Neural Mechanisms Underlying the Computation of
Hierarchical Tree Structures in Mathematics
Tomoya Nakai1,2, Kuniyoshi L. Sakai1,3*
1 Department of Basic Science, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo, Japan, 2 Japan Society for the Promotion of
Science, Kojimachi, Chiyoda-ku, Tokyo, Japan, 3 CREST, Japan Science and Technology Agency, Goban-cho, Chiyoda-ku, Tokyo, Japan

Abstract
Whether mathematical and linguistic processes share the same neural mechanisms has been a matter of controversy. By
examining various sentence structures, we recently demonstrated that activations in the left inferior frontal gyrus (L. IFG)
and left supramarginal gyrus (L. SMG) were modulated by the Degree of Merger (DoM), a measure for the complexity of tree
structures. In the present study, we hypothesize that the DoM is also critical in mathematical calculations, and clarify
whether the DoM in the hierarchical tree structures modulates activations in these regions. We tested an arithmetic taskthat involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, wefound significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among thetested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, andthat activations in the bilateral IPS and precuneus were independent from other possible factors. Moreover, by fitting
parametric models of eight factors, we found that the model of DoM in the hierarchical tree structures was the best to
explain the modulation of activations in these five regions. Using dynamic causal modeling, we showed that the model witha modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, was highly
probable. The intrinsic, i.e., task-independent, connection from the L. IFG to the L. IPS, as well as that from the L. IPS to the R.
IPS, would provide a feedforward signal, together with negative feedback connections. We indicate that mathematics andlanguage share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and thatmathematics recruits the additional network of the L. IPS and R. IPS.
Citation: Nakai T, Sakai KL (2014) Neural Mechanisms Underlying the Computation of Hierarchical Tree Structures in Mathematics. PLoS ONE 9(11): e111439.
doi:10.1371/journal.pone.0111439
Editor: Peter Howell, University College London, United Kingdom
Received May 23, 2014; Accepted September 27, 2014; Published November 7, 2014
Copyright: ß 2014 Nakai, Sakai. 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.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its
Supporting Information files.
Funding: This work was supported by a Core Research for Evolutional Science and Technology (CREST) grant from the Japan Science and Technology Agency(JST) (to KLS), by a Grant-in-Aid for Scientific Research (S) (No. 20220005) from the Ministry of Education, Culture, Sports, Science and Technology (to KLS), and by
a Grant-in-Aid for JSPS Fellows (No. 26.9945) from the Japan Society for the Promotion of Science (JSPS) (to TN). The funders had no role in study design, datacollection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors declare that no competing interests exist.
* Email: sakai@mind.c.u-tokyo.ac.jp

structures, ‘‘number of nodes’’ would be a straight-forward model,
simply counting the total number of non-terminal nodes (branching points) and terminal nodes of a tree structure. However, thismodel cannot capture hierarchical levels within the tree (sister
relations in linguistic terms), whereas the DoM plays a critical rolein measuring hierarchical levels of tree structures, such that thesame numbers are assigned to the nodes with an identicalhierarchical level [8]. Therefore, the model of DoM can properly
capture recursiveness in the whole tree structures. Here we apply
the computational concept of DoM to tree structures inmathematical calculations, and we hypothesize that the DoM
actually represents specific loads in the computation of hierarchical tree structures also in mathematics.
Generally speaking, mathematical calculations consist of at least
two components. One component is ‘‘mathematical syntax’’ that
determines how terms and operators are combined together toform mathematical expressions. We hypothesize that mathematical syntax is shared with linguistic syntax in a deeper sense, andthat both syntax can be automatically processed without
consciousness or verbalization. The other component is ‘‘mathe-

Introduction
One of the fundamental properties common to language andmathematics is the critical involvement of tree structures in thosecomprehension and production processes. Indeed, sentences
consist of hierarchical tree structures with recursive branches[1,2], and mathematical calculations can be also expressed byhierarchical tree structures [3], which may derive from the uniqueand universal property of recursive computation in humans [4].
This point provides a good motivation for the theoretical andexperimental approaches advocated in the present study. According to modern linguistics, the construction of any grammaticalphrase or sentence is based on the fundamental linguistic
operation of ‘‘Merge,’’ which combines two syntactic objects toform a larger structure [5]. To properly measure the complexity oftree structures with a formal property of Merge and iterativity(recursiveness) [6], we recently introduced ‘‘the Degree of Merger(DoM),’’ which was defined as the maximum depth of merged
subtrees (i.e., Mergers) within an entire sentence [7]. Among
various models that may possibly quantify the complexity of tree

