Abstract

Mental calculation is a complex cognitive operation that is composed of a set of distinct functional processes. Using functional magnetic resonance imaging (fMRI), we mapped brain activity in healthy subjects performing arithmetical tasks and control tasks evoking a comparable load on visuo-constructive, linguistic, attentional and mnemonic functions. During calculation, as well as non-mathematical tasks, similar cortical networks consisting of bilateral prefrontal, premotor and parietal regions were activated, suggesting that most of these cortical areas do not exclusively represent modules for calculation but support more general cognitive operations that are instrumental but not specific to mental arithmetic. Significant differences between calculation and the non-mathematical tasks were found in parietal sub-regions, where non-arithmetic number or letter substitution tasks preferentially activated the superior parietal lobules whereas calculation predominantly elicited activation of the left dorsal angular gyrus and the medial parietal cortices. We interpret the latter activations to reflect sub-processes of mental calculation that are related to the processing of numerical representations during exact calculation and to arithmetical fact retrieval. Finally, we found that more complex calculation tasks involving the application of calculation rules increased activity in left inferior frontal areas that are known to subserve linguistic and working memory functions. Taken together, these findings help to embed the specific cognitive operation of calculation into a neural framework that provides the required set of instrumental components. This result may further inform the cognitive modeling of calculation and adds to the understanding of neuropsychological deficit patterns in patients.

Introduction

Since Henschen coined the term ‘acalculia’ in his classical overview of calculation impairments in brain-damaged patients (Henschen, 1919), there has been a large amount of work on both characterizing the deficits in performing calculations and on relating these to sites of lesions in neurological patients. While disturbances of calculation may occasionally be found in relative isolation from other cognitive impairments (Warrington, 1982), they are usually associated with deficits in wider functional contexts than the performance of arithmetical operations alone (Collignon et al., 1977). Neuropsychological assessment has guided researchers to define different types of acalculia. In particular, specific disturbances of arithmetical skills, called pure anarithmetia, were distinguished from other functional impairments (alexia, agraphia, visuospatial disorders and constructional apraxia) that affect calculation without a genuine deficit in number processing and calculation (Levin et al., 1993). In other words, linguistic and visuo-spatial malfunctions have inevitable repercussions on calculation, but are not specific to it.

Mirroring the diversity of clinical manifestations, the conceptual models developed for calculation assume a composite multi-level process. In their abstract–modular model of number processing McCloskey et al. distinguish input mechanisms subserving the recognition of numbers and operation symbols from output mechanisms for writing or uttering numbers (McCloskey et al., 1985). In the case of written numerical material these mechanisms involve visuo-spatial and visuo-constructive computations and linguistic processes. To perform arithmetical tasks a third component is postulated that stores basic number facts and rules, as well as procedures for more complex arithmetical operations. A different cognit- ive model of number processing is the ‘triple code theory’ (Dehaene, 1992), which proposes two representational forms for input and output devoid of semantic information (visual arabic and auditory verbal code) and a third, analogical ‘magnitude’ or ‘quantity’ representation. While an asemantic route is sufficient for reading or writing arabic numerals, mental calculation requires access to numerical meaning via a semantic route. Both models share in common the notion that calculation relies on components of a more general nature that are also required for other cognitive functions. These include sensory feature input processing, visuo-constructive capacity, symbolic operational and particularly linguistic processing, as well as memory and attention.

Parallelling the diversity of functions recruited for the execution of mental arithmetic, deficits in calculation have been associated with a variety of lesion locations in patients. From the early work of Henschen onwards, a key role has repeatedly, but not exclusively, been assigned to the left angular gyrus (Boller and Grafman, 1985). However, instead of pinpointing a specific area as a ‘calculation center’, an overview of lesion studies allows merely a vague conclusion, namely that left hemispheric lesions are more likely than right hemispheric lesions to affect calcu- lation and posterior damage more likely than anterior damage (Dahmen et al., 1982; Grafman et al., 1982). These findings suggest not only that several distinct and spatially segregated areas subserve calculation, but also that the neuronal processes performed in these areas support a broad set of cognitive oper- ations and are not confined to calculation.

