# Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells

arXiv: https://arxiv.org/abs/2311.02316 · HTML: https://mlsanity.com/paper/2311.02316/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 69 references that are in the map.
- First posted on arXiv 4 Nov 2023.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Contrastive Representation Distillation](https://mlsanity.com/paper/1910.10699/) (2019)
- [Learning Representations by Maximizing Mutual Information Across Views](https://mlsanity.com/paper/1906.00910/) (2019)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [A Cookbook of Self-Supervised Learning](https://mlsanity.com/paper/2304.12210/) (2023)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [On Mutual Information in Contrastive Learning for Visual Representations](https://mlsanity.com/paper/2005.13149/) (2020)
- [Emergence of grid-like representations by training recurrent neural networks to perform spatial localization](https://mlsanity.com/paper/1803.07770/) (2018)
- [Self-Supervised Learning with an Information Maximization Criterion](https://mlsanity.com/paper/2209.07999/) (2022)
- [Self-Supervised Learning via Maximum Entropy Coding](https://mlsanity.com/paper/2210.11464/) (2022)
- [An Information-Theoretic Perspective on Variance-Invariance-Covariance Regularization](https://mlsanity.com/paper/2303.00633/) (2023)
- [Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion](https://mlsanity.com/paper/1810.05597/) (2018)
- [The Sense of Place: Grid Cells in the Brain and the Transcendental Number e](https://mlsanity.com/paper/1304.0031/) (2013)
- [Winning the lottery with neural connectivity constraints: faster learning across cognitive tasks with spatially constrained sparse RNNs](https://mlsanity.com/paper/2207.03523/) (2022)
- [Actionable Neural Representations: Grid Cells from Minimal Constraints](https://mlsanity.com/paper/2209.15563/) (2022)
- [On Path Integration of Grid Cells: Group Representation and Isotropic Scaling](https://mlsanity.com/paper/2006.10259/) (2020)
- [Growing Brains: Co-emergence of Anatomical and Functional Modularity in Recurrent Neural Networks](https://mlsanity.com/paper/2310.07711/) (2023)
- [Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells](https://mlsanity.com/paper/2210.02684/) (2022)

## Built on by (most cited first, 8 of 8)

- [Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks](https://mlsanity.com/paper/2312.08550/) (2023)
- [Cognitive Overload Attack:Prompt Injection for Long Context](https://mlsanity.com/paper/2410.11272/) (2024)
- [Position: Machine Learning Conferences Should Establish a "Refutations and Critiques" Track](https://mlsanity.com/paper/2506.19882/) (2025)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [REMI: Reconstructing Episodic Memory During Internally Driven Path Planning](https://mlsanity.com/paper/2507.02064/) (2025)
- [Geometric sparsification in recurrent neural networks](https://mlsanity.com/paper/2406.06290/) (2024)
- [Spectral-Stimulus Information for Self-Supervised Stimulus Encoding](https://mlsanity.com/paper/2407.06195/) (2024)
- [Venice-H1: Failure-Aware Query Re-Ranking with Multi-Scale Grid Signatures for Referring Image Segmentation](https://mlsanity.com/paper/2606.22546/) (2026)

## Abstract (arXiv)

To solve the spatial problems of mapping, localization and navigation, the mammalian lineage has developed striking spatial representations. One important spatial representation is the Nobel-prize winning grid cells: neurons that represent self-location, a local and aperiodic quantity, with seemingly bizarre non-local and spatially periodic activity patterns of a few discrete periods. Why has the mammalian lineage learnt this peculiar grid representation? Mathematical analysis suggests that this multi-periodic representation has excellent properties as an algebraic code with high capacity and intrinsic error-correction, but to date, there is no satisfactory synthesis of core principles that lead to multi-modular grid cells in deep recurrent neural networks. In this work, we begin by identifying key insights from four families of approaches to answering the grid cell question: coding theory, dynamical systems, function optimization and supervised deep learning. We then leverage our insights to propose a new approach that combines the strengths of all four approaches. Our approach is a self-supervised learning (SSL) framework - including data, data augmentations, loss functions and a network architecture - motivated from a normative perspective, without access to supervised position information or engineering of particular readout representations as needed in previous approaches. We show that multiple grid cell modules can emerge in networks trained on our SSL framework and that the networks and emergent representations generalize well outside their training distribution. This work contains insights for neuroscientists interested in the origins of grid cells as well as machine learning researchers interested in novel SSL frameworks.