# Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells

arXiv: https://arxiv.org/abs/2210.02684 · HTML: https://mlsanity.com/paper/2210.02684/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 46 references that are in the map.
- First posted on arXiv 6 Oct 2022.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Emergence of grid-like representations by training recurrent neural networks to perform spatial localization](https://mlsanity.com/paper/1803.07770/) (2018)
- [Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion](https://mlsanity.com/paper/1810.05597/) (2018)
- [Emergence of functional and structural properties of the head direction system by optimization of recurrent neural networks](https://mlsanity.com/paper/1912.10189/) (2019)
- [On Path Integration of Grid Cells: Group Representation and Isotropic Scaling](https://mlsanity.com/paper/2006.10259/) (2020)

## Built on by (most cited first, 5 of 5)

- [Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells](https://mlsanity.com/paper/2311.02316/) (2023)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [A minimalistic representation model for head direction system](https://mlsanity.com/paper/2411.10596/) (2024)
- [Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning](https://mlsanity.com/paper/2505.14806/) (2025)

## Abstract (arXiv)

The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at urlhttps://github.com/DehongXu/grid-cell-rnn.