Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Neural Mechanisms Underlying the Computation of Hierarchical Tree Structures in Mathematics

Neural Mechanisms Underlying the Computation of Hierarchical Tree Structures in Mathematics 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 task that involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, we found significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among the tested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, and that 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 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. 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 and language share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and that mathematics 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:  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, data collection 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, Introduction simply counting the total number of non-terminal nodes (branch- One of the fundamental properties common to language and ing points) and terminal nodes of a tree structure. However, this mathematics is the critical involvement of tree structures in those model cannot capture hierarchical levels within the tree (sister comprehension and production processes. Indeed, sentences relations in linguistic terms), whereas the DoM plays a critical role consist of hierarchical tree structures with recursive branches in measuring hierarchical levels of tree structures, such that the [1,2], and mathematical calculations can be also expressed by same numbers are assigned to the nodes with an identical hierarchical tree structures [3], which may derive from the unique hierarchical level [8]. Therefore, the model of DoM can properly and universal property of recursive computation in humans [4]. capture recursiveness in the whole tree structures. Here we apply This point provides a good motivation for the theoretical and the computational concept of DoM to tree structures in experimental approaches advocated in the present study. Accord- mathematical calculations, and we hypothesize that the DoM ing to modern linguistics, the construction of any grammatical actually represents specific loads in the computation of hierarchi- phrase or sentence is based on the fundamental linguistic cal tree structures also in mathematics. operation of ‘‘Merge,’’ which combines two syntactic objects to Generally speaking, mathematical calculations consist of at least form a larger structure [5]. To properly measure the complexity of two components. One component is ‘‘mathematical syntax’’ that tree structures with a formal property of Merge and iterativity determines how terms and operators are combined together to (recursiveness) [6], we recently introduced ‘‘the Degree of Merger form mathematical expressions. We hypothesize that mathemat- (DoM),’’ which was defined as the maximum depth of merged ical syntax is shared with linguistic syntax in a deeper sense, and subtrees (i.e., Mergers) within an entire sentence [7]. Among that both syntax can be automatically processed without various models that may possibly quantify the complexity of tree consciousness or verbalization. The other component is ‘‘mathe- PLOS ONE | www.plosone.org 1 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math matical semantics’’ that deals with the meaning of terms, performed without generating such nested constructions to operators, and mathematical expressions. There have been a integrate new terms in a linear sequence. Under the Quad number of lesion studies and imaging studies, which claimed that condition, the participants were asked to regard the lower three linguistic and arithmetic abilities were separable in the brain [9– digits (e.g., 2, 4, and 9) as part of a quadratic sequence in 16]. However, they examined participants’abilities in elementary increasing order (2, 4, 9, 17, 28,… in this case), in which the processing of numbers or variables, i.e., mathematical semantics, differences between each pair of adjacent terms resulted in a but not those in mathematical syntax. On the other hand, subordinate linear sequence (2, 5, 8, 11,… in this case) (Figure 1C). common activation between linguistic and arithmetic tasks may The participants were instructed to calculate the third term of the not necessarily mean that the same neural system subserves the subordinate linear sequence. We theoretically predicted that this same processes included in both tasks, because even a single region generative process imposed effective computation and actual may have multiple functions. A superficial comparison of activated construction of hierarchical tree structures (Figure 2A), just like the regions between linguistic and arithmetic tasks cannot resolve this generative process of integrating new words in a sentence. As a critical issue. Here we propose a direct test, based on our previous basic control, we used a digit-matching (Match) task, in which the finding that activations in the left inferior frontal gyrus (L. IFG) participants simply stored five digits in memory (Figure 1D). and the left supramarginal gyrus (L. SMG) were parametrically Under each of the three conditions in the arithmetic task, we modulated by the DoM in a linguistic task; indeed, the DoM examined which of a hierarchical tree structure model or a flat tree turned out to be the best factor among the 19 models tested [7]. If structure model could properly explain the results. A linear activations in the same regions are exactly modulated by the DoM sequence tested under the Linear condition internally combined alone in an arithmetic task, then the results would indicate arithmetic calculations of addition and subtraction tested under functional commonality between linguistic and arithmetic com- the Simple condition, whereas a quadratic sequence (e.g., 1, 2, 5, putation. We thus focus on mathematical syntax, and our goal is to 10, 17,…) tested under the Quad condition internally involved a clarify whether the computation of hierarchical tree structures in subordinate linear sequence (e.g., 1, 3, 5, 7,…). In addition, the mathematics and language share the same neural network. Match task had no arithmetic calculation. Based on the Controversial issues still remain concerning the involvement of hierarchical tree structure model (Figure 2A), the idea behind the L. IFG in mathematics. It has been proposed in a lesion study such nested task designs was to linearly increase the DoM in the that the L. IFG is a shared substrate between sentence hierarchical tree structures, i.e., ‘‘1, 2, and 3’’ under the Simple, comprehension and arithmetic calculations [17], while arithmetic Linear, and Quad conditions, respectively (see the row of the DoM and algebraic calculations seemed to be preserved in some under ‘‘Factors in the hierarchical tree structures’’ in Table 1). On agrammatic patients [18,19]. However, the knowledge about the other hand, it is possible that the participants covertly brackets and operators for determining the order of calculations verbalized each individual calculation process with digits and does not guarantee effective computation of hierarchical tree operations (e.g., ‘‘723 = 4’’). If such individual calculations were structures in mathematical expressions. Moreover, when the represented by the flat tree structure model (Figure 2B), the DoMs patients were allowed to explicitly write partial results of individual became all ‘‘1’’ under the three conditions (see the row of the DoM calculations for either given or generated brackets, as in the case of under ‘‘Factors in the flat tree structures’’ in Table 1). However, the latter studies, individual calculations could be performed the relationships among the generated digits became unnecessarily without actual construction of hierarchical tree structures for the complex in the flat tree structure model (compare gray arrows whole expression. Likewise, overlapping activation in the L. IFG under the Quad condition in Figure 2A and 2B). We predict that has been reported between sentence comprehension and arithme- recursive computation in both linguistic and arithmetic processes tic calculations [20], while other researchers opposed the automatically employs hierarchical tree structures; this argument involvement of the L. IFG in mathematics [21]. Regarding these closely resembles an argument over hierarchical versus linear- imaging experiments, various factors including memory-related order representation of a sentence, e.g., ‘‘[[To be happy] is fun]’’ factors and applications of arithmetic operations (here denoted as similar to [|327|+7]. ‘‘number of operations’’; e.g., +, 6, etc.) may have influenced We fitted parametric models of eight factors (Table 1) to cortical activation. Therefore, the effect of the DoM should be activations in each identified region, and determined the most segregated from other factors involved. crucial factor accounting for activations. The operational defini- In the present functional magnetic resonance imaging (fMRI) tions of these factors other than the DoM are as follows. ‘‘Number study, we prepared a novel arithmetic task, in which participants of nodes’’ was equal to the total number of digits that appeared in performed a series of specified arithmetic calculations without each tree structure. For the stimuli used in the Match task, there writing or overtly verbalizing partial results, and they stored two were five terminal nodes without nonterminal nodes. The factor of digits in memory for matching. We presented five digits, and tested ‘‘verbal encoding’’ in the flat tree structures was the total number three conditions in the arithmetic task: simple calculation (Simple, of all possible digits and operations verbalized for individual capitalized to indicate a condition name), linear sequence (Linear), calculations, which corresponded to the addition of two factors: and quadratic sequence (Quad) conditions. Under the Simple ‘‘number of nodes’’ in the flat tree structures and ‘‘number of condition, the participants were asked to mentally perform the operations.’’ We also considered three common factors that were addition (e.g., 3+7 = 10), as well as the subtraction, of the upper applicable to both types of tree structures. First, ‘‘number of two digits (Figure 1A). Under the Linear condition, the partici- operations’’ was equated between the Simple and Linear pants were asked to regard the upper two digits (e.g., 3 and 7) as a conditions. Secondly, ‘‘number of generated digits for calculation’’ part of a linear sequence in increasing order (3, 7, 11, 15, 19,… in was the number of digits generated temporarily in a calculation this case), in which the differences between each pair of adjacent (i.e., not available on screen). For an example under the Quad terms were constant (Figure 1B). The participants were instructed condition shown in Figure 2B, there were four generated digits to calculate the third term of the linear sequence. We employed used in this calculation, ‘‘2, 5, 3, and 5.’’ Thirdly, ‘‘number of linear sequences, because it naturally imposed recursive compu- stored digits for matching’’ was the number of memorized digits tation; e.g., 3, 7, 11, 15, 19,…, obtained by nested constructions that were used for digit-matching. Its estimate was two and five for ((((3+4) +4) +4) +4 …). The arithmetic task could not be correctly PLOS ONE | www.plosone.org 2 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Figure 1. Examples of the stimuli used in the arithmetic task and Match task. Five yellow digits, together with a green character as a cue, were presented for 5 s, followed by the presentation of one white digit (matching stimulus) for 3 s. For each example, the task- relevant digits and a brief outline of the calculation processes are shown 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 Figure 2. Hierarchical and flat tree structure models for shown here are correct examples. (A) Under the Simple condition, arithmetic calculations. For each example shown in Figure 1, an indicated by the presentation of ‘‘6’’ as a cue, the addition and entire calculation process is represented by either hierarchical (A) or flat subtraction of the upper two digits (i.e., two of the five yellow digits) (B) tree structure models, where the lowest and leftmost branch were mentally performed by the participants. We instructed the corresponds to the first arithmetic calculation performed. A digit in participants to always subtract a smaller value from a larger value. (B) black at each nonterminal node is obtained from an arithmetic Under the Linear condition, indicated by the presentation of ‘‘a ’’ as a operation (e.g., + or 2) indicated just below each node, where ‘–’ cue, the third term of a linear sequence initiated by the upper two denotes a single operation of subtracting a smaller value from a larger digits was calculated by the participants. The first row at the top of the value. The digits in red denote the DoM at individual nodes, where a right panel represents the linear sequence, and the second row with reference point of zero is at the top node. The digits in blue denote branches represents the constant differences between each pair of ‘‘number of operations.’’ A gray arrow denotes corresponding digits adjacent terms in the linear sequence, as often used in math textbooks. generated temporarily in a calculation. The stored digits are circled in (C) Under the Quad condition, indicated by the presentation of ‘‘b ’’ as a red, as shown in Figure 1. In the hierarchical tree structure model, each cue, the following arithmetic calculations were performed by using the tree structure is based on recursive computation. Under the Linear lower three digits (i.e., three of the five yellow digits). The first row in condition, the hierarchical representation of the given example is: [|32 the right panel represents the given quadratic sequence. The second 7| +7] = 11. Under the Quad condition, the hierarchical representation row represents the differences between each pair of adjacent terms in of the given example is: [||224|2 |4–9|| +5] = 8. the given sequence. The participants were asked to regard the resultant doi:10.1371/journal.pone.0111439.g002 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, null or negative (see Table 1). Moreover, the Linear – Simple indicated by the presentation of ‘‘m’’ as a cue, the participants stored all contrast was also free from the following factors: ‘‘number of of the five digits in memory. nodes’’ in either tree structures, the DoM and ‘‘verbal encoding’’ doi:10.1371/journal.pone.0111439.g001 in the flat tree structures, ‘‘number of operations,’’ and ‘‘number of stored digits for matching.’’ The Quad – Linear contrast the arithmetic task and the Match task, respectively (the digits provides no further information, since no additional factors can be circled in red in Figure 1 and 2). controlled. To examine the functional specialization of cortical By taking the Match task as a basic control for the arithmetic regions in an unbiased manner [22], we adopted whole-brain task, we eliminated any task-related cognitive factors, such as analyses in such stringent contrasts as Simple – Match and Linear visual processing of stimuli, the identification of presented digits, – Simple. digit-matching, and motor responses. Both ‘‘number of generated Recent neuroimaging studies have examined functional con- digits for calculation’’ and ‘‘number of stored digits for matching’’ nectivity underlying elementary calculations. For example, it has might contribute to loads of short-term memory or ‘‘working been reported that the magnitude of functional connectivity memory,’’ but the Simple – Match contrast was free from these between the bilateral intraparietal sulcus (IPS) correlated with the memory-related factors, because their subtracted estimates were performances of a subtraction task [23]. Another fMRI study with PLOS ONE | www.plosone.org 3 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Table 1. Estimates of various factors to account for activations. Factors in the hierarchical tree structures Factor Simple Linear Quad Match DoM 1 2 3 0 No. of nodes 6 5 9 5 Simple – Match Linear – Match Quad – Match Linear – Simple DoM 1 2 3 1 No. of nodes 1 0 4 21 Factors in the flat tree structures Factor Simple Linear Quad Match DoM 1 1 1 0 No. of nodes 6 6 12 5 Verbal encoding 8 8 16 5 Simple – Match Linear – Match Quad – Match Linear – Simple DoM 1 1 1 0 No. of nodes 1 1 7 0 Verbal encoding 3 3 11 0 Common factors Factor Simple Linear Quad Match No. of operations 2 2 4 0 No. of generated 01 4 0 digits for calculation No. of stored 22 2 5 digits for matching Simple – Match Linear – Match Quad – Match Linear – Simple No. of operations 2 2 4 0 No. of generated 0 14 1 digits for calculation No. of stored digits 23 23 230 for matching 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 of operations’’ 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 Granger causality mapping reported that a participant group with (laterality quotient: 50–100), according to the Edinburgh inventory high arithmetic scores in a multiplication task had stronger [27]. Prior to their participation in the study, written informed bidirectional connections between the L. IPS and R. IPS than the consent was obtained from each participant after the nature and group with low arithmetic scores [24]. Both of those connectivity possible consequences of the studies were explained. Approval for studies with elementary calculation tasks did not involve a the experiments was obtained from the institutional review board recursive application of arithmetic operations. The effective of the University of Tokyo, Komaba. connectivity during arithmetic tasks with recursive computation, as well as the connectivity between the L. IFG and bilateral IPS, Stimuli should be thus clarified. In the present study, we adopted dynamic In each trial (Figure 1), we presented a main stimulus consisting causal modeling instead of Granger causality mapping, because of five yellow digits (from 1 to 9) and a green character (6,a ,b , n n dynamic causal modeling has been shown to be more effective for or m). Each green character was used as a cue to indicate one of fMRI data [25,26]. Our findings would elucidate the crucial the four conditions: ‘‘6’’ for the Simple condition, ‘‘a ’’ network of the L. IFG, L. IPS, and R. IPS for computing representing a linear sequence for the Linear condition (e.g., hierarchical tree structures in mathematics. a = 1, 3, 5, 7,…), ‘‘b ’’ representing a quadratic sequence for the n n Quad condition (e.g., b = 1, 2, 5, 10,…), and ‘‘m’’ for the Match Materials and Methods task. According to a pilot study, the duration of 5 s for the main stimulus was long enough for the participants to correctly perform Participants the task. At the center of the screen, one white digit (from 0 to 9) Twenty college students (15 males, aged 18–30 years), who had was subsequently presented for 3 s as a matching stimulus, not majored in mathematics but studied high school mathematics followed by a 200 ms blank to make the duration of the trial twice including linear and quadratic sequences, participated in the fMRI as long as the repetition time of the fMRI scans. experiment. All participants were healthy and right-handed PLOS ONE | www.plosone.org 4 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math To make the stimuli physically identical among the three scanning. Eight scanning sessions were performed in one day. A single scanning session contained 36 trials (six trials for each of the conditions in the arithmetic task, except for the cue characters, we used the same set of main stimuli, in which the five digits were Simple, Linear, and Quad conditions; 18 trials for the Match task). The arithmetic and Match tasks were alternately performed, so always arranged in two rows (48 different combinations of digits). The upper two digits were relevant to the Simple and Linear that MR signals for each condition in the arithmetic task were conditions, while the lower three digits were relevant to the Quad sufficiently separated. In the arithmetic task, no stimuli appeared condition. The digits in each row were arranged in increasing twice or more times across all scanning sessions, whereas each order from left to right, while no digit appeared twice in an entire stimulus appeared twice in the Match task. To prevent any stimulus. For the upper two digits, we excluded such ubiquitous condition-specific strategy, the order of the Simple, Linear, and combinations as ‘‘2 and 4,’’ ‘‘3 and 6,’’ or ‘‘4 and 8,’’ as well as Quad conditions was pseudorandomized; the orders of tasks and trivial combinations with the constant difference of one (e.g., ‘‘2 conditions were also counter-balanced across participants. and 3’’). For the lower three digits, we excluded certain combinations (e.g., ‘‘1, 2, and 4’’), in which the subordinate linear MRI Data Acquisition sequence became trivial (1, 2, 3, 4,… in this case). In the Match The fMRI scans were conducted on a 3.0 T scanner (Signa task, all digits in both rows were memorized; we prepared 72 HDxt; GE Healthcare, Milwaukee, WI) with a bird-cage head coil. different combinations of digits, including all combinations used in For the fMRI, we scanned 40 axial slices that were 3-mm thick the arithmetic task. with a 0.3-mm gap, covering from 252.8 to 78.9 mm from the The participants wore earplugs and an eyeglass-like MRI- anterior to posterior commissure line in the vertical direction, compatible display (resolution, 8006600 pixels; VisuaStim Digital, using a gradient-echo echo-planar imaging sequence [repetition Resonance Technology Inc., Northridge, CA). For fixation, a time = 4.1 s, echo time = 60 ms, flip angle = 90u, field of view = 2 2 small red cross was always shown at the center of the screen to 1926192 mm , resolution = 363mm ]. In a single scanning initiate eye movements from the same fixed position, and the session, we obtained 72 volumes following four dummy images, participants were instructed to return their eyes to this position for which allowed for the rise of the MR signals. For each participant, the matching stimulus. Reaction times were measured from the sessions without head movement were used for analyses; the onset of the matching stimulus. The stimulus presentation and number of abandoned sessions was less than four for all collection of behavioral data (accuracy and reaction times) were participants. After completion of the fMRI sessions, high- controlled using the LabVIEW software and interface (National resolution T1-weighted images of the whole brain (192 axial Instruments, Austin, TX). slices, 16161mm ) were acquired from all participants with a fast spoiled gradient recalled acquisition in the steady state sequence Tasks (repetition time = 8.4 ms, echo time = 2.6 ms, flip angle = 25u, field of view = 2566256 mm ). In each trial of the arithmetic task, the participants were asked to silently perform a series of specified arithmetic calculations, and to store two digits in memory obtained from the arithmetic fMRI Data Analyses calculations without using their fingers. The participants then We performed data analyses with fMRI using SPM8 statistical judged whether or not one of the stored digits matched the white parametric mapping software (Wellcome Trust Centre for digit, and responded by pressing one of two nonmagnetic buttons Neuroimaging, London, UK; http://www.fil.ion.ucl.ac.uk/spm/) (Current Designs, Inc., Philadelphia, PA): a right button if [28], implemented on a MATLAB platform (MathWorks, Natick, matched (in half of the trials), or a left button if mismatched (in MA). The acquisition timing of each slice was corrected using the the other half). middle slice as a reference for the echo-planar imaging data. We Under the Simple condition (Figure 1A), the participants stored realigned the echo-planar imaging data in multiple sessions to the the results of addition and subtraction in memory. When the result first volume in all sessions, and removed sessions that included was a ‘‘two-figure number’’ (e.g., 10), the participants simply data with a translation of.2 mm in any of the three directions and stored the last digit (i.e., the digit in ‘‘the one’s place’’ in math with a rotation of.1.4u around any of the three axes; these terms, 0 in this case) in memory. We instructed them to always thresholds were empirically determined from our previous studies subtract a smaller value from a larger value (e.g., |327| = 4), as [29–31]. represented by the sign of absolute value here. Under the Linear Each participant’s T1-weighted structural image was coregis- condition (Figure 1B), the participants were instructed to obtain tered to the mean functional image generated during realignment. the constant difference first by the subtraction (e.g., |327| = 4), The coregistered structural image was spatially normalized to the and then to calculate the third term by adding the constant standard brain space as defined by the Montreal Neurological difference to the second term (4+7 = 11 in this case); they stored Institute using the ‘‘unified segmentation’’ algorithm with medium the results of these arithmetic calculations (4 and 1 in this case) in regularization, which is a generative model that combines tissue memory. Under the Quad condition (Figure 1C), the participants segmentation, bias correction, and spatial normalization in the were instructed to perform a series of subtractions (e.g., |224| inversion of a single unified model [32]. After spatial normaliza- = 2, and then |429| = 5) in order to obtain the constant tion, the resultant deformation field was applied to the realigned difference of the subordinate linear sequence by the subtraction functional imaging data, which was resampled every 3 mm using (|225| = 3 in this case). They were further instructed to calculate seventh-degree B-spline interpolation. All normalized functional the third term of the subordinate linear sequence by adding the images were then smoothed by using an isotropic Gaussian kernel constant difference to the second term (3+5 = 8). They stored the of 9 mm full-width at half maximum. Low-frequency noise was results of these arithmetic calculations (3 and 8 in this case) in removed by high-pass filtering at 1/128 Hz. memory. In a first-level analysis (i.e., a fixed-effects analysis), each In the Match task (Figure 1D), the participants simply stored all participant’s hemodynamic responses induced by the trials were of five yellow digits in memory, and judged whether one of those modeled with a box-car function (band-pass during 1–3 s after the digits matched the white digit. The participants underwent main stimulus onset) convolved with a hemodynamic function. practice sessions for these arithmetic and Match tasks before The 0–1 s period from the main stimulus onset was related with PLOS ONE | www.plosone.org 5 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math the identification of a cue, and the 3–5 s period was likely to be Results confounded with preparatory processes of digit-matching. We thus Behavioral Results selected the 1–3 s period in order to focus on arithmetic The behavioral data are shown in Table 2, indicating that the calculations. Only event-related responses of correct trials were tasks were performed almost perfectly. With respect to accuracy, a analyzed. The images of the three conditions in the arithmetic task one-way repeated measures analysis of variance showed a and Match tasks were then generated in a general linear model for significant main effect of condition including the Match task each participant, and used for intersubject comparisons in a [F(3, 57) = 20, P,0.0001]. Post-hoc paired t-tests among all second-level analysis (i.e., random-effects analysis). To discount conditions (significance level at a = 0.0083, Bonferroni-corrected) any general effects associated with task difficulty or performance showed that the accuracy under the Quad condition was differences among the participants, individual error rates (100 – significantly lower than that under the other conditions (P, accuracy), which were more sensitive than reaction times in our 0.001), while the differences of accuracy among the other three tasks, were entered for each task as a nuisance factor. For all fMRI conditions were not significant (P.0.1). Regarding reaction times data analyses, the statistical threshold was set to P,0.05 for the (RTs), a main effect of condition was significant [F(3, 57) = 12, voxel level, corrected for multiple comparisons [family-wise error P,0.0001], and the RTs under the Quad condition was correction] across the whole brain. significantly longer than those under the Simple condition (P, For the anatomical identification of activated regions, we 0.001). The Match task was also more demanding than the Simple basically used the Anatomical Automatic Labeling method [33]. In and Linear conditions, leading to significantly longer RTs (P, region of interest analyses, we extracted the percent signal changes 0.001). averaged among participants at each local maximum using the MarsBaR-toolbox (http://marsbar.sourceforge.net/). To statisti- Selective Activation in the Arithmetic Task cally evaluate the fitness of a single factor’s parametric model to To identify cortical regions involved in the arithmetic task, we activations, we calculated the coefficient of determination (r ) and first tested the Linear – Match contrast with the liberal control of a residual sum of squares with MATLAB; we obtained the fitted Match. We then tested the direct Quad – Simple contrast, values by multiplying the estimates of the factor (see Table 1) by a 2 2 2 inclusively masked with Linear – Match to guarantee the fitting scale. For a no-intercept model, r =12 S(y2ˆy) /Sy consistency of activation patterns. In both contrasts, overall should be calculated, where y and y denote the fitted values and activation was clearly left-dominant. Significant activation was the observed signal changes for each contrast, respectively [34]. observed in the L. IFG (Brodmann’s areas 44/45), bilateral SMG For this calculation, we used R software (http://www.r-project. (Brodmann’s area 40), bilateral IPS (Brodmann’s areas 7/39/40), org/). To further take account of individual variability, we used a and precuneus (Brodmann’s area 7), as well as in the bilateral restricted maximum-likelihood method, fitting ‘‘linear mixed- lateral premotor cortex (LPMC, Brodmann’s areas 6/8), left effects models’’ with individual activations as dependent variables, anterior short insular gyrus (ASG), and pre-supplementary motor with the estimates of each factor as a regressor, and with the area (pre-SMA, Brodmann’s areas 6/8) (Figure 3A, 3B, and participants as random effects. For this calculation, we used an Table 3). The left middle temporal gyrus (L. MTG, Brodmann’s nlme (linear and nonlinear mixed-effects models) package (http:// area 21) was significantly activated only in Linear – Match. cran.r-project.org/web/packages/nlme/) on R software. We tried to narrow down the critical regions for the computation of hierarchical tree structures, by using more Dynamic Causal Modeling Data Analyses stringent contrasts. In Simple – Match, where the memory-related We performed data analyses with dynamic causal modeling factors of ‘‘number of generated digits for calculation’’ and using DCM10 on SPM8 [35]. We concatenated the scans from the ‘‘number of stored digits for matching’’ were eliminated, separate sessions, and reanalyzed the preprocessed data with the significant activation was localized in the L. IFG and L. SMG general linear model, which contained two regressors representing (Figure 3C and Table 3), indicating that activations in these the Quad condition and Match task for making a meaningful regions were free from the memory-related factors. Moreover, in contrast, as well as a regressor representing both the Quad and Linear – Simple, significant activation was observed in the Linear conditions for driving inputs. The regressor representing bilateral IPS and precuneus (Figure 3D and Table 3), indicating the Quad condition was also used for a modulatory effect. The that activations in these regions were independent from other effects of transition between sessions were taken into account with possible factors. Although activations in the L. IFG and L. SMG regressors of sessions. With the volume-of-interest tool in SPM8, were below the threshold in Linear – Simple, they were significant the time series was extracted by taking the first eigenvariate across with small volume correction (corrected P,0.05, 9 mm radius all suprathreshold voxels in Quad – Match for each participant from each local maximum determined by Simple – Match). In (uncorrected P,0.05), confined within 6 mm from the group local Simple – Match with small volume correction, activations in the L. maximum of a single region. IPS, but not those in the R. IPS or precuneus, were significant (9 mm radius from each local maximum determined by Linear – We specified 18 models with systematic variations in a modulatory effect and driving inputs (Figure S1). After estimating Simple). These results suggest that the L. IFG, L. SMG, and L. IPS, together with the limited contribution of the R. IPS and all models for each participant, we identified the most likely model precuneus, are well specialized in arithmetic calculations. by 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 the Cortical Activations Specifically Modulated by the DoM in model for any randomly selected participants, or on the the Hierarchical Tree Structures exceedance probability, i.e., the probability that the model is a At the local maxima of these five regions, we further examined better fit to the data than any other models tested. After percent signal changes under each condition in the arithmetic task determining the best model, we evaluated the parameter estimates with reference to the Match task (Figure 4). A linear modulation of of this particular model by one-sample t-tests [36]. activations was observed in the L. IFG, L. SMG, and bilateral IPS among these three conditions, indicating that the results were PLOS ONE | www.plosone.org 6 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Table 2. Behavioral data under each condition. Simple Linear 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 consistent with the DoM in the hierarchical tree structures. The estimates multiplied by the fitting scale) against corresponding precuneus showed weaker activations under the Simple condition, signal changes averaged across participants (Table 5, Tables S1- exhibiting relatively larger responses under the more demanding S4 in File S1). Linear and Quad conditions (Figure 4E). Among the parametric models of eight factors tested, the model Next we examined how well activations in each of the L. IFG, of DoM in the hierarchical tree structures produced by far the L. SMG, bilateral IPS, and precuneus correlated with the DoM in lowest residual sum of squares value (#0.0043) and the largest the hierarchical tree structures and other factors. All of the Simple coefficient of determination (r )($0.99) for the L. IFG, L. SMG, – Match, Linear – Match, and Quad – Match contrasts predicted and bilateral IPS. We further evaluated the goodness of fit for each that activations should be exactly zero when a factor produced no model by using one-sample t-tests (significance level at a = 0.0167, effect or load relative to the Match task. We thus adopted a no- Bonferroni-corrected) between the fitted value for each contrast intercept model, in which the percent signal changes of each and individual activations. The model of DoM for these four region were fitted with a single (thus minimal) scale parameter to a regions produced no significant deviation for the three contrasts model of each factor using its subtracted estimates (Table 4). For (P$0.13). In the precuneus, the goodness of fit was marginal for the three contrasts, a least-squares method was used to minimize one P-value (P = 0.027) under the Simple condition. By fitting the residual sum of squares for the three fitted values (i.e., the three ‘‘linear mixed-effects models,’’ we found that the model of DoM 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 lateral surfaces 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 PLOS ONE | www.plosone.org 7 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math PLOS ONE | www.plosone.org 8 November 2014 | Volume 9 | Issue 11 | e111439 Table 3. Significantly activated regions in the arithmetic task. Contrast Brain region BA Side xyz Z-Value Voxels Linear – Match LPMC 6/8 L 227 5 55 5.5 82 R 33 5 58 4.5 5 IFG 44/45 L 245 11 25 6.8 309 251 32 22 5.2 * ASG – L 230 26 1 4.6 5 pre-SMA 6/8 M 2620 46 6.0 53 SMG 40 L 245 240 43 6.9 483 IPS 7/39/40 L 233 252 40 6.8 * SMG 40 R 48 231 46 4.5 5 IPS 7/39/40 R 36 255 43 6.0 169 Precuneus 7 M 26 270 46 6.1 155 MTG 21 L 251 255 211 5.2 15 Quad – Simple, masked with Linear – Match LPMC 6/8 L 224 5 55 5.4 62 R 30 5 58 5.6 5 IFG 44/45 L 245 5 37 5.9 131 245 11 25 5.4 * ASG – L 230 23 1 6.1 5 pre-SMA 6/8 M 2620 43 6.4 51 0 11 55 6.1 * SMG 40 L 248 237 43 5.4 352 IPS 7/39/40 L 233 249 40 5.8 * 227 270 43 5.5 * SMG 40 R 45 231 43 5.9 5 IPS 7/39/40 R 39 252 43 5.4 110 33 261 34 4.8 * Precuneus 7 M 212 261 52 6.3 155 9 270 49 6.2 * Simple – Match IFG 44/45 L 245 11 25 4.6 5 SMG 40 L 248 237 46 4.8 12 Linear – Simple IPS 7/39/40 L 230 270 46 4.6 3 R39 255 43 4.5 4 Precuneus 7 M 23 270 43 4.7 28 Stereotactic 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 short insular 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 an asterisk is included within the same cluster shown one row above. doi:10.1371/journal.pone.0111439.t003 Neural Basis of Hierarchical Processes in Math Figure 4. Quantified activations modulated by the DoM in the hierarchical 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 (see Table 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 the hierarchical tree structures. Effective Connectivity among the L. IFG and Bilateral IPS Because the L. SMG is relatively close to the L. IPS, and the precuneus locates midway between the bilateral IPS, we focused on the L. IFG and bilateral IPS alone for modeling the effective connectivity in the dynamic causal modeling analyses. Here we assumed 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 we tested six models with a modulatory effect under the Quad condition on the unidirectional or bidirectional connections (see Figure S1 for all models tested). Using a random-effects Bayesian model selection, we showed that the model 1 (Figure 5A), with a modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, yielded by far the highest 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 the participants. 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 a parameter 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 structures in mathematics, and we obtained three striking results. First, we found significant activation in the L. IFG, L. SMG, bilateral IPS, and precuneus selectively among the three conditions in the arithmetic task (Figure 3). Secondly, by examining percent signal changes in each region, a linear modulation of activations was observed in the L. IFG, L. SMG, and bilateral IPS among these three conditions (Figure 4). Moreover, by fitting the parametric models of eight factors, we found that the DoM in the hierarchical tree 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 of mathematical syntax processed in these regions, excluding the load on ‘‘working memory.’’ The dominance of the hierarchical PLOS ONE | www.plosone.org 9 November 2014 | Volume 9 | Issue 11 | e111439 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 tree structure model is consistent with our previous results of the L. a matter of controversy whether the L. IFG is also critically IFG and L. SMG activation in the direct comparison between involved in mathematics. It was claimed in a previous study that sentences and letter strings, which were assumed to have ‘‘it [the L. IFG] does not appear to play a dominant role [in hierarchical tree structures and flat tree structures, respectively mathematics], which instead is taken up by fusiform, parietal and [7]. Thirdly, using dynamic causal modeling, we showed that the precentral cortices,’’ i.e., visuo-spatial areas [21]. In this previous model with a modulatory effect for the connection from the L. IPS study, the participants performed a short-term matching task, to the L. IFG, and with driving inputs into the L. IFG, was highly which was solved without requiring any arithmetic calculation. In probable (Figure 5). For this best model, the top-down intrinsic contrast, Makuuchi et al. [20] showed overlapped activation across connection from the L. IFG to the L. IPS, as well as that from the sentence comprehension and arithmetic calculations, with an L. IPS to the R. IPS, would provide a feedforward signal with their increase of activations in the L. IFG for the higher level of reverse connections representing a negative feedback signal. These structural hierarchy, although the modulation of activations in the results indicate that the network of the L. IFG and bilateral IPS bilateral IPS were not fully examined. The present results clearly subserves the computation of hierarchical tree structures in showed that the DoM in the hierarchical tree structures was by far mathematics. more effective than the load on ‘‘working memory,’’ ‘‘number of Previous imaging studies have established that syntactic operations,’’ and other factors for explaining the L. IFG and bilateral IPS activations. processes during sentence comprehension selectively activate the L. IFG and/or the L. LPMC [29,37–41]. By directly contrasting a According to some previous imaging studies, the domain- demanding condition for ‘‘working memory,’’ we have previously specificity for the arithmetic calculation in the L. IFG and other demonstrated that both regions are indeed independent from such regions has remained unclear. For example, in recent fMRI studies domain-general cognitive factors [29], indicating that these regions [43,44], two conditions with large and small numbers were have a critical role as a putative grammar center [42]. It has been compared in an addition task, and such domain-general factors as Table 5. Fittings and likelihood of various models tested in the L. IFG. Factors in the hierarchical tree structures Factor RSS r P-values Log-likelihood Likelihood ratio *DoM 0.0024 . 