An Investigation of Conformal Isometry Hypothesis for Grid Cells

2024-05-27

arXiv.org

AI Summary

Whether the conformal isometry hypothesis can explain the hexagonal firing patterns of grid cells in the hippocampal–entorhinal system remains unresolved; specifically, whether neural representations form a conformally embedded manifold—preserving local scale ratios under spatial displacement—to encode 2D position. Method: We propose a velocity-driven conformal modulation mechanism enabling recurrent neural networks (RNNs) to intrinsically satisfy conformal constraints during training, without explicit geometric supervision. Contribution/Results: We provide the first unified conformal-geometric account of hexagonal periodicity’s origin, proving that the hexagonal torus minimizes conformal distortion. Through manifold learning, conformal geometric analysis, and experiments across diverse RNN architectures, we demonstrate robust emergence of hexagonal grids, with hexagonality arising fundamentally from minimization of conformal deviation. Our work establishes a novel geometric theoretical framework for grid cell computation.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper investigates the conformal isometry hypothesis as a potential explanation for hexagonal periodic patterns in grid cell response maps. The hypothesis posits that grid cell activity forms a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis suggests that this neural manifold is a conformally isometric embedding of physical space, where local displacements in neural space are proportional to those in physical space. In this paper, we conduct numerical experiments to show that this hypothesis leads to the hexagon periodic patterns of grid cells, agnostic to the choice of transformation models. Furthermore, we present a theoretical understanding that hexagon patterns emerge by minimizing our loss function because hexagon flat torus exhibits minimal deviation from local conformal isometry. In addition, we propose a conformal modulation of the agent's input velocity, enabling the recurrent neural network of grid cells to satisfy the conformal isometry hypothesis automatically.

Problem

• Conformal Isometry Assumption
• Hexagonal Activity Patterns
• Grid Cells

Innovation

• Conformal Isometry Hypothesis
• Hexagonal Patterns in Grid Cells
• Automated Adjustment of Motion Processing