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Throughout the history of mathematics, concepts of
number and space have been tightly intertwined. We
tested the hypothesis that cortical circuits for spatial
attention contribute to mental arithmetic. We trained a
multivariate classifier to infer the direction of an eye
movement, left or right, from the brain activation
measured in posterior parietal cortex. Without further
training, the classifier then generalized to an arithmetic
task. Its left versus right classification could be used to
sort out subtraction versus addition trials, whether
performed with symbols or with sets of dots. These
findings are consistent with the suggestion that mental
arithmetic co-opts parietal circuitry associated with
spatial coding.
The human species is unique in its capacity to create
revolutionary cultural inventions such as writing and
mathematics, which dramatically enhance its native
competence. From a neurobiological standpoint, such
inventions are too recent for natural selection to have
dedicated them specific brain mechanisms. It has therefore
been suggested that they co-opt or “recycle” evolutionarily
older circuits with a related function (1), thus enriching
(without necessarily replacing) their domain of use. For
instance, learning to read recruits a left infero-temporal area
originally engaged in object recognition, and even the
seemingly arbitrary shapes of our letters may originate in a
neural repertoire of junctions detectors useful for scene
recognition and available to all primates (2). In the case of
mathematics, although foundational intuitions such as number
sense (3) and spatial maps (4) are present in many animal
species and in humans prior to education, mathematical
constructions vastly exceed these initial domains of inherited
competence. It has been argued that analogies between
number and space play a crucial role in the expansion of
mathematical concepts (5). Here, we investigate the role of
brain areas for spatial coding in mental arithmetic.
Many behavioral experiments have demonstrated
automatic links between number and space. Even young
children and uneducated adults readily conceive of numbers
as forming an internal spatial continuum or “mental number
line” (6). Merely perceiving an Arabic digit suffices to elicit a
spatial bias in both attentional orienting (7) and manual
responses (8), with small numbers inducing a left-sided and
large numbers a right-sided advantage in left-to-right readers.
When adults perform approximate additions and subtractions,
they overshoot towards larger numbers for addition and
towards smaller numbers for subtraction, as if carried along
by spatial momentum (9). Perhaps the most conclusive
evidence for numerical-spatial links comes from the
syndrome of spatial hemineglect, in which brain-lesioned
patients fail to attend to one side of space, usually the left
side. When such patients attempt to bisect a numerical
interval, their responses are shifted towards larger numbers,
as if neglecting the left half of the numerical segment where
small numbers are represented (10).
The brain mechanisms of these numerical-spatial
interactions, however, remain largely unknown. In both
monkeys and humans, number processing recruits a brain
area deep within the intraparietal sulcus (hIPS) (11, 12). This
site partially overlaps with area ventral intraparietal cortex
(VIP), an area coding for multimodal spatial movement and
tightly interconnected with nearby area lateral intraparietal
cortex (LIP) involved in saccadic and attention control (
13–
15). A model of the VIP-LIP circuitry proposes that it
implements a form of vector addition of eye and retinal
position information (16). We therefore reasoned that this
circuit might be co-opted for a similar function in the
arithmetic domain. Given the cultural link between small
numbers and the left side of space, and right numbers and the
right side of space in left-to-right readers, we predicted that
mental addition, which increases number size, would be
associated with a rightward shift of attention, and subtraction
with a leftward shift. Hence, the activation pattern in parietal
cortex during addition would resemble the activation pattern
associated with a rightward eye movement, while subtraction
would resemble a leftward eye movement.
In a 3 Tesla fMRI scanner, participants first performed a
localizer task for eye movements. By contrasting eye
Recruitment of an Area Involved in Eye Movements During Mental Arithmetic
André Knops,
1,2,3
* Bertrand Thirion,
2,4
Edward M. Hubbard,
1,2,3
Vincent Michel,
2,3,4
Stanislas Dehaene
1,2,3,5
1
INSERM, Cognitive Neuroimaging Unit, F-91191 Gif-sur-Yvette, France.
2
CEA, I2BM, NeuroSpin, F-91191 Gif-sur-Yvette,
France.
3
Université Paris-Sud, F-91405 Orsay, France.
4
INRIA Saclay – Île de France, Orsay, France.
5
Collège de France, Paris,
France.
*To whom correspondence should be addressed. E-mail: knops.andre@gmail.com
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movements against fixation, we isolated a set of six cortical
regions classically associated with saccades and used in all
subsequent classifier-based analyses (17): bilateral posterior
superior parietal lobule (PSPL), at a site overlapping with the
proposed human homolog of monkey area LIP (18); bilateral
frontal eye fields proper (FEF); and two clusters of activation
lateral to FEF (lFEF; Fig. 1B).
Participants performed a second set of fMRI runs during
which they either moved their eyes rightward or leftward on
randomly intermixed trials. We adopted a machine learning
approach to search for a linear combination of these voxel-
based activation signals that reliably separated leftward and
rightward saccades (19). We trained a linear support vector
machine (SVM) using a ten-fold cross-trial validation
approach in which the classifier is first trained on a random
subset of 90% of activation images (one image per trial), and
then performance is evaluated on the remaining 10% of trials.
