An attractor account of numerosity readout

Numerical perception is often described as the product of two systems: a near-exact mechanism for small sets and an approximate one for larger sets, whose variability follows Weber's law. Gallistel and Gelman proposed instead that both behaviours could arise from a single representation of discrete numerosities whose precision degrades progressively with magnitude. Here we give this idea a mechanistic form, where a noisy estimate of numerosity, already extracted by upstream perceptual processing, is turned into a discrete numerical response. Integers are represented as memories along a one-dimensional number line, jointly creating a Hopfield-inspired attractor landscape. This estimate provides the initial state, and recurrent dynamics clean it up by pulling it towards the stored integer representations. Because the memory components become broader and overlap more strongly with magnitude, the same landscape produces near-categorical responses for small numerosities and increasingly coarse, Weber-like estimates for larger ones. We tested these predictions by reanalysing 52,995 individual responses from an openly available visual-enumeration dataset. The attractor model outperformed matched static and constant-width controls, a purely driven accumulator, and Weber-style and bounded-optimal alternatives. Response time followed naturally from the number of clean-up steps, with a finite stopping budget accounting for the large-number plateau. The fitted response distributions approached a stable Weber fraction at larger numerosities, and participants with sharper low-number landscapes showed lower Weber noise on a non-overlapping set of large-number trials. Together, these results suggest that magnitude-dependent landscape geometry and recurrent clean-up make separable contributions to a shared numerical readout.