The calculating brain: an fMRI study

Abstract

To explore brain areas involved in basic numerical computation, functional magnetic imaging (fMRI) scanning was performed on college students during performance of three tasks; simple arithmetic, numerical magnitude judgment, and a perceptual-motor control task. For the arithmetic relative to the other tasks, results for all eight subjects revealed bilateral activation in Brodmann’s area 44, in dorsolateral prefrontal cortex (areas 9 and 10), in inferior and superior parietal areas, and in lingual and fusiform gyri. Activation was stronger on the left for all subjects, but only at Brodmann’s area 44 and the parietal cortices. No activation was observed in the arithmetic task in several other areas previously implicated for arithmetic, including the angular and supramarginal gyri and the basal ganglia. In fact, angular and supramarginal gyri were significantly deactivated by the verification task relative to both the magnitude judgment and control tasks for every subject. Areas activated by the magnitude task relative to the control were more variable, but in five subjects included bilateral inferior parietal cortex. These results confirm some existing hypotheses regarding the neural basis of numerical processes, invite revision of others, and suggest productive lines for future investigation.

Introduction

Two central components of numerical cognition are simple arithmetic (e.g., 4×7=?) and magnitude judgment (e.g., 24 or 25, which is larger?). We used fMRI to explore patterns of neural activation for these tasks, with two related goals. First, we sought to confirm and extend findings of the neuropsychological literature. A wealth of patient data suggests that arithmetic is mediated by left or possibly bilateral inferior parietal areas [1], [2], [14], [15], [16], [23], [29], [38], [40]. Some evidence suggests that the region around the left angular and supramarginal gyri may be particularly important for these tasks [15], [17], [18], [20], [21], [39], [40]. Other studies have tentatively implicated the left frontal lobe [24], [35], the basal ganglia [5], [19], [41], and the thalamus [27]. However, the complete set and precise localization of critical structures has yet to be conclusively established. In contrast to the apparent left hemisphere bias for arithmetic, there is some evidence suggesting that simple magnitude processing involves right or perhaps bilateral parietal areas [7], [8], [9], [10], [22]. A number of functional neuroimaging experiments exploring arithmetic in normals have also been conducted [3], [10], [30], [31], [33], most of these investigated relatively complex calculations such as counting backward by sevens from a three digit number [30]. Activation has typically been observed in inferior parietal and (or) prefrontal areas in these studies, providing a rough convergence with at least some patterns in the patient data. However, complex arithmetic tasks likely engage a number of cognitive processes not directly associated with arithmetic and thus can not address the issue of which areas of brain activation correspond to specific component processes of interest, such as arithmetic fact retrieval or magnitude comparison.

One exception is a positron emission topography (PET) study by Deheane and colleagues, which was designed explicitly to study arithmetic and magnitude processing [10]. Their subjects were shown two single-digit numbers and they either multiplied them together or identified the larger number. The multiplication task compared to a pure rest condition (i.e., involving no active stimulus processing) revealed right cuneus, bilateral inferior parietal, and left fusiform and lingual gyrus activation. There was also lateral occipital activation, which was interpreted as reflecting visual processing, and precentral and supplementary motor area activation, interpreted as reflecting motor responses associated with subvocalization of the answer. Deheane et al. observed no reliable activation in other areas previously implicated in arithmetic, including prefrontal cortex, thalamus and basal ganglia. Of particular note, there was no reliable angular gyrus activation, and the left supramarginal gyrus was significantly deactivated during multiplication compared to rest. The magnitude task compared to rest did not reliably activate any areas beyond the perceptual and motor areas, but there was a trend toward bilateral inferior parietal activation. Deheane et al. suggested that the left inferio–mesial–occipito–temporal area (including fusiform and lingual gyri) is involved in identifying digits and transmitting their identity to other areas, that basal ganglia may be involved in retrieving rotely memorized arithmetic facts (they were drawing here on the neuropsychological literature; they found no activation in this area), and that the inferior parietal areas represent abstract magnitude information which may be brought to bear for magnitude comparison and also for arithmetic when rote retrieval fails [9].

If the areas suggested in the above summary of patient and neuroimaging data are accurate and exhaustive, then we should find activation in the following areas for simple arithmetic; left or bilateral inferior parietal cortex (possibly centered around the angular gyrus), bilateral thalamus, left basal ganglia, left fusiform and lingual gyri, and perhaps left prefrontal cortex. Predictions for magnitude judgment relative to control are less well established, but the available evidence suggests bilateral inferior parietal cortex with a possible bias toward the right.

A second and related goal of this study was to provide new insight into possible relations between cognitive representations for arithmetic and for numerical magnitude. One possibility, consistent with modular and connectionist models of number processing [25], [26], [37], is that arithmetic facts are stored and represented as part of an abstract network dedicated to representing numerical magnitude. This perspective suggests that substantially overlapping brain areas should be activated by arithmetic fact retrieval and magnitude judgment. An alternative framework advanced by Dehaene posits separate systems for representing analog magnitude, verbal, and visual number forms [6]. In this model, arithmetic is mediated primarily by the verbal code, with semantic, or magnitude, codes providing a supportive role if the verbal pathway fails. This perspective would be consistent with a finding that arithmetic and magnitude comparison activate largely distinct brain regions. One viable possibility is a left hemisphere bias for arithmetic and a right hemisphere bias for magnitude processes.

The experimental design involved three conditions: (1) multiplication verification, in which a problem was presented along with a candidate answer, and subjects responded whether the candidate answer was true or false (e.g., 4×7=35; true or false?), (2) magnitude judgment, in which two two-digit numbers were presented (e.g., 24 25) and subjects responded whether the left or right side number was larger, (3) and a detect ones control condition, in which four digits were presented and subjects were required to determine whether one of them was a 1 (e.g., 4 2 1 7; is there a 1 present?). Each type of stimulus required a dichotomous response decision (true/false for verification, left/right for magnitude judgment, and yes/no for detect ones). Areas more activated by verification relative to the detect ones control task should primarily reflect processes involved in solving arithmetic problems (i.e., factoring out perceptual, digit, and motor processes), and areas more activated by magnitude judgment relative to detect ones condition should reflect primarily processes uniquely involved in magnitude judgment. The direct comparison of verification and magnitude judgment should reveal areas more activated by each task of interest relative to the other.

These tasks have several properties that make them well suited for initial exploration of basic arithmetic and magnitude processes. First, each task condition consists only of digits and arithmetic symbols and subtends identical vertical and horizontal visual angles. Thus, basic perceptual differences among tasks are negligible and are not likely to result in significant activation differences outside of primary visual areas which are of little interest in this study. Second, each type of task is nevertheless clearly distinct from the others. Subjects will therefore have no confusion about which task is to be performed on the stimuli during scanning. In pilot behavioral work, we observed that, if tasks are perceptually identical and only task goals are changed over conditions, subjects sometimes become confused about which task is to be performed. For example, in a pilot behavioral study we presented items such as ‘4 7 35’ under alternating instructions either to multiply the first two digits and compare the result to the second two (i.e., a verification task) or to determine the larger magnitude of the two two-digit numbers. Subjects occasionally confused these two task goals during performance. Further, even if overt confusion did not occur, exact perceptual equivalence of stimuli might promote automatic processing of the inappropriate task. These factors clearly would cause serious problems with interpretability of fMRI results. By incorporating relatively minor perceptual differences in the items, we completely avoid the problem of task goal confusions, at the sacrifice of introducing only modest perceptual differences.

A third desirable property of the tasks is that they all involve numerical stimuli. Our primary interest in this study was isolation of processes involved in arithmetic and magnitude processing, rather than processing of digits per se. The presence of four digits for each stimuli roughly equates basic digit processing in each condition and thus activation reflecting basic digit processing should not be present in the task comparisons. The importance of using a digit control task can be demonstrated by considering an alternative design in which the control is either a pure rest condition [10] or a non-numerical perceptual-motor control. In such a design, activation in the magnitude task relative to the control task might reflect the magnitude processes of interest, or, alternatively, either basic perceptual processes (in the case of the pure rest control), or non-magnitude aspects of visual digit processing (in the case of a non-numerical perceptual-motor control). Magnitude processes can only be isolated by using a control condition which involves every process except magnitude processing. Note that the detect one’s control may also involve some magnitude processing, provided that such processing is automatically triggered by the perception of any numerical stimuli. However, it is reasonable a priori to expect magnitude processing to be of greater magnitude in the number comparison task, which explicitly requires accessing of magnitude information to execute the task goal. Thus, we would still expect to see activation in magnitude processing areas in the magnitude judgment vs detect 1’s control task.

A final property of the tasks is that they all involve roughly equivalent, dichotomous motor responses that provide accuracy data. Our decision to use the verification task rather than a simpler production task in which subjects produce the answer (e.g. 4×7=?) was motivated by an overriding goal of collecting accuracy data. Not only could accuracy data not be collected for the somewhat simpler production task using the dichotomous response buttons at our disposal, but subject vocalization of the answer in that task would result in head movement which might seriously compromise image quality.

If verification is to reflect the underlying task of primary interest (production), we need to assure that subjects use a produce-compare strategy, in which they first produce the answer (probably by subvocalization), and then compare it to the candidate answer given, to solve the problems. To maximize use of a produce-compare strategy, we (a) selected subjects who reported that they were easily able to retrieve answers to problems (this is important because ease of retrieval is likely to increase use of a produce-compare strategy in verification) [32], (b) used problem-answer combinations which elicited a high frequency of produce-compare strategy reports in previous research [32], and (c) instructed subjects to use a produce-compare strategy. These steps do not guarantee that subjects used produce-compare during scanning for all verification problems, but they do allow for reasonable confidence that this was the dominant strategy.