0.99 0.34, 0.75, 0.97 15.0 1 –16 No. of nodes 0.18 0.70 , 0.0001, 0.054, 0.65 –20.5 3.9 6 10 Factors in the flat tree structures Factor RSS r P-values Log-likelihood Likelihood ratio –8 DoM 0.062 0.90 0.0016, 0.0025, 0.85 –2.5 2.6 6 10 –14 No. of nodes 0.13 0.78 , 0.0001, 0.0033, 0.20 –15.8 4.46 10 –10 Verbal encoding 0.063 0.89 , 0.0001, 0.14, 0.17 –6.7 3.7 6 10 Common factors Factor RSS r P-values Log-likelihood Likelihood ratio No. of operations 0.015 0.98 0.033, 0.21, 0.68 9.6 0.0044 –14 No. of generated digits for 0.13 0.78 , 0.0001, , 0.0001, 0.24 –15.0 9.1 6 10 calculation No. of stored digits for 0.60 0 , 0.0001, , 0.0001, , n/a n/a matching 0.0001 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. Note that 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 (r ), and larger P-values. The likelihood of models with all null estimates was not calculable (n/a). A likelihood ratio is the ratio of each model’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 PLOS ONE | www.plosone.org 10 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math naturally extended to ‘‘musical syntax’’ as well, since harmonic progressions 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 of numbers [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 repeti- tion priming of numbers (e.g., from ‘‘NINE’’ to ‘‘9’’) caused the activation 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 and adjacent L. SMG are also involved in language, especially in vocabulary knowledge or lexical processing [54,55], as well as in searching syntactic features [7]. Taken these and present results together, the L. IPS/SMG would be also involved in both mathematics 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 Quad conditions. Using a visual picture-sentence matching task, we have recently tested twenty-one patients with a left frontal glioma, and found abnormal 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, and cerebellar nuclei, which were overactivated in the patients with a glioma 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 underactivated in the patients with a glioma in the opercular/triangular parts of Figure 5. Effective connectivity among the L. IFG and bilateral the L. IFG. Among the activated regions in the present study IPS. (A) The best model with a positive modulatory effect for the bottom-up connection from the L. IPS to the L. IFG, and with driving (Table 3), the L. IFG, L. IPS, R. LPMC, R. IFG, and pre-SMA inputs into the L. IFG. Mean parameter estimates that exceeded the are included in the network I, whereas the L. LPMC and L. MTG statistical threshold (corrected P,0.05) are indicated alongside the are included in the network II and network III, respectively. The intrinsic connections. Bar graphs show expected probabilities (B) and overall activated regions for the arithmetic calculations thus share exceedance probabilities (C) of all models tested (Figure S1). their functional roles with the syntax-related regions in language. doi:10.1371/journal.pone.0111439.g005 Our previous fMRI study revealed that the functional connectivity between the L. IFG and L. SMG was selectively task difficulty might explain the enhanced activations. It should be enhanced during sentence processing [59]. A recent dynamic noted that any hierarchical processes associated with ‘‘two-figure causal modeling study with a cross-modal picture-sentence numbers,’’ as well as more verbal encoding, are also involved in matching task has suggested that the L. IFG received driving the task with large numbers. In the present study, we discounted inputs and transferred that information to the temporal regions any general effects associated with task difficulty by entering [60]. Our recent dynamic causal modeling study has suggested a individual error rates for each task as a nuisance factor. We have top-down intrinsic information flow of syntactic processing from previously demonstrated that the L. IFG is a domain-specific the L. IFG to the L. SMG, with driving inputs into the L. IFG [7]. neural system for syntactic computation in language, which is This model is consistent with our present results of dynamic causal separable from other domain-general cognitive systems [42]. In modeling, which further indicate that L. IPS activations mirrored the present study, we indicate that the hierarchical tree structures a top-down influence regarding the DoM in the hierarchical tree in mathematics are also computed by the same domain-specific structures computed in the L. IFG. For the bottom-up connection system. Our successful approach on mathematical syntax can be from the L. IPS/SMG to the L. IFG, the modulatory effect under PLOS ONE | www.plosone.org 11 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math the conditions with the largest DoM was negative in this previous bidirectional connections between the L. IFG and L. IPS, as well study, whereas the modulatory effect under the Quad condition as between the L. IPS and R. IPS. Eighteen models were (with the largest DoM) was positive in the present study. While systematically constructed with driving inputs into one of the three lexical feedback was minimum for processing jabberwocky regions. For each input type, we tested six models for the sentences in the previous study, a positive feedback about modulatory effect under the Quad condition. operations would be utilized for constructing hierarchical tree (TIF) structures in the present paradigm. File S1 Table S1: Fittings and likelihood of various models tested The present results of dynamic causal modeling suggest that the in the L. SMG. Table S2: Fittings and likelihood of various models syntactic information on hierarchical tree structures provided in tested in the L. IPS. Table S3: Fittings and likelihood of various the L. IFG would be further processed through the top-down models tested in the R. IPS. Table S4: Fittings and likelihood of intrinsic connection from the L. IPS to the R. IPS (Figure 5). The various models tested in the precuneus. L. IPS and R. IPS may have different roles in processing (DOC) arithmetic calculations, but their individual roles in mathematical syntax should be clarified in the future studies. In addition, it is Acknowledgments possible that the L. MTG, significantly activated in Linear – Match, is involved in mathematical semantics, as this region would We thank S. Ohta and H. Miyashita for their helpful discussions, N. subserve semantics in language. We indicate that mathematics and Komoro for her technical assistance, and H. Matsuda for her administra- language share the network of the L. IFG and L. IPS/SMG for the tive assistance. computation of hierarchical tree structures, and that mathematics recruits the additional network of the L. IPS and R. IPS, with an Author Contributions information flow from the former to the latter. Conceived and designed the experiments: TN KLS. Performed the experiments: TN KLS. Analyzed the data: TN KLS. Wrote the paper: TN Supporting Information KLS. Figure S1 Models tested in the dynamic causal model- ing analyses. We assumed intrinsic, i.e., task-independent, References 1. Chomsky N (1957) Syntactic Structures. The Hague: Mouton Publishers. 117 p. 22. Friston KJ, Henson RN (2006) Commentary on: Divide and conquer; A defence of functional localisers. Neuroimage 30: 1097–1099. 2. Chomsky N (1965) Aspects of the Theory of Syntax. Cambridge, MA: The MIT Press. 251 p. 23. Park J, Park DC, Polk TA (2013) Parietal functional connectivity in numerical 3. Ernest P (1987) A model of the cognitive meaning of mathematical expressions. cognition. Cereb Cortex 23: 2127–2135. Br J Educ Psychol 57: 343–370. 24. Krueger F, Landgraf S, van der Meer E, Deshpande G, Hu X (2011) Effective 4. Chomsky N (2007) Biolinguistic explorations: Design, development, evolution. connectivity of the multiplication network: A functional MRI and multivariate Int J Philos Stud 15: 1–21. Granger causality mapping study. Hum Brain Mapp32: 1419–1431. 5. Chomsky N (1995) The Minimalist Program. Cambridge, MA: The MIT Press. 25. David O, Guillemain I, Saillet S, Reyt S, Deransart C, et al. (2008) Identifying 420 p. neural drivers with functional MRI: An electrophysiological validation. PLOS 6. Fukui N (2011) Merge and Bare Phrase Structure. In: Boeckx C, editor. The Biol 6, e315: 2683–2697. Oxford Handbook of Linguistic Minimalism.Oxford, UK: Oxford University 26. Friston K (2009) Causal modelling and brain connectivity in functional magnetic Press. pp. 73–95. resonance imaging. PLOS Biol 7, e1000033: 220–225. 7. Ohta S, Fukui N, Sakai KL (2013) Syntactic computation in the human brain: 27. Oldfield RC (1971) The assessment and analysis of handedness: The Edinburgh The Degree of Merger as a key factor. PLOS ONE 8, e56230: 1–16. inventory. Neuropsychologia 9: 97–113. 8. Ohta S, Fukui N, Sakai KL (2013) Computational principles of syntax in the 28. Friston KJ, Holmes AP, Worsley KJ, Poline J-P, Frith CD, et al. (1995) Statistical regions specialized for language: Integrating theoretical linguistics and functional parametric maps in functional imaging: A general linear approach. Hum Brain neuroimaging. Front Behav Neurosci 7, 204: 1–13. Mapp 2: 189–210. 9. Cappelletti M, Butterworth B, Kopelman M (2001) Spared numerical abilities in 29. Hashimoto R, Sakai KL (2002) Specialization in the left prefrontal cortex for a case of semantic dementia. Neuropsychologia 39: 1224–1239. sentence comprehension. Neuron 35: 589–597. 10. Zago L, Pesenti M, Mellet E, Crivello F, Mazoyer B, et al. (2001) Neural 30. Suzuki K, Sakai KL (2003) An event-related fMRI study of explicit syntactic correlates of simple and complex mental calculation. Neuroimage 13: 314–327. processing of normal/anomalous sentences in contrast to implicit syntactic 11. Butterworth B (2005) Developmental Dyscalculia. In: Campbell JID, editor. processing. Cereb Cortex 13: 517–526. Handbook of Mathematical Cognition.New York, NY: Psychology Press. pp. 455– 31. Kinno R, Kawamura M, Shioda S, Sakai KL (2008) Neural correlates of 467. noncanonical syntactic processing revealed by a picture-sentence matching task. 12. Fedorenko E, Behr MK, Kanwisher N (2011) Functional specificity for high- Hum Brain Mapp 29: 1015–1027. level linguistic processing in the human brain. Proc Natl Acad Sci USA 108: 32. Ashburner J, Friston KJ (2005) Unified segmentation. Neuroimage 26: 839–851. 16428–16433. 33. Tzourio-Mazoyer N, Landeau B, Papathanassiou D, Crivello F, Etard O, et al. 13. Benn Y, Zheng Y, Wilkinson ID, Siegal M, Varley R (2012) Language in (2002) Automated anatomical labeling of activations in SPM using a calculation: A core mechanism? Neuropsychologia 50: 1–10. macroscopic anatomical parcellation of the MNI MRI single-subject brain. 14. Monti MM, Parsons LM, Osherson DN (2012) Thought beyond language: Neuroimage 15: 273–289. Neural dissociation of algebra and natural language. Psychol Sci 23: 914–922. 34. Kva ˚ lseth TO (1985) Cautionary note about R . Am Stat 39: 279–285. 15. Klessinger N, Szczerbinski M, Varley R (2012) The role of number words: The 35. Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuro- phonological length effect in multidigit addition. Mem Cogn 40: 1289–1302. image 19: 1273–1302. 16. Benn Y, Wilkinson ID, Zheng Y, Kadosh KC, Romanowski CAJ, et al. (2013) 36. Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, et al. Differentiating core and co-opted mechanisms in calculation: The neuroimaging (2010) Ten simple rules for dynamic causal modeling. Neuroimage 49: 3099– of calculation in aphasia. Brain Cogn 82: 254–264. 17. Baldo JV, Dronkers NF (2007) Neural correlates of arithmetic and language 37. Stromswold K, Caplan D, Alpert N, Rauch S (1996) Localization of syntactic comprehension: A common substrate? Neuropsychologia 45: 229–235. comprehension by positron emission tomography. Brain Lang 52: 452–473. 18. Varley RA, Klessinger NJC, Romanowski CAJ, Siegal M (2005) Agrammatic 38. Dapretto M, Bookheimer SY (1999) Form and content: Dissociating syntax and but numerate. Proc Natl Acad Sci USA 102: 3519–3524. semantics in sentence comprehension. Neuron 24: 427–432. 19. Klessinger N, Szczerbinski M, Varley R (2007) Algebra in a man with severe 39. Embick D, Marantz A, Miyashita Y, O’Neil W, Sakai KL (2000) A syntactic aphasia. Neuropsychologia 45: 1642–1648. specialization for Broca’s area. Proc Natl Acad Sci USA 97: 6150–6154. 20. Makuuchi M, Bahlmann J, Friederici AD (2012) An approach to separating the 40. Musso M, Moro A, Glauche V, Rijntjes M, Reichenbach J, et al. (2003) Broca’s levels of hierarchical structure building in language and mathematics. Phil area and the language instinct. Nat Neurosci 6: 774–781. Trans R Soc B 367: 2033–2045. 41. Friederici AD, Ru ¨ schemeyer S-A, Hahne A, Fiebach CJ (2003) The role of left 21. Maruyama M, Pallier C, Jobert A, Sigman M, Dehaene S (2012) The cortical inferior frontal and superior temporal cortex in sentence comprehension: representation of simple mathematical expressions. Neuroimage 61: 1444–1460. Localizing syntactic and semantic processes. Cereb Cortex 13: 170–177. PLOS ONE | www.plosone.org 12 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math 42. Sakai KL (2005) Language acquisition and brain development. Science 310: 52. Piazza M, Pinel P, Le Bihan D, Dehaene S (2007) A magnitude code common to 815–819. numerosities and number symbols in human intraparietal cortex. Neuron 53: 43. Fedorenko E, Duncan J, Kanwisher N (2012) Language-selective and domain- 293–305. general regions lie side by side within Broca’s area. Curr Biol 22: 2059–2062. 53. Price GR, Mazzocco MMM, Ansari D (2013) Why mental arithmetic counts: 44. Fedorenko E, Duncan J, Kanwisher N (2013) Broad domain generality in focal Brain activation during single digit arithmetic predicts high school math scores. regions of frontal and parietal cortex. Proc Natl Acad Sci USA 110: 16616– J Neurosci 33: 156–163. 16621. 54. Lee H, Devlin JT, Shakeshaft C, Stewart LH, Brennan A, et al. (2007) 45. Rohrmeier M (2011) Towards a generative syntax of tonal harmony. J Math Anatomical traces of vocabulary acquisition in the adolescent brain. J Neurosci Music 5: 35–53. 27: 1184–1189. 46. Cipolotti L, Butterworth B, Denes G (1991) A specific deficit for numbers in a 55. Pattamadilok C, Knierim IN, Duncan KJK, Devlin JT (2010) How does case of dense acalculia. Brain 114: 2619–2637. learning to read affect speech perception? J Neurosci 30: 8435–8444. 47. Dehaene S, Cohen L (1997) Cerebral pathways for calculation: Double 56. Pinel P, Dehaene S, Rivie `re D, LeBihan D (2001) Modulation of parietal dissociation between rote verbal and quantitative knowledge of arithmetic. activation by semantic distance in a number comparison task. Neuroimage 14: Cortex 33: 219–250. 1013–1026. 48. Dehaene S, Spelke E, Pinel P, Stanescu R, Tsivkin S (1999) Sources of 57. Ischebeck A, Zamarian L, Egger K, Schocke M, Delazer M (2007) Imaging early mathematical thinking: Behavioral and brain-imaging evidence. Science 284: practice effects in arithmetic. Neuroimage 36: 993–1003. 970–974. 58. Kinno R, Ohta S, Muragaki Y, Maruyama T, Sakai KL (2014) Differential 49. Eger E, Sterzer P, Russ MO, Giraud A-L, Kleinschmidt A (2003) A supramodal reorganization of three syntax-related networks induced by a left frontal glioma. number representation in human intraparietal cortex. Neuron 37: 719–725. Brain 137: 1193–1212. 50. Piazza M, Izard V, Pinel P, Le Bihan D, Dehaene S (2004) Tuning curves for 59. Homae F, Yahata N, Sakai KL (2003) Selective enhancement of functional approximate numerosity in the human intraparietal sulcus. Neuron 44: 547–555. connectivity in the left prefrontal cortex during sentence processing. Neuroimage 51. Naccache L, Dehaene S (2001) The priming method: Imaging unconscious 20: 578–586. repetition priming reveals an abstract representation of number in the parietal 60. den Ouden D-B, Saur D, Mader W, Schelter B, Lukic S, et al. (2012) Network lobes. Cereb Cortex 11: 966–974. modulation during complex syntactic processing. Neuroimage 59: 815–823. PLOS ONE | www.plosone.org 13 November 2014 | Volume 9 | Issue 11 | e111439 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png PLoS ONE Pubmed Central

Neural Mechanisms Underlying the Computation of Hierarchical Tree Structures in Mathematics

PLoS ONE , Volume 9 (11) – Nov 7, 2014

Loading next page...