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Neural Basis of Hierarchical Processes in Mathmatical semantics’’ that deals with the meaning of terms,
operators, and mathematical expressions. There have been anumber of lesion studies and imaging studies, which claimed that
linguistic and arithmetic abilities were separable in the brain [9–16]. However, they examined participants’abilities in elementary
processing of numbers or variables, i.e., mathematical semantics,
but not those in mathematical syntax. On the other hand,common activation between linguistic and arithmetic tasks may
not necessarily mean that the same neural system subserves thesame processes included in both tasks, because even a single region
may have multiple functions. A superficial comparison of activatedregions between linguistic and arithmetic tasks cannot resolve thiscritical issue. Here we propose a direct test, based on our previous
finding that activations in the left inferior frontal gyrus (L. IFG)
and the left supramarginal gyrus (L. SMG) were parametricallymodulated by the DoM in a linguistic task; indeed, the DoM
turned out to be the best factor among the 19 models tested [7]. If
activations in the same regions are exactly modulated by the DoM
alone in an arithmetic task, then the results would indicate
functional commonality between linguistic and arithmetic computation. We thus focus on mathematical syntax, and our goal is toclarify whether the computation of hierarchical tree structures inmathematics and language share the same neural network.
Controversial issues still remain concerning the involvement of
the L. IFG in mathematics. It has been proposed in a lesion study
that the L. IFG is a shared substrate between sentencecomprehension and arithmetic calculations [17], while arithmetic
and algebraic calculations seemed to be preserved in someagrammatic patients [18,19]. However, the knowledge aboutbrackets and operators for determining the order of calculations
does not guarantee effective computation of hierarchical tree
structures in mathematical expressions. Moreover, when thepatients were allowed to explicitly write partial results of individual
calculations for either given or generated brackets, as in the case of
the latter studies, individual calculations could be performed
without actual construction of hierarchical tree structures for thewhole expression. Likewise, overlapping activation in the L. IFG
has been reported between sentence comprehension and arithmetic calculations [20], while other researchers opposed the
involvement of the L. IFG in mathematics [21]. Regarding these
imaging experiments, various factors including memory-related
factors and applications of arithmetic operations (here denoted as‘‘number of operations’’; e.g., +, 6, etc.) may have influenced
cortical activation. Therefore, the effect of the DoM should besegregated from other factors involved.
In the present functional magnetic resonance imaging (fMRI)
study, we prepared a novel arithmetic task, in which participants
performed a series of specified arithmetic calculations without
writing or overtly verbalizing partial results, and they stored two
digits in memory for matching. We presented five digits, and tested
three conditions in the arithmetic task: simple calculation (Simple,
capitalized to indicate a condition name), linear sequence (Linear),and quadratic sequence (Quad) conditions. Under the Simple
condition, the participants were asked to mentally perform theaddition (e.g., 3+7 = 10), as well as the subtraction, of the upper
two digits (Figure 1A). Under the Linear condition, the participants were asked to regard the upper two digits (e.g., 3 and 7) as apart of a linear sequence in increasing order (3, 7, 11, 15, 19,… inthis case), in which the differences between each pair of adjacentterms were constant (Figure 1B). The participants were instructedto calculate the third term of the linear sequence. We employedlinear sequences, because it naturally imposed recursive computation; e.g., 3, 7, 11, 15, 19,…, obtained by nested constructions((((3+4) +4) +4) +4 …). The arithmetic task could not be correctly
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performed without generating such nested constructions tointegrate new terms in a linear sequence. Under the Quadcondition, the participants were asked to regard the lower three
digits (e.g., 2, 4, and 9) as part of a quadratic sequence inincreasing order (2, 4, 9, 17, 28,… in this case), in which thedifferences between each pair of adjacent terms resulted in asubordinate linear sequence (2, 5, 8, 11,… in this case) (Figure 1C).
The participants were instructed to calculate the third term of thesubordinate linear sequence. We theoretically predicted that thisgenerative process imposed effective computation and actual
construction of hierarchical tree structures (Figure 2A), just like thegenerative process of integrating new words in a sentence. As abasic control, we used a digit-matching (Match) task, in which theparticipants simply stored five digits in memory (Figure 1D).
Under each of the three conditions in the arithmetic task, we
examined which of a hierarchical tree structure model or a flat tree
structure model could properly explain the results. A linear
sequence tested under the Linear condition internally combinedarithmetic calculations of addition and subtraction tested under
the Simple condition, whereas a quadratic sequence (e.g., 1, 2, 5,
10, 17,…) tested under the Quad condition internally involved asubordinate linear sequence (e.g., 1, 3, 5, 7,…). In addition, theMatch task had no arithmetic calculation. Based on thehierarchical tree structure model (Figure 2A), the idea behindsuch nested task designs was to linearly increase the DoM in thehierarchical tree structures, i.e., ‘‘1, 2, and 3’’ under the Simple,
Linear, and Quad conditions, respectively (see the row of the DoM
under ‘‘Factors in the hierarchical tree structures’’ in Table 1). On
the other hand, it is possible that the participants covertly
verbalized each individual calculation process with digits andoperations (e.g., ‘‘723 = 4’’). If such individual calculations wererepresented by the flat tree structure model (Figure 2B), the DoMs
became all ‘‘1’’ under the three conditions (see the row of the DoM
under ‘‘Factors in the flat tree structures’’ in Table 1). However,
the relationships among the generated digits became unnecessarily
complex in the flat tree structure model (compare gray arrows
under the Quad condition in Figure 2A and 2B). We predict thatrecursive computation in both linguistic and arithmetic processes
automatically employs hierarchical tree structures; this argument
closely resembles an argument over hierarchical versus linearorder representation of a sentence, e.g., ‘‘[[To be happy] is fun]’’similar to [|327|+7].
We fitted parametric models of eight factors (Table 1) toactivations in each identified region, and determined the most
crucial factor accounting for activations. The operational definitions of these factors other than the DoM are as follows. ‘‘Numberof nodes’’ was equal to the total number of digits that appeared ineach tree structure. For the stimuli used in the Match task, there
were five terminal nodes without nonterminal nodes. The factor of‘‘verbal encoding’’ in the flat tree structures was the total numberof all possible digits and operations verbalized for individual
calculations, which corresponded to the addition of two factors:‘‘number of nodes’’ in the flat tree structures and ‘‘number ofoperations.’’ We also considered three common factors that wereapplicable to both types of tree structures. First, ‘‘number ofoperations’’ was equated between the Simple and Linear
conditions. Secondly, ‘‘number of generated digits for calculation’’was the number of digits generated temporarily in a calculation(i.e., not available on screen). For an example under the Quadcondition shown in Figure 2B, there were four generated digits
used in this calculation, ‘‘2, 5, 3, and 5.’’ Thirdly, ‘‘number ofstored digits for matching’’ was the number of memorized digitsthat were used for digit-matching. Its estimate was two and five for2

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Neural Basis of Hierarchical Processes in MathFigure 1. Examples of the stimuli used in the arithmetic task
and Match task. Five yellow digits, together with a green character asa cue, were presented for 5 s, followed by the presentation of one
white digit (matching stimulus) for 3 s. For each example, the taskrelevant digits and a brief outline of the calculation processes areshown in the right panels, where the digits circled in red are the digits
stored in memory for digit-matching. The participants judged whether
one of those stored digits appeared as a white digit; all white digits
shown here are correct examples. (A) Under the Simple condition,
indicated by the presentation of ‘‘6’’ as a cue, the addition andsubtraction of the upper two digits (i.e., two of the five yellow digits)
were mentally performed by the participants. We instructed theparticipants to always subtract a smaller value from a larger value. (B)Under the Linear condition, indicated by the presentation of ‘‘an’’ as acue, the third term of a linear sequence initiated by the upper two
digits was calculated by the participants. The first row at the top of theright panel represents the linear sequence, and the second row with
branches represents the constant differences between each pair ofadjacent terms in the linear sequence, as often used in math textbooks.
(C) Under the Quad condition, indicated by the presentation of ‘‘bn’’ as acue, the following arithmetic calculations were performed by using thelower three digits (i.e., three of the five yellow digits). The first row inthe right panel represents the given quadratic sequence. The second
row represents the differences between each pair of adjacent terms in
the given sequence. The participants were asked to regard the resultant
second row as a linear sequence, whose third term was then calculated
in the same manner as the Linear condition. (D) In the Match task,
indicated by the presentation of ‘‘m’’ as a cue, the participants stored allof the five digits in memory.
doi:10.1371/journal.pone.0111439.g001

Figure 2. Hierarchical and flat tree structure models forarithmetic calculations. For each example shown in Figure 1, an
entire calculation process is represented by either hierarchical (A) or flat(B) tree structure models, where the lowest and leftmost branchcorresponds to the first arithmetic calculation performed. A digit inblack at each nonterminal node is obtained from an arithmetic
operation (e.g., + or 2) indicated just below each node, where ‘–’
denotes a single operation of subtracting a smaller value from a larger
value. The digits in red denote the DoM at individual nodes, where areference point of zero is at the top node. The digits in blue denote‘‘number of operations.’’ A gray arrow denotes corresponding digits
generated temporarily in a calculation. The stored digits are circled inred, as shown in Figure 1. In the hierarchical tree structure model, eachtree structure is based on recursive computation. Under the Linear
condition, the hierarchical representation of the given example is: [|327| +7] = 11. Under the Quad condition, the hierarchical representationof the given example is: [||224|2 |4–9|| +5] = 8.
doi:10.1371/journal.pone.0111439.g002

null or negative (see Table 1). Moreover, the Linear – Simple
contrast was also free from the following factors: ‘‘number ofnodes’’ in either tree structures, the DoM and ‘‘verbal encoding’’
in the flat tree structures, ‘‘number of operations,’’ and ‘‘numberof stored digits for matching.’’ The Quad – Linear contrast
provides no further information, since no additional factors can becontrolled. To examine the functional specialization of corticalregions in an unbiased manner [22], we adopted whole-brain
analyses in such stringent contrasts as Simple – Match and Linear– Simple.
Recent neuroimaging studies have examined functional connectivity underlying elementary calculations. For example, it has
been reported that the magnitude of functional connectivity
between the bilateral intraparietal sulcus (IPS) correlated with theperformances of a subtraction task [23]. Another fMRI study withthe arithmetic task and the Match task, respectively (the digits
circled in red in Figure 1 and 2).
By taking the Match task as a basic control for the arithmetictask, we eliminated any task-related cognitive factors, such as
visual processing of stimuli, the identification of presented digits,
digit-matching, and motor responses. Both ‘‘number of generated
digits for calculation’’ and ‘‘number of stored digits for matching’’
might contribute to loads of short-term memory or ‘‘working
memory,’’ but the Simple – Match contrast was free from thesememory-related factors, because their subtracted estimates were

