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

Neural population geometry plays a crucial role in determining how task variables are represented and translated into behavior. The authors have introduced a variational principle that seeks to strike a balance between task accuracy and representational smoothness, all while considering stimulus-dependent weighting. This principle establishes a connection between the statistics of stimuli and the shape of the latent neural coordinate.

The principle makes specific predictions regarding distortions in bounded ensembles. For magnitude estimation, it anticipates distortions near the edges of these ensembles, global affine contraction for Gaussian ensembles, and warping towards high-probability regions in bimodal ensembles. To test the bounded-uniform case, the researchers trained recurrent neural networks to reproduce time intervals.

As the boundary distortions evolved, the neural networks' learned geometries exhibited three distinct regimes: too narrow to resolve, measurable, and extending across the stimulus range.

The authors propose that a shape parameter derived from an empirical neural coordinate can capture the nonlinear component of mean behavior through an affine readout. Crucially, only behavioral gain and offset need to be fitted afterward. Remarkably, in the bounded-uniform setting, the variational coordinate and the Bayesian least-squares mean response share strikingly similar shapes, even after accounting for affine normalization.

These findings provide compelling evidence for a variational framework that explains how stimulus statistics shape neural geometry and give rise to Bayesian-like behavior.