Language and calculation within the parietal lobe: a combined cognitive, anatomical and fMRI study

Abstract

We report the case of a patient (ATH) who suffered from aphasia, deep dyslexia, and acalculia, following a lesion in her left perisylvian area. She showed a severe impairment in all tasks involving numbers in a verbal format, such as reading aloud, writing to dictation, or responding verbally to questions of numerical knowledge. In contrast, her ability to manipulate non-verbal representations of numbers, i.e., Arabic numerals and quantities, was comparatively well preserved, as evidenced for instance in number comparison or number bisection tasks. This dissociated impairment of verbal and non-verbal numerical abilities entailed a differential impairment of the four arithmetic operations. ATH performed much better with subtraction and addition, that can be solved on the basis of quantity manipulation, than with multiplication and division problems, that are commonly solved by retrieving stored verbal sequences. The brain lesion affected the classical language areas, but spared a subset of the left inferior parietal lobule that was active during calculation tasks, as demonstrated with functional MRI. Finally, the relative preservation of subtraction versus multiplication may be related to the fact that subtraction activated the intact right parietal lobe, while multiplication activated predominantly left-sided areas.

Introduction

In 1920, Henschen, in the same publication in which the term acalculia was coined, claimed that “the calculation ability is a highly composite cerebral function that results from the collaboration of various posterior areas of the left hemisphere” [26]. A wealth of recent studies of numerical abilities in animals, infants, healthy and brain-lesioned adults largely supports this modular approach at the cognitive and at the anatomical level, and confirms that parietal areas are crucial to number processing (see review in Refs. [18], [20]). A major distinction that cuts across numerical abilities is whether they are contingent or not upon the language faculty. It has been shown that animals and preverbal infants possess a variety of number processing abilities, such as the capacity to match numerosities within and across perceptual modalities [8], [37] and perform elementary arithmetic computations [5], [41]. However, the range of numerical abilities widens dramatically as soon as children acquire language, which allows them to associate verbal labels to any precisely defined quantity. Children are then able to develop rich procedures for manipulating numbers in symbolic form, in particular through the mastery of the written language, and following formal mathematical training in school. However, the precise functional and anatomical interplay of numbers and language in human adults is still in many respects poorly understood.

It is clear that some aspects of number processing are only by-products of general verbal abilities. Such is the case of the abilities to utter, read, write, or repeat number words, which are not expected to differ much from ordinary words as far as such input, output, or transcoding processes are concerned (see discussion in Refs. [11], [12]). A more debated issue, which is the focus of the present study, is the exact role played by verbal processes in elementary arithmetic. We do not refer here to the verbal processes involved in understanding the operands of arithmetic problems or in eventually producing a response, but to verbal processes putatively involved in the computation or retrieval of the appropriate result. We suggested previously, as part of the triple-code model of number processing, that number facts that have been learned by rote at school, first and foremost the overlearned multiplication table, are retrieved as automatic verbal associations [18]. In Henschen’s terms, “the simplest operations, for instance, addition and multiplication of single digits, are generally solved in a completely automatic fashion. We hear the multiplication table internally, and we can utter the result without reflection” [26]. In contrast, subtraction problems, which are not commonly learned by rote, must be solved through mental manipulations of the quantities represented by the operands, or ‘semantic elaboration’. The status of addition and division is in principle more ambiguous. While many simple addition problems are memorized in a verbal form like multiplication problems, they can also be solved rapidly using counting and other quantity-driven backup strategies such as referring to 10 (e.g. 6 + 5 = 6 + 4 + 1 = 10 + 1 = 11).

This conception generates explicit predictions concerning: (1) the possible patterns of dissociation between operations in brain-damaged patients; and (2) the relationships between, on the one hand, arithmetic abilities and, on the other hand, general verbal and quantity manipulation abilities. In contrast, most other neuropsychological models of mental arithmetic are actually neutral regarding these two points. Thus the main alternative model, proposed by McCloskey and his colleagues [14], [30], postulates that all arithmetic problems are solved on the basis of a single abstract representation of number meaning, and that the four arithmetic operations are supported by distinct and potentially dissociable processes [14]. This hypothesis was initially proposed because there seemed to be no clear pattern in the observed dissociations between preserved and impaired operations in brain-lesioned patients.

