A “Deep Network” Model of Numerosity Perception: Development, Skilled Performance, and Dyscalculia.

Zorzi, M. & Stoianov, I.

University of Padova, Italy

Numerosity estimation is an evolutionarily ancient ability that is thought to be foundational to mathematical learning in humans. It is widely believed that this ability is supported by a specialized mechanism, known as the Approximate Number System (ANS), which in primates has a specific neural substrate in the intraparietal sulcus. The ANS would represent numerosity in an abstract way and its representational precision, also known as “number acuity”, is thought to be causally linked to both typical and atypical pathways of numeracy acquisition. We recently showed that visual numerosity emerges as a high-order statistical feature of images in “deep” neural networks that learn a hierarchical generative model of the sensory input without supervision (Stoianov & Zorzi, 2012, Nat. Neurosc.). Here we demonstrate that the model can readily account for the modulation of numerosity estimation performance induced by manipulations of continuous visual properties of the stimuli, such as cumulative surface and occupied area - a finding that was interpreted as evidence against a mechanism that computes abstract numerosity (i.e., the ANS). Second, we show that the model’s number acuity during learning improves in a way that mirrors the human developmental trajectory from infancy throughout adulthood. Third, we show that the atypical developmental trajectory of number acuity observed in dyscalculic children can be simulated in the model by limiting the computational resources (i.e., number of hidden neurons in the deep layers) that are available for learning the generative model, which is in line with the finding of reduced gray matter density in the intraparietal sulcus of dyscalculic subjects.