). For each neuron, number and standard were
included as factors to determine whether there were any differences
in firing rate that depended on whether a particular numerical
stimulus served as standard or deviant.
For each neuron, we tested whether neural responses were
systematically related to numerosity as predicted using the model:
Y ¼ b1 þ b2 logN þ e
ð2Þ
where Y is the spike rate in the period 50–450 ms after numerical cue
onset, logN is the logarithmic transform of the numerosities 2–32,
and e is random error. The regression coefficients, b1 and b2, were
estimated using weighted least squares. This model was also used to
estimate modulation of neural response across the subpopulations of
neurons preselected for large or small number preference (shown in
Figure 7). F and p-values for the regression are reported in Results as
well as b2 and its confidence intervals. Neurons were initially analyzed
using both linear numerical value and the logarithmic transform of
number. For neurons significantly modulated by number regardless
of the scale used, we compared the r-values from each neuron’s fit of
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Monotonic Coding of Numerosity in LIP
response to linear number, and the logarithmic transform of number.
To determine whether there was a statistically better fit of the data
using a linear or logarithmic scale, we computed an F statistic, and
corresponding p-value, from the ratio of the sum of squared
deviances from the best fits of the average population data using
both numerical scales (SSlin/SSlog).
To determine whether firing rate depended on whether the
numerical stimulus was presented as standard or deviant (Figure 6),
we used the model:
Y ¼ b1 þ b2 logN þ b3 S þ e
Y ¼ b1 þ b2 S þ b3 logN þ e
where Y is the firing rate in the epoch 50–450 ms after stimulus onset,
S is an attribute listed above (Equation 4a), and logN is the
logarithmic transform of the numerosities 2–32. The null hypothesis
that numerosity did not affect the neural response when other
stimulus attributes were included (b3 ¼ 0) was tested using nested
regression as described above.
Finally, to test whether the effect of number persisted when all of
the stimulus attributes listed above were included, we modeled the
neural response in the period 50–450 ms after stimulus onset (Y) as:
ð3Þ
where Y is the spike rate in the period 50–450 ms after numerical cue
onset, logN is the logarithmic transform of the numerosities 2–32, S is
a dummy variable set to 1 on standard trials and 0 on deviant, and e is
random error. The null hypothesis that S did not affect the neural
response (b3 ¼ 0) was tested using nested regression as described in
Equation 1.
To determine whether firing rate depended on stimulus attributes
(Figure 9) that covary with numerosity we used the model:
Y ¼ b1 þ b2 S þ e
Y ¼ b1 þ b2 D þ b3 Ir þ b4 Ig þ b5 A þ b6 E þ b7 logN þ e
ð5Þ
in which D is density (0 ¼ low, 1 ¼ high), Ir and Ig are dummy variables
for color (as in Equation 4a), A is cumulative number of pixels, E is
element size, and logN is the logarithmic transform of the
numerosities 2–32. We used nested regression to test the null
hypotheses that there was no significant effect of number on the
neural response (b7 ¼ 0) when the other stimulus attributes were
known.
ð4aÞ
where Y is the spike rate in the period 50–450 ms after numerical cue
onset, S is one of the variables that describe the particular stimulus
presented on each trial, and e is random error. Attributes tested were
total pixels (cumulative number of pixels of the stimulus), element
size (radius in pixels), and density (0 ¼ low and 1 ¼ high). Color was
represented by two dummy variables, Ir (1 ¼ red, 0 otherwise) and Ig (1
¼ green, 0 otherwise). The fit yielded estimates of the modulation of
firing rate due to the stimulus attribute modeled as well as a test of
the null hypothesis that b2 ¼ 0.
To determine whether modulation of neural responses due to
number persisted in addition to modulation by other stimulus
attributes, Equation 4a was modified to include number as a factor.
We used the model:
Acknowledgments
Author contributions. All authors conceived and designed the
experiments and wrote the paper. JDR performed the experiments
and analyzed the data. EMB and MLP contributed reagents/materials/
analysis tools.
Funding. This work was supported by EY014742 (JDR), the John
Merck Fund (EMB), and the Klingenstein Foundation (MLP).
Competing interests. The authors have declared that no competing
interests exist.
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