# Progressive Development of the Number Sense in a Deep Neural Network

- Will Y. Zou,
*Stanford University*
- James L. McClelland,
*Stanford University*

## Abstract

What are the developmental bases of the number sense? This ability
could arise through evolution or experience. Stoianov & Zorzi (2012, Nature
Neuroscience, 8, 194-196) showed that a neural network could learn number sense
from visual examples containing varying numbers of elements. However, the
layer-wise training regime is unrealistic from a developmental standpoint. A key
observation is that number acuity progressively develops from infancy to
adulthood (as reflected by a decreasing Weber fraction). This development
involves accumulation of single examples, each of which updates the connection
weights in a hierarchical system. We present an unsupervised deep network that
learns all weights as it observes one `number example’ at a time. As
on-line training progresses, neurons representing numerosity start to emerge in
the deeper layers, and the Weber fraction progressively sharpens. These results
establish that a generic learning algorithm in a deep network gives rise to a
clear developmental trajectory of the number sense.

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