The process was repeated one hundred times, each time with
a new random assignment of trials. Using only voxels from
the bilateral PSPL region, we obtained a mean accuracy
across all participants of 70.3% ± 2.4% (1 standard error),
which is significantly above the chance level of 50% (t(14) =
8.39, P < .001). Analysis by signal detection theory gave
similar results (average d’ across subjects = 1.1 ± 0.15, t(14)
= 7.58, P < .001). Thus, saccade direction, which is known to
be coded by neurons in monkey area LIP, could be inferred
from fMRI of human posterior parietal cortex.
Crucially, we then examined whether the same classifier,
without further training, would generalize to approximate
arithmetic. In new fMRI runs, participants saw two
successive numbers (presented as Arabic numerals or as sets
of dots), mentally calculated their approximate sum or
difference, and subsequently chose the closest number among
7 possible outcomes. We concentrated on brain activation just
after the presentation of the second operand, at which time
the participants performed the calculation (Fig. 1A).
Calculation activated a network of brain areas comprising
bilateral hIPS, prefrontal and premotor areas, with
considerable overlap between both notations (Fig. 1B).
Calculation overlapped only partially with saccades in
bilateral PSPL, but, as predicted, the classifier trained with
bilateral PSPL activations during saccades generalized to
calculation images. Equating addition with rightward
saccades and subtraction with leftward saccades, the mean
accuracy for inferring whether an addition or subtraction was
performed, averaged over all participants, was 55.0% ± 1.8%,
which is significantly greater than chance (t(14) = 2.78, P =
.015; d’=0.31 ± 0.10, t(14) = 2.85, P = .013).
Further analyses showed that, when the saccades classifier
was tested with addition images, it classified them as
rightward saccades 61% of the time (Fig. 2D), which is above
chance level (t(14) = 2.35 , P = .03). For subtraction,
however, only 49.1 % of images were classified as leftward
saccades (t(14) = -0.16, n.s.). This asymmetry, although
unexpected, is congruent with earlier reports of larger
rightward saccades in response to large numbers, relative to
leftward saccades with small numbers (20) and might reflect
reading habits in Western cultures.
A key aspect of the cortical recycling view is that saccadic
areas of the posterior parietal lobule should contribute to
calculation, not only when performed with concrete sets of
objects, but even with Arabic numerals, which are a recent
product of human culture. We therefore tested the
generalization from saccades to calculation in each notation
separately. The saccade-trained classifier could distinguish
addition from subtraction with an average accuracy of 54.3%
± 2% for Arabic numerals (t(14) = 2.26, P = .02; d’= 0.38 ±
0.11, t(14) = 2.1, P = .054) and with an average accuracy of
55.8% ± 2% for non-symbolic notation (t(14) = 2.93, P =
.005; d’= 0.38 ± 0.14, t(14) = 2.74, P = .016). Thus, both
symbolic and non-symbolic calculations rely in part on brain
circuits for saccadic eye movements.
As a further test of this sharing of resources for non-
symbolic and symbolic arithmetic, we also examined the
ability to predict which operation was being performed in one
notation, on the basis of a classifier trained to sort additions
versus subtractions in the other notation. This cross-notation
generalization yielded good results, both for the prediction of
non-symbolic calculation from the symbolic notation (mean
accuracy: 60.7% ± 2.5%, t(14) = 4.37, P < .001; d’= 0.53 ±
0.16, t(14) = 3.32, P = .005) and vice-versa (mean accuracy:
62.2% ± 2.1%, t(14) = 5.71, P < .001; d’ = 0.75 ± 0.14, t(14)
= 5.39, P < .001). This finding indicates that the PSPL region
is comparably involved in solving mental arithmetic problems
in both notations. Approximate arithmetic with sets of dots is
part of an inherited ‘number sense’ available to infants (21)
and non-human primates (22), but the cross-notation
generalization proves that the corresponding brain circuitry is
also used by arithmetic with culturally specific Arabic
numerals.
Given the observed parietal cross-talk, one may wonder
whether the arithmetic task, although involving only central
visual presentations and a constantly present fixation point,
led to overt eye movements. Eye position was continuously
monitored throughout fMRI, and we found no detectable
change in horizontal fixation at or around the time of the
arithmetic calculation (17). Furthermore, the observed cross-
talk was specific to posterior parietal cortex. Activation
patterns in FEF and lFEF could be reliably used to classify
left versus right saccades (respectively 56.9% ± 2.4%, t(14) =
2.87 , P = .012, and 57.8% ± 2.4%, t(14) = 3.3, P = .005), but
this classification did not generalize to addition versus
subtraction (respectively 49% ± 5.3% (t(14) = -0.18, P = .86)
and 49.2% ± 6.3% (t(14) = -0.12, P = .9)). The absence of
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decodable FEF activation during arithmetic confirms that
calculation specifically engages parietal rather than frontal
spatial mechanisms and involves covert visuo-spatial
mechanisms, not overt eye movements. As a final test of the
specificity of our results to area PSPL, we repeated the major
analyses with two control regions (hand motor area M1 and
hIPS). None of these regions yielded better-than chance
generalization from saccades to calculation (17).
