Modeling the approximate number system to quantify the contribution of visual stimulus features

Abstract

The approximate number system (ANS) subserves estimation of the number of items in a set. Typically, ANS function is assessed by requiring participants to compare the number of dots in two arrays. Accuracy is determined by the numerical ratio of the sets being compared, and each participant’s Weber fraction (w) provides a quantitative index of ANS acuity. When making numerical comparisons, however, performance is also influenced by non-numerical features of the stimuli, such as the size and spacing of dots. Current models of numerosity comparison do not account for these effects and consequently lead to different estimates of w depending on the methods used to control for non-numerical features. Here we proffer a new model that teases apart the effects of ANS acuity from the effects of non-numerical stimulus features. The result is an estimate of w that is a more theoretically valid representation of numerical acuity and novel terms that denote the degree to which a participant’s perception of number is affected by non-numerical features. We tested this model in a sample of 20 adults and found that, by correctly attributing errors due to non-numerical stimulus features, the w obtained was more reliable across different stimulus conditions. We found that although non-numerical features biased numerosity discriminations in all participants, number was the primary feature driving discriminations in most of them. Our findings support the idea that, while numerosity is a distinct visual quantity, the internal representation of number is tightly bound to the representation of other magnitudes. This tool for identifying the different effects of the numerical and non-numerical features of a stimulus has important implications not only for the behavioral investigation of the ANS, but also for the collection and analyses of neural data sets associated with ANS function.

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

“…Density judgements were biased towards numerosity by about 78% and area by 53%, whereas number judgements were biased towards area by only 15%. This is consistent with several studies that showed small biases of numerosity estimates towards area (DeWind, Adams, Platt, & Brannon, 2015;Gebuis & Reynvoet, 2012a, 2012b, but stronger effects of numerosity on area judgements (Hurewitz, Gelman, & Schnitzer, 2006;Nys & Content, 2012) and density judgments (Dakin et al, 2011). Importantly, the selective sensitivity for numerosity over density is far less pronounced with dense stimuli, where the results are consistent with independent analyses of density and area .…”

“…This result not only shows that numerosity is not derived indirectly from density, texture or other low-level features, but also casts doubts on the idea that density is a primary visual feature: rather, density seems to be derived indirectly from numerosity, reinforcing previous evidence showing that at low numerosities, density judgments are particularly unreliable and are often surrogated by number judgments . Overall our results add to a growing body of literature showing that visual numerosity is perceived directly, rather than being recalculated from area and density (Anobile, Arrighi, et al, 2016;Anobile, Castaldi, Turi, Tinelli, & Burr, 2016;Anobile et al, 2014;Anobile, Cicchini, et al, 2016;Anobile et al, 2015;Arrighi et al, 2014;Burr & Ross, 2008;Cicchini et al, 2016;DeWind et al, 2015;Hurewitz et al, 2006;Kramer, Di Bono, & Zorzi, 2011;Ross & Burr, 2010Stoianov & Zorzi, 2012).…”

Subtypes and comorbidity in mathematical learning disabilities

“…However, while reviews supporting the number sense theory do not discuss confounds and evidential weaknesses, both behavioral and neuroimaging evidence for this proposal are very weak, or even negative: There are practically no fMRI studies which deliver clear evidence and negative findings seem to outweigh positive ones (see Szűcs et al, 2013a,b, for a review). Further, past behavioral data referring the putative impairment of a so-called nonsymbolic number representation are disqualified by major confounds in the stimulus material (Bugden and Ansari, 2015;DeWind et al, 2015;Fuhs and McNeil, 2013;Gilmore et al, 2013;Mix et al, 1997;Szűcs et al, 2013a,b). In addition, it is doubtful whether there is a connection between the magnitude representation and school mathematics achievement at all (see Szűcs et al, 2014 for analysis).…”

Numerosity representation is encoded in human subcortex

“…Thus, besides number, size, and spacing, at least seven nonnumerical dimensions can be expressed as a function of those three dimensions: total perimeter, total surface area, individual surface area, field area, sparsity, coverage, and apparent closeness (see ref. 64 for precise mathematical definitions and derivations of the choice model). After estimating the degree to which number, size, and spacing influence each participant's choice behavior (b number , b size , and b spacing respectively), these β estimates were created to form a β vector for behavior.…”

“…In recent work, an innovative approach has been developed to provide a statistical test of the effects of numerical and nonnumerical magnitudes on behavior or on neural activity (62)(63)(64)(65). These studies have demonstrated a large effect of numerosity and minimal, if any, effects of nonnumerical cues (such as surface area, sparsity, or spacing) on behavioral measures of numerosity judgment or on neural activity evoked by passively viewing dot arrays of a wide range of numerosities with no explicit task demands on magnitude.…”