On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding

TL;DR

The paper addresses why grid cells exhibit hexagonal firing patterns by proposing that 2D physical space is embedded into a high-dimensional neural space via a conformal isometry, preserving local distances up to a constant factor $s$. It employs a minimalistic single-module setting with an explicit metric and a loss $L = L_1 + \lambda L_2$, where $L_1$ enforces local distance preservation ${\|{\mathbf v}({\mathbf x}+{\Delta \mathbf x}) - {\mathbf v}({\mathbf x})\| = s\,{\|{\Delta \mathbf x}\|}$ and $L_2$ enforces the transformation $F({\mathbf v}({\mathbf x}), {\Delta \mathbf x})$. Through numerical experiments with linear and nonlinear transformation models, the authors show hexagonal grid firing patterns emerge as maximally distance-preserving embeddings, and they theoretically demonstrate that a hexagon flat torus minimizes higher-order deviations from local isometry, yielding $D(\Delta {\mathbf x}) = c\,||\Delta {\mathbf x}||^4$. Supporting neural data analysis reveals a roughly linear relation between displacements in neural space and physical space, consistent with the conformal-isometry picture and suggesting approximate constancy of the neural-vector norm across positions. The work further extends to multiple modules with independent scaling, implying multi-scale, conformal grid representations and connecting to path-planning advantages. Overall, the paper provides a geometric, normative principle for hexagonal grid patterns and their role in planning and navigation, robust to the specific form of the underlying transformation.

Abstract

This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form 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 proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.

Figures (15)

• Figure 1: (a) The self-position ${\bm{x}} = ({x}_1, {x}_2)$ in 2D Euclidean space is represented by a vector ${\bm{v}}({\bm{x}})$ in the $d$-dimensional neural space. When the agent moves by $\Delta {\bm{x}}$, the vector is transformed to ${\bm{v}}({\bm{x}}+\Delta {\bm{x}}) = F({\bm{v}}({\bm{x}}), \Delta {\bm{x}})$. (b) $F(\cdot, \Delta {\bm{x}})$ is a representation of the self-motion $\Delta {\bm{x}}$. (c) $\mathbb{M} = ({\bm{v}}({\bm{x}}), {\bm{x}} \in D)$ is a 2D manifold in the neural space, and is an embedding of the 2D Euclidean domain $\mathbb{D}$.
• Figure 2: Hexagonal periodic patterns learned in linear models. (a) Learned patterns of linear models with different scales. (b) Toroidal structure spectral analysis of the activities of grid cells.
• Figure 3: Left(a-d): Learned patterns for nonlinear models with different rectified functions. Right(e-h): Ablation for the linear model.
• Figure 4: The relationship between $\|{\bm{v}}({\bm{x}}+\Delta {\bm{x}}) - {\bm{v}}({\bm{x}})\|$ and $\|\Delta {\bm{x}}\|$ in the learned representations.
• Figure 5: Analysis of real neural data. (a) A clear linear relationship between $\|{\bm{v}}({\bm{x}}+\Delta {\bm{x}})-{\bm{v}}({\bm{x}})\|$ and $\|\Delta {\bm{x}}\|$ is shown in the real neural data. The unit of $\Delta {\bm{x}}$ is meter. (b) Distribution of $\|{\bm{v}}({\bm{x}})\|$ in neural data.

Theorems & Definitions (20)

• Proposition 1
• proof
• Definition 2
• Definition 3
• Proposition 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof