neural-manifolds-crystallized-embeddings

name

neural-manifolds-crystallized-embeddings

description

Neural manifolds as crystallized embeddings: a synthesis of free energy principle, generalized synchronization, and Hebbian plasticity. Proposes that neural manifolds emerge developmentally through three interacting processes: dynamical contraction (free energy minimization), generalized synchronization (reservoir computing embedding), and correlation-based Hebbian plasticity that crystallizes embedded manifolds into recurrent connectivity. Use when studying neural manifold formation, head-direction cells, grid cells, free energy principle in neural coding, Hebbian learning of attractor networks, or developmental emergence of neural representations. arXiv: 2605.04200

Neural Manifolds as Crystallized Embeddings

Paper: Vikas N. O'Reilly-Shah (2026). "Neural Manifolds as Crystallized Embeddings: A Synthesis of the Free Energy Principle, Generalized Synchronization, and Hebbian Plasticity" arXiv: 2605.04200 Categories: q-bio.NC

Core Thesis

Mature neural manifolds (head-direction, grid-cell, stimulus-driven visual manifolds) are not genetically prespecified templates, but developmental products of three interacting processes:

1. Dynamical contraction (free energy principle)
2. Generalized synchronization (reservoir computing embedding theorems)
3. Correlation-based Hebbian plasticity (crystallization into recurrent connectivity)

Three-Process Framework

1. Dynamical Contraction

• Free energy minimization drives the system toward low-dimensional manifolds
• Neural dynamics contract onto embedded submanifolds in state space
• Provides the geometric scaffold for representation

2. Generalized Synchronization

• Sensory-driven synchronization generates correlations across neural populations
• Reservoir computing embedding theorems ensure faithful representation
• Input statistics determine the geometry of the synchronization manifold

3. Hebbian Crystallization

• Hebbian plasticity acts on correlations from synchronization
• Crystallizes the embedded manifold into recurrent connectivity
• Yields autonomous continuous attractor network (when fixed point exists)

Key Synthesis

Links three theoretical frameworks:

• Free Energy Principle: explains why dynamics contract onto manifolds
• Reservoir Computing Embedding Theorems: guarantees representational capacity
• Contraction Theory of Hebbian Networks: explains developmental crystallization

Testable Predictions

1. Dimensional thresholds: Specific thresholds for topological recovery of manifolds
2. Developmental sensitivity: Critical periods for plasticity-dependent manifold formation
3. Input statistics dependence: Attractor geometry depends on sensory input statistics
4. Fixed point existence: Central open problem — whether Hebbian fixed point preserves embedding quality

Applications

• Head-direction cell system development
• Grid cell formation mechanisms
• Stimulus-driven visual manifold emergence
• Understanding developmental disorders of neural representation
• Brain-computer interface design (leveraging natural manifold structure)

Central Open Problem

Whether the Hebbian fixed point exists and preserves the embedding quality of the synchronization manifold. This determines whether the three-process synthesis fully accounts for observed neural manifold properties.

Activation Keywords

• neural manifold, crystallized embedding, free energy principle
• generalized synchronization, Hebbian plasticity, attractor network
• head-direction cells, grid cells, developmental neuroscience
• reservoir computing, contraction theory, continuous attractor