In fact, however, the functional analysis proposed by the triple-code model has received support from a number of reports of brain-damaged patients with dissociations between operations. In these cases, arithmetic impairments often affect multiplication more severely than subtraction [14], [19], [27], [33], or subtraction more severely than multiplication [19], [23], while the performance of all these patients with addition was intermediate between their performance with multiplication and subtraction. The triple-code model accounts naturally for this pattern, and predicts that it should never be possible to find a patient with impaired multiplication and subtraction, yet with relatively preserved addition; nor should it be possible to have a selective impairment of addition relative to multiplication and subtraction. Also regarding the relationships between calculation impairments and deficits outside of the arithmetic domain, other models remain generally silent, while there is some empirical support to the predictions derived from the triple-code model. For instance, we studied two patients who presented the following double dissociation pattern [19]. On the one hand, patient BOO showed a general deficit of verbal automatisms, entailing a severe impairment of multiplication fact retrieval, while subtraction problem solving was relatively spared. On the other hand, patient MAR showed a general deficit of quantity manipulation, entailing a severe impairment of subtraction, while the retrieval of memorized multiplication facts was better preserved. In the present case, the fact that patient ATH’s performance was impaired in a variety of verbal tasks (including manipulation of number words) and preserved in quantity manipulation tasks, led us to expect a specific dissociation between multiplication and subtraction. Note that the contrast between operations in terms of their underlying mechanisms (retrieval vs. algorithm) is not an absolute one. Some familiar subtraction facts may in principle be learned by rote and, conversely, multiplication facts may be solved or checked through algorithmic manipulations. Still, the crucial claim derived from the triple-code model is that, in normal subjects, subtraction and multiplication rely differentially on these two types of mechanisms, and therefore, can be doubly dissociated following brain lesions that differentially affect areas devoted to verbal vs. quantity manipulation processes.

This cognitive debate has a counterpart at the level of brain functional anatomy. Roughly stated, the issue is that of the relationships between the brain areas that subserve verbal abilities and those that subserve quantitative number processing, and of the contribution of these structures to arithmetic problem solving. The core language areas, those whose lesion entails a variety of aphasic deficits, associate left perisylvian regions belonging to the frontal lobe (specially Broca’s area), to the supero-lateral temporal lobe (specially Wernicke’s area), and to the inferior parietal lobule (specially the supramarginal gyrus) [29]. Additional infero-temporal structures are devoted to the visual processing of written words, and their lesion induces pure alexia [4], [15]. Finally, a network of subcortical structures and pathways connected to the cortical language areas are also necessary to verbal processes [1], [3]. The extent of the areas that subserve number processing is less precisely defined. First, verbal representations of numbers probably rely on the same structures as words in general, i.e., the classical language system outlined earlier. We have postulated that a special role is devoted to cortico-subcortical loops in retrieving verbal automatic associations such as rote arithmetic facts. Second, functional imaging studies in normal subjects and data from brain-damaged patients suggest that bilateral cortical regions centred on the intraparietal sulcus play a central role in number processing [7], [16], [22], [34], [35]. Studies of patients with acalculia following lesions in the left intraparietal region, frequently in the context of Gerstmann’s syndrome, show that the numerical deficit affects primarily quantity manipulations, while verbal routines such as reading aloud or writing to dictation, may be entirely preserved [19], [20], [24], [38], [39].

In summary, the left parietal lobe is likely to be crucial in the two major aspects of number processing, i.e., verbal and non-verbal. Furthermore, the absence of any impairment of language and of number transcoding in Gerstmann’s acalculia, following lesions in the intraparietal region, indicates that within the parietal lobe verbal and non-verbal processes involve areas that are at least partially nonoverlapping.

We study here the numerical abilities of a patient who presented with aphasia and deep dyslexia following a lesion affecting the classical language areas, including part of the inferior parietal lobule. Firstly, we show that the patient’s non-verbal numerical abilities were largely spared, while most tasks involving numbers in a verbal format were impaired. Secondly, we show that, as predicted, this predominantly verbal deficit entailed a disproportionate deficit of multiplication, as compared with subtraction. Thirdly, through a precise study of the lesion’s topography, we try to delineate, within the parietal lobe, the areas subserving verbal and non-verbal number processing. Finally, we try to clarify the mechanisms of the patient’s intact and impaired performance using functional MRI.