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Neural Basis of Hierarchical Processes in Math

Table 1. Estimates of various factors to account for activations.

Factors in the hierarchicaltree structures

Factors in the flat tree
structures

Common factors

Factor

SimpleLinear

Quad

Match

DoM

1

2

3

0

No. of nodes

6

5

9

5

Simple – MatchLinear – Match

Quad – MatchLinear – Simple

DoM

1

2

3

1

No. of nodes

1

0

4

21

Factor

SimpleLinear

Quad

Match

DoM

1

1

1

0

No. of nodes

6

6

12

5

Verbal encoding

8

8

16

5

Simple – MatchLinear – Match

Quad – MatchLinear – Simple

DoM

1

1

1

0

No. of nodes

1

1

7

0

Verbal encoding

3

3

11

0

Factor

SimpleLinear

Quad

Match

No. of operations

2

2

4

0

No. of generated
digits for calculation

0

1

4

0

No. of storeddigits for matching

2

2

2

5

Simple – MatchLinear – Match

Quad – MatchLinear – Simple

No. of operations

2

2

4

0

No. of generated
digits for calculation

0

1

4

1

No. of stored digits
for matching

23

23

23

0

We defined the estimate of a factor as the largest value that the factor can take for each condition: e.g., ‘‘1, 2, 3, and 0’’ for the DoM in the hierarchical tree structures. For
each factor, its unit load should be invariable among all conditions, making an independent subtraction between estimates of the same factor possible. We assumed
that positive and negative values of the subtracted estimates corresponded to activations and deactivations, respectively. Under all tested conditions, ‘‘number ofoperations’’ was equal to ‘‘number of Merge,’’ which was the total number of binary branches in the tree structures (Figure 2). Null or negative subtracted estimates
were denoted in bold.
doi:10.1371/journal.pone.0111439.t001(laterality quotient: 50–100), according to the Edinburgh inventory[27]. Prior to their participation in the study, written informed
consent was obtained from each participant after the nature andpossible consequences of the studies were explained. Approval forthe experiments was obtained from the institutional review boardof the University of Tokyo, Komaba.

Granger causality mapping reported that a participant group withhigh arithmetic scores in a multiplication task had stronger
bidirectional connections between the L. IPS and R. IPS than thegroup with low arithmetic scores [24]. Both of those connectivity
studies with elementary calculation tasks did not involve a
recursive application of arithmetic operations. The effective
connectivity during arithmetic tasks with recursive computation,
as well as the connectivity between the L. IFG and bilateral IPS,
should be thus clarified. In the present study, we adopted dynamic
causal modeling instead of Granger causality mapping, because
dynamic causal modeling has been shown to be more effective for
fMRI data [25,26]. Our findings would elucidate the crucial
network of the L. IFG, L. IPS, and R. IPS for computing
hierarchical tree structures in mathematics.

Stimuli
In each trial (Figure 1), we presented a main stimulus consistingof five yellow digits (from 1 to 9) and a green character (6, an, bn,or m). Each green character was used as a cue to indicate one of
the four conditions: ‘‘6’’ for the Simple condition, ‘‘an’’representing a linear sequence for the Linear condition (e.g.,
an = 1, 3, 5, 7,…), ‘‘bn’’ representing a quadratic sequence for the
Quad condition (e.g., bn = 1, 2, 5, 10,…), and ‘‘m’’ for the Matchtask. According to a pilot study, the duration of 5 s for the main
stimulus was long enough for the participants to correctly performthe task. At the center of the screen, one white digit (from 0 to 9)
was subsequently presented for 3 s as a matching stimulus,
followed by a 200 ms blank to make the duration of the trial twice
as long as the repetition time of the fMRI scans.

Materials and Methods
Participants
Twenty college students (15 males, aged 18–30 years), who had
not majored in mathematics but studied high school mathematics
including linear and quadratic sequences, participated in the fMRI
experiment. All participants were healthy and right-handed
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Neural Basis of Hierarchical Processes in Mathscanning. Eight scanning sessions were performed in one day. A
single scanning session contained 36 trials (six trials for each of the
Simple, Linear, and Quad conditions; 18 trials for the Match task).
The arithmetic and Match tasks were alternately performed, sothat MR signals for each condition in the arithmetic task were
sufficiently separated. In the arithmetic task, no stimuli appeared
twice or more times across all scanning sessions, whereas eachstimulus appeared twice in the Match task. To prevent any
condition-specific strategy, the order of the Simple, Linear, andQuad conditions was pseudorandomized; the orders of tasks andconditions were also counter-balanced across participants.

To make the stimuli physically identical among the three
conditions in the arithmetic task, except for the cue characters, weused the same set of main stimuli, in which the five digits werealways arranged in two rows (48 different combinations of digits).
The upper two digits were relevant to the Simple and Linear
conditions, while the lower three digits were relevant to the Quadcondition. The digits in each row were arranged in increasingorder from left to right, while no digit appeared twice in an entire
stimulus. For the upper two digits, we excluded such ubiquitouscombinations as ‘‘2 and 4,’’ ‘‘3 and 6,’’ or ‘‘4 and 8,’’ as well as
trivial combinations with the constant difference of one (e.g., ‘‘2
and 3’’). For the lower three digits, we excluded certain
combinations (e.g., ‘‘1, 2, and 4’’), in which the subordinate linear
sequence became trivial (1, 2, 3, 4,… in this case). In the Matchtask, all digits in both rows were memorized; we prepared 72
different combinations of digits, including all combinations used in
the arithmetic task.
The participants wore earplugs and an eyeglass-like MRIcompatible display (resolution, 8006600 pixels; VisuaStim Digital,
Resonance Technology Inc., Northridge, CA). For fixation, asmall red cross was always shown at the center of the screen to
initiate eye movements from the same fixed position, and theparticipants were instructed to return their eyes to this position forthe matching stimulus. Reaction times were measured from theonset of the matching stimulus. The stimulus presentation andcollection of behavioral data (accuracy and reaction times) were
controlled using the LabVIEW software and interface (National
Instruments, Austin, TX).

