A neural disconnection hypothesis on impaired numerical processing

In recent years the number of fMRI studies aiming to pinpoint the neural substrates of number processing increased steadily (e.g., Dehaene et al., 2003; see Dehaene, 2009; Nieder and Dehaene, 2009 for reviews; see Arsalidou and Taylor, 2010 for a meta-analysis). Nevertheless, while considerable progress has been made, the neural correlates and cognitive mechanisms involved in the retrieval of arithmetic facts such as for multiplication (e.g., 2 × 3 = 6) or for addition (e.g., 2 + 3 = 5) fact knowledge is still a matter of debate. According to the currently most influential model of numerical cognition, the Triple Code Model by Dehaene and colleagues (Dehaene and Cohen, 1995, 1997; Dehaene et al., 2003) the retrieval of arithmetic fact knowledge is verbally mediated and subserved by a neural system relying on left-hemispheric perisylvian language areas and the angular gyrus. It is thus assumed that arithmetic facts are retrieved directly from long-term memory without the need of any magnitude manipulations. Instead, the verbally mediated representation format is claimed to trigger the retrieval of a word sequence from memory, which is associated with the AG (e.g., “two times three equals six”). Consistent with this prediction of the Triple Code Model, evidence from neuroimaging (e.g., Rickard et al., 2000; Delazer et al., 2003, 2005; Ischebeck et al., 2006, 2007; Grabner et al., 2009; Zamarian et al., 2009) as well as single-case studies with brain damaged patients (e.g., Hittmair-Delazer et al., 1994; Lee, 2000; Cohen et al., 2000) point to an involvement of the left angular gyrus (AG) in the retrieval of arithmetic facts. Compelling evidence for a functional involvement of the left AG comes from a training study by Delazer et al. (2003). In their study adults were trained on complex multiplication problems (e.g., 17 × 26 =) over 5 days. In a subsequent fMRI session, contrasts between trained and untrained problems revealed stronger activation in the left angular gyrus for the trained than the untrained set interpreted to indicate increased reliance on fact-retrieval to solve trained problems. In contrast, higher activation in frontal cortices and in the lPS for the untrained problems was suggested to reflect higher demands on executive functions and magnitude manipulations, respectively. In sum, these activation differences, in particular between the IPS and the angular gyrus for trained vs. untrained problems indicated a transition from quantity-based processing to direct fact retrieval (for similar results, see Ischebeck et al., 2007). However, there is also contradictory evidence in the literature. For instance, Van Harskamp et al. (2005) reported a patient with a severe multiplication fact-retrieval deficit, although his brain lesion did not involve the left AG. More particularly, a detailed review of single-case studies of patients, for whom neuroradiological images are available, revealed that also the lesion of patient BOO (Dehaene and Cohen, 1997), who presented with a selective impairment for multiplication, did not involve the AG, the supramarginal gyrus (SMG) or perisylvian language areas. This can be seen from the axial lesion drawings in Figure 3 (p. 226). Moreover, in patient ATH who also exhibited a severe impairment for rote multiplication (Cohen and Dehaene, 2000), the posterior part of the angular gyrus seemed spared. Additionally, Van Harskamp and Cipolotti (2001) reported a patient VP who presented with a selective impairment for simple multiplication problems and extended atrophy due to Alzheimer's disease, while the left angular gyrus was relatively spared. Finally, Zaunmueller et al. (2009) reported a patient with a severe multiplication fact-retrieval deficit although his brain lesion did neither involve the left AG or even left-hemispheric language areas. Thus, at a first glance these results do not seem to be in concordance with the notion that the left AG is critical for the retrieval of arithmetic (multiplication) facts nor that this structure in conjunction with temporal, frontal and subcortical areas is indeed critical for arithmetic (multiplication) fact knowledge as suggested by the Triple Code Model.