Topology-defined units in numerosity perception

Abstract

What is a number? The number sense hypothesis suggests that numerosity is "a primary visual property" like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/ enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.W hat is a number? The answer to this age-old and fundamental question of philosophy has increasingly benefited from recent scientific investigation using psychology and neuroscience. The number sense hypothesis (1, 2) suggests that a number is "a basic property of the environment" (3) and particularly, because of its remarkable adaptation effect, "a primary visual property" (2), like color, contrast, or orientation (2-8). However, exactly what attribute in the environment is the primary property and determines numbers in the number sense, or more concretely, what is counted in numerosity perception? Consider the invariant nature of numerosity perception. It is self-evident that numerosity is invariant to specific features (e.g., orientation, size, shape, and color) of individual items to be counted. In other words, the primitive units to be counted must be invariant with variation in form dimensions and other visual features (2, 3, 7, 9-11). Then, the critical question becomes how to define precisely such abstract and invariant attributes. ResultsGeneralizing Connection to a Topological Invariant: Connectivity. We designed arbitrary and irregular shapes of connecting line segments (Fig. 1A, Upper) to test the invariant effect of connection on numerosity judgement independent of the concrete forms or manners of connection (12, 13). Three conditions of connection were constructed: zero, one, and two connected pairs of dots in the test patterns (Fig. 1A). A typical numerosity discrimination task was adopted, in which two visual patterns of dots were briefly presented on opposite sides of fixation (one serving as a reference and the other one serving as a test), and subj...

Connecting visual objects reduces perceived numerosity and density for sparse but not dense patterns

“…The upper panels (A and B) show data for 15-dot stimuli (0.5 dots/deg 2 ), bottom panels (C and D) for 100-dot stimuli (3.33 dots/deg 2 ). For the sparse patterns, the PSEs for numerosity judgements are clearly shifted to the left for the connected stimuli, implying that they seem to contain fewer elements, agreeing with all the previous literature (Fornaciai et al, 2016;Franconeri et al, 2009;He et al, 2009;He et al, 2015;Kirjakovski & Matsumoto, 2016). For the dense pattern there is still a difference in PSEs, but the difference is much reduced, from about 40% for 15 dots to about 17% for 100 dots.…”

“…We confirmed previous studies showing that connecting items in the visual scene results in a strong and reliable underestimation of numerosity (Fornaciai et al, 2016;Franconeri et al, 2009;He et al, 2009;He et al, 2015). We expanded these results to show that: 1) this effect is strongly reduced at higher numerosities; 2) when asked to judge density rather than numerosity, participants underestimated the density as well as the numerosity of stimuli with connectors for sparse patterns; 3) as for numerosity, the effect of connectivity on apparent density depends on item spacing: for sparse items, density was underestimated, switching to overestimation for denser stimuli.…”

“…Does the Approximate Number System operate on unsegmented visual scenes, as suggested by Durgin (2008) and Dakin et al (2011) or does it operate on segmented objects Castelli, Glaser, & Butterworth, 2006;Fornaciai et al, 2016;Franconeri et al, 2009;He et al, 2009;He et al, 2015;Kirjakovski & Matsumoto, 2016)? This study strengthens the idea that the Approximate Number System operates only Anobile,Cicchini,Pomè,& Burr 141 when items are sparse enough to permit spatial segregation.…”

“…Connecting dots not only changes the perceived numerosity of a pattern, but also the selectivity of the fMRI BOLD response. Connecting three dot-pairs in a pattern of ten dots causes the maximum repetition adaptation to occur at eight rather than ten dots (He et al, 2015). Similarly, while adapting to a 20-dot field does not change the apparent numerosity of a 20 isolated dots (as adaptation to the same number has no net effect), it does reduce the apparent numerosity of 10 dot-pairs, which appear less numerous than 10 isolated dots, and even less numerous after adaptation (Fornaciai, Cicchini, & Burr, 2016).…”

“…A consequence of this is that if two items are connected, simply by joining them with lines, the apparent numerosity of the connected items is greatly reduced (Franconeri, Bemis, & Alvarez, 2009;He, Zhang, Zhou, & Chen, 2009;He, Zhou, Zhou, He, & Chen, 2015) -see examples of Figure 1, and also the Supplementary Movie. In fact the lines do not need to be physically present: the effect works well with illusory contours as well (Kirjakovski & Matsumoto, 2016).…”