MRI Data Acquisition
The fMRI scans were conducted on a 3.0 T scanner (Signa
HDxt; GE Healthcare, Milwaukee, WI) with a bird-cage head coil.
For the fMRI, we scanned 40 axial slices that were 3-mm thickwith a 0.3-mm gap, covering from 252.8 to 78.9 mm from theanterior to posterior commissure line in the vertical direction,using a gradient-echo echo-planar imaging sequence [repetition
time = 4.1 s, echo time = 60 ms, flip angle = 90u, field of view =1926192 mm2, resolution = 363 mm2]. In a single scanningsession, we obtained 72 volumes following four dummy images,
which allowed for the rise of the MR signals. For each participant,sessions without head movement were used for analyses; thenumber of abandoned sessions was less than four for allparticipants. After completion of the fMRI sessions, highresolution T1-weighted images of the whole brain (192 axial
slices, 16161 mm3) were acquired from all participants with a fast
spoiled gradient recalled acquisition in the steady state sequence(repetition time = 8.4 ms, echo time = 2.6 ms, flip angle = 25u,
field of view = 2566256 mm2).

Tasks
In each trial of the arithmetic task, the participants were asked
to silently perform a series of specified arithmetic calculations, andto store two digits in memory obtained from the arithmetic
calculations without using their fingers. The participants thenjudged whether or not one of the stored digits matched the white
digit, and responded by pressing one of two nonmagnetic buttons(Current Designs, Inc., Philadelphia, PA): a right button ifmatched (in half of the trials), or a left button if mismatched (inthe other half).
Under the Simple condition (Figure 1A), the participants stored
the results of addition and subtraction in memory. When the result
was a ‘‘two-figure number’’ (e.g., 10), the participants simply
stored the last digit (i.e., the digit in ‘‘the one’s place’’ in mathterms, 0 in this case) in memory. We instructed them to always
subtract a smaller value from a larger value (e.g., |327| = 4), as
represented by the sign of absolute value here. Under the Linear
condition (Figure 1B), the participants were instructed to obtainthe constant difference first by the subtraction (e.g., |327| = 4),
and then to calculate the third term by adding the constantdifference to the second term (4+7 = 11 in this case); they stored
the results of these arithmetic calculations (4 and 1 in this case) inmemory. Under the Quad condition (Figure 1C), the participants
were instructed to perform a series of subtractions (e.g., |224|= 2, and then |429| = 5) in order to obtain the constantdifference of the subordinate linear sequence by the subtraction(|225| = 3 in this case). They were further instructed to calculate
the third term of the subordinate linear sequence by adding theconstant difference to the second term (3+5 = 8). They stored theresults of these arithmetic calculations (3 and 8 in this case) inmemory.
In the Match task (Figure 1D), the participants simply stored allof five yellow digits in memory, and judged whether one of thosedigits matched the white digit. The participants underwent
practice sessions for these arithmetic and Match tasks before
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fMRI Data Analyses
We performed data analyses with fMRI using SPM8 statisticalparametric mapping software (Wellcome Trust Centre for
Neuroimaging, London, UK; http://www.fil.ion.ucl.ac.uk/spm/)[28], implemented on a MATLAB platform (MathWorks, Natick,MA). The acquisition timing of each slice was corrected using the
middle slice as a reference for the echo-planar imaging data. We
realigned the echo-planar imaging data in multiple sessions to thefirst volume in all sessions, and removed sessions that included
data with a translation of.2 mm in any of the three directions andwith a rotation of.1.4u around any of the three axes; thesethresholds were empirically determined from our previous studies[29–31].
Each participant’s T1-weighted structural image was coregistered to the mean functional image generated during realignment.
The coregistered structural image was spatially normalized to thestandard brain space as defined by the Montreal Neurological
Institute using the ‘‘unified segmentation’’ algorithm with medium
regularization, which is a generative model that combines tissue
segmentation, bias correction, and spatial normalization in the
inversion of a single unified model [32]. After spatial normalization, the resultant deformation field was applied to the realigned
functional imaging data, which was resampled every 3 mm using
seventh-degree B-spline interpolation. All normalized functional
images were then smoothed by using an isotropic Gaussian kernelof 9 mm full-width at half maximum. Low-frequency noise wasremoved by high-pass filtering at 1/128 Hz.
In a first-level analysis (i.e., a fixed-effects analysis), eachparticipant’s hemodynamic responses induced by the trials weremodeled with a box-car function (band-pass during 1–3 s after themain stimulus onset) convolved with a hemodynamic function.
The 0–1 s period from the main stimulus onset was related with
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Neural Basis of Hierarchical Processes in Maththe identification of a cue, and the 3–5 s period was likely to be
confounded with preparatory processes of digit-matching. We thus
selected the 1–3 s period in order to focus on arithmetic
calculations. Only event-related responses of correct trials wereanalyzed. The images of the three conditions in the arithmetic task
and Match tasks were then generated in a general linear model foreach participant, and used for intersubject comparisons in asecond-level analysis (i.e., random-effects analysis). To discountany general effects associated with task difficulty or performancedifferences among the participants, individual error rates (100 –accuracy), which were more sensitive than reaction times in ourtasks, were entered for each task as a nuisance factor. For all fMRI
data analyses, the statistical threshold was set to P,0.05 for thevoxel level, corrected for multiple comparisons [family-wise error
correction] across the whole brain.
For the anatomical identification of activated regions, webasically used the Anatomical Automatic Labeling method [33]. In
region of interest analyses, we extracted the percent signal changes
averaged among participants at each local maximum using theMarsBaR-toolbox (http://marsbar.sourceforge.net/). To statistically evaluate the fitness of a single factor’s parametric model toactivations, we calculated the coefficient of determination (r2) anda residual sum of squares with MATLAB; we obtained the fitted
values by multiplying the estimates of the factor (see Table 1) by afitting scale. For a no-intercept model, r2 = 12 S(y2ŷ)2/Sy2
should be calculated, where ŷ and y denote the fitted values andthe observed signal changes for each contrast, respectively [34].
For this calculation, we used R software (http://www.r-project.
org/). To further take account of individual variability, we used a
restricted maximum-likelihood method, fitting ‘‘linear mixedeffects models’’ with individual activations as dependent variables,with the estimates of each factor as a regressor, and with theparticipants as random effects. For this calculation, we used annlme (linear and nonlinear mixed-effects models) package (http://cran.r-project.org/web/packages/nlme/) on R software.

Results
Behavioral Results
The behavioral data are shown in Table 2, indicating that thetasks were performed almost perfectly. With respect to accuracy, aone-way repeated measures analysis of variance showed asignificant main effect of condition including the Match task
[F(3, 57) = 20, P,0.0001]. Post-hoc paired t-tests among all
conditions (significance level at a = 0.0083, Bonferroni-corrected)
showed that the accuracy under the Quad condition was
significantly lower than that under the other conditions (P,0.001), while the differences of accuracy among the other three
conditions were not significant (P.0.1). Regarding reaction times(RTs), a main effect of condition was significant [F(3, 57) = 12,
P,0.0001], and the RTs under the Quad condition was
significantly longer than those under the Simple condition (P,0.001). The Match task was also more demanding than the Simple
and Linear conditions, leading to significantly longer RTs (P,0.001).

