A Perceptual-Like Population-Coding Mechanism of Approximate Numerical Averaging

Abstract

Humans possess a remarkable ability to rapidly form coarse estimations of numerical averages. This ability is important for making decisions that are based on streams of numerical or value-based information, as well as for preference formation. Nonetheless, the mechanism underlying rapid approximate numerical averaging remains unknown, and several competing mechanism may account for it. Here, we tested the hypothesis that approximate numerical averaging relies on perceptual-like processes, instantiated by population coding. Participants were presented with rapid sequences of numerical values (four items per second) and were asked to convey the sequence average. We manipulated the sequences' length, variance, and mean magnitude and found that similar to perceptual averaging, the precision of the estimations improves with the length and deteriorates with (higher) variance or (higher) magnitude. To account for the results, we developed a biologically plausible population-coding model and showed that it is mathematically equivalent to a population vector. Using both quantitative and qualitative model comparison methods, we compared the population-coding model to several competing models, such as a step-by-step running average (based on leaky integration) and a midrange model. We found that the data support the population-coding model. We conclude that humans' ability to rapidly form estimations of numerical averages has many properties of the perceptual (intuitive) system rather than the arithmetic, linguistic-based (analytic) system and that population coding is likely to be its underlying mechanism.

The averaging of numerosities: A psychometric investigation of the mental line

“…This could not happen if the participants form the estimates on the basis of few samples, say subject to WM capacity limitations (2-4). Instead, the results are consistent with a population code model, in which the responses to each stimuli is aggregated on the numerosity representation, and the estimate is obtained via a population code that estimates the center of mass (Brezis et al, 2018). In some of the participants, we also found small temporal biases.…”

“…Rather, the evidence indicates that the estimation mechanism corresponds to a frequency-based estimation (the estimation of the center of mass of a noisy frequency distribution of the numbers), which is somewhat similar to the one suggested by the ANS representation system (see Brezis, Bronfman, Jacoby, Lavidor, & Usher, 2016;Brezis et al, 2015Brezis et al, , 2018. In particular, Brezis et al (2015Brezis et al ( , 2018 have proposed an ANS type of population code model, which accounts for a characteristic signature of the population code: Precision improves with the length of the sequence (see Fig. 1, blue line).…”

“…While, analytically, averaging can be viewed as being equivalent to addition (followed by division by the number of terms), there are reasons to believe that this is not the way that participants estimate the average of rapid sequences of (symbolic) numbers (Brezis, Bronfman, & Usher, 2015Malmi & Samson, 1983;Mitrani-Rosenbaum, Glickman, & Usher, 2020), as they can provide accurate and rapid estimations of the average, even when they do not know the number of elements, or when elements in a specific range are to be discarded after the sequence presentation (Malmi & Samson, 1983). Rather, the evidence indicates that the estimation mechanism corresponds to a frequency-based estimation (the estimation of the center of mass of a noisy frequency distribution of the numbers), which is somewhat similar to the one suggested by the ANS representation system (see Brezis, Bronfman, Jacoby, Lavidor, & Usher, 2016;Brezis et al, 2015Brezis et al, , 2018. In particular, Brezis et al (2015Brezis et al ( , 2018 have proposed an ANS type of population code model, which accounts for a characteristic signature of the population code: Precision improves with the length of the sequence (see Fig.…”

“…To our knowledge, there is less research on the averaging of (nonsymbolic) numerosities. While there is research on averaging of numerical (symbolic) numbers (Brezis et al, 2015(Brezis et al, , 2018Spitzer et al, 2017;Vandormael, Castañón, Balaguer, Li, & Summerfield, 2017), testing averaging of nonsymbolic numbers has the added bonus of excluding symbolic computations, and thus exclusively targeting the ANS system.…”

Fast and effective: Intuitive processes in complex decisions

“…As our task involves some practice, the assumption that the decision mechanism includes learned weights (reflecting the attributes' importance) is not implausible. 2 Alternatively, the mechanism of weighted averaging could be mediated by a population code model (Brezis et al, 2016;Brezis et al, 2015;Brezis, Bronfman & Usher, 2017), which operates using numerosity detectors (Dehaene, Molko, Cohen & Wilson, 2004;Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2 This does not require to endorse all the assumptions of the PCS model, such as RT being based on convergence to asymptotic activation; an alternative assumption is an integration to boundary. The property of PCS that is important to our results is the parallel integration of values from all attributes.…”