So far, only a few functional magnetic resonance imaging (fMRI) studies have addressed the neural correlates of mental arithmetic (Burbaud et al., 1995, 1999; Rueckert et al., 1996; Chochon et al., 1999; Dehaene et al., 1999; Rickard et al., 2000). Together, they as well as mostly earlier studies using positron emission tomography (Roland and Friberg, 1985; Dehaene et al., 1996; Sakurai et al., 1996; Pesenti et al., 2000) describe a distributed network that subserves the performance of calcu- lation tasks, including prefrontal, premotor and parietal cortices. However, the exact interpretation of the observed activations appears difficult for at least two reasons. Firstly, the experi- mental conditions and statistical comparisons in most of these studies non-selectively highlighted structures partaking in calculation as well as lower levels of numerical processing. Therefore, the different functional contributions of these various brain sites could not be dissociated. Only recently, Dehaene and co-workers attempted to dissociate different arithmetical processes through direct comparisons of exact and approximate calculation tasks (Dehaene et al., 1999). They observed relative differences in brain activation produced by these tasks, in particular an increase in activation in the left inferior frontal lobe during exact calculations and bilaterally along the intraparietal sulci during approximation. In another fMRI study from the same laboratory, Chochon and co-workers found evidence for a partial functional–neuroanatomical dissociation of different numerical operations (digit naming, comparison, multiplication and subtraction) (Chochon et al., 1999), which is in line with neuropsychological observations in brain-damaged patients. The limitations of both studies arise from the fact that while they do compare different arithmetical processing of numerical material, they do not (and cannot within this framework) account for concomitant changes in load on underlying cognitive processes recruited for task execution. In other words, the differences observed between different more or less arithmetical tasks are confounded with task-related differences in tapping, for instance, working memory or covert verbalisation.

In the present study we therefore used fMRI to investigate more closely the determinants of activity in the brain areas cooperatively mediating mathematical performance. We chose an alternative approach to those described above in comparing brain activity during one type of mental arithmetic, i.e. exact calculation, with activity evoked by construed control tasks that were non-mathematical but imposed a similar load on the instrumental cognitive processes. Firstly, we included a result matching task to follow each series of calculations. This control situation maintained a functional load on anarithmetical num- erical processing while not necessitating any actual calculation. Furthermore, to investigate a possible task-related specificity of cerebral activation, we introduced several experimental tasks at high cognitive levels that controlled for some of the other instrumental components required for calculation (Figs 1 and 2).

Materials and Methods

Subjects and Imaging

We studied in depth six healthy male native German speakers (mean age 25.8 ± 2.9 years) who had given their written informed consent. All were consistent right-handers as assessed with a modified German version of the Edinburgh Inventory (Oldfield, 1971). All subjects were students or employees at the Research Center Jülich. We did not formally assess educational level but all had finished high school and had never experienced manifest problems with arithmetic at school. Imaging was performed on a 1.5 T MRI system (Siemens Vision, Erlangen, Germany) using a standard, circularly polarized head coil. We obtained structural images with a MPRAGE sequence (repetition time TR = 11.4 ms, echo time TE = 4.4 ms, flip angle = 15°, field of view FOV = 230 × 230 mm2, matrix size = 256 × 256, voxel size = 0.9 × 0.9 × 1.25 mm3). For functional studies we dynamically acquired series of gradient echo planar image volumes oriented in parallel with the AC–PC plane and spanning almost the entire cerebrum (TR per slice = 90 ms, TE = 66 ms, flip angle = 90°). An image volume was recorded every 3 s and comprised 16 contiguous slices (FOV = 200 × 200 mm2, matrix size = 64 × 64, voxel size = 3.1 × 3.1 × 5.0 mm3 with 0.5 mm gaps between adjacent sections). Five functional series with 96 volumes each (112 volumes in the compound calculation condition) were measured per subject, every series covering six repetitions of the respective condition. The order of conditions was randomized across subjects, each condition being announced prior to the onset of the imaging series.

Experimental Conditions

Calculation Tasks

We used two calculation conditions. Simple calculation involved multiplication and division tasks that did not go beyond stored table facts (e.g. 3 × 8, 30/6). Compound calculation involved multiplication and division tasks with larger operands (e.g. 3 × 18, 85/17) but remained in the same number domain (1–99).

The assignment of a given calculation task to mere fact retrieval or to a more complex computational process depends in part on the subject's proficiency. To verify our subjects' arithmetical performance level we carried out off-line tests by presenting 32 number multiplication and division tasks on a computer screen (e.g. 4 × 6). The subjects were instructed to respond both correctly and rapidly and the experimenter pushed a response button as soon as the oral response was initiated. Half of the tasks we assumed to require only fact retrieval, i.e. simple number multiplication or division. Identical calculations also occurred during the scanning period. Since numerical fact retrieval is highly overlearned, the preceding off-line presentation should not pose a problem for the results obtained during imaging. Carry-over of stimulus–response associations from off-line testing to the scanning period was excluded by the use of different stimulus presentation formats (whole task versus sequential, see below). The other half were compound number multiplication or division tasks, which were merely similar to those used during the subsequent scanning period. This procedure was chosen to minimize the possibility of result memorization but nonetheless to obtain valid performance time differences for the conditions tested during imaging. The order of tasks from the two classes was randomized.