 
/lp/pubmed-central/neural-mechanisms-underlying-the-computation-of-hierarchical-tree-q3jtaxm00u

References (123)

Publisher
Pubmed Central
Copyright
© 2014 Nakai, Sakai
ISSN
1932-6203
eISSN
1932-6203
DOI
10.1371/journal.pone.0111439
Publisher site
See Article on Publisher Site

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 task that involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, we found significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among the tested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, and that 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 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. 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 and language share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and that mathematics 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:  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, data collection 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, Introduction simply counting the total number of non-terminal nodes (branch- One of the fundamental properties common to language and ing points) and terminal nodes of a tree structure. However, this mathematics is the critical involvement of tree structures in those model cannot capture hierarchical levels within the tree (sister comprehension and production processes. Indeed, sentences relations in linguistic terms), whereas the DoM plays a critical role consist of hierarchical tree structures with recursive branches in measuring hierarchical levels of tree structures, such that the [1,2], and mathematical calculations can be also expressed by same numbers are assigned to the nodes with an identical hierarchical tree structures [3], which may derive from the unique hierarchical level [8]. Therefore, the model of DoM can properly and universal property of recursive computation in humans [4]. capture recursiveness in the whole tree structures. Here we apply This point provides a good motivation for the theoretical and the computational concept of DoM to tree structures in experimental approaches advocated in the present study. Accord- mathematical calculations, and we hypothesize that the DoM ing to modern linguistics, the construction of any grammatical actually represents specific loads in the computation of hierarchi- phrase or sentence is based on the fundamental linguistic cal tree structures also in mathematics. operation of ‘‘Merge,’’ which combines two syntactic objects to Generally speaking, mathematical calculations consist of at least form a larger structure [5]. To properly measure the complexity of two components. One component is ‘‘mathematical syntax’’ that tree structures with a formal property of Merge and iterativity determines how terms and operators are combined together to (recursiveness) [6], we recently introduced ‘‘the Degree of Merger form mathematical expressions. We hypothesize that mathemat- (DoM),’’ which was defined as the maximum depth of merged ical syntax is shared with linguistic syntax in a deeper sense, and subtrees (i.e., Mergers) within an entire sentence [7]. Among that both syntax can be automatically processed without various models that may possibly quantify the complexity of tree consciousness or verbalization. The other component is ‘‘mathe- PLOS ONE | www.plosone.org 1 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math matical semantics’’ that deals with the meaning of terms, performed without generating such nested constructions to operators, and mathematical expressions. There have been a integrate new terms in a linear sequence. Under the Quad number of lesion studies and imaging studies, which claimed that condition, the participants were asked to regard the lower three linguistic and arithmetic abilities were separable in the brain [9– digits (e.g., 2, 4, and 9) as part of a quadratic sequence in 16]. However, they examined participants’abilities in elementary increasing order (2, 4, 9, 17, 28,… in this case), in which the processing of numbers or variables, i.e., mathematical semantics, differences between each pair of adjacent terms resulted in a but not those in mathematical syntax. On the other hand, subordinate linear sequence (2, 5, 8, 11,… in this case) (Figure 1C). common activation between linguistic and arithmetic tasks may The participants were instructed to calculate the third term of the not necessarily mean that the same neural system subserves the subordinate linear sequence. We theoretically predicted that this same processes included in both tasks, because even a single region generative process imposed effective computation and actual may have multiple functions. A superficial comparison of activated construction of hierarchical tree structures (Figure 2A), just like the regions between linguistic and arithmetic tasks cannot resolve this generative process of integrating new words in a sentence. As a critical issue. Here we propose a direct test, based on our previous basic control, we used a digit-matching (Match) task, in which the finding that activations in the left inferior frontal gyrus (L. IFG) participants simply stored five digits in memory (Figure 1D). and the left supramarginal gyrus (L. SMG) were parametrically Under each of the three conditions in the arithmetic task, we modulated by the DoM in a linguistic task; indeed, the DoM examined which of a hierarchical tree structure model or a flat tree turned out to be the best factor among the 19 models tested [7]. If structure model could properly explain the results. A linear activations in the same regions are exactly modulated by the DoM sequence tested under the Linear condition internally combined alone in an arithmetic task, then the results would indicate arithmetic calculations of addition and subtraction tested under functional commonality between linguistic and arithmetic com- the Simple condition, whereas a quadratic sequence (e.g., 1, 2, 5, putation. We thus focus on mathematical syntax, and our goal is to 10, 17,…) tested under the Quad condition internally involved a clarify whether the computation of hierarchical tree structures in subordinate linear sequence (e.g., 1, 3, 5, 7,…). In addition, the mathematics and language share the same neural network. Match task had no arithmetic calculation. Based on the Controversial issues still remain concerning the involvement of hierarchical tree structure model (Figure 2A), the idea behind the L. IFG in mathematics. It has been proposed in a lesion study such nested task designs was to linearly increase the DoM in the that the L. IFG is a shared substrate between sentence hierarchical tree structures, i.e., ‘‘1, 2, and 3’’ under the Simple, comprehension and arithmetic calculations [17], while arithmetic Linear, and Quad conditions, respectively (see the row of the DoM and algebraic calculations seemed to be preserved in some under ‘‘Factors in the hierarchical tree structures’’ in Table 1). On agrammatic patients [18,19]. However, the knowledge about the other hand, it is possible that the participants covertly brackets and operators for determining the order of calculations verbalized each individual calculation process with digits and does not guarantee effective computation of hierarchical tree operations (e.g., ‘‘723 = 4’’). If such individual calculations were structures in mathematical expressions. Moreover, when the represented by the flat tree structure model (Figure 2B), the DoMs patients were allowed to explicitly write partial results of individual became all ‘‘1’’ under the three conditions (see the row of the DoM calculations for either given or generated brackets, as in the case of under ‘‘Factors in the flat tree structures’’ in Table 1). However, the latter studies, individual calculations could be performed the relationships among the generated digits became unnecessarily without actual construction of hierarchical tree structures for the complex in the flat tree structure model (compare gray arrows whole expression. Likewise, overlapping activation in the L. IFG under the Quad condition in Figure 2A and 2B). We predict that has been reported between sentence comprehension and arithme- recursive computation in both linguistic and arithmetic processes tic calculations [20], while other researchers opposed the automatically employs hierarchical tree structures; this argument involvement of the L. IFG in mathematics [21]. Regarding these closely resembles an argument over hierarchical versus linear- imaging experiments, various factors including memory-related order representation of a sentence, e.g., ‘‘[[To be happy] is fun]’’ factors and applications of arithmetic operations (here denoted as similar to [|327|+7]. ‘‘number of operations’’; e.g., +, 6, etc.) may have influenced We fitted parametric models of eight factors (Table 1) to cortical activation. Therefore, the effect of the DoM should be activations in each identified region, and determined the most segregated from other factors involved. crucial factor accounting for activations. The operational defini- In the present functional magnetic resonance imaging (fMRI) tions of these factors other than the DoM are as follows. ‘‘Number study, we prepared a novel arithmetic task, in which participants of nodes’’ was equal to the total number of digits that appeared in performed a series of specified arithmetic calculations without each tree structure. For the stimuli used in the Match task, there writing or overtly verbalizing partial results, and they stored two were five terminal nodes without nonterminal nodes. The factor of digits in memory for matching. We presented five digits, and tested ‘‘verbal encoding’’ in the flat tree structures was the total number three conditions in the arithmetic task: simple calculation (Simple, of all possible digits and operations verbalized for individual capitalized to indicate a condition name), linear sequence (Linear), calculations, which corresponded to the addition of two factors: and quadratic sequence (Quad) conditions. Under the Simple ‘‘number of nodes’’ in the flat tree structures and ‘‘number of condition, the participants were asked to mentally perform the operations.’’ We also considered three common factors that were addition (e.g., 3+7 = 10), as well as the subtraction, of the upper applicable to both types of tree structures. First, ‘‘number of two digits (Figure 1A). Under the Linear condition, the partici- operations’’ was equated between the Simple and Linear pants were asked to regard the upper two digits (e.g., 3 and 7) as a conditions. Secondly, ‘‘number of generated digits for calculation’’ part of a linear sequence in increasing order (3, 7, 11, 15, 19,… in was the number of digits generated temporarily in a calculation this case), in which the differences between each pair of adjacent (i.e., not available on screen). For an example under the Quad terms were constant (Figure 1B). The participants were instructed condition shown in Figure 2B, there were four generated digits to calculate the third term of the linear sequence. We employed used in this calculation, ‘‘2, 5, 3, and 5.’’ Thirdly, ‘‘number of linear sequences, because it naturally imposed recursive compu- stored digits for matching’’ was the number of memorized digits tation; e.g., 3, 7, 11, 15, 19,…, obtained by nested constructions that were used for digit-matching. Its estimate was two and five for ((((3+4) +4) +4) +4 …). The arithmetic task could not be correctly PLOS ONE | www.plosone.org 2 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Figure 1. Examples of the stimuli used in the arithmetic task and Match task. Five yellow digits, together with a green character as a cue, were presented for 5 s, followed by the presentation of one white digit (matching stimulus) for 3 s. For each example, the task- relevant digits and a brief outline of the calculation processes are shown 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 Figure 2. Hierarchical and flat tree structure models for shown here are correct examples. (A) Under the Simple condition, arithmetic calculations. For each example shown in Figure 1, an indicated by the presentation of ‘‘6’’ as a cue, the addition and entire calculation process is represented by either hierarchical (A) or flat subtraction of the upper two digits (i.e., two of the five yellow digits) (B) tree structure models, where the lowest and leftmost branch were mentally performed by the participants. We instructed the corresponds to the first arithmetic calculation performed. A digit in participants to always subtract a smaller value from a larger value. (B) black at each nonterminal node is obtained from an arithmetic Under the Linear condition, indicated by the presentation of ‘‘a ’’ as a operation (e.g., + or 2) indicated just below each node, where ‘–’ cue, the third term of a linear sequence initiated by the upper two denotes a single operation of subtracting a smaller value from a larger digits was calculated by the participants. The first row at the top of the value. The digits in red denote the DoM at individual nodes, where a right panel represents the linear sequence, and the second row with reference point of zero is at the top node. The digits in blue denote branches represents the constant differences between each pair of ‘‘number of operations.’’ A gray arrow denotes corresponding digits adjacent terms in the linear sequence, as often used in math textbooks. generated temporarily in a calculation. The stored digits are circled in (C) Under the Quad condition, indicated by the presentation of ‘‘b ’’ as a red, as shown in Figure 1. In the hierarchical tree structure model, each cue, the following arithmetic calculations were performed by using the tree structure is based on recursive computation. Under the Linear lower three digits (i.e., three of the five yellow digits). The first row in condition, the hierarchical representation of the given example is: [|32 the right panel represents the given quadratic sequence. The second 7| +7] = 11. Under the Quad condition, the hierarchical representation row represents the differences between each pair of adjacent terms in of the given example is: [||224|2 |4–9|| +5] = 8. the given sequence. The participants were asked to regard the resultant doi:10.1371/journal.pone.0111439.g002 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, null or negative (see Table 1). Moreover, the Linear – Simple indicated by the presentation of ‘‘m’’ as a cue, the participants stored all contrast was also free from the following factors: ‘‘number of of the five digits in memory. nodes’’ in either tree structures, the DoM and ‘‘verbal encoding’’ doi:10.1371/journal.pone.0111439.g001 in the flat tree structures, ‘‘number of operations,’’ and ‘‘number of stored digits for matching.’’ The Quad – Linear contrast the arithmetic task and the Match task, respectively (the digits provides no further information, since no additional factors can be circled in red in Figure 1 and 2). controlled. To examine the functional specialization of cortical By taking the Match task as a basic control for the arithmetic regions in an unbiased manner [22], we adopted whole-brain task, we eliminated any task-related cognitive factors, such as analyses in such stringent contrasts as Simple – Match and Linear visual processing of stimuli, the identification of presented digits, – Simple. digit-matching, and motor responses. Both ‘‘number of generated Recent neuroimaging studies have examined functional con- digits for calculation’’ and ‘‘number of stored digits for matching’’ nectivity underlying elementary calculations. For example, it has might contribute to loads of short-term memory or ‘‘working been reported that the magnitude of functional connectivity memory,’’ but the Simple – Match contrast was free from these between the bilateral intraparietal sulcus (IPS) correlated with the memory-related factors, because their subtracted estimates were performances of a subtraction task [23]. Another fMRI study with PLOS ONE | www.plosone.org 3 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Table 1. Estimates of various factors to account for activations. Factors in the hierarchical tree structures Factor Simple Linear Quad Match DoM 1 2 3 0 No. of nodes 6 5 9 5 Simple – Match Linear – Match Quad – Match Linear – Simple DoM 1 2 3 1 No. of nodes 1 0 4 21 Factors in the flat tree structures Factor Simple Linear Quad Match DoM 1 1 1 0 No. of nodes 6 6 12 5 Verbal encoding 8 8 16 5 Simple – Match Linear – Match Quad – Match Linear – Simple DoM 1 1 1 0 No. of nodes 1 1 7 0 Verbal encoding 3 3 11 0 Common factors Factor Simple Linear Quad Match No. of operations 2 2 4 0 No. of generated 01 4 0 digits for calculation No. of stored 22 2 5 digits for matching Simple – Match Linear – Match Quad – Match Linear – Simple No. of operations 2 2 4 0 No. of generated 0 14 1 digits for calculation No. of stored digits 23 23 230 for matching 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 of operations’’ 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 Granger causality mapping reported that a participant group with (laterality quotient: 50–100), according to the Edinburgh inventory high arithmetic scores in a multiplication task had stronger [27]. Prior to their participation in the study, written informed bidirectional connections between the L. IPS and R. IPS than the consent was obtained from each participant after the nature and group with low arithmetic scores [24]. Both of those connectivity possible consequences of the studies were explained. Approval for studies with elementary calculation tasks did not involve a the experiments was obtained from the institutional review board recursive application of arithmetic operations. The effective of the University of Tokyo, Komaba. connectivity during arithmetic tasks with recursive computation, as well as the connectivity between the L. IFG and bilateral IPS, Stimuli should be thus clarified. In the present study, we adopted dynamic In each trial (Figure 1), we presented a main stimulus consisting causal modeling instead of Granger causality mapping, because of five yellow digits (from 1 to 9) and a green character (6,a ,b , n n dynamic causal modeling has been shown to be more effective for or m). Each green character was used as a cue to indicate one of fMRI data [25,26]. Our findings would elucidate the crucial the four conditions: ‘‘6’’ for the Simple condition, ‘‘a ’’ network of the L. IFG, L. IPS, and R. IPS for computing representing a linear sequence for the Linear condition (e.g., hierarchical tree structures in mathematics. a = 1, 3, 5, 7,…), ‘‘b ’’ representing a quadratic sequence for the n n Quad condition (e.g., b = 1, 2, 5, 10,…), and ‘‘m’’ for the Match Materials and Methods task. According to a pilot study, the duration of 5 s for the main stimulus was long enough for the participants to correctly perform Participants the task. At the center of the screen, one white digit (from 0 to 9) Twenty college students (15 males, aged 18–30 years), who had was subsequently presented for 3 s as a matching stimulus, not majored in mathematics but studied high school mathematics followed by a 200 ms blank to make the duration of the trial twice including linear and quadratic sequences, participated in the fMRI as long as the repetition time of the fMRI scans. experiment. All participants were healthy and right-handed PLOS ONE | www.plosone.org 4 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math To make the stimuli physically identical among the three scanning. Eight scanning sessions were performed in one day. A single scanning session contained 36 trials (six trials for each of the conditions in the arithmetic task, except for the cue characters, we used the same set of main stimuli, in which the five digits were Simple, Linear, and Quad conditions; 18 trials for the Match task). The arithmetic and Match tasks were alternately performed, so always arranged in two rows (48 different combinations of digits). The upper two digits were relevant to the Simple and Linear that MR signals for each condition in the arithmetic task were conditions, while the lower three digits were relevant to the Quad sufficiently separated. In the arithmetic task, no stimuli appeared condition. The digits in each row were arranged in increasing twice or more times across all scanning sessions, whereas each order from left to right, while no digit appeared twice in an entire stimulus appeared twice in the Match task. To prevent any stimulus. For the upper two digits, we excluded such ubiquitous condition-specific strategy, the order of the Simple, Linear, and combinations as ‘‘2 and 4,’’ ‘‘3 and 6,’’ or ‘‘4 and 8,’’ as well as Quad conditions was pseudorandomized; the orders of tasks and trivial combinations with the constant difference of one (e.g., ‘‘2 conditions were also counter-balanced across participants. and 3’’). For the lower three digits, we excluded certain combinations (e.g., ‘‘1, 2, and 4’’), in which the subordinate linear MRI Data Acquisition sequence became trivial (1, 2, 3, 4,… in this case). In the Match The fMRI scans were conducted on a 3.0 T scanner (Signa task, all digits in both rows were memorized; we prepared 72 HDxt; GE Healthcare, Milwaukee, WI) with a bird-cage head coil. different combinations of digits, including all combinations used in For the fMRI, we scanned 40 axial slices that were 3-mm thick the arithmetic task. with a 0.3-mm gap, covering from 252.8 to 78.9 mm from the The participants wore earplugs and an eyeglass-like MRI- anterior to posterior commissure line in the vertical direction, compatible display (resolution, 8006600 pixels; VisuaStim Digital, using a gradient-echo echo-planar imaging sequence [repetition Resonance Technology Inc., Northridge, CA). For fixation, a time = 4.1 s, echo time = 60 ms, flip angle = 90u, field of view = 2 2 small red cross was always shown at the center of the screen to 1926192 mm , resolution = 363mm ]. In a single scanning initiate eye movements from the same fixed position, and the session, we obtained 72 volumes following four dummy images, participants were instructed to return their eyes to this position for which allowed for the rise of the MR signals. For each participant, the matching stimulus. Reaction times were measured from the sessions without head movement were used for analyses; the onset of the matching stimulus. The stimulus presentation and number of abandoned sessions was less than four for all collection of behavioral data (accuracy and reaction times) were participants. After completion of the fMRI sessions, high- controlled using the LabVIEW software and interface (National resolution T1-weighted images of the whole brain (192 axial Instruments, Austin, TX). slices, 16161mm ) were acquired from all participants with a fast spoiled gradient recalled acquisition in the steady state sequence Tasks (repetition time = 8.4 ms, echo time = 2.6 ms, flip angle = 25u, field of view = 2566256 mm ). In each trial of the arithmetic task, the participants were asked to silently perform a series of specified arithmetic calculations, and to store two digits in memory obtained from the arithmetic fMRI Data Analyses calculations without using their fingers. The participants then We performed data analyses with fMRI using SPM8 statistical judged whether or not one of the stored digits matched the white parametric mapping software (Wellcome Trust Centre for digit, and responded by pressing one of two nonmagnetic buttons Neuroimaging, London, UK; http://www.fil.ion.ucl.ac.uk/spm/) (Current Designs, Inc., Philadelphia, PA): a right button if [28], implemented on a MATLAB platform (MathWorks, Natick, matched (in half of the trials), or a left button if mismatched (in MA). The acquisition timing of each slice was corrected using the the other half). middle slice as a reference for the echo-planar imaging data. We Under the Simple condition (Figure 1A), the participants stored realigned the echo-planar imaging data in multiple sessions to the the results of addition and subtraction in memory. When the result first volume in all sessions, and removed sessions that included was a ‘‘two-figure number’’ (e.g., 10), the participants simply data with a translation of.2 mm in any of the three directions and stored the last digit (i.e., the digit in ‘‘the one’s place’’ in math with a rotation of.1.4u around any of the three axes; these terms, 0 in this case) in memory. We instructed them to always thresholds were empirically determined from our previous studies subtract a smaller value from a larger value (e.g., |327| = 4), as [29–31]. represented by the sign of absolute value here. Under the Linear Each participant’s T1-weighted structural image was coregis- condition (Figure 1B), the participants were instructed to obtain tered to the mean functional image generated during realignment. the constant difference first by the subtraction (e.g., |327| = 4), The coregistered structural image was spatially normalized to the and then to calculate the third term by adding the constant standard brain space as defined by the Montreal Neurological difference to the second term (4+7 = 11 in this case); they stored Institute using the ‘‘unified segmentation’’ algorithm with medium the results of these arithmetic calculations (4 and 1 in this case) in regularization, which is a generative model that combines tissue memory. Under the Quad condition (Figure 1C), the participants segmentation, bias correction, and spatial normalization in the were instructed to perform a series of subtractions (e.g., |224| inversion of a single unified model [32]. After spatial normaliza- = 2, and then |429| = 5) in order to obtain the constant tion, the resultant deformation field was applied to the realigned difference of the subordinate linear sequence by the subtraction functional imaging data, which was resampled every 3 mm using (|225| = 3 in this case). They were further instructed to calculate seventh-degree B-spline interpolation. All normalized functional the third term of the subordinate linear sequence by adding the images were then smoothed by using an isotropic Gaussian kernel constant difference to the second term (3+5 = 8). They stored the of 9 mm full-width at half maximum. Low-frequency noise was results of these arithmetic calculations (3 and 8 in this case) in removed by high-pass filtering at 1/128 Hz. memory. In a first-level analysis (i.e., a fixed-effects analysis), each In the Match task (Figure 1D), the participants simply stored all participant’s hemodynamic responses induced by the trials were of five yellow digits in memory, and judged whether one of those modeled with a box-car function (band-pass during 1–3 s after the digits matched the white digit. The participants underwent main stimulus onset) convolved with a hemodynamic function. practice sessions for these arithmetic and Match tasks before The 0–1 s period from the main stimulus onset was related with PLOS ONE | www.plosone.org 5 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math the identification of a cue, and the 3–5 s period was likely to be Results confounded with preparatory processes of digit-matching. We thus Behavioral Results selected the 1–3 s period in order to focus on arithmetic The behavioral data are shown in Table 2, indicating that the calculations. Only event-related responses of correct trials were tasks were performed almost perfectly. With respect to accuracy, a analyzed. The images of the three conditions in the arithmetic task one-way repeated measures analysis of variance showed a and Match tasks were then generated in a general linear model for significant main effect of condition including the Match task each participant, and used for intersubject comparisons in a [F(3, 57) = 20, P,0.0001]. Post-hoc paired t-tests among all second-level analysis (i.e., random-effects analysis). To discount conditions (significance level at a = 0.0083, Bonferroni-corrected) any general effects associated with task difficulty or performance showed that the accuracy under the Quad condition was differences among the participants, individual error rates (100 – significantly lower than that under the other conditions (P, accuracy), which were more sensitive than reaction times in our 0.001), while the differences of accuracy among the other three tasks, were entered for each task as a nuisance factor. For all fMRI conditions were not significant (P.0.1). Regarding reaction times data analyses, the statistical threshold was set to P,0.05 for the (RTs), a main effect of condition was significant [F(3, 57) = 12, voxel level, corrected for multiple comparisons [family-wise error P,0.0001], and the RTs under the Quad condition was correction] across the whole brain. significantly longer than those under the Simple condition (P, For the anatomical identification of activated regions, we 0.001). The Match task was also more demanding than the Simple basically used the Anatomical Automatic Labeling method [33]. In and Linear conditions, leading to significantly longer RTs (P, region of interest analyses, we extracted the percent signal changes 0.001). averaged among participants at each local maximum using the MarsBaR-toolbox (http://marsbar.sourceforge.net/). To statisti- Selective Activation in the Arithmetic Task cally evaluate the fitness of a single factor’s parametric model to To identify cortical regions involved in the