Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group Independent Component Analysis of the mental addition and subtraction of fractions

Abstract

The suitability of a previously hypothesized triple-code model of numerical processing, involving analog magnitude, auditory verbal, and visual Arabic codes of representation, was investigated for the complex mathematical task of the mental addition and subtraction of fractions. Functional magnetic resonance imaging (fMRI) data from 15 normal adult subjects were processed using exploratory group Independent Component Analysis (ICA). Separate task-related components were found with activation in bilateral inferior parietal, left perisylvian, and ventral occipitotemporal areas. These results support the hypothesized triple-code model corresponding to the activated regions found in the individual components and indicate that the triple-code model may be a suitable framework for analyzing the neuropsychological bases of the performance of complex mathematical tasks.

Introduction

The precise neuropsychological model and neural substrates associated with math cognition continue to be a subject of investigation. The triple-code model of numerical processing by Dehaene (1992) and Dehaene and Cohen, 1995, Dehaene and Cohen, 1997 proposes that numbers are represented in three codes that serve different functions, have distinct functional neuroarchitectures, and are related to performance on specific tasks (see review in (van Harskamp and Cipolotti, 2001)). The analog magnitude code represents numerical quantities on a mental number line Dehaene, 1989, Dehaene et al., 1990, Restle, 1970, includes semantic knowledge regarding proximity (e.g., 5 is close to 6) and relative size (e.g., 5 is smaller than 6), is used in magnitude comparison Dehaene, 1989, Dehaene et al., 1990, Moyer and Landauer, 1967 and approximation tasks, among others, and is predicted to engage the bilateral inferior parietal regions Chochon et al., 1999, Dehaene et al., 1999, Stanescu-Cosson et al., 2000. The auditory verbal code (or word frame) manipulates sequences of number words, is used for retrieving well-learned, rote, arithmetic facts such as addition and multiplication tables (Gonzalez and Kolers, 1982), and is predicted to engage general-purpose language modules, including the left perisylvian network and the left basal ganglia and thalamic nuclei, which have been associated with memory and sequence execution Dehaene, 1997, Houk and Wise, 1995. The visual Arabic code (or number form) represents and spatially manipulates numbers in Arabic format Ashcraft and Stazyk, 1981, Cohen and Dehaene, 1991, Dahmen et al., 1982, Weddell and Davidoff, 1991, is used for multidigit calculation and parity judgments (Dehaene and Cohen, 1991), and is predicted to engage bilateral inferior ventral occipitotemporal regions belonging to the ventral visual pathway (Dehaene, 1992), with the left used for visual identification of words and digits, and the right used only for simple Arabic numbers (Dehaene, 1997).

The triple-code model proposes various transcoding paths between the three representational codes. Overall, two major coordinated routes are proposed by Dehaene and Cohen (1995, 1997): a direct asemantic route that transcodes written numerals to auditory verbal representations in the left perisylvian language areas to guide retrieval of rote knowledge of arithmetic facts (e.g., presented with 2 × 3, processed as “two times three, six”) without semantic mediation, and an indirect semantic route specialized for quantitative processing that manipulates analog magnitude representations in bilateral parietal areas to compare operands and uses back-up strategies by manipulating visual Arabic representations when rote knowledge is not available in verbal memory, such as decomposing complex problems into new problems for which facts can be retrieved (e.g., 13 + 5 = 10 + 5 + 3 = 15 + 3 = 18) (LeFevre et al., 1996), and monitors the plausibility of the direct route using approximate means Ashcraft and Stazyk, 1981, Dehaene and Cohen, 1991. Furthermore, prefrontal areas and the anterior cingulate associated with a global workspace are proposed to coordinate the sequencing of processing through the modules in the appropriate order, holding intermediate results in working memory, and detecting errors Dehaene, 1997, Dehaene and Naccache, 2001, Dehaene et al., 1996, Kopera-Frye et al., 1996, Shallice and Evans, 1978.

Other theorists have proposed differing models. For example, McCloskey Dagenbach and McCloskey, 1992, McCloskey, 1992, McCloskey et al., 1986 proposes modules for comprehension, calculation, and number production. Specifically, the comprehension module translates word and Arabic numbers into abstract internal representations of numbers, calculations are performed on these representations, and then the abstract representations are converted to verbal or Arabic numbers using specific number production modules. Thus, in McCloskey's model, amodal abstract internal representations of numbers are operated on, rather than numbers represented in specific codes (i.e., quantity, verbal, or Arabic) as in Dehaene's model. Furthermore, McCloskey's model assumes impairment affects individual representations of stored arithmetic facts, segregated by type of operations (e.g., addition vs. division), and thus predicts arbitrary rather than systematic dissociation between operations; whereas Dehaene's Dehaene and Cohen, 1995, Dehaene and Cohen, 1997 model predicts specific impairments to operations associated with distinct functional neuroarchitectures (e.g., impairment to indirect semantic route associated with complex addition, subtraction, and division) (van Harskamp and Cipolotti, 2001). Dehaene's model is conceptually similar to the encoding complex model by Campbell and Clark (1988), which also assumes that numbers are operated on using specific codes, rather than abstract representations, which is also consistent with the preferred entry code hypothesis by Noel and Seron (1992).

