# Emergence of grid-like representations by training recurrent neural networks to perform spatial localization

arXiv: https://arxiv.org/abs/1803.07770 · HTML: https://mlsanity.com/paper/1803.07770/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2018.

## Built on by (most cited first, 30 of 52)

- [Relating transformers to models and neural representations of the hippocampal formation](https://mlsanity.com/paper/2112.04035/) (2021)
- [Sphere2Vec: A General-Purpose Location Representation Learning over a Spherical Surface for Large-Scale Geospatial Predictions](https://mlsanity.com/paper/2306.17624/) (2023)
- [Emergence of Maps in the Memories of Blind Navigation Agents](https://mlsanity.com/paper/2301.13261/) (2023)
- [Disentanglement with Biological Constraints: A Theory of Functional Cell Types](https://mlsanity.com/paper/2210.01768/) (2022)
- [Brain-inspired learning in artificial neural networks: a review](https://mlsanity.com/paper/2305.11252/) (2023)
- [Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks](https://mlsanity.com/paper/2312.08550/) (2023)
- [Multi-Object Navigation with dynamically learned neural implicit representations](https://mlsanity.com/paper/2210.05129/) (2022)
- [Reverse engineering recurrent neural networks with Jacobian switching linear dynamical systems](https://mlsanity.com/paper/2111.01256/) (2021)
- [Linking In-context Learning in Transformers to Human Episodic Memory](https://mlsanity.com/paper/2405.14992/) (2024)
- [Better artificial intelligence does not mean better models of biology](https://mlsanity.com/paper/2504.16940/) (2025)
- [Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells](https://mlsanity.com/paper/2311.02316/) (2023)
- [A Topological Deep Learning Framework for Neural Spike Decoding](https://mlsanity.com/paper/2212.05037/) (2022)
- [Learning with a Mole: Transferable latent spatial representations for navigation without reconstruction](https://mlsanity.com/paper/2306.03857/) (2023)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [Hippocampal formation-inspired probabilistic generative model](https://mlsanity.com/paper/2103.07356/) (2021)
- [Graph augmented Deep Reinforcement Learning in the GameRLand3D environment](https://mlsanity.com/paper/2112.11731/) (2021)
- [How to build a cognitive map: insights from models of the hippocampal formation](https://mlsanity.com/paper/2202.01682/) (2022)
- [Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells](https://mlsanity.com/paper/2210.02684/) (2022)
- [SpikePropamine: Differentiable Plasticity in Spiking Neural Networks](https://mlsanity.com/paper/2106.02681/) (2021)
- [Human-AI Shared Control via Policy Dissection](https://mlsanity.com/paper/2206.00152/) (2022)
- [Time Makes Space: Emergence of Place Fields in Networks Encoding Temporally Continuous Sensory Experiences](https://mlsanity.com/paper/2408.05798/) (2024)
- [EDEN: Entorhinal Driven Egocentric Navigation Toward Robotic Deployment](https://mlsanity.com/paper/2506.03046/) (2025)
- [Emergent behavior and neural dynamics in artificial agents tracking turbulent plumes](https://mlsanity.com/paper/2109.12434/) (2021)
- [Sphere2Vec: Multi-Scale Representation Learning over a Spherical Surface for Geospatial Predictions](https://mlsanity.com/paper/2201.10489/) (2022)
- [The Frost Hollow Experiments: Pavlovian Signalling as a Path to Coordination and Communication Between Agents](https://mlsanity.com/paper/2203.09498/) (2022)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [Sufficient conditions for offline reactivation in recurrent neural networks](https://mlsanity.com/paper/2505.17003/) (2025)
- [REMI: Reconstructing Episodic Memory During Internally Driven Path Planning](https://mlsanity.com/paper/2507.02064/) (2025)
- [Emergence of Double-slit Interference by Representing Visual Space in Artificial Neural Networks](https://mlsanity.com/paper/2205.10081/) (2022)
- [From Data-Fitting to Discovery: Interpreting the Neural Dynamics of Motor Control through Reinforcement Learning](https://mlsanity.com/paper/2305.11107/) (2023)

## Abstract (arXiv)

Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and functional significance of these spatial representations remain largely mysterious. As a new way to understand these neural representations, we trained recurrent neural networks (RNNs) to perform navigation tasks in 2D arenas based on velocity inputs. Surprisingly, we find that grid-like spatial response patterns emerge in trained networks, along with units that exhibit other spatial correlates, including border cells and band-like cells. All these different functional types of neurons have been observed experimentally. The order of the emergence of grid-like and border cells is also consistent with observations from developmental studies. Together, our results suggest that grid cells, border cells and others as observed in EC may be a natural solution for representing space efficiently given the predominant recurrent connections in the neural circuits.