Selective Activation in the Arithmetic Task
To identify cortical regions involved in the arithmetic task, we
first tested the Linear – Match contrast with the liberal control ofMatch. We then tested the direct Quad – Simple contrast,
inclusively masked with Linear – Match to guarantee theconsistency of activation patterns. In both contrasts, overall
activation was clearly left-dominant. Significant activation wasobserved in the L. IFG (Brodmann’s areas 44/45), bilateral SMG(Brodmann’s area 40), bilateral IPS (Brodmann’s areas 7/39/40),
and precuneus (Brodmann’s area 7), as well as in the bilaterallateral premotor cortex (LPMC, Brodmann’s areas 6/8), left
anterior short insular gyrus (ASG), and pre-supplementary motor
area (pre-SMA, Brodmann’s areas 6/8) (Figure 3A, 3B, andTable 3). The left middle temporal gyrus (L. MTG, Brodmann’sarea 21) was significantly activated only in Linear – Match.
We tried to narrow down the critical regions for thecomputation of hierarchical tree structures, by using more
stringent contrasts. In Simple – Match, where the memory-related
factors of ‘‘number of generated digits for calculation’’ and‘‘number of stored digits for matching’’ were eliminated,significant activation was localized in the L. IFG and L. SMG(Figure 3C and Table 3), indicating that activations in theseregions were free from the memory-related factors. Moreover, in
Linear – Simple, significant activation was observed in thebilateral IPS and precuneus (Figure 3D and Table 3), indicatingthat activations in these regions were independent from otherpossible factors. Although activations in the L. IFG and L. SMG
were below the threshold in Linear – Simple, they were significant
with small volume correction (corrected P,0.05, 9 mm radiusfrom each local maximum determined by Simple – Match). In
Simple – Match with small volume correction, activations in the L.
IPS, but not those in the R. IPS or precuneus, were significant(9 mm radius from each local maximum determined by Linear –Simple). These results suggest that the L. IFG, L. SMG, and L.
IPS, together with the limited contribution of the R. IPS andprecuneus, are well specialized in arithmetic calculations.

Dynamic Causal Modeling Data Analyses
We performed data analyses with dynamic causal modelingusing DCM10 on SPM8 [35]. We concatenated the scans from theseparate sessions, and reanalyzed the preprocessed data with thegeneral linear model, which contained two regressors representing
the Quad condition and Match task for making a meaningful
contrast, as well as a regressor representing both the Quad andLinear conditions for driving inputs. The regressor representing
the Quad condition was also used for a modulatory effect. The
effects of transition between sessions were taken into account withregressors of sessions. With the volume-of-interest tool in SPM8,
the time series was extracted by taking the first eigenvariate acrossall suprathreshold voxels in Quad – Match for each participant(uncorrected P,0.05), confined within 6 mm from the group local
maximum of a single region.
We specified 18 models with systematic variations in amodulatory effect and driving inputs (Figure S1). After estimating
all models for each participant, we identified the most likely modelby using random-effects Bayesian model selection on DCM10.
Inferences from Bayesian model selection can be based on the
expected probability, i.e., the expected likelihood of obtaining themodel for any randomly selected participants, or on theexceedance probability, i.e., the probability that the model is abetter fit to the data than any other models tested. After
determining the best model, we evaluated the parameter estimatesof this particular model by one-sample t-tests [36].

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Cortical Activations Specifically Modulated by the DoM in
the Hierarchical Tree Structures
At the local maxima of these five regions, we further examined
percent signal changes under each condition in the arithmetic taskwith reference to the Match task (Figure 4). A linear modulation ofactivations was observed in the L. IFG, L. SMG, and bilateral IPS
among these three conditions, indicating that the results were6

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Neural Basis of Hierarchical Processes in Math

Table 2. Behavioral data under each condition.

SimpleLinear

Quad

Match

Accuracy (%)

97.662.7

98.562.9

92.665.0

98.861.4

RTs (ms)

8386151

8506150

8956174

9316172

Behavioral data (mean 6 standard deviation) of the accuracy and reaction times are shown for each condition. Only correct trials were included for reaction times,
which were measured from the onset of the matching stimulus.
doi:10.1371/journal.pone.0111439.t002

estimates multiplied by the fitting scale) against corresponding
signal changes averaged across participants (Table 5, Tables S1S4 in File S1).
Among the parametric models of eight factors tested, the modelof DoM in the hierarchical tree structures produced by far thelowest residual sum of squares value (#0.0043) and the largest
coefficient of determination (r2) ($0.99) for the L. IFG, L. SMG,
and bilateral IPS. We further evaluated the goodness of fit for eachmodel by using one-sample t-tests (significance level at a = 0.0167,
Bonferroni-corrected) between the fitted value for each contrast
and individual activations. The model of DoM for these four
regions produced no significant deviation for the three contrasts(P$0.13). In the precuneus, the goodness of fit was marginal for
one P-value (P = 0.027) under the Simple condition. By fitting‘‘linear mixed-effects models,’’ we found that the model of DoM

consistent with the DoM in the hierarchical tree structures. The
precuneus showed weaker activations under the Simple condition,
exhibiting relatively larger responses under the more demanding
Linear and Quad conditions (Figure 4E).
Next we examined how well activations in each of the L. IFG,
L. SMG, bilateral IPS, and precuneus correlated with the DoM in
the hierarchical tree structures and other factors. All of the Simple– Match, Linear – Match, and Quad – Match contrasts predictedthat activations should be exactly zero when a factor produced noeffect or load relative to the Match task. We thus adopted a nointercept model, in which the percent signal changes of eachregion were fitted with a single (thus minimal) scale parameter to amodel of each factor using its subtracted estimates (Table 4). For
the three contrasts, a least-squares method was used to minimize
the residual sum of squares for the three fitted values (i.e., the three

Figure 3. Significantly activated regions in the arithmetic task. Cortical activation maps are shown for the contrasts of Linear – Match (A),
Quad – Simple, masked with Linear – Match (B), Simple – Match (C), and Linear – Simple (D). Activation is projected onto the left (L) and right lateralsurfaces of a standard brain (family-wise error corrected P,0.05), as well as onto the dorsal surface where there was significant activation. A
transverse plane at z = 1 shows activation in the left anterior short insular gyrus. See Tables 3 for the stereotactic coordinates.
doi:10.1371/journal.pone.0111439.g003

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PLOS ONE | www.plosone.org
7/39/4040
7/39/40
7
21