arithmetic task, we activations, we calculated the coefficient of determination (r ) and first tested the Linear – Match contrast with the liberal control of a residual sum of squares with MATLAB; we obtained the fitted Match. We then tested the direct Quad – Simple contrast, values by multiplying the estimates of the factor (see Table 1) by a 2 2 2 inclusively masked with Linear – Match to guarantee the fitting scale. For a no-intercept model, r =12 S(y2ˆy) /Sy consistency of activation patterns. In both contrasts, overall should be calculated, where y and y denote the fitted values and activation was clearly left-dominant. Significant activation was the observed signal changes for each contrast, respectively [34]. observed in the L. IFG (Brodmann’s areas 44/45), bilateral SMG For this calculation, we used R software (http://www.r-project. (Brodmann’s area 40), bilateral IPS (Brodmann’s areas 7/39/40), org/). To further take account of individual variability, we used a and precuneus (Brodmann’s area 7), as well as in the bilateral restricted maximum-likelihood method, fitting ‘‘linear mixed- lateral premotor cortex (LPMC, Brodmann’s areas 6/8), left effects models’’ with individual activations as dependent variables, anterior short insular gyrus (ASG), and pre-supplementary motor with the estimates of each factor as a regressor, and with the area (pre-SMA, Brodmann’s areas 6/8) (Figure 3A, 3B, and participants as random effects. For this calculation, we used an Table 3). The left middle temporal gyrus (L. MTG, Brodmann’s nlme (linear and nonlinear mixed-effects models) package (http:// area 21) was significantly activated only in Linear – Match. cran.r-project.org/web/packages/nlme/) on R software. We tried to narrow down the critical regions for the computation of hierarchical tree structures, by using more Dynamic Causal Modeling Data Analyses stringent contrasts. In Simple – Match, where the memory-related We performed data analyses with dynamic causal modeling factors of ‘‘number of generated digits for calculation’’ and using DCM10 on SPM8 [35]. We concatenated the scans from the ‘‘number of stored digits for matching’’ were eliminated, separate sessions, and reanalyzed the preprocessed data with the significant activation was localized in the L. IFG and L. SMG general linear model, which contained two regressors representing (Figure 3C and Table 3), indicating that activations in these the Quad condition and Match task for making a meaningful regions were free from the memory-related factors. Moreover, in contrast, as well as a regressor representing both the Quad and Linear – Simple, significant activation was observed in the Linear conditions for driving inputs. The regressor representing bilateral IPS and precuneus (Figure 3D and Table 3), indicating the Quad condition was also used for a modulatory effect. The that activations in these regions were independent from other effects of transition between sessions were taken into account with possible factors. Although activations in the L. IFG and L. SMG regressors of sessions. With the volume-of-interest tool in SPM8, were below the threshold in Linear – Simple, they were significant the time series was extracted by taking the first eigenvariate across with small volume correction (corrected P,0.05, 9 mm radius all suprathreshold voxels in Quad – Match for each participant from each local maximum determined by Simple – Match). In (uncorrected P,0.05), confined within 6 mm from the group local Simple – Match with small volume correction, activations in the L. maximum of a single region. IPS, but not those in the R. IPS or precuneus, were significant (9 mm radius from each local maximum determined by Linear – We specified 18 models with systematic variations in a modulatory effect and driving inputs (Figure S1). After estimating Simple). These results suggest that the L. IFG, L. SMG, and L. IPS, together with the limited contribution of the R. IPS and all models for each participant, we identified the most likely model precuneus, are well specialized in arithmetic calculations. by 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 the Cortical Activations Specifically Modulated by the DoM in model for any randomly selected participants, or on the the Hierarchical Tree Structures exceedance probability, i.e., the probability that the model is a At the local maxima of these five regions, we further examined better fit to the data than any other models tested. After percent signal changes under each condition in the arithmetic task determining the best model, we evaluated the parameter estimates with reference to the Match task (Figure 4). A linear modulation of of this particular model by one-sample t-tests [36]. activations was observed in the L. IFG, L. SMG, and bilateral IPS among these three conditions, indicating that the results were PLOS ONE | www.plosone.org 6 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math Table 2. Behavioral data under each condition. Simple Linear 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 consistent with the DoM in the hierarchical tree structures. The estimates multiplied by the fitting scale) against corresponding precuneus showed weaker activations under the Simple condition, signal changes averaged across participants (Table 5, Tables S1- exhibiting relatively larger responses under the more demanding S4 in File S1). Linear and Quad conditions (Figure 4E). Among the parametric models of eight factors tested, the model Next we examined how well activations in each of the L. IFG, of DoM in the hierarchical tree structures produced by far the L. SMG, bilateral IPS, and precuneus correlated with the DoM in lowest residual sum of squares value (#0.0043) and the largest the hierarchical tree structures and other factors. All of the Simple coefficient of determination (r )($0.99) for the L. IFG, L. SMG, – Match, Linear – Match, and Quad – Match contrasts predicted and bilateral IPS. We further evaluated the goodness of fit for each that activations should be exactly zero when a factor produced no model by using one-sample t-tests (significance level at a = 0.0167, effect or load relative to the Match task. We thus adopted a no- Bonferroni-corrected) between the fitted value for each contrast intercept model, in which the percent signal changes of each and individual activations. The model of DoM for these four region were fitted with a single (thus minimal) scale parameter to a regions produced no significant deviation for the three contrasts model of each factor using its subtracted estimates (Table 4). For (P$0.13). In the precuneus, the goodness of fit was marginal for the three contrasts, a least-squares method was used to minimize one P-value (P = 0.027) under the Simple condition. By fitting the residual sum of squares for the three fitted values (i.e., the three ‘‘linear mixed-effects models,’’ we found that the model of DoM 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 lateral surfaces 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 PLOS ONE | www.plosone.org 7 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math PLOS ONE | www.plosone.org 8 November 2014 | Volume 9 | Issue 11 | e111439 Table 3. Significantly activated regions in the arithmetic task. Contrast Brain region BA Side xyz Z-Value Voxels Linear – Match LPMC 6/8 L 227 5 55 5.5 82 R 33 5 58 4.5 5 IFG 44/45 L 245 11 25 6.8 309 251 32 22 5.2 * ASG – L 230 26 1 4.6 5 pre-SMA 6/8 M 2620 46 6.0 53 SMG 40 L 245 240 43 6.9 483 IPS 7/39/40 L 233 252 40 6.8 * SMG 40 R 48 231 46 4.5 5 IPS 7/39/40 R 36 255 43 6.0 169 Precuneus 7 M 26 270 46 6.1 155 MTG 21 L 251 255 211 5.2 15 Quad – Simple, masked with Linear – Match LPMC 6/8 L 224 5 55 5.4 62 R 30 5 58 5.6 5 IFG 44/45 L 245 5 37 5.9 131 245 11 25 5.4 * ASG – L 230 23 1 6.1 5 pre-SMA 6/8 M 2620 43 6.4 51 0 11 55 6.1 * SMG 40 L 248 237 43 5.4 352 IPS 7/39/40 L 233 249 40 5.8 * 227 270 43 5.5 * SMG 40 R 45 231 43 5.9 5 IPS 7/39/40 R 39 252 43 5.4 110 33 261 34 4.8 * Precuneus 7 M 212 261 52 6.3 155 9 270 49 6.2 * Simple – Match IFG 44/45 L 245 11 25 4.6 5 SMG 40 L 248 237 46 4.8 12 Linear – Simple IPS 7/39/40 L 230 270 46 4.6 3 R39 255 43 4.5 4 Precuneus 7 M 23 270 43 4.7 28 Stereotactic 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 short insular 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 an asterisk is included within the same cluster shown one row above. doi:10.1371/journal.pone.0111439.t003 Neural Basis of Hierarchical Processes in Math Figure 4. Quantified activations modulated by the DoM in the hierarchical 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 (see Table 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 the hierarchical tree structures. Effective Connectivity among the L. IFG and Bilateral IPS Because the L. SMG is relatively close to the L. IPS, and the precuneus locates midway between the bilateral IPS, we focused on the L. IFG and bilateral IPS alone for modeling the effective connectivity in the dynamic causal modeling analyses. Here we assumed 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 we tested six models with a modulatory effect under the Quad condition on the unidirectional or bidirectional connections (see Figure S1 for all models tested). Using a random-effects Bayesian model selection, we showed that the model 1 (Figure 5A), with a modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, yielded by far the highest 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 the participants. 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 a parameter 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 structures in mathematics, and we obtained three striking results. First, we found significant activation in the L. IFG, L. SMG, bilateral IPS, and precuneus selectively among the three conditions in the arithmetic task (Figure 3). Secondly, by examining percent signal changes in each region, a linear modulation of activations was observed in the L. IFG, L. SMG, and bilateral IPS among these three conditions (Figure 4). Moreover, by fitting the parametric models of eight factors, we found that the DoM in the hierarchical tree 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 of mathematical syntax processed in these regions, excluding the load on ‘‘working memory.’’ The dominance of the hierarchical PLOS ONE | www.plosone.org 9 November 2014 | Volume 9 | Issue 11 | e111439 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 tree structure model is consistent with our previous results of the L. a matter of controversy whether the L. IFG is also critically IFG and L. SMG activation in the direct comparison between involved in mathematics. It was claimed in a previous study that sentences and letter strings, which were assumed to have ‘‘it [the L. IFG] does not appear to play a dominant role [in hierarchical tree structures and flat tree structures, respectively mathematics], which instead is taken up by fusiform, parietal and [7]. Thirdly, using dynamic causal modeling, we showed that the precentral cortices,’’ i.e., visuo-spatial areas [21]. In this previous model with a modulatory effect for the connection from the L. IPS study, the participants performed a short-term matching task, to the L. IFG, and with driving inputs into the L. IFG, was highly which was solved without requiring any arithmetic calculation. In probable (Figure 5). For this best model, the top-down intrinsic contrast, Makuuchi et al. [20] showed overlapped activation across connection from the L. IFG to the L. IPS, as well as that from the sentence comprehension and arithmetic calculations, with an L. IPS to the R. IPS, would provide a feedforward signal with their increase of activations in the L. IFG for the higher level of reverse connections representing a negative feedback signal. These structural hierarchy, although the modulation of activations in the results indicate that the network of the L. IFG and bilateral IPS bilateral IPS were not fully examined. The present results clearly subserves the computation of hierarchical tree structures in showed that the DoM in the hierarchical tree structures was by far mathematics. more effective than the load on ‘‘working memory,’’ ‘‘number of Previous imaging studies have established that syntactic operations,’’ and other factors for explaining the L. IFG and bilateral IPS activations. processes during sentence comprehension selectively activate the L. IFG and/or the L. LPMC [29,37–41]. By directly contrasting a According to some previous imaging studies, the domain- demanding condition for ‘‘working memory,’’ we have previously specificity for the arithmetic calculation in the L. IFG and other demonstrated that both regions are indeed independent from such regions has remained unclear. For example, in recent fMRI studies domain-general cognitive factors [29], indicating that these regions [43,44], two conditions with large and small numbers were have a critical role as a putative grammar center [42]. It has been compared in an addition task, and such domain-general factors as Table 5. Fittings and likelihood of various models tested in the L. IFG. Factors in the hierarchical tree structures Factor RSS r P-values Log-likelihood Likelihood ratio *DoM 0.0024 . 