During functional imaging, stimuli (see Fig. 1) were serially presented on a screen which was positioned outside the magnet coil and could be seen via a mirror placed above the subject's eyes. The instruction was to perform the calculations silently, i.e. without speaking and avoiding lip or tongue movements. Since calculation with large multiplication factors required more time than with small operands in prior testing, the rate of item presentation in compound number calculation was slowed. This resulted in fewer calculation steps in compound number calculation (three operations of 9 s each) than in the other conditions (five operations of 3 s each). Because the relative time that subjects are performing a task influences the signal intensity changes in fMRI, we preferred to engage subjects continuously on each task instead of choosing rate matching with confounding long waiting periods in all tasks except for the compound calculation task.

For similar reasons we did not employ an event-related design, as this would mean modeling each response as a function of the time required for a given computation by a given subject. Such a procedure is prone to confound intensity and duration of a neural activity change due to the intrinsic low-pass properties of hemodynamic signals.

Control Tasks

In order to selectively address those cognitive components of calculation tasks that were not related to calculation proper, we chose to create novel tasks by replacing numbers with letters and operation symbols with difference of magnitude signs (> and <). When using letters instead of numbers the resulting task was termed letter pseudo-multiplication and pseudo-division, the instruction being to covertly verbalize the required operation but to maintain the initial item as the result of every computational step (DT divided by F equals DT; see Fig. 1E). When using magnitude symbols instead of multiplication and division symbols the result was a substitution task in which the tip of the symbol indicated whether the left or the right element in the processed item was to be replaced (Fig. 1C). While letter pseudo-multiplication and pseudo-division is a nonsense task that matches for linguistic input and output processing, the number substitution task not only evokes anarithmetical number processing but also involves meaningful transformations that impose a high load on visuo-constructive capacity. Finally, a combination of both manipulations resulted in a letter substitution task analogous to the number substitution task (Fig. 1D).

Baseline Condition

Both calculation and substitution tasks were followed by a result matching period that was considered as the baseline condition. During this period the correct result from a series of sequentially presented possible results was to be indicated by a button press (Fig. 1). The position of the correct item within the series was varied across trials to prevent predictability. This procedure removed any ambiguity between activations related to the result matching period and activations related to the preceding experimental task.

In addition to the brief motor activation related to pushing the button, the result matching period maintained visual input processing and a working memory load. Although not required for correct performance, access to semantic representations of letters and numbers, including a representation of magnitude in the case of numbers, most probably also took place during this period. In short, the result matching period controlled in a meaningful context for all aspects of the activation conditions that did not involve the calculation or substitution procedures as such. The hypothesized processing components of all conditions are given in Figure 2.

Data Analysis

Data processing used software for statistical parametric mapping (SPM v.96b; Wellcome Department of Cognitive Neurology, London, http://www.fil.ion.ucl.ac.uk/spm/). The first six images of each time series were discarded to allow establishment of steady-state magnet- ization. The image volumes were realigned, corrected for motion, global signal intensity variation (proportional scaling) and low frequency fluctuations (high pass filters of 60 and 90 s cut-off), co-registered, normalized onto standard stereotactic space (using a template provided courtesy of the Montreal Neurological Institute) and smoothed with an 8 mm full-width at half-maximum Gaussian kernel (Friston et al., 1995). Image volumes obtained during the visual presentation of items after the correct response (‘baseline’) and images of series with an incorrect or missing response were modeled as separate conditions. These were not included in the contrasts tested by the statistical analyses. Further- more, one subject had to be discarded from statistical analysis due to image artifacts which we were unable to explain or correct for.

Statistical tests were based on a fixed effects model and performed for single experimental conditions by contrasting them with the individually modeled ensuing result matching periods of variable length. Across conditions we tested for component-specific effects by direct comparison of the condition-specific effects (thus formally corresponding to inter- actions). For condition-specific effects we report results for areas that achieved a significance threshold of P < 0.05 (corrected) in either of the two calculation conditions. To allow comparison we also list findings in these areas for the other conditions, provided these foci passed a significance level of P < 0.001 (uncorrected). For the same reason we list responses exceeding the significance level of P < 0.001 (uncorrected) in cases where the contralateral homologous area was activated at P < 0.05 (corrected). For comparisons between conditions we report responses at the level of P < 0.05 (corrected). Areas for which we had a prior hypothesis, e.g. the left angular gyrus, are reported if they were activated at P < 0.001 (uncorrected). For visualization, activations are displayed at P < 0.001 (uncorrected) to enhance sensitivity in delineating spatial extent.