Previous lesion and neuroimaging studies have provided some support for Dehaene's distinction between the asemantic language- and culture-dependent system used for exact math and the semantic language-independent system used for approximate math (Ansari and Karmiloff-Smith, 2002). Neuropsychological evidence indicates a double dissociation between the two major routes. Patients with left perisylvian lesions, but spared inferior parietal regions, demonstrate impairment in tasks involving verbal representations of number, but can perform tasks involving nonverbal representations of number (i.e., quantity and Arabic) Cipolotti and Butterworth, 1995, Cohen et al., 2000, Dagenbach and McCloskey, 1992, Dehaene and Cohen, 1997, Lampl et al., 1994, Pesenti et al., 2000a; whereas patients with parietal lesions show acalculia or deficits in understanding quantity meaning Cipolotti et al., 1991, Dehaene and Cohen, 1997, Delazer and Benke, 1997. Neuroimaging studies indicate that parietal regions are activated during number processing and calculation Gruber et al., 2001, Naccache and Dehaene, 2001, Pinel et al., 2001, such as digit comparison (Chochon et al., 1999), single digit multiplication (Dehaene et al., 1996), and approximation Dehaene et al., 1999, Stanescu-Cosson et al., 2000, to a greater extent than for rote math and are not activated in phonological or lexical tasks Dehaene et al., 1999, Pesenti et al., 2000b, Stanescu-Cosson et al., 2000. More specifically, Chochon et al. (1999) implicated a parieto-fronto-cingular network (intraparietal sulcus, postcentral gyrus, inferior frontal-BA 44/45, dorsolateral frontal-BA 46/9, superior frontal-BA 6/8, SMA, and premotor cortex) related to the performance of digit naming and comparison and multiplication and subtraction tasks. However, unexpected results were found for multiplication, including the absence of activation in the predicted language areas and the presence of activation in left intraparietal regions, which is consistent with other studies (Dehaene et al., 1996) and was discussed as possibly reflecting the combinatorial use of direct retrieval and quantity-based strategies. Stanescu-Cosson et al. (2000) found that left prefrontal and bilateral angular regions showed greater activation during rote addition tasks, especially for small numbers, while the bilateral intraparietal, precentral, dorsolateral, and superior prefrontal regions showed greater activation during approximate addition tasks. Larger single digit exact math problems were associated with increased activation in the same bilateral intraparietal regions as approximate math as well as left inferior and superior frontal gyri activation.

To explore the applicability of the triple-code model to complex mathematical operations, we designed a functional magnetic resonance imaging (fMRI) paradigm involving the mental addition and subtraction of fractions (e.g., 2/3 − 1/4). The task is expected to recruit all three elements of the triple-code model: the analog magnitude code for information regarding relative size and for the proportional math techniques used to change denominators (e.g., transforming 2/3 into 8/12 and 1/4 into 3/12), the auditory verbal code for retrieval of rote facts (e.g., subtraction of the numerators 8 − 3 = 5), and the visual Arabic code for the representation and mental manipulation of numerals in Arabic format (e.g., recognizing numerators vs. denominators and performing addition and subtraction when rote facts are unknown).

Because the hemodynamic response functions (HRFs) of the three components are not known precisely a priori, and may in fact have considerable variance across subjects (due to differences in speed of performing the problems, as well as possible differences in processing strategies), we selected group Independent Component Analysis (ICA) for analysis of the data. ICA has been previously proposed as a data-driven approach for analysis of fMRI data (McKeown et al., 1998). ICA operates by linearly unmixing the fMRI data into spatially independent component maps (details given in McKeown et al., 1998). The method has been extended for multisubject analyses (Calhoun et al., 2001b) and the generation of across-subjects random-effects statistical inferences. ICA offers the advantage of not requiring accurate modeling of the HRF for each subject and cognitive component. The group ICA technique has been shown to provide similar results to standard model-based approaches (Calhoun et al., 2001a) and has been used recently in studies investigating simulated driving (Calhoun et al., 2002), visual perception (Calhoun et al., 2001a), and language processing (Schmithorst and Holland, 2003).