0.99 0.34, 0.75, 0.97 15.0 1 –16 No. of nodes 0.18 0.70 , 0.0001, 0.054, 0.65 –20.5 3.9 6 10 Factors in the flat tree structures Factor RSS r P-values Log-likelihood Likelihood ratio –8 DoM 0.062 0.90 0.0016, 0.0025, 0.85 –2.5 2.6 6 10 –14 No. of nodes 0.13 0.78 , 0.0001, 0.0033, 0.20 –15.8 4.46 10 –10 Verbal encoding 0.063 0.89 , 0.0001, 0.14, 0.17 –6.7 3.7 6 10 Common factors Factor RSS r P-values Log-likelihood Likelihood ratio No. of operations 0.015 0.98 0.033, 0.21, 0.68 9.6 0.0044 –14 No. of generated digits for 0.13 0.78 , 0.0001, , 0.0001, 0.24 –15.0 9.1 6 10 calculation No. of stored digits for 0.60 0 , 0.0001, , 0.0001, , n/a n/a matching 0.0001 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. Note that 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 (r ), and larger P-values. The likelihood of models with all null estimates was not calculable (n/a). A likelihood ratio is the ratio of each model’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 PLOS ONE | www.plosone.org 10 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math naturally extended to ‘‘musical syntax’’ as well, since harmonic progressions 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 of numbers [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 repeti- tion priming of numbers (e.g., from ‘‘NINE’’ to ‘‘9’’) caused the activation 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 and adjacent L. SMG are also involved in language, especially in vocabulary knowledge or lexical processing [54,55], as well as in searching syntactic features [7]. Taken these and present results together, the L. IPS/SMG would be also involved in both mathematics 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 Quad conditions. Using a visual picture-sentence matching task, we have recently tested twenty-one patients with a left frontal glioma, and found abnormal 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, and cerebellar nuclei, which were overactivated in the patients with a glioma 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 underactivated in the patients with a glioma in the opercular/triangular parts of Figure 5. Effective connectivity among the L. IFG and bilateral the L. IFG. Among the activated regions in the present study IPS. (A) The best model with a positive modulatory effect for the bottom-up connection from the L. IPS to the L. IFG, and with driving (Table 3), the L. IFG, L. IPS, R. LPMC, R. IFG, and pre-SMA inputs into the L. IFG. Mean parameter estimates that exceeded the are included in the network I, whereas the L. LPMC and L. MTG statistical threshold (corrected P,0.05) are indicated alongside the are included in the network II and network III, respectively. The intrinsic connections. Bar graphs show expected probabilities (B) and overall activated regions for the arithmetic calculations thus share exceedance probabilities (C) of all models tested (Figure S1). their functional roles with the syntax-related regions in language. doi:10.1371/journal.pone.0111439.g005 Our previous fMRI study revealed that the functional connectivity between the L. IFG and L. SMG was selectively task difficulty might explain the enhanced activations. It should be enhanced during sentence processing [59]. A recent dynamic noted that any hierarchical processes associated with ‘‘two-figure causal modeling study with a cross-modal picture-sentence numbers,’’ as well as more verbal encoding, are also involved in matching task has suggested that the L. IFG received driving the task with large numbers. In the present study, we discounted inputs and transferred that information to the temporal regions any general effects associated with task difficulty by entering [60]. Our recent dynamic causal modeling study has suggested a individual error rates for each task as a nuisance factor. We have top-down intrinsic information flow of syntactic processing from previously demonstrated that the L. IFG is a domain-specific the L. IFG to the L. SMG, with driving inputs into the L. IFG [7]. neural system for syntactic computation in language, which is This model is consistent with our present results of dynamic causal separable from other domain-general cognitive systems [42]. In modeling, which further indicate that L. IPS activations mirrored the present study, we indicate that the hierarchical tree structures a top-down influence regarding the DoM in the hierarchical tree in mathematics are also computed by the same domain-specific structures computed in the L. IFG. For the bottom-up connection system. Our successful approach on mathematical syntax can be from the L. IPS/SMG to the L. IFG, the modulatory effect under PLOS ONE | www.plosone.org 11 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math the conditions with the largest DoM was negative in this previous bidirectional connections between the L. IFG and L. IPS, as well study, whereas the modulatory effect under the Quad condition as between the L. IPS and R. IPS. Eighteen models were (with the largest DoM) was positive in the present study. While systematically constructed with driving inputs into one of the three lexical feedback was minimum for processing jabberwocky regions. For each input type, we tested six models for the sentences in the previous study, a positive feedback about modulatory effect under the Quad condition. operations would be utilized for constructing hierarchical tree (TIF) structures in the present paradigm. File S1 Table S1: Fittings and likelihood of various models tested The present results of dynamic causal modeling suggest that the in the L. SMG. Table S2: Fittings and likelihood of various models syntactic information on hierarchical tree structures provided in tested in the L. IPS. Table S3: Fittings and likelihood of various the L. IFG would be further processed through the top-down models tested in the R. IPS. Table S4: Fittings and likelihood of intrinsic connection from the L. IPS to the R. IPS (Figure 5). The various models tested in the precuneus. L. IPS and R. IPS may have different roles in processing (DOC) arithmetic calculations, but their individual roles in mathematical syntax should be clarified in the future studies. In addition, it is Acknowledgments possible that the L. MTG, significantly activated in Linear – Match, is involved in mathematical semantics, as this region would We thank S. Ohta and H. Miyashita for their helpful discussions, N. subserve semantics in language. We indicate that mathematics and Komoro for her technical assistance, and H. Matsuda for her administra- language share the network of the L. IFG and L. IPS/SMG for the tive assistance. computation of hierarchical tree structures, and that mathematics recruits the additional network of the L. IPS and R. IPS, with an Author Contributions information flow from the former to the latter. Conceived and designed the experiments: TN KLS. Performed the experiments: TN KLS. Analyzed the data: TN KLS. Wrote the paper: TN Supporting Information KLS. Figure S1 Models tested in the dynamic causal model- ing analyses. We assumed intrinsic, i.e., task-independent, References 1. Chomsky N (1957) Syntactic Structures. The Hague: Mouton Publishers. 117 p. 22. Friston KJ, Henson RN (2006) Commentary on: Divide and conquer; A defence of functional localisers. Neuroimage 30: 1097–1099. 2. Chomsky N (1965) Aspects of the Theory of Syntax. Cambridge, MA: The MIT Press. 251 p. 23. Park J, Park DC, Polk TA (2013) Parietal functional connectivity in numerical 3. Ernest P (1987) A model of the cognitive meaning of mathematical expressions. cognition. Cereb Cortex 23: 2127–2135. Br J Educ Psychol 57: 343–370. 24. Krueger F, Landgraf S, van der Meer E, Deshpande G, Hu X (2011) Effective 4. Chomsky N (2007) Biolinguistic explorations: Design, development, evolution. connectivity of the multiplication network: A functional MRI and multivariate Int J Philos Stud 15: 1–21. Granger causality mapping study. Hum Brain Mapp32: 1419–1431. 5. Chomsky N (1995) The Minimalist Program. Cambridge, MA: The MIT Press. 25. David O, Guillemain I, Saillet S, Reyt S, Deransart C, et al. (2008) Identifying 420 p. neural drivers with functional MRI: An electrophysiological validation. PLOS 6. Fukui N (2011) Merge and Bare Phrase Structure. In: Boeckx C, editor. The Biol 6, e315: 2683–2697. Oxford Handbook of Linguistic Minimalism.Oxford, UK: Oxford University 26. Friston K (2009) Causal modelling and brain connectivity in functional magnetic Press. pp. 73–95. resonance imaging. PLOS Biol 7, e1000033: 220–225. 7. Ohta S, Fukui N, Sakai KL (2013) Syntactic computation in the human brain: 27. Oldfield RC (1971) The assessment and analysis of handedness: The Edinburgh The Degree of Merger as a key factor. PLOS ONE 8, e56230: 1–16. inventory. Neuropsychologia 9: 97–113. 8. Ohta S, Fukui N, Sakai KL (2013) Computational principles of syntax in the 28. Friston KJ, Holmes AP, Worsley KJ, Poline J-P, Frith CD, et al. (1995) Statistical regions specialized for language: Integrating theoretical linguistics and functional parametric maps in functional imaging: A general linear approach. Hum Brain neuroimaging. Front Behav Neurosci 7, 204: 1–13. Mapp 2: 189–210. 9. Cappelletti M, Butterworth B, Kopelman M (2001) Spared numerical abilities in 29. Hashimoto R, Sakai KL (2002) Specialization in the left prefrontal cortex for a case of semantic dementia. Neuropsychologia 39: 1224–1239. sentence comprehension. Neuron 35: 589–597. 10. Zago L, Pesenti M, Mellet E, Crivello F, Mazoyer B, et al. (2001) Neural 30. Suzuki K, Sakai KL (2003) An event-related fMRI study of explicit syntactic correlates of simple and complex mental calculation. Neuroimage 13: 314–327. processing of normal/anomalous sentences in contrast to implicit syntactic 11. Butterworth B (2005) Developmental Dyscalculia. In: Campbell JID, editor. processing. Cereb Cortex 13: 517–526. Handbook of Mathematical Cognition.New York, NY: Psychology Press. pp. 455– 31. Kinno R, Kawamura M, Shioda S, Sakai KL (2008) Neural correlates of 467. noncanonical syntactic processing revealed by a picture-sentence matching task. 12. Fedorenko E, Behr MK, Kanwisher N (2011) Functional specificity for high- Hum Brain Mapp 29: 1015–1027. level linguistic processing in the human brain. Proc Natl Acad Sci USA 108: 32. Ashburner J, Friston KJ (2005) Unified segmentation. Neuroimage 26: 839–851. 16428–16433. 33. Tzourio-Mazoyer N, Landeau B, Papathanassiou D, Crivello F, Etard O, et al. 13. Benn Y, Zheng Y, Wilkinson ID, Siegal M, Varley R (2012) Language in (2002) Automated anatomical labeling of activations in SPM using a calculation: A core mechanism? Neuropsychologia 50: 1–10. macroscopic anatomical parcellation of the MNI MRI single-subject brain. 14. Monti MM, Parsons LM, Osherson DN (2012) Thought beyond language: Neuroimage 15: 273–289. Neural dissociation of algebra and natural language. Psychol Sci 23: 914–922. 34. Kva ˚ lseth TO (1985) Cautionary note about R . Am Stat 39: 279–285. 15. Klessinger N, Szczerbinski M, Varley R (2012) The role of number words: The 35. Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuro- phonological length effect in multidigit addition. Mem Cogn 40: 1289–1302. image 19: 1273–1302. 16. Benn Y, Wilkinson ID, Zheng Y, Kadosh KC, Romanowski CAJ, et al. (2013) 36. Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, et al. Differentiating core and co-opted mechanisms in calculation: The neuroimaging (2010) Ten simple rules for dynamic causal modeling. Neuroimage 49: 3099– of calculation in aphasia. Brain Cogn 82: 254–264. 17. Baldo JV, Dronkers NF (2007) Neural correlates of arithmetic and language 37. Stromswold K, Caplan D, Alpert N, Rauch S (1996) Localization of syntactic comprehension: A common substrate? Neuropsychologia 45: 229–235. comprehension by positron emission tomography. Brain Lang 52: 452–473. 18. Varley RA, Klessinger NJC, Romanowski CAJ, Siegal M (2005) Agrammatic 38. Dapretto M, Bookheimer SY (1999) Form and content: Dissociating syntax and but numerate. Proc Natl Acad Sci USA 102: 3519–3524. semantics in sentence comprehension. Neuron 24: 427–432. 19. Klessinger N, Szczerbinski M, Varley R (2007) Algebra in a man with severe 39. Embick D, Marantz A, Miyashita Y, O’Neil W, Sakai KL (2000) A syntactic aphasia. Neuropsychologia 45: 1642–1648. specialization for Broca’s area. Proc Natl Acad Sci USA 97: 6150–6154. 20. Makuuchi M, Bahlmann J, Friederici AD (2012) An approach to separating the 40. Musso M, Moro A, Glauche V, Rijntjes M, Reichenbach J, et al. (2003) Broca’s levels of hierarchical structure building in language and mathematics. Phil area and the language instinct. Nat Neurosci 6: 774–781. Trans R Soc B 367: 2033–2045. 41. Friederici AD, Ru ¨ schemeyer S-A, Hahne A, Fiebach CJ (2003) The role of left 21. Maruyama M, Pallier C, Jobert A, Sigman M, Dehaene S (2012) The cortical inferior frontal and superior temporal cortex in sentence comprehension: representation of simple mathematical expressions. Neuroimage 61: 1444–1460. Localizing syntactic and semantic processes. Cereb Cortex 13: 170–177. PLOS ONE | www.plosone.org 12 November 2014 | Volume 9 | Issue 11 | e111439 Neural Basis of Hierarchical Processes in Math 42. Sakai KL (2005) Language acquisition and brain development. Science 310: 52. Piazza M, Pinel P, Le Bihan D, Dehaene S (2007) A magnitude code common to 815–819. numerosities and number symbols in human intraparietal cortex. Neuron 53: 43. Fedorenko E, Duncan J, Kanwisher N (2012) Language-selective and domain- 293–305. general regions lie side by side within Broca’s area. Curr Biol 22: 2059–2062. 53. Price GR, Mazzocco MMM, Ansari D (2013) Why mental arithmetic counts: 44. Fedorenko E, Duncan J, Kanwisher N (2013) Broad domain generality in focal Brain activation during single digit arithmetic predicts high school math scores. regions of frontal and parietal cortex. Proc Natl Acad Sci USA 110: 16616– J Neurosci 33: 156–163. 16621. 54. Lee H, Devlin JT, Shakeshaft C, Stewart LH, Brennan A, et al. (2007) 45. Rohrmeier M (2011) Towards a generative syntax of tonal harmony. J Math Anatomical traces of vocabulary acquisition in the adolescent brain. J Neurosci Music 5: 35–53. 27: 1184–1189. 46. Cipolotti L, Butterworth B, Denes G (1991) A specific deficit for numbers in a 55. Pattamadilok C, Knierim IN, Duncan KJK, Devlin JT (2010) How does case of dense acalculia. Brain 114: 2619–2637. learning to read affect speech perception? J Neurosci 30: 8435–8444. 47. Dehaene S, Cohen L (1997) Cerebral pathways for calculation: Double 56. Pinel P, Dehaene S, Rivie `re D, LeBihan D (2001) Modulation of parietal dissociation between rote verbal and quantitative knowledge of arithmetic. activation by semantic distance in a number comparison task. Neuroimage 14: Cortex 33: 219–250. 1013–1026. 48. Dehaene S, Spelke E, Pinel P, Stanescu R, Tsivkin S (1999) Sources of 57. Ischebeck A, Zamarian L, Egger K, Schocke M, Delazer M (2007) Imaging early mathematical thinking: Behavioral and brain-imaging evidence. Science 284: practice effects in arithmetic. Neuroimage 36: 993–1003. 970–974. 58. Kinno R, Ohta S, Muragaki Y, Maruyama T, Sakai KL (2014) Differential 49. Eger E, Sterzer P, Russ MO, Giraud A-L, Kleinschmidt A (2003) A supramodal reorganization of three syntax-related networks induced by a left frontal glioma. number representation in human intraparietal cortex. Neuron 37: 719–725. Brain 137: 1193–1212. 50. Piazza M, Izard V, Pinel P, Le Bihan D, Dehaene S (2004) Tuning curves for 59. Homae F, Yahata N, Sakai KL (2003) Selective enhancement of functional approximate numerosity in the human intraparietal sulcus. Neuron 44: 547–555. connectivity in the left prefrontal cortex during sentence processing. Neuroimage 51. Naccache L, Dehaene S (2001) The priming method: Imaging unconscious 20: 578–586. repetition priming reveals an abstract representation of number in the parietal 60. den Ouden D-B, Saur D, Mader W, Schelter B, Lukic S, et al. (2012) Network lobes. Cereb Cortex 11: 966–974. modulation during complex syntactic processing. Neuroimage 59: 815–823. PLOS ONE | www.plosone.org 13 November 2014 | Volume 9 | Issue 11 | e111439

Journal

PLoS ONEPubmed Central

Published: Nov 7, 2014

There are no references for this article.