Results

Off-line Testing of Performance

After removal of 17 trials with incorrect solutions and five outliers (> mean calculation time + 3 SD) the grand mean calculation time of the remaining 138 trials was 2.50 ± 1.20 s. Specifically, simple calculation took 1.89 ± 0.86 s for multiplication and 1.93 ± 0.80 s for division, whereas compound calculation took 3.05 ± 1.21 s for multiplication and 3.03 ± 1.38 s for division. Despite the small sample size, there was a significant effect when comparing simple with compound number calculation (ANOVA, subjects F = 21.07, P < 0.01, items F = 19.72, P < 0.001), simple calculations being faster for multiplication and division. We found no significant effect of calculation type (multiplication or division) and no interaction between the two factors. Thus, the behavioral data give evid- ence for additional costs in compound calculation compared with simple calculation (presumably reflecting additional cognitive processes), whereas there was no such difference between multiplication and division tasks in the subjects of this study.

Functional Imaging

During functional imaging we could only obtain condition- related error rates (and not reaction times) because our experimental design avoided overt responses during calculation as a potential source of artifacts. Analyzing these error rates we found no significant differences between conditions (P < 0.05) at the behavioral level, indicating that after compensation for the additional time requirements of compound calculation our experimental conditions were balanced for task difficulty. Statistical comparisons of condition-specific brain activity levels were performed in several steps. We first identified cortical regions that were activated by each condition relative to the respective result matching period. All five conditions showed a similar left-dominant bilateral prefrontal, premotor and parietal response pattern (Fig. 3 and Table 1).

In more detail, the commonly activated brain regions comprised the prefrontal cortex lining the anterior inferior frontal sulcus, the opercular part of the inferior frontal gyrus (BA 44), the premotor cortex (BA 6), the intraparietal sulcus and the adjacent inferior and superior parietal cortices and the pre- supplementary motor area, partially extending into the anterior cingulate. The lateralization of prefrontal and parietal foci to the left hemisphere during calculation shifted to a more sym- metrical distribution in the substitution conditions. Letter pseudo-calculation yielded the smallest extent of significant activation. Our further analyses were aimed at revealing differences between conditions. First, we contrasted simple number calculation with letter pseudo-calculation and number substitution. Each condition was considered with regard to the activation it produced compared to its control period. Letter pseudo-calculation matches number calculation for covert linguistic processing of the presented items and operation symbols and for maintenance of a result over a succession of items.

The comparison between simple number calculation and letter pseudo-calculation revealed left-sided activations at P < 0.001 (uncorrected) in dorsal parts of the inferior frontal gyrus (BA 44/45), in the middle third of the inferior frontal sulcus, in the ascending and descending parts of the intraparietal sulcus as well as in the posterior cingulate cortex (BA 23/31) and the adjacent precuneus (BA 31/7). A similar response pattern was seen when contrasting compound number calculation with letter pseudo-calculation (Fig. 4A), with responses in the left frontal areas and the ascending intraparietal sulcus (P < 0.05, corrected). Given that the item presentation rate was lower in compound calculation than in pseudo-calculation, these areas seem to be actively engaged in the computations performed and not merely in processing input and output.

In further comparisons we tried to differentiate activations related to visuo-spatial symbol manipulation from activations related to calculation proper. In number substitution the adequate item category is maintained but the operation symbols changed. Although < and > were explicitly chosen as in the realm of mathematically meaningful symbols, in the substitution task they did not indicate arithmetical operations but reordering of symbols engaging visuo-spatial and visuo-constructive cap- acities. Testing for greater activity during number substitution than during number calculation, we found bilateral activation of the superior parietal lobule (Fig. 4B), suggesting that number substitution had an even greater demand on such capacities than number calculation.

Comparing the parietal activation foci in Figure 4A,B, it is apparent that the parietal activation observed across the conditions shown in Figure 3AD spans the superior and inferior parietal lobules separated by the intraparietal sulcus, i.e. several functionally distinct brain areas. The opposite comparison, i.e. probing for greater activity in number calculation than substitution (Fig. 4C), revealed a focus in the left dorsal angular gyrus (P < 0.001, uncorrected). Together with the similarly localized activation focus shown in Figure 4A, these data underline the key role of the left angular gyrus in mental calculation. Moreover, this comparison confirmed the finding of strong posterior cingulate cortex activation extending into the precuneus (P < 0.05, corrected) that had already been observed when contrasting number calculation with letter pseudo- calculation.

We also tested whether our experiment could identify correlates of automatic access to a number-specific magnitude representation in an anarithmetic context. However, we observed no significant activity differences when directly contrasting number with letter substitution. This indicates that when embedded in tasks as devised here, number (as opposed to letter) processing does not detectably engage different neuronal structures.