IPS
SMG
IPS
PrecuneusMTG

8
7/39/407Precuneus

40

SMG
IPS

44/45

IFG

7

Precuneus

M

23

230
39

R

248

270

255

270

237

11

270

9
245

261

261

33
212

252

39

231

270

227
45

249

233

237

11

0
248

20
26

23

43

43

4.7

4.5

4.6

4.8

46
46

4.6

6.2

6.3

4.8

5.4

25

49

52

34

43

5.9

5.5

43
43

5.8

5.4

6.1

6.4

6.1

5.4

5.9

5.6

5.4

5.2

6.1

6.0

4.5

6.8

6.9

6.0

4.6

5.2

6.8

4.5

5.5

Z-Value

40

43

55

43

1

25

230

37
11

58

55

211

46

43

46

40

5

L

L

L

M

R

R

L

L

M

L

46
43

245

5

5

255

270

255

231

252

240

20

1

245

30

R
L

224

251

26

36

48

233

245

26

26

22

32

251
230

25

58

55

z

11

5

5

y

245

L

L

M

R

R

L

L

M

L

L

227
33

R

x

L

Side

28

4

3

12

5

*

155

*

110

5

*

*

352

*

51

5

*

131

5

62

15

155

169

5

*

483

53

5

*

309

5

82

VoxelsStereotactic coordinates (x, y, z) in the Montreal Neurological Institute space (mm) are shown for each activation peak of Z-values (corrected P,0.05). LPMC, lateral premotor cortex; IFG, inferior frontal gyrus; ASG, anterior shortinsular gyrus; pre-SMA, pre-supplementary motor area; SMG, supramarginal gyrus; IPS, intraparietal sulcus; MTG, middle temporal gyrus; BA, Brodmann’s area; L, left hemisphere; R, right hemisphere; M, medial. The region with anasterisk is included within the same cluster shown one row above.
doi:10.1371/journal.pone.0111439.t003

Linear – Simple

Simple – Match

7/39/40

7/39/40

IPS

IPS

40

SMG

40

6/8

pre-SMA

SMG

–

ASG

44/45

40

SMG

IFG

6/8

pre-SMA

6/8

–

ASG

LPMC

44/45

IFG

Quad – Simple, masked with Linear – Match

6/8

LPMC

Linear – MatchBA

Brain region

Contrast

Table 3. Significantly activated regions in the arithmetic task.

Neural Basis of Hierarchical Processes in Math

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Neural Basis of Hierarchical Processes in MathFigure 4. Quantified activations modulated by the DoM in thehierarchical tree structures. The percent signal changes in the L. IFG(A) and L. SMG (B) were taken from the local maxima in Simple – Match(circled in yellow in Figure 3C), whereas those in the L. IPS (C), R. IPS (D),and precuneus (E) were taken from the local maxima in Linear – Simple(circled in yellow in Figure 3D). Activations are shown for the Simple,
Linear, and Quad conditions with reference to the Match task. Error bars
indicate the standard error of the mean for the participants (N = 20).
Overlaid red dots and lines denote the values fitted with the estimates(digits in red) for the DoM in the hierarchical tree structures (seeTable 4).
doi:10.1371/journal.pone.0111439.g004

was by far most likely to explain the modulation of activations for
all five regions (Table 5, Tables S1–S4 in File S1). Evidently, all of
the other factors were clearly less effective than the DoM in thehierarchical tree structures.

Effective Connectivity among the L. IFG and Bilateral IPS
Because the L. SMG is relatively close to the L. IPS, and theprecuneus locates midway between the bilateral IPS, we focusedon the L. IFG and bilateral IPS alone for modeling the effective
connectivity in the dynamic causal modeling analyses. Here weassumed intrinsic, i.e., task-independent, bidirectional connections
between the L. IFG and L. IPS, as well as between the bilateral
IPS. We systematically constructed 18 models with driving inputs
into one of these three regions, such that for each input type wetested six models with a modulatory effect under the Quadcondition on the unidirectional or bidirectional connections (seeFigure S1 for all models tested). Using a random-effects Bayesianmodel selection, we showed that the model 1 (Figure 5A), with amodulatory effect for the connection from the L. IPS to the L.
IFG, and with driving inputs into the L. IFG, yielded by far thehighest expected probability and exceedance probability(Figure 5B and 5C).
For this best model, we further tested whether the parameter
estimates were significantly different from zero among theparticipants. The intrinsic connection from the L. IFG to the L.
IPS [1.02, t(19) = 8.9, P,0.001], and that from the L. IPS to the
R. IPS were significantly positive [0.62, t(19) = 7.0, P,0.001].
In contrast, the intrinsic connection from the L. IPS to the L. IFG[20.82, t(19) = 5.4, P,0.0001], and that from the R. IPS to the L.
IPS were significantly negative [20.55, t(19) = 3.4, P,0.005](significance level at a = 0.0125; Bonferroni-corrected within aparameter class of intrinsic connections). The positive modulatory
effect for the connection from the L. IPS to the L. IFG [1.03,
t(19) = 6.2, P,0.001], as well as the driving inputs into the L. IFG[0.71, t(19) = 6.0, P,0.001], were also significant.

Discussion
Here we introduced the DoM to the hierarchical tree structuresin mathematics, and we obtained three striking results. First, wefound significant activation in the L. IFG, L. SMG, bilateral IPS,
and precuneus selectively among the three conditions in thearithmetic task (Figure 3). Secondly, by examining percent signal
changes in each region, a linear modulation of activations wasobserved in the L. IFG, L. SMG, and bilateral IPS among thesethree conditions (Figure 4). Moreover, by fitting the parametricmodels of eight factors, we found that the DoM in the hierarchicaltree structures best explained the modulation of activations in the
L. IFG, L. SMG, bilateral IPS, and precuneus (Table 5, Tables
S1–S4 in File S1). These results indicate the existence ofmathematical syntax processed in these regions, excluding theload on ‘‘working memory.’’ The dominance of the hierarchical
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Neural Basis of Hierarchical Processes in Math

Table 4. The results of fitted scale and values for each activated region, by using ‘‘DoM in the hierarchical tree structures’’.

Brain region

Fitted scale

Fitted values

L. IFG

0.21

0.21, 0.41, 0.62

L. SMG

0.21

0.21, 0.42, 0.63

L. IPS

0.17

0.17, 0.34, 0.51

R. IPS

0.13

0.13, 0.26, 0.38

Precuneus

0.22

0.22, 0.45, 0.67

We obtained fitted values by multiplying the estimates of ‘‘1, 2, and 3’’ (see Table 1) by the fitting scale. The three fitted values correspond to activations observed
under the Simple, Linear, and Quad conditions (see Figure 4).
doi:10.1371/journal.pone.0111439.t004

a matter of controversy whether the L. IFG is also criticallyinvolved in mathematics. It was claimed in a previous study that‘‘it [the L. IFG] does not appear to play a dominant role [inmathematics], which instead is taken up by fusiform, parietal andprecentral cortices,’’ i.e., visuo-spatial areas [21]. In this previous
study, the participants performed a short-term matching task,
which was solved without requiring any arithmetic calculation. In
contrast, Makuuchi et al. [20] showed overlapped activation acrosssentence comprehension and arithmetic calculations, with anincrease of activations in the L. IFG for the higher level ofstructural hierarchy, although the modulation of activations in thebilateral IPS were not fully examined. The present results clearly
showed that the DoM in the hierarchical tree structures was by far
more effective than the load on ‘‘working memory,’’ ‘‘number ofoperations,’’ and other factors for explaining the L. IFG andbilateral IPS activations.
According to some previous imaging studies, the domainspecificity for the arithmetic calculation in the L. IFG and otherregions has remained unclear. For example, in recent fMRI studies[43,44], two conditions with large and small numbers werecompared in an addition task, and such domain-general factors astree structure model is consistent with our previous results of the L.
IFG and L. SMG activation in the direct comparison between
sentences and letter strings, which were assumed to havehierarchical tree structures and flat tree structures, respectively[7]. Thirdly, using dynamic causal modeling, we showed that themodel with a modulatory effect for the connection from the L. IPS
to the L. IFG, and with driving inputs into the L. IFG, was highly
probable (Figure 5). For this best model, the top-down intrinsic
connection from the L. IFG to the L. IPS, as well as that from the
L. IPS to the R. IPS, would provide a feedforward signal with theirreverse connections representing a negative feedback signal. These
results indicate that the network of the L. IFG and bilateral IPS
subserves the computation of hierarchical tree structures inmathematics.
Previous imaging studies have established that syntactic
processes during sentence comprehension selectively activate the
L. IFG and/or the L. LPMC [29,37–41]. By directly contrasting ademanding condition for ‘‘working memory,’’ we have previously
demonstrated that both regions are indeed independent from suchdomain-general cognitive factors [29], indicating that these regions
have a critical role as a putative grammar center [42]. It has been

Table 5. Fittings and likelihood of various models tested in the L. IFG.

