André Knops

On the functional role of human parietal cortex in number processing: How gender mediates the impact of a 'virtual lesion' induced by rTMS.

Areas around the horizontal part of the intraparietal sulcus (hIPS) have repeatedly been reported to participate in processing numerical magnitude. Using transcranial magnetic stimulation (TMS), we investigated the functional role of the hIPS by examining two effects from the domain of numerical cognition: in magnitude comparison tasks response latencies are inversely related to the numerical distance between two numbers. This distance effect indexes access to the mental number representation. In magnitude comparison tasks responses are faster when decade and unit comparison would lead to the same decision (e.g. 42_57, 4 < 5 and 2 < 7) than when they would not (e.g. 47_62, 4 < 6 but 7 > 2). This compatibility effect reflects unit-decade integration processes. Differential susceptibility of (fe)male participants to TMS was examined. We applied repetitive TMS (rTMS; 1Hz for 10 min) over the left hIPS in 12 participants (6 female). No stimulation and vertex stimulation served as control conditions. The effect of rTMS was mediated by gender: in male participants, the distance effect decreased after TMS over hIPS. For female participants distance and compatibility effect both increased. This modulation of the compatibility effect was limited in duration to no more than 4 min. The hIPS seems to be functionally involved both in number magnitude processing and in integrating unit-decade magnitude information of two-digit numbers. Relative hemispheric specialization of the hIPS with respect to two-digit magnitude comparison is discussed.

A special role for numbers in working memory? An fMRI study

Although numbers carry averbal semantics (i.e., magnitude), they are often utilized in verbal Working Memory (vWM) tasks. However, vWM is thought to rely on a purely phonological code. Here, we tested the influence of (a) averbal semantics and (b) different tasks on performance in a vWM context by examining stimulus and task-specific variation of activity in the horizontal parts of the intraparietal sulcus (hIPS). The hIPS has previously been shown to subserve magnitude processing modulated by (a) specific stimuli and (b) specific tasks. Two variants of an n-back paradigm (comparison and identity match tasks) utilizing letters and numbers as stimulus material at different levels of vWM load were administered in 16 participants. Behavioral and functional imaging data reveal stimulus-specific modulation of activity in the hIPS suggesting a semantic influence of numbers. In the identity match task, numbers induced additional hIPS activation compared to letters while letters never induced additional hIPS activation when compared to numbers. Letters as compared to numbers only induced additional hIPS activation in the comparison task. These results question the assumption of a purely phonological code in vWM because hIPS activation subserving magnitude processing is modulated by stimulus semantics and task demands.

Considering structural connectivity in the triple code model of numerical cognition: differential connectivity for magnitude processing and arithmetic facts

The current study provides a generalizable account of the anatomo-functional associations as well as the connectivity of representational codes underlying numerical processing as suggested by the triple code model (TCM) of numerical cognition. By evaluating the neural networks subserving numerical cognition in two specific and substantially different numerical tasks with regard to both grey matter localizations as well as white matter tracts we (1) considered the possibility of additional memory-related cortex areas crucial for arithmetic fact retrieval (e.g., the hippocampus); (2) specified the functional involvement of prefrontal areas in number magnitude processing, and, finally; (3) identified the connections between these anatomo-functional instantiations of the representations involved in number magnitude processing and arithmetic fact retrieval employing probabilistic fiber tracking. The resulting amendments to the TCM are summarized in a schematic update, and ideas concerning the possible functional interplay between number magnitude processing and arithmetic fact retrieval are discussed.

The Exact vs. Approximate Distinction in Numerical Cognition May Not Be Exact, but Only Approximate: How Different Processes Work Together in Multi-Digit Addition.

Two types of calculation processes have been distinguished in the literature: approximate processes are supposed to rely heavily on the non-verbal quantity system, whereas exact processes are assumed to crucially involve the verbal system. These two calculation processes were commonly distinguished by manipulation of two factors in addition problems: the identity of the target and the distance of the distractor. However, in all previous studies, these two factors were not manipulated independently. In this fMRI study, we could disentangle the two factors by using a different (two-digit) number stimulus set. Both behavioral and neurofunctional data suggest that the cognitive processes involved could be best explained by the (independent) factors target and distractor distance. Based on these data we suggest that the exact/approximate distinction does not seem to be as generally valid as previously assumed. We conclude that this study may be a starting point for a closer examination of the experimental, procedural and strategic conditions of when the exact/approximate distinction is valid and when it is not.

A Shared, Flexible Neural Map Architecture Reflects Capacity Limits in Both Visual Short-Term Memory and Enumeration

Human cognition is characterized by severe capacity limits: we can accurately track, enumerate, or hold in mind only a small number of items at a time. It remains debated whether capacity limitations across tasks are determined by a common system. Here we measure brain activation of adult subjects performing either a visual short-term memory (vSTM) task consisting of holding in mind precise information about the orientation and position of a variable number of items, or an enumeration task consisting of assessing the number of items in those sets. We show that task-specific capacity limits (three to four items in enumeration and two to three in vSTM) are neurally reflected in the activity of the posterior parietal cortex (PPC): an identical set of voxels in this region, commonly activated during the two tasks, changed its overall response profile reflecting task-specific capacity limitations. These results, replicated in a second experiment, were further supported by multivariate pattern analysis in which we could decode the number of items presented over a larger range during enumeration than during vSTM. Finally, we simulated our results with a computational model of PPC using a saliency map architecture in which the level of mutual inhibition between nodes gives rise to capacity limitations and reflects the task-dependent precision with which objects need to be encoded (high precision for vSTM, lower precision for enumeration). Together, our work supports the existence of a common, flexible system underlying capacity limits across tasks in PPC that may take the form of a saliency map.