In a final step of the analysis we attempted to differentiate between the two types of calculation tasks applied, i.e. simple and compound number calculation. When testing for greater activity in the simple compared to the compound number calculation task we found a response in the aforementioned medial parietal zone covering the precuneus and posterior cingulate cortex (P < 0.001, uncorrected). Since item presenta- tion rate was slowed for compound number calculation, this finding may simply reflect a higher rate of mathematical fact retrieval in the more rapid succession of calculations with simple numbers. Conversely, the opposite comparison probed for activation differences due to the more complex calculation processes required in compound than in simple calculation. We found left-sided activity changes at P < 0.001 (uncorrected) in dorsal parts of the inferior frontal gyrus (BA 44/45), in ventrolateral prefrontal cortex adjacent to the anterior inferior frontal sulcus and in the anterior cingulate cortex (at the border of BA 8, 24 and 32; see Fig. 4D). Interestingly, the focus in the anterior cingulate cortex was identical to that we observed when comparing number substitution with simple number calculation, i.e. it was sensitive to non-arithmetical challenge.

Discussion

The present study aimed at a better understanding of functional contributions from the diverse activation foci detected for mental calculation in more general comparisons (Rueckert et al., 1996; Sakurai et al., 1996; Burbaud et al., 1999). We therefore construed experimental control conditions that were matched for relevant components in the set of operations required for mental arithmetic (see Fig. 2). In a first step the different activation conditions, including the calculation tasks, were contrasted with result matching to reveal activation areas related to common task components. Subsequently, direct comparisons across conditions were made to detect significant differences between the tasks.

The common activations across arithmetical tasks and non- arithmetical control tasks cover a distributed cortical network that is in accordance with results from previous studies cited above and comprises mostly left-sided prefrontal, premotor and parietal regions (Table 1 and Fig. 3). These brain regions are well known to contribute to linguistic and visuo-spatial processing and to related aspects of working memory function, i.e. the phonological loop and the visuo-spatial sketchpad [see for example Baddeley, Paulesu and Swartz (Baddeley, 1992; Paulesu et al., 1993; Swartz et al., 1995)]. Hence, these results support the notion that most of the cortical areas involved in mathemat- ical tasks represent neural substrates not only for calculation proper but also for other, related cognitive operations. Correspondingly, it has been suggested that most calculation disturbances subsequent to brain lesions arise secondary to one or several of three so-called instrumental problems: spatial deficit, visuo-constructive impairment and aphasia (Collignon et al., 1977).

Besides the processes already controlled for in the baseline condition of result matching, letter pseudo-calculation shared inner speech and lexical processing of operation symbols with the other conditions. It was devised as an intermediate level control task to identify those activations in the other four conditions that were related to execution of a meaningful operation, i.e. active processing of item representations held in working memory (see Fig. 2). Comparing calculation with pseudo-calculation, increased activity was found to be left-sided in the middle third and dorsal posterior parts of the inferior frontal gyrus and medial and inferior lateral parietal regions (Fig. 4A). Similar frontal activation patterns were seen when substitution tasks were compared with letter pseudo-calculation. In sum, additional activation in the left inferior frontal cortex was induced by all tasks involving meaningful operations on working memory representations of letters or numbers. The known involvement of the dorsal inferior frontal gyrus in phonological [for an extensive review see Indefrey and Levelt (Indefrey and Levelt, 2000)], syntactic (Caplan et al., 1998, 1999) and semantic (Poldrack et al., 1998, Friederici et al., 2000) processing of language suggests that the operations required by the tasks also evoked linguistic processes. Activations in the middle third of the left inferior frontal sulcus, on the other hand, were within the variation range of Brodmann areas 9 and 46, as previously reported (Rajkowska and Goldman-Rakic, 1995). Considering that both areas are known regions of working memory functions [for reviews see Ungerleider and Goldman-Rakic (Ungerleider, 1995; Goldman-Rakic, 1996] and that, in contrast to pseudo-calculation, the meaningful tasks required frequent exchanges and manipulations of items to be held in working memory, it seems plausible to assume that the observed activations were due to additional working memory recruitment.

