# A Grid Cell-Inspired Structured Vector Algebra for Cognitive Maps

arXiv: https://arxiv.org/abs/2503.08608 · HTML: https://mlsanity.com/paper/2503.08608/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 48 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 15 of 15)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [In-memory hyperdimensional computing](https://mlsanity.com/paper/1906.01548/) (2019)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [Vector Symbolic Architectures as a Computing Framework for Emerging Hardware](https://mlsanity.com/paper/2106.05268/) (2021)
- [Variable Binding for Sparse Distributed Representations: Theory and Applications](https://mlsanity.com/paper/2009.06734/) (2020)
- [Robust computation with rhythmic spike patterns](https://mlsanity.com/paper/1901.07718/) (2019)
- [Neuromorphic Visual Scene Understanding with Resonator Networks](https://mlsanity.com/paper/2208.12880/) (2022)
- [The Sense of Place: Grid Cells in the Brain and the Transcendental Number e](https://mlsanity.com/paper/1304.0031/) (2013)
- [On Path Integration of Grid Cells: Group Representation and Isotropic Scaling](https://mlsanity.com/paper/2006.10259/) (2020)
- [Binding in hippocampal-entorhinal circuits enables compositionality in cognitive maps](https://mlsanity.com/paper/2406.18808/) (2024)
- [Generalized Holographic Reduced Representations](https://mlsanity.com/paper/2405.09689/) (2024)
- [Visual Odometry with Neuromorphic Resonator Networks](https://mlsanity.com/paper/2209.02000/) (2022)
- [Factorizers for Distributed Sparse Block Codes](https://mlsanity.com/paper/2303.13957/) (2023)
- [Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns](https://mlsanity.com/paper/1411.2136/) (2014)
- [Distributed Representations Enable Robust Multi-Timescale Symbolic Computation in Neuromorphic Hardware](https://mlsanity.com/paper/2405.01305/) (2024)

## Abstract (arXiv)

The entorhinal-hippocampal formation is the mammalian brain's navigation system, encoding both physical and abstract spaces via grid cells. This system is well-studied in neuroscience, and its efficiency and versatility make it attractive for applications in robotics and machine learning. While continuous attractor networks (CANs) successfully model entorhinal grid cells for encoding physical space, integrating both continuous spatial and abstract spatial computations into a unified framework remains challenging. Here, we attempt to bridge this gap by proposing a mechanistic model for versatile information processing in the entorhinal-hippocampal formation inspired by CANs and Vector Symbolic Architectures (VSAs), a neuro-symbolic computing framework. The novel grid-cell VSA (GC-VSA) model employs a spatially structured encoding scheme with 3D neuronal modules mimicking the discrete scales and orientations of grid cell modules, reproducing their characteristic hexagonal receptive fields. In experiments, the model demonstrates versatility in spatial and abstract tasks: (1) accurate path integration for tracking locations, (2) spatio-temporal representation for querying object locations and temporal relations, and (3) symbolic reasoning using family trees as a structured test case for hierarchical relationships.