Scientific Understanding of Consciousness
Consciousness as an Emergent Property of Thalamocortical Activity

Topographic Numerosity in Parietal Cortex


Science 6 September 2013: Vol. 341 no. 6150 pp. 1123-1126

Topographic Representation of Numerosity in the Human Parietal Cortex

B. M. Harvey, B. P. Klein, N. Petridou, S. O. Dumoulin

1Department of Experimental Psychology, Helmholtz Institute, Utrecht University, Utrecht, 3584 CS, Netherlands.

2Department of Radiology, Rudolf Magnus Institute of Neuroscience, University Medical Center Utrecht, Utrecht, 3584 CX, Netherlands.


Numerosity, the set size of a group of items, is processed by the association cortex, but certain aspects mirror the properties of primary senses. Sensory cortices contain topographic maps reflecting the structure of sensory organs. Are the cortical representation and processing of numerosity organized topographically, even though no sensory organ has a numerical structure? Using high-field functional magnetic resonance imaging (at a field strength of 7 teslas), we described neural populations tuned to small numerosities in the human parietal cortex. They are organized topographically, forming a numerosity map that is robust to changes in low-level stimulus features. The cortical surface area devoted to specific numerosities decreases with increasing numerosity, and the tuning width increases with preferred numerosity. These organizational properties     extend topographic principles to the representation of higher-order abstract features in the association cortex.

Humans and many other animals use numerosity to guide behavior and decisions. Numerosity perception becomes less precise as the size of numbers increases and is particularly effective for small numbers. Animals, infants, and tribes with no numerical language perceive numerosity although they cannot count or use symbolic representations of number. Thus, numerosity processing is an evolutionarily preserved cognitive function, distinct from counting and humans’ unique symbolic and mathematical abilities. Because aspects of numerosity processing     mirror primary sensory perception, it has been referred to as a “number sense.”

The primary sensory and motor cortices in the brain are organized topographically. Is the neural organization for numerosity similarly topographic? The neural representation of numerosity resides in higher-order association cortices, including the posterior parietal cortex. Human functional magnetic resonance imaging (fMRI) consistently identifies this region as particularly responsive to numerosity manipulations, and in similar regions, macaque neurophysiology describes neurons tuned to visual numerosity. Both human fMRI and macaque neurophysiological response properties are closely linked to behavioral numerosity performance.

We elicited responses to visual patterns with varying numerosity in study participants, while acquiring high-field (7 teslas) fMRI data. Changing numerosity in a visual display affects visual features such as luminance, contrast, density, and total edge length. Therefore, establishing numerosity selectivity requires several control conditions. Consequently, we included conditions in which total dot area (“constant area” condition), individual dot size (“constant dot size”), or total dot circumference (“constant circumference”) were constant. A further condition contained much higher dot pattern density (“high density”). Finally, to check generalization to other objects, we replaced dots with different shapes (“variable features”). During stimulus presentation, participants reported when dots were shown in white rather than black (10% of presentations). No numerosity judgments were required. Participants performed above 90% correct.

The displayed numerosity varied systematically within an fMRI scan. This stimulus elicited remarkably different response profiles at different recording sites. We summarized these fMRI signals using numerosity-tuned neural models. These describe Gaussian functions in logarithmic numerosity space, following behavioral, computational, neuroimaging, and neurophysiological results. The models have two parameters: preferred numerosity and tuning width (the numerosity range to which the population responds). This analysis is analogous to conventional population receptive field analysis in the visual cortex. These models explain much of the signal variance (R2), summarizing fMRI responses with two parameters. They capture similar amounts of variance for both example response profiles, explaining time course differences by different numerosity tunings.

A specific region in the posterior parietal cortex was highlighted, where the models captured much response variance in all stimulus conditions. This region’s position was consistent between the eight participants, in the posterior superior parietal lobule, centered at mean (SD) Montreal Neurological Institute x,y,z coordinates of 23, –60, 60 and closely matches previous reports of a region responding strongly to numerosity manipulations.

Projecting each recording site’s preferred numerosity onto the unfolded cortical surface revealed an orderly topographic map. Medial and lateral regions preferred low and high numerosities, respectively. The topographic progression and its direction were consistent between participants and stimulus conditions. Numerosity selectivity was also present in the left hemisphere and in neighboring regions of the right hemisphere, but with lower variance explained and less clear, less consistent topographic structure.

Neuroimaging studies consistently show that this part of the parietal cortex responds to numerosity manipulations, and parietal lesions can cause number-processing deficits.

Macaque neurophysiology demonstrates numerosity tuning in single neurons in a similar parietal region. Based on similar behavioral performance and cortical location of numerosity-selective populations in humans and macaques, we expect similar topographic organization in macaques. The spatial scale of the topographic organization is several centimeters. Consequently, methodological limitations at single-neuron resolution, topography may be obscured by the scatter of response properties, broad single-neuron tuning, neurons with other response properties, and an unknown direction of topography change. However, both methodologies are complementary, and our measurements are consistent with neurophysiology. Both support numerosity tuning, albeit at different scales, in similar parts of the brain, with more neurons tuned to smaller numerosities and increases in tuning width with preferred numerosity. These properties are analogous to organization properties of the sensory and motor cortices and may underlie the decreased precision at higher numerosities that is commonly seen in human and animal behavior

What is the nature of the numerosity representation? We found no number-tuned responses for Arabic numerals, suggesting that neurons here do not respond to symbolic number representations. We propose that current biologically plausible computational models of numerosity processing, driven by visual features, can produce the numerosity selectivity we see. Some models suggest that (as we find) numerosity selectivity depends on stimulus features, such as dot size. Computational models of numerosity extraction may thus explain these differences in numerosity tuning, consistent with behavioral results

Numerosity processing and its cortical organization may be fundamental to human abilities in mathematics and economics. Although numerosity judgments and complex mathematical abilities rely on different processes, individual differences in these abilities are correlated. Macaques and young children can perform simple, approximate addition and subtraction. In macaques, the parietal and prefrontal cortices contain neurons responding specifically during simple mathematical tasks, together with numerosity-selective neurons. Associations between visual numerosity and symbolic number representations develop early in life. Numerosity, number, and size   are fundamental to our understanding of magnitude and quantity and underlie higher-level concepts of value

Our results demonstrate that topographic representations, common in the sensory and motor cortices, can emerge within the brain to represent abstract features such as numerosity. Similarities in cortical organization suggest that the computational benefits of topographic representations, for example efficiency in wiring, apply to higher-order cognitive functions and sensory-motor functions alike. As such, topographic organization may be common in higher cognitive functions. On the other hand, topographic organization supports the view that numerosity perception resembles a primary sense. These views are not mutually exclusive, but both challenge the established distinction between primary topographic representations and abstracted representations of higher cognitive functions.

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