Embedding of low-dimensional sensory dynamics in recurrent networks: Implications for the geometry of neural representation.

Neural population activity in sensory cortex is organized on low-dimensional manifolds, but it is unclear why such manifolds should arise and what determines their geometry. We address this sensory representation problem by modeling cortical populations as recurrent circuits driven by low-dimensional, regular sensory dynamics (e.g. motion on a circle, head direction, multi-frequency tones on tori). By combining tools from generalized synchronization and delay-embedding theory, specialized to this quasiperiodic regime, we show that contracting recurrent networks generically develop smooth internal manifolds that embed the sensory dynamics. The dimensional requirement is modest and depends only on the intrinsic dimension [Formula: see text] of the effective sensory manifold, not on the complexity of the external world: a hidden dimension [Formula: see text] generically suffices (e.g. [Formula: see text] for a circle, [Formula: see text] for a two-frequency torus; bounds compatible with Whitney and Takens' embedding theorems). We then prove a prediction-separation result that links representational geometry directly to predictive performance, without assuming knowledge of contraction rates: if the circuit can predict future sensory inputs with small error, then states with different futures must be separated in neural state space, up to a resolution set by the prediction error. The resulting scale-limited embeddings naturally give rise to categorical boundaries, metameric equivalence of distinct stimuli, and discrimination thresholds. Numerical experiments with trained [Formula: see text] recurrent networks driven by head-direction-like and multi-frequency signals recover ring- and torus-shaped hidden manifolds with the expected topology; state separation improves most rapidly near the [Formula: see text] threshold. Training typically pushes the networks beyond the strict contraction regime where the theory guarantees faithful embedding, yet convergence consistent with generalized synchronization and manifold recovery persist, indicating that our conditions are sufficient but not necessary. Together, these results provide a mechanistic account of why low-dimensional sensory manifolds emerge in recurrent circuits and how prediction constrains their resolution, grounded in dynamical systems embedding theory and consistent with empirical findings on cortical population dynamics.