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

The free energy principle casts perception as variational inference, but its biological implementation remains underspecified. In particular, the generalized-coordinate formalism should not be read as a literal claim that neurons compute arbitrary Taylor expansions. This paper argues that generalized synchronization provides the missing bottom-up mechanism. A contractive recurrent circuit driven by structured sensory input can synchronize to the driving dynamics. Under generic embedding conditions developed in the reservoir-computing literature, the resulting synchronization map can embed the low-dimensional sensory manifold into neural state space. Thus, the geometry predicted by the free energy principle need not be imposed from above by an explicitly Bayesian neural calculus; it can arise from ordinary recurrent dynamics driven by the world. I then propose a developmental extension. Hebbian plasticity acting on the correlations generated by sensory-driven synchronization may crystallize the embedded manifold into recurrent connectivity, yielding an autonomous continuous attractor network when the required fixed point exists. On this view, mature head-direction, grid-cell, and stimulus-driven visual manifolds are not genetically prespecified templates, but developmental products of three interacting processes: dynamical contraction, generalized synchronization, and correlation-based plasticity. The synthesis links the free energy principle, reservoir-computing embedding theorems, and contraction-theoretic models of Hebbian recurrent networks. It also yields testable predictions about dimensional thresholds for topological recovery, developmental sensitivity to plasticity, and the dependence of attractor geometry on input statistics. The central open problem is whether the Hebbian fixed point exists and preserves the embedding quality of the synchronization manifold.