Endogenous Precision of the Number Sense

The behavioral variability in psychophysical experiments and the stochasticity of sensory neurons have revealed the inherent imprecision in the brain’s representations of environmental variables^1–6. Numerosity studies yield similar results, pointing to an imprecise ‘number sense’ in the brain^7–13. If the imprecision in representations reflects an optimal allocation of limited cognitive resources, as suggested by efficient-coding models^14–26, then it should depend on the context in which representations are elicited^25,27. Through an estimation task and a discrimination task, both involving numerosities, we show that the scale of subjects’ imprecision increases, but sublinearly, with the width of the prior distribution from which numbers are sampled. This sublinear relation is notably different in the two tasks. The double dependence of the imprecision — both on the prior and on the task — is consistent with the optimization of a tradeoff between the expected reward, different for each task, and a resource cost of the encoding neurons’ activity. Comparing the two tasks allows us to clarify the form of the resource constraint. Our results suggest that perceptual noise is endogenously determined, and that the precision of percepts varies both with the context in which they are elicited, and with the observer’s objective.