Numerical ordering and symbolic arithmetic share frontal and parietal circuits in the right hemisphere

A prominent proposal in numerical cognition states that our mental calculation abilities are grounded in the approximate number system (ANS). Recently, it was proposed that this association is mediated by numerical ordering abilities. As a first step in elucidating the neural correlates of this link this study tested which areas in the human brain carry information common to both calculation and numerical ordering. While lying in an MR scanner 17 healthy participants (a) decided whether or not a given number triplet was presented in numerically ascending order, and (b) solved simple addition and subtraction problems. Standard general linear model analyses revealed a largely overlapping network in fronto-parietal regions for both tasks. By analyzing the spatial information over voxels using a whole-brain searchlight algorithm we identified a right hemispheric network comprising areas along the intraparietal sulcus and in the inferior frontal cortex which was similarly involved in order judgments and symbolic arithmetic. Functional and anatomical characteristics of this network make it a candidate for linking the ANS to mental arithmetic.

Spatial representations of numbers in children and their connection with calculation abilities

Evidence for a connection between number and space processing comes from behavioural, patient, and brain imaging data, but only a few studies have addressed this issue in children. We asked children (n=118) at the age of 8-9 years to decide which one of the two numerical distances in a visually presented number triplet was numerically larger. Numerical and spatial distances were manipulated independently, resulting in congruent, neutral, and incongruent conditions. The spatial distances between the numbers clearly affected the comparison of numerical distances: reactions times were faster and error rates smaller for congruent than for incongruent trials. These findings are in line with the assumption of a spatial layout of mental number representations in third graders. Correlations between the size of the congruity effect and calculation abilities were found to be differently marked for girls and boys: a positive correlation was found for boys, while a marginally negative correlation was obtained for girls.

In How Many Ways is the Approximate Number System Associated with Exact Calculation

The approximate number system (ANS) has been consistently found to be associated with math achievement. However, little is known about the interactions between the different instantiations of the ANS and in how many ways they are related to exact calculation. In a cross-sectional design, we investigated the relationship between three measures of ANS acuity (non-symbolic comparison, non-symbolic estimation and non-symbolic addition), their cross-sectional trajectories and specific contributions to exact calculation. Children with mathematical difficulties (MD) and typically achieving (TA) controls attending the first six years of formal schooling participated in the study. The MD group exhibited impairments in multiple instantiations of the ANS compared to their TA peers. The ANS acuity measured by all three tasks positively correlated with age in TA children, while no correlation was found between non-symbolic comparison and age in the MD group. The measures of ANS acuity significantly correlated with each other, reflecting at least in part a common numerosity code. Crucially, we found that non-symbolic estimation partially and non-symbolic addition fully mediated the effects of non-symbolic comparison in exact calculation.

Micro and Macro Pattern Analyses of fMRI Data Support Both Early and Late Interaction of Numerical and Spatial Information

Numbers and space are two semantic primitives that interact with each other. Both recruit brain regions along the dorsal pathway, notably parietal cortex. This makes parietal cortex a candidate for the origin of numerical–spatial interaction. The underlying cognitive architecture of the interaction is still under scrutiny. Two classes of explanations can be distinguished. The early interaction approach assumes that numerical and spatial information are integrated into a single representation at a semantic level. A second approach postulates independent semantic representations. Only at the stage of response selection and preparation these two streams interact. In this study we used a numerical landmark task to identify the locus of the interaction between numbers and space. While lying in an MR scanner participants decided on the smaller of two numerical intervals in a visually presented number triplet. The spatial position of the middle number was varied; hence spatial intervals were congruent or incongruent with the numerical intervals. Responses in incongruent trials were slower and less accurate than in congruent trials. By combining across-vertex correlations (micro pattern) with a cluster analysis (macro pattern) we identified large-scale networks that were devoted to number processing, eye movements, and sensory–motor functions. Using support vector classification in different regions of interest along the intraparietal sulcus, the frontal eye fields, and supplementary motor area we were able to distinguish between congruent and incongruent trials in each of the networks. We suggest that the identified networks participate in the integration of numerical and spatial information and that the exclusive assumption of either an early or a late interaction between numerical and spatial information does not do justice to the complex interaction between both dimensions.

Dissociating estimation from comparison and response eliminates parietal involvement in sequential numerosity perception

It has been widely debated whether the parietal cortex stores an abstract representation of numerosity that is activated for Arabic digits as well as for non-symbolic stimuli in a sensory modality independent fashion. Some studies suggest that numerical information in time-invariant (simultaneous) symbolic and non-symbolic visual stimuli is represented in the parietal cortex. In humans, whether the same representation is activated for time-variant (sequential) stimuli and for stimuli coming from different modalities has not been determined. To investigate this idea, we measured the brain activation of healthy adults performing estimation and/or comparison of sequential visual (series of dots) and auditory (series of beeps) numerosities. Our experimental design allowed us to separate numerosity estimation from comparison and response related factors. The BOLD response in the parietal cortex increased only when participants were engaged in the comparison of two consecutive numerosities that required a response. Using multivariate pattern analysis we trained a classifier to decode numerosity in various regions of interest (ROI). We failed to find any parietal ROI where the classifier could decode numerosities during the estimation phase. Rather, when participants were not engaged in comparison we were able to decode numerosity in an auditory cortex ROI for auditory stimuli and in a visual cortex ROI for visual stimuli. On the other hand, during the response period the classifier successfully decoded numerosity information in a parietal ROI for both visual and auditory numerosities. These results were further confirmed by support vector regression. In sum, our study does not support the involvement of the parietal cortex during estimation of sequential numerosity in the absence of an active task with a response requirement.