In the substitution tasks the operations to be performed preferentially relied on visuo-spatial processing resources (i.e. replacing the left or right digit), whereas the calculation tasks required access to number meaning and magnitude repres- entation, arithmetical processing of the memorized items and retrieval of arithmetical facts. We further assessed activations related to these different operations through direct comparisons of the respective tasks. The greater activity in the superior parietal lobule during substitution of numbers than during calculation (Fig. 4B) indicated that the substitution conditions successfully controlled for the demands on visuo-spatial and/or visuo-constructive processing. In the reverse comparison we found enhanced activity during calculation in the left dorsal angular gyrus as well as in the posterior cingulate cortex and precuneus (Fig. 4C). Ever since the publications of Peritz and Henschen (Peritz, 1918; Henschen, 1919) there have been attempts to ascertain a possible role of the (left) angular gyrus as a specific calculation center. Converging evidence from neuro- psychology, electrophysiology and functional neuroimaging points to an involvement of inferior parietal regions in math- ematical tasks, although their exact functional contributions remain ill defined. In three patients with left parietal ischemic infarctions overlapping along the intraparietal sulcus calculation disturbances were apparent, especially in more complex arithmetical problems, and were ascribed to a disruption of working memory (Takayama et al., 1994). A working memory deficit was also proposed in a more recent case report of a parietal lesion including the angular gyrus, where number transcoding but not calculation was disturbed (Markowitsch et al., 1999). In support of this view, certain components of event-related potentials (ERP) that were recorded at parietal sites during mental arithmetic have been associated with task diffi- culty and were thought to reflect the load imposed on working memory (Pauli et al., 1996). A different functional description was given for a patient with a left parietal intra-cerebral hematoma (Warrington, 1982), attributing his acalculia to faulty access to semantic entries of arithmetical facts. Correspondingly, a recent study using cortical electrostimulation in the left parietal lobe demonstrated a disruption of arithmetical fact retrieval (Whalen et al., 1997). Finally, Dehaene and Cohen assessed two patients with a double dissociation of lesion sites and behavioral effects (Dehaene and Cohen, 1997). They con- cluded that the inferior parietal lobe might contain an abstract semantic representation of numbers that is necessary to guide arithmetical fact retrieval and to perform calculations (Dehaene et al., 1998). A different interpretation of the cause of acalculia was reached in a recent very detailed case report of pure Gerstmann syndrome subsequent to a lesion of left parietal white matter (Mayer et al., 1999). Following the concept of a ‘Grundstörung’, the common cognitive denominator of the disturbed functions was hypothesized to be an impairment in mental manipulation of images. Together, these studies indicate that several operational levels of the processes required for mental calculation may depend on functional integrity of lateral inferior parietal cortices and the underlying white matter.

The most recent neuroimaging studies on number processing and mental calculation have also examined determinants of parietal activation in greater detail. ERP and fMRI data suggest that the intraparietal sulci are preferentially involved in approxi- mation and the angular gyri in exact calculation (Dehaene et al., 1999). While Rickard et al. found activations restricted to the intraparietal sulci alone (Rickard et al., 2000), a study by Chochon et al. supports the assumption that both the intra- parietal sulci and the inferior parietal lobules, i.e. dorsal angular and/or supramarginal gyri, subserve the execution of number processing tasks (Chochon et al., 1999).

Several of our findings further the understanding of the role(s) that different parietal brain areas play in the processing of numbers and in mental calculation. First of all, we failed to corroborate the hypothesis of Dehaene's triple code theory that the bilateral intraparietal sulcus corresponds to a specific semantic representation of numerical quantities (Dehaene et al., 1998). In contrast, this brain region was non-specifically activated under all experimental conditions when compared with result matching, irrespective of the material (numbers or letters) and the actual nature of the task performed. This suggests that the intraparietal sulcus participates in processes that are instrumental but not specific to mental calculation.

On the other hand, our results extend the previous findings of (left) dorsal angular gyrus activation during number processing by showing that there is more activation in this region when the brain processes numbers in an arithmetical context than in a different cognitive task. This indicates that the angular gyrus is sensitive to the arithmetical nature of the cognitive operation performed and supports the assumption of a role of the angular gyrus in excact calculation (Dehaene et al., 1999). The most recent, albeit indirect, evidence for the role of the angular gyrus comes from another fMRI study that, when comparing activation patterns in perfect and imperfect calculation performers, only found differences in the left angular gyrus, with less activation in perfect performers, and interpreted this finding in the context of proficiency (Menon et al., 2000).

Another important novel finding in our study relates to the functional contribution of the medial parietal structures activated during number calculation, i.e. the posterior cingulate cortex and the precuneus. Both neuropsychological investi- gations and functional neuroimaging studies have established the importance of these areas in memory and, in particular, retrieval functions. In semantically cued word retrieval tasks the precuneus has been shown to be sensitive to item imageability (Fletcher et al., 1996). On the other hand, a more recent study by Krause and co-workers indicates that the precuneus may be involved in episodic associative memory retrieval independent of presentation modality and imagery content of the presented material (Krause et al., 1999). Our results provide experimental support for a role of this area in arithmetic fact retrieval because the number calculation and substitution tasks differed with respect to the necessity of this process. This finding does not exclude the involvement of additional brain areas during arithmetical fact retrieval in exact calculation tasks. Along these lines, our results are in accordance with the notion that exact calculation involves structures subserving language functions (Dehaene et al., 1999) (see Fig. 3B and 4A). However, activations in left inferior frontal areas which presumably underlie these functions were canceled out when compared with tasks without calculation but with similar demands on language functions (Fig. 4C). Not surprisingly, the recruitment of language func- tions, even in number processing contexts, is not restricted to arithmetic fact retrieval. This view receives additional support from data presented by Chochon and co-workers (Chochon et al., 1999).

