# Actionable Neural Representations: Grid Cells from Minimal Constraints

arXiv: https://arxiv.org/abs/2209.15563 · HTML: https://mlsanity.com/paper/2209.15563/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2022.

## Built on by (most cited first, 10 of 10)

- [Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks](https://mlsanity.com/paper/2312.08550/) (2023)
- [Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells](https://mlsanity.com/paper/2311.02316/) (2023)
- [Flow Equivariant World Models: Memory for Partially Observed Dynamic Environments](https://mlsanity.com/paper/2601.01075/) (2026)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [Predictive representations: building blocks of intelligence](https://mlsanity.com/paper/2402.06590/) (2024)
- [Range, not Independence, Drives Modularity in Biologically Inspired Representations](https://mlsanity.com/paper/2410.06232/) (2024)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [Spectral-Stimulus Information for Self-Supervised Stimulus Encoding](https://mlsanity.com/paper/2407.06195/) (2024)
- [MapFormer: Self-Supervised Learning of Cognitive Maps with Input-Dependent Positional Embeddings](https://mlsanity.com/paper/2511.19279/) (2025)
- [Distributional Active Inference](https://mlsanity.com/paper/2601.20985/) (2026)

## Abstract (arXiv)

To afford flexible behaviour, the brain must build internal representations that mirror the structure of variables in the external world. For example, 2D space obeys rules: the same set of actions combine in the same way everywhere (step north, then south, and you won't have moved, wherever you start). We suggest the brain must represent this consistent meaning of actions across space, as it allows you to find new short-cuts and navigate in unfamiliar settings. We term this representation an `actionable representation'. We formulate actionable representations using group and representation theory, and show that, when combined with biological and functional constraints - non-negative firing, bounded neural activity, and precise coding - multiple modules of hexagonal grid cells are the optimal representation of 2D space. We support this claim with intuition, analytic justification, and simulations. Our analytic results normatively explain a set of surprising grid cell phenomena, and make testable predictions for future experiments. Lastly, we highlight the generality of our approach beyond just understanding 2D space. Our work characterises a new principle for understanding and designing flexible internal representations: they should be actionable, allowing animals and machines to predict the consequences of their actions, rather than just encode.