In summary, we demonstrated that a multivariate classifier
can distinguish between brain activations during mental
addition and subtraction, after having been trained on images
from a separate experiment requiring saccades to the right or
left. This generalization was observed with numbers
presented either as Arabic symbols or as non-symbolic sets of
dots, which implies shared cognitive processes between both
notations. The observed generalization goes beyond previous
demonstrations of classifier-based decoding of line
orientation and other pictorial contents from early visual areas
(23–26), object identity and category from ventral visual
cortex (27), noun identity from distributed cortical regions
(28), or intentions from premotor, prefrontal and striatal sites
(29). Although generalization was found across different
image sizes (27), from real to imagined images (26) or from
trained nouns to novel nouns (28), inference remained
confined to the trained domain. By contrast, the present
research demonstrates generalization from a low-level
sensori-motor task to a high-level cognitive task involving
learned cultural symbols.
Our results confirm a prediction first made by Hubbard et
al. (13) that mental calculation can be likened to a spatial
shift along a mental “number line”. In a certain sense, when a
Western participant calculates 18+5, the activation moves
“rightward” from 18 to 23. This spatial shift relies on neural
circuitry in PSPL shared with those involved in updating
spatial information during saccadic eye movements. The
findings are reminiscent of the ‘embodied cognition’
perspective which stipulates that perceptual and action
mechanisms lie at the core of human abstract thinking (30).
However, the ‘recycling’ view that we propose does not
imply that abstract concepts originate in sensori-motor
learning. Indeed, there is ample evidence that abstract
numerical concepts have a long evolutionary history and a
dedicated neuronal circuitry in intraparietal cortex, partially
distinct from neighboring visuo-spatial circuits (31). Our
proposal is that human mathematics builds from foundational
concepts (space, time, and number) by progressively co-
opting cortical areas whose prior organization fits with the
cultural need. The PSPL area, perhaps because of its capacity
for vector addition during eye movement computation (16),
appears to have a connectivity or internal structure relevant to
arithmetic.
The contribution of PSPL appears to be fundamentally
different from the function of other regions such as FEF or
hIPS, where no generalization from saccades to calculation
was found. The PSPL is active, not only during saccades, but
during a broad variety of tasks involving as a common
denominator the representation, updating, or attention to
spatial locations. This makes it an ideal site for explaining the
broad variety of numerical-spatial interactions that have been
observed behaviorally with eye, hand, or attention
movements (13).
Like any fMRI study, the present work is correlative and
cannot establish whether the observed PSPL activation plays
a causal role in calculation. One interpretation is that the
PSPL is causally recruited during the actual computation of
the result of arithmetic operations. Another is that calculation
is effected by other means and that the PSPL activation
merely reflects a subsequent spread of activation to visuo-
spatial areas, perhaps because the final numerical result
attracts attention on the mental number line. To separate those
alternatives, future work should evaluate the impact of
temporary or permanent lesions, for instance using
transcranial magnetic stimulation of dorsal parietal areas,
which has already been show to causes joint impairments in
attentive visual search and arithmetic (32).
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Supporting Online Material
www.sciencemag.org/cgi/content/full/1171599/DC1
Materials and Methods
Figs. S1 to S3
References
29 January 2009; accepted 27 April 2009
Published online 7 May 2009; 10.1126/science.1171599
Include this information when citing this paper.
Fig. 1. (A) Schematic depiction of a calculation trial. After
the initial presentation of an instructional cue (letters A, S or
C for addition, subtraction or color task, respectively) two
quantities were presented successively, either as dot patterns
or Arabic digits. After a variable delay period, seven
responses alternatives appeared on screen and participants
had to choose the alternative closest to the actual outcome.
(B) Brain activation in the calculation task and the saccades
localizer task projected on lateral and top views of the brain.
The images shown result from contrasting symbolic (red) or
non-symbolic (green) calculation to the color task, and from
contrasting saccades to rest (blue) (P = .005, uncorrected).
Fig. 2. (A) Classification performance (d-prime) for each
participant in the saccades task (participants sorted according
to d-prime). (B) Classification performance (d-prime) per
participant for generalization of the classifier trained on
left/right saccades to subtraction/addition trials. (C) Voxel
clusters in left and right PSPL region that resulted from the
saccade localizer task and served as ROI for the classifier,
rendered on white matter/grey matter boundary. (D)
Percentages of trials classified as right saccades for
subtraction (orange), addition (light blue) and left and right
saccades (red and blue, respectively).
