Neural representations of two-digit numbers: A parametric fMRI study

Abstract

When participants are asked to decide which of a pair of two-digit Arabic numbers is larger, they compare units and decades, even when units are irrelevant. Typically, behavioral responses are slower when units are incongruent with the decade comparison (e.g. 81_26, because 8 > 2 but 1 < 6; unit–decade compatibility effect, [Nuerk, H.-C., Weger, U., Willmes, K. 2001. Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition 82, B25–B33.]). We defined parametric regressors to examine the effect of decade digit distance and unit distance-based compatibility processing on the fMRI signal to investigate the neural correlates of two-digit magnitude processing. Fourteen male right-handed volunteers (mean age = 27, range 21–38 years) took part in the study. In a rapid event-related design, participants had to decide which of two two-digit Arabic numbers was larger. Data were preprocessed and analyzed statistically in SPM2. Activation in the anterior portion of the right IPS was significantly modulated by compatibility-based unit distance processing. Furthermore, decade distance predicted an increase in the fMRI signal from cortex around the left IPS, right anterior and right posterior IPS. These results indicate that magnitude representations of unit–decade compatibility and decade distance are subserved by the intraparietal cortex and that the symbolic structure of the Arabic number system is an important determinant of multi-digit number magnitude processing. Implications of the present results in terms of general symbolic information processing are discussed.

Introduction

Number magnitude is one of the most salient semantic number representations activated in calculation and magnitude comparison tasks (Nieder, 2005, Goebel and Rushworth, 2005). The neural correlates of number magnitude processing have been shown to be localized in the cortex around the intraparietal sulcus (IPS) bilaterally (Pinel et al., 1999, Pinel et al., 2004, Piazza et al., 2004, Simon et al., 2002, Dehaene et al., 2003 for a review). Number magnitude modulates the fMRI signal in the intraparietal cortex even when magnitude processing is irrelevant for the task (Eger et al., 2003) and when numerical stimuli are presented unconsciously (Naccache and Dehaene, 2001). Examination of brain damaged patients (e.g. Cohen et al., 2000) and single-cell recordings in monkeys (Nieder and Miller, 2004) also corroborate the view that the intraparietal cortex is decisive for processing number magnitude.

Not only small numbers activate magnitude representations. Behavioral studies show that two-digit (Dehaene et al., 1990, Nuerk et al., 2001), and even three-digit numbers (Meeuwissen et al., 2003, Tlauka, 2002) are represented as number magnitudes. In the last years, some imaging studies have investigated the neural correlates of two-digit number magnitude processing (Pinel et al., 2001, Goebel et al., 2004). Pinel et al. (2004) found activated voxels in the intraparietal cortex bilaterally when participants compared two-digit numbers with the standard 65. More specifically, a recent fMRI study by Goebel et al. (2004) contrasted directly the effect of single-digit and two-digit numbers on brain activation and found evidence for some specialization in the right intraparietal cortex. Goebel and colleagues presented single-digit and two-digit numbers, which the participants had to compare with the standards 5 and 65, respectively. The authors found that single-digit numbers activated the anterior part of the right intraparietal cortex more strongly than two-digit numbers, which in turn activated more the posterior part of the intraparietal cortex, bilaterally. An EEG study by Whalen and Morelli (2002) also provided evidence for some specialization in the intraparietal cortex. Whalen and Morelli (2002) have examined the ERP correlates of unit and decade magnitude representations in a two-digit magnitude comparison task with standard 65. They estimated the location of dipole sources separately for decade and unit digits and found separated dipoles in right parietal regions specific for unit and decade digits. The dipole for the units was located in the anterior parietal cortex and the dipole for decades in the posterior parietal cortex. The studies of Goebel et al. (2004) and Whalen and Morelli (2002) seem to indicate that two-digit numbers are preferentially processed in the posterior portion of the IPS, while one-digit numbers are processed in its anterior portion.

Interestingly, behavioral studies show that two-digit numbers seem to simultaneously activate separated magnitude representations for units and decades (Nuerk et al., 2001, Nuerk et al., 2002a, Nuerk et al., 2002b, Nuerk et al., 2004b, Nuerk et al., 2005, Ratinckx et al., in press). Recently, Nuerk and Willmes (2005) have developed a cognitive model of two-digit number comparison (the hybrid model), which tries to capture empirical findings concerning activation of digital and analog magnitude representations of two-digit numbers. The hybrid model predicts that when two-digit numbers (e.g. 57) are processed with regard to their magnitude, digit-based magnitude representations for units and decades are activated complementarily with activation of the analog representation (57 = {5}, {7}, {57}). The most direct evidence supporting the hybrid model comes from experiments about the unit–decade compatibility effect. The unit–decade compatibility effect indicates activation of separate magnitude representations for units and decades (Nuerk et al., 2001). In several behavioral experiments, Nuerk and colleagues have shown that unit and decade distances are computed separately and that units interfere with the processing of decade numbers in a way incompatible with a purely analog two-digit magnitude representation (Nuerk et al., 2001, Nuerk et al., 2002b, Nuerk et al., 2004b, Nuerk et al., 2004a, Nuerk et al., 2005, Wood et al., 2005). For instance, comparison of two-digit numbers is faster and more accurate when the larger number contains the larger unit digit (e.g. 76_21; since 7 > 2 and 6 > 1 are compatible) than when the smaller number contains the larger unit digit (e.g. 81_26; because 8 > 2, and 1 < 6 are incompatible; Fig. 1). The impact of unit–decade compatibility on behavioral responses is called unit–decade compatibility effect (Nuerk et al., 2001) and it is significant even (1) when overall distance is matched in compatible and incompatible conditions (55 in the example above) and (2) unit digits are irrelevant for selecting the larger number.

Recently, Zhang and Wang (2005) investigated the role of external symbolic and internal number representations on two-digit number processing in a behavioral study. The authors argue that the external symbolic number representation determines the activation of separate magnitude representations of unit and decade digits in two-digit Arabic numbers. The influence of the symbolic base-10 structure of Arabic multi-digit numbers on magnitude processing increases when the perceptual load increases. Tasks involving the encoding of external multi-digit numbers seem to favor the parallel activation of separate magnitude representations for units and decades.

When behavioral performance points to the activation of separated magnitude representations for units and decades, one may suppose that examination of the neural correlates of decade distance and compatibility-based unit distance processing may reveal activation of specialized brain regions, which cannot be identified when concentrating exclusively on the analog magnitude of two-digit numbers. Behavioral data indicate that the magnitude representation of two-digit numbers is more complex than that of one-digit numbers. Hence, it may activate specialized neural correlates for representing decade distances and the interference of the unit comparison on the decades.

In order to dissociate the effects of decade distance and unit–decade compatibility processing on the fMRI signal, we employed a set of stimuli developed by Nuerk et al. (2001), and used the decade distance and unit–decade compatibility of stimuli as quantitative predictors for analyzing the changes in fMRI signal. Pinel et al. (2001) and Nuerk et al. (2001) as well as Nieder and Miller (2004) chose a similar approach for analyzing the fMRI signal, behavioral performance or single-cell firing rates, respectively. If there are portions of intraparietal cortex specialized in processing decade distances and unit–decade compatibility, their fMRI signal should correlate more strongly with the quantitative predictors representing decade distances and unit–decade compatibility than neighboring regions. In contrast, if the different portions of intraparietal cortex all correlate with the quantitative predictors to the same extent, we may conclude that the neural correlates of decade distances and unit–decade compatibility processing are not functionally specialized.