In sum, our findings provide evidence that the angular gyrus and the medial parietal areas play essential roles in a functional circuit ensuring mathematical performance. They support a role of the angular gyrus in exact calculation by showing that this structure is particularly sensitive to number processing in an arithmetical context. They furthermore suggest a role of medial parietal structures in the retrieval of arithmetical facts. Although these two processes inevitably interact during calculation, the possibility of a selective impairment has been shown in a patient who, subsequent to operation on a left parietal tumor, presented acalculia, agraphia, finger agnosia, right–left disorientation and apraxia but was found to display preserved arithmetical fact retrieval (Delazer and Benke, 1997).

A well-known behavioral feature of mental calculation that was also observed in our subjects is the so-called problem size effect, meaning that responses are slower on problems with larger operands (Campbell, 1987; Siegler, 1988). More difficult calculations require planned sequential usage of stored math- ematical rules. These processes take additional time compared to direct retrieval of arithmetical facts from memory. In other words, whenever direct retrieval of simple table facts is in- sufficient to solve an arithmetical problem, it will be necessary to decompose the problem, by the use of mathematical rules, into sub-steps for which single table facts can be successfully retrieved. For example, 6 × 12 may be solved through trans- formation, decomposition and recomposition in the following way: 6 × 12 = 6 × (10 + 2) = 6 × 10 + 6 × 2 = 60 + 12 = 72.

When comparing the respective activations in our study, compound calculation elicited higher activity in dorsal parts of the left inferior frontal gyrus, in the anterior inferior frontal sulcus and in the anterior cingulate cortex (see Fig. 4D). The anterior cingulate focus was also activated in the number substitution task when compared with simple number calcu- lation and can therefore not be specifically related to calculation. Its activity might indicate factors such as attentional effort or task difficulty in monitoring result generation. Activations in the inferior frontal cortex were also detected in more basic comparisons (cf. Figs 4A and 3AE, respectively) and attributed to linguistic processing (BA 44/45) as well as to the on-line maintenance of information in the presence of additional, possibly interfering, cognitive processes (anterior inferior frontal sulcus). Activity in these regions once again increased during compound calculation as compared with simple calcu- lation, suggesting a further involvement of language and working memory functions during the use of rule-based decomposition and recomposition strategies.

In conclusion, this study has confirmed that an extended bilateral prefrontal–premotor–parietal network subserves mental calculation. However, to a large extent these cortical areas do not seem to be exclusively involved in arithmetical procedures, but also in other cognitive contexts relying on similar instrumental components, such as working memory, processing symbolic information, mental image transformations and inner speech. In contrast, we have demonstrated specific functional contributions of lateral (angular gyrus) and medial (posterior cingulate and precuneus) parietal cortices to the processing of numerical representations during exact calcula- tion, including arithmetical fact retrieval. Finally, our findings suggest that the use of decomposition and recomposition strategies in more complex calculation problems does not rely on neuronal resources specific to mental arithmetic, but recruits left inferior frontal areas subserving language and working memory functions.

Notes

We would like to thank the Research Centre Jülich for providing the facilities that made this work possible and especially Dr Stefan Posse, Dr Jon Shah and Maisa Grosse-Ruyken for help in data acquisition. A.K. is supported by the Volkswagen-Stiftung.

Address correspondence to Dr Oliver Gruber, Max-Planck-Institute of Cognitive Neuroscience, PO Box 500 355, D-04303 Leipzig, Germany.

Table 1

Brain regions commonly activated across conditions (compared to result matching) as determined by statistical parametric mapping (see Materials and Methods)