Factors in the hierarchicaltree structures

Factors in the flat tree
structures

Common factors

FactorRSS

r2

P-values

Log-likelihood

Likelihood ratio*DoM

0.0024. 0.990.34, 0.75, 0.97

15.0

1

No. of nodes

0.18

0.70

, 0.0001, 0.054, 0.65

–20.5

3.9 6 10–16

FactorRSS

r2

P-values

Log-likelihood

Likelihood ratio

DoM

0.062

0.90

0.0016, 0.0025, 0.85

–2.5

2.6 6 10–8

No. of nodes

0.13

0.78

, 0.0001, 0.0033, 0.20

–15.8

4.46 10–14

Verbal encoding

0.063

0.89

, 0.0001, 0.14, 0.17

–6.7

3.7 6 10–10

FactorRSS

r2

P-values

Log-likelihood

Likelihood ratio

No. of operations

9.6

0.015

0.98

0.033, 0.21, 0.68

No. of generated digits for 0.13
calculation

0.78, 0.0001, , 0.0001, 0.24 –15.0

9.1 6 10–14

No. of stored digits formatching

0, 0.0001, , 0.0001, ,0.0001

n/a

0.60

n/a

0.0044

Percent signal changes in the L. IFG were fitted with a single scale parameter to a model of each factor using its subtracted estimates (Table 1) for Simple – Match,Linear – Match, and Quad – Match. The P-values for the t-tests between the fitted value for each contrast and individual activations are shown in ascending order. Notethat the model of DoM in the hierarchical tree structures (with an asterisk) resulted in the best fit for this region, i.e., with the least residual sum of squares (RSS), largest
coefficient of determination (r2), and larger P-values. The likelihood of models with all null estimates was not calculable (n/a). A likelihood ratio is the ratio of eachmodel’s likelihood to the best model’s likelihood. The best model of DoM in the hierarchical tree structures was by far more likely than the other models.
doi:10.1371/journal.pone.0111439.t005

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Neural Basis of Hierarchical Processes in Mathnaturally extended to ‘‘musical syntax’’ as well, since harmonicprogressions are expressed by hierarchical tree structures [45].
Lesion studies have previously reported that the damage to the
L. IPS or R. IPS caused deficits in elementary processes ofnumbers [46,47]. For example, a lesion in the R. IPS was
associated with deficits in performing even simple subtraction with
one-figure numbers [47]. Moreover, activation in the bilateral IPS
has been frequently observed in fMRI studies on numerosities,
digits, and elementary calculations [48–50]. Unconscious repetition priming of numbers (e.g., from ‘‘NINE’’ to ‘‘9’’) caused theactivation suppression in the bilateral IPS [51]. The repetition
suppression and recovery for the deviant number (e.g., ‘‘50’’ versus
repeated numbers around ‘‘18’’) in these regions were also
independent from notation changes (e.g., dots to digits or digits
to dots) [52]. A recent fMRI study has reported that high school
arithmetic scores correlated negatively with activations in the R.
IPS during an elementary calculation task, while a positive
correlation was observed in the L. SMG [53]. The L. IPS andadjacent L. SMG are also involved in language, especially invocabulary knowledge or lexical processing [54,55], as well as insearching syntactic features [7]. Taken these and present results
together, the L. IPS/SMG would be also involved in bothmathematics and language. As regards the precuneus, its
activation has been reported in the previous fMRI studies on
number comparisons or arithmetic calculations [56,57]. Our
results suggest that the R. IPS and precuneus support the L. IPS/
SMG under such demanding conditions as the Linear and Quadconditions.
Using a visual picture-sentence matching task, we have recently
tested twenty-one patients with a left frontal glioma, and foundabnormal overactivity and/or underactivity in 14 syntax-related
regions [58]. By examining the functional and anatomical
connectivity among those regions, we have clarified three
syntax-related networks. The network I (syntax and its supportive
system) consists of the opercular/triangular parts of the L. IFG, L.
IPS, right lateral frontal regions, pre-SMA, and right temporal
regions, which were overactivated in the patients with a glioma in
the L. LPMC. The network II (syntax and input/output interface)consists of the L. LPMC, left angular gyrus, lingual gyrus, andcerebellar nuclei, which were overactivated in the patients with aglioma in the opercular/triangular parts of the L. IFG. The
network III (syntax and semantics) consists of the left ventral
frontal and posterior temporal regions, which were underactivatedin the patients with a glioma in the opercular/triangular parts of
the L. IFG. Among the activated regions in the present study(Table 3), the L. IFG, L. IPS, R. LPMC, R. IFG, and pre-SMA
are included in the network I, whereas the L. LPMC and L. MTG
are included in the network II and network III, respectively. The
overall activated regions for the arithmetic calculations thus share
their functional roles with the syntax-related regions in language.
Our previous fMRI study revealed that the functional
connectivity between the L. IFG and L. SMG was selectively
enhanced during sentence processing [59]. A recent dynamic
causal modeling study with a cross-modal picture-sentence
matching task has suggested that the L. IFG received driving
inputs and transferred that information to the temporal regions[60]. Our recent dynamic causal modeling study has suggested a
top-down intrinsic information flow of syntactic processing fromthe L. IFG to the L. SMG, with driving inputs into the L. IFG [7].
This model is consistent with our present results of dynamic causal
modeling, which further indicate that L. IPS activations mirrored
a top-down influence regarding the DoM in the hierarchical tree
structures computed in the L. IFG. For the bottom-up connection
from the L. IPS/SMG to the L. IFG, the modulatory effect under

Figure 5. Effective connectivity among the L. IFG and bilateral
IPS. (A) The best model with a positive modulatory effect for thebottom-up connection from the L. IPS to the L. IFG, and with driving
inputs into the L. IFG. Mean parameter estimates that exceeded thestatistical threshold (corrected P,0.05) are indicated alongside theintrinsic connections. Bar graphs show expected probabilities (B) andexceedance probabilities (C) of all models tested (Figure S1).
doi:10.1371/journal.pone.0111439.g005

task difficulty might explain the enhanced activations. It should be
noted that any hierarchical processes associated with ‘‘two-figure
numbers,’’ as well as more verbal encoding, are also involved in
the task with large numbers. In the present study, we discounted
any general effects associated with task difficulty by entering
individual error rates for each task as a nuisance factor. We havepreviously demonstrated that the L. IFG is a domain-specificneural system for syntactic computation in language, which isseparable from other domain-general cognitive systems [42]. In
the present study, we indicate that the hierarchical tree structuresin mathematics are also computed by the same domain-specificsystem. Our successful approach on mathematical syntax can be

