Neural Manifolds as Crystallized Embeddings: A Synthesis of the Free Energy Principle, Generalized Synchronization, and Hebbian Plasticity

Abstract

The free energy principle casts perception as variational inference, but its biological implementation is underspecified. The generalized-coordinate formalism is not a literal claim that neurons compute arbitrary Taylor expansions. This paper argues that generalized synchronization (GS) provides the missing bottom-up mechanism. Certain recurrent circuits satisfy a contraction property: nearby trajectories converge exponentially. A contracting circuit driven by structured sensory input synchronizes to driving dynamics. Under generic embedding conditions, the resulting synchronization map embeds the low-dimensional sensory manifold into neural state space. The geometry predicted by the free energy principle is not imposed from above by an explicitly Bayesian neural calculus. It arises from ordinary recurrent dynamics. I then propose a developmental extension. Hebbian plasticity acting on the correlations generated by sensory-driven synchronization shapes the embedded manifold into recurrent connectivity, producing a continuous attractor network that approximates the embedded sensory manifold. Prediction-separation results bound the representational fidelity of the resulting circuit by prediction accuracy: where the network predicts future observations well, the synchronization map separates underlying states; where prediction fails, the representation collapses. The collapses are observable as categorical perception, metameric equivalence, and discrimination thresholds. On this view, mature head-direction, grid-cell, and stimulus-driven visual manifolds are developmental products of three interacting processes: dynamical contraction, generalized synchronization, and correlation-based plasticity. The central open problems are whether the Hebbian fixed point exists and whether Hebbian dynamics produce a sufficiently accurate predictor on the relevant input distribution.