A Variational Principle for the Emergence of Neural Geometry Underlying Bayesian-Like Behavior

Neural population geometry constrains how task variables are represented and read out into behavior. We propose a variational principle in which learning balances task accuracy and representational smoothness under stimulus-dependent weighting. This balance links stimulus statistics to the shape of a latent neural coordinate. For magnitude estimation, the principle predicts distortions near the edges of bounded ensembles, global affine contraction for Gaussian ensembles, and warping toward high-probability regions for bimodal ensembles. We test the bounded-uniform case in recurrent neural networks trained to reproduce time intervals. Their learned geometries occupy three regimes as the boundary distortions change from too narrow to resolve, to measurable, to extending across the stimulus range. A shape parameter selected from an empirical neural coordinate predicts the nonlinear component of mean behavior through an affine readout, with only behavioral gain and offset fitted afterward. In this bounded-uniform setting, the variational coordinate and Bayesian least-squares mean response have closely matching shapes after affine normalization. These results suggest a variational account of how stimulus statistics shape neural geometry and yield Bayesian-like behavior.