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Neural Basis of Hierarchical Processes in Maththe conditions with the largest DoM was negative in this previous
study, whereas the modulatory effect under the Quad condition(with the largest DoM) was positive in the present study. While
lexical feedback was minimum for processing jabberwocky
sentences in the previous study, a positive feedback about
operations would be utilized for constructing hierarchical tree
structures in the present paradigm.
The present results of dynamic causal modeling suggest that thesyntactic information on hierarchical tree structures provided in
the L. IFG would be further processed through the top-down
intrinsic connection from the L. IPS to the R. IPS (Figure 5). The
L. IPS and R. IPS may have different roles in processing
arithmetic calculations, but their individual roles in mathematicalsyntax should be clarified in the future studies. In addition, it ispossible that the L. MTG, significantly activated in Linear –Match, is involved in mathematical semantics, as this region would
subserve semantics in language. We indicate that mathematics andlanguage share the network of the L. IFG and L. IPS/SMG for thecomputation of hierarchical tree structures, and that mathematics
recruits the additional network of the L. IPS and R. IPS, with aninformation flow from the former to the latter.

bidirectional connections between the L. IFG and L. IPS, as well
as between the L. IPS and R. IPS. Eighteen models weresystematically constructed with driving inputs into one of the three
regions. For each input type, we tested six models for themodulatory effect under the Quad condition.
(TIF)
File S1 Table S1: Fittings and likelihood of various models tested
in the L. SMG. Table S2: Fittings and likelihood of various models
tested in the L. IPS. Table S3: Fittings and likelihood of various
models tested in the R. IPS. Table S4: Fittings and likelihood ofvarious models tested in the precuneus.
(DOC)

Acknowledgments
We thank S. Ohta and H. Miyashita for their helpful discussions, N.
Komoro for her technical assistance, and H. Matsuda for her administrative assistance.

Author Contributions
Conceived and designed the experiments: TN KLS. Performed theexperiments: TN KLS. Analyzed the data: TN KLS. Wrote the paper: TN
KLS.

Supporting Information
Figure S1 Models tested in the dynamic causal modeling analyses. We assumed intrinsic, i.e., task-independent,

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Neural Mechanisms Underlying the Computationof Hierarchical Tree Structures in Mathematics
Tomoya Nakai, Kuniyoshi L. Sakai
Figure S1.

Table S1. Fittings and likelihood of various models tested in the L. SMG.

Factors in thehierarchicaltree structures

Factors in theflat tree
structures

Commonfactors

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

*DoM

0.00430.99

0.20, 0.69, 0.92

15.1

1

No. of nodes

0.17

0.72

< 0.0001, 0.028, 0.68

–23.3

2.2 × 10–17

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

DoM

0.064

0.90

0.0025, 0.0078, 0.54

–6.0

6.9 × 10–10

No. of nodes

0.13

0.80

< 0.0001, 0.0013, 0.31

–19.3

1.2 × 10–15

Verbal encoding

0.056

0.91

0.0006, 0.097, 0.24

–8.7

5.0 × 10–11

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

No. of operations

0.00900.98

0.13, 0.31, 0.81

12.3

0.061

No. of generated
digits for calculation

0.13

0.79

< 0.0001, 0.0002, 0.39

–19.3

1.2 × 10–15

No. of stored digits
for matching

0.63

0

< 0.0001, < 0.0001,< 0.0001n/an/a

In the L. SMG, the model of DoM in the hierarchical tree structures was by far more likely than the other models.

2

Table S2. Fittings and likelihood of various models tested in the L. IPS.

Factors in thehierarchicaltree structures

Factors in theflat tree
structures

Commonfactors

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

*DoM

0.00320.99

0.36, 0.57, 0.92

7.1

1

No. of nodes

0.14

0.66

< 0.0001, 0.98, > 0.99

–23.4

6.1 × 10–14

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

DoM

0.075

0.82

0.0003, 0.027, 0.50

–11.7

6.9 × 10–9

No. of nodes

0.092

0.78

< 0.0001, 0.35, 0.50

–18.8

5.7 × 10–12

Verbal encoding

0.053

0.87

0.0011, 0.46, 0.58

–12.2

4.1 × 10–9

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

No. of operations

0.031

0.92

0.0043, 0.082, 0.21

–3.2

3.5 × 10–5

No. of generated
digits for calculation

0.074

0.82

0.0007, 0.017, 0.42

–16.2

8.0 × 10–11

No. of stored digits
for matching

0.41

0

< 0.0001, < 0.0001,
0.017

n/a

n/a

In the L. IPS, the model of DoM in the hierarchical tree structures was by far more likely than the other models.

3

Table S3. Fittings and likelihood of various models tested in the R. IPS.

Factors in thehierarchicaltree structures

Factors in theflat tree
structures

Commonfactors

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

*DoM

0.00290.99

0.14, 0.67, 0.93

25.1

1

No. of nodes

0.075

0.67

< 0.0001, 0.58, 0.93

–3.4

4.0 × 10–13

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

DoM

0.049

0.79

< 0.0001, 0.098, 0.52

3.9

6.0 × 10–10

No. of nodes

0.047

0.80

< 0.0001, 0.50, 0.61

1.5

5.5 × 10–11

Verbal encoding

0.028

0.88

0.0014, 0.26, 0.51

7.2

1.7 × 10–8

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

No. of operations

0.020

0.91

0.0031, 0.057, 0.81

13.1

5.7 × 10–6

No. of generated
digits for calculation

0.035

0.85

0.0010, 0.029, 0.41

5.1

2.1 × 10–9

No. of stored digits
for matching

0.23

0

< 0.0001, < 0.0001,
0.029

n/a

n/a

In the R. IPS, the model of DoM in the hierarchical tree structures was by far more likely than the other models.

4

Table S4. Fittings and likelihood of various models tested in the precuneus.

Factors in thehierarchicaltree structures

Factors in theflat tree
structures

Commonfactors

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

*DoM

0.014

0.98

0.027, 0.69, 0.93

–5.90

1

No. of nodes

0.21

0.71

< 0.0001, 0.20, 0.86

–31.2

9.9 × 10–12

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

DoM

0.18

0.75

< 0.0001, 0.0038, 0.60

–28.1

2.2 × 10–10

No. of nodes

0.12

0.83

< 0.0001, 0.57, 0.96

–24.5

8.3 × 10–9

Verbal encoding

0.072

0.90

0.0003, 0.063, 0.63

–18.5

3.4 × 10–6

FactorRSS

r2

P-values

Log-likelihood

Likelihood
ratio

No. of operations

0.066

0.91

0.0002, 0.040, 0.58

–16.1

3.8 × 10–5

No. of generated
digits for calculation

0.084

0.88

0.0002, 0.031, 0.46

–20.1

1.9 × 10–9

No. of stored digits
for matching

0.72

0

< 0.0001, < 0.0001,
0.031

n/a

n/a

In the precuneus, the model of DoM in the hierarchical tree structures was by far more likely than the other models.

5