Area Compound no. calculation Simple no. calculation No. substitution Letter substitution Letter pseudo-calculation 
The values given are the spatial coordinates of activation maxima and the Z scores obtained in these areas in the various conditions (cf. Fig. 1). 
aP < 0.001, uncorrected; bP < 0.05, corrected; ns, not significant. 
Anterior inferior frontal sulcus 
 Left –48,44,8; 6.92b –44,44,12; 4.58b –44,44,8; 5.86b –44,48,8; 3.75a –56,44,0; 5.55b 
 Right  48,40,12; 4.31a  52,40,8; 3.37a  48,44,4; 5.82b  48,40,4; 3.97a  ns 
Premotor cortex 
 Left –48,–8,40; 6.62b –48,0,32; 6.28b –48,0,36; 6.04b –48,4,32; 4.31a –48,–8,40; 5.30b 
 Right ns  44,–8,32; 4.85b  52,–8,36; 3.55a  52,–12,48; 3.71a  52,–8,40; 3.60a 
Inferior frontal gyrus (opercular part) 
 Left –52,8,20; 5.78b –52,8,16; 4.76b –52,8,16; 7.27b –56,8,12; 6.16b –64,12,12; 5.87b 
 Right  ns  ns  ns  ns  ns 
Intraparietal sulcus and adjacent parietal lobules 
 Left –28,–68,48; 7.02b –36,–68,48; 5.80b –32,–64,52; 7.81b –24,–68,52; 7.50b –20,–72,48; 3.85a 
 Right  12,–76,52; 4.40a  24,–64,40; 5.72b  24,–68,52; 7.74b  24,–68,52; 7.49b  32,–60,60; 4.00a 
Presupplementary motor area  –4,16,48; 3.97a  –4,8,56; 3.60a  –4,8,52; 5.46b  –4,8,52; 3.98a –12,12,52; 4.08a 
Area Compound no. calculation Simple no. calculation No. substitution Letter substitution Letter pseudo-calculation 
The values given are the spatial coordinates of activation maxima and the Z scores obtained in these areas in the various conditions (cf. Fig. 1). 
aP < 0.001, uncorrected; bP < 0.05, corrected; ns, not significant. 
Anterior inferior frontal sulcus 
 Left –48,44,8; 6.92b –44,44,12; 4.58b –44,44,8; 5.86b –44,48,8; 3.75a –56,44,0; 5.55b 
 Right  48,40,12; 4.31a  52,40,8; 3.37a  48,44,4; 5.82b  48,40,4; 3.97a  ns 
Premotor cortex 
 Left –48,–8,40; 6.62b –48,0,32; 6.28b –48,0,36; 6.04b –48,4,32; 4.31a –48,–8,40; 5.30b 
 Right ns  44,–8,32; 4.85b  52,–8,36; 3.55a  52,–12,48; 3.71a  52,–8,40; 3.60a 
Inferior frontal gyrus (opercular part) 
 Left –52,8,20; 5.78b –52,8,16; 4.76b –52,8,16; 7.27b –56,8,12; 6.16b –64,12,12; 5.87b 
 Right  ns  ns  ns  ns  ns 
Intraparietal sulcus and adjacent parietal lobules 
 Left –28,–68,48; 7.02b –36,–68,48; 5.80b –32,–64,52; 7.81b –24,–68,52; 7.50b –20,–72,48; 3.85a 
 Right  12,–76,52; 4.40a  24,–64,40; 5.72b  24,–68,52; 7.74b  24,–68,52; 7.49b  32,–60,60; 4.00a 
Presupplementary motor area  –4,16,48; 3.97a  –4,8,56; 3.60a  –4,8,52; 5.46b  –4,8,52; 3.98a –12,12,52; 4.08a 
Figure 1.

Experimental design (black arrow indicates right hand key press).

Figure 1.

Experimental design (black arrow indicates right hand key press).

Figure 2.

Hypothesized processing components of the experimental conditions: result-matching (RM), letter pesudo-calculation (LPC), letter substitution (LS), number substitution (NS), simple number calculation (SNC), compound number calculation (CNC). The darker shade of cells representing visuospatial processing in substitution tasks indicates a relatively stronger recruitment of this processing component compared to calculation tasks.

Figure 2.

Hypothesized processing components of the experimental conditions: result-matching (RM), letter pesudo-calculation (LPC), letter substitution (LS), number substitution (NS), simple number calculation (SNC), compound number calculation (CNC). The darker shade of cells representing visuospatial processing in substitution tasks indicates a relatively stronger recruitment of this processing component compared to calculation tasks.

Figure 3.

Activations during mental calculation and related control tasks. The results from statistical parametric mapping in five subjects were rendered onto the lateral and medial surfaces of a standard anatomical (T1-weighted) reference brain template. The activations show brain areas more active during experimental conditions A–E (cf. Fig. 1) than during ensuing result matching. For spatial coordinates see Table 1.

Activations during mental calculation and related control tasks. The results from statistical parametric mapping in five subjects were rendered onto the lateral and medial surfaces of a standard anatomical (T1-weighted) reference brain template. The activations show brain areas more active during experimental conditions A–E (cf. Fig. 1) than during ensuing result matching. For spatial coordinates see Table 1.

Figure 4.

Activations related to cognitive components of mental calculation. These activations were determined by contrasting the results from the calculation and control tasks shown in Figure 3 and are displayed on brain surface renderings. Calculation refers to compound number calculation in (A) and to simple number calculation in (B) and (C), substitution to number substitution (cf. Fig. 1).

Activations related to cognitive components of mental calculation. These activations were determined by contrasting the results from the calculation and control tasks shown in Figure 3 and are displayed on brain surface renderings. Calculation refers to compound number calculation in (A) and to simple number calculation in (B) and (C), substitution to number substitution (cf. Fig. 1).

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