On Path Integration of Grid Cells: Isotropic Metric, Conformal Embedding and Group Representation Article Swipe
The purpose of this paper is to understand how the grid cells may perform path integration calculations. We study a general representational model of path integration in which the self-position is represented by a vector formed by the activities of a population of grid cells, and the self-motion is represented by the change of this vector which is transformed by a general recurrent network for path integration. For local infinitesimal self-motion, the change of the vector is determined by the directional derivative of the recurrent network, and the norm of the directional derivative captures the metric of the path integration model. We identify an isotropic condition on the norm of the directional derivative of the recurrent network, so that the local change of this vector is a conformal embedding of the local self-motion. We then study a minimally simple prototype model where the local change is a linear transformation of the vector. This linear model gives rise to explicit algebraic structure in terms of matrix Lie group representation of 2D self-motion, as well as explicit geometric structure where the self-motion is represented by the rotation of the vector. We connect the isotropic condition under the linear model to the hexagon grid patterns of the response maps of grid cells. Our numerical experiments demonstrate that our model learns hexagon grid patterns which share various observed properties of the grid cells in the rodent brain. Furthermore, the learned model is capable of near exact path integration.
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Metadata
- Type
- article
- Language
- en
- Landing Page
- https://arxiv.org/abs/2006.10259v4
- OA Status
- green
- Cited By
- 2
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W3115516881
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W3115516881Canonical identifier for this work in OpenAlex
- Title
-
On Path Integration of Grid Cells: Isotropic Metric, Conformal Embedding and Group RepresentationWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2020Year of publication
- Publication date
-
2020-06-18Full publication date if available
- Authors
-
Ruiqi Gao, Jianwen Xie, Xue-Xin Wei, Song‐Chun Zhu, Ying WuList of authors in order
- Landing page
-
https://arxiv.org/abs/2006.10259v4Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/abs/2006.10259v4Direct OA link when available
- Concepts
-
Path integration, Mathematics, Grid, Conformal map, Topology (electrical circuits), Norm (philosophy), Mathematical analysis, Embedding, Geometry, Computer science, Artificial intelligence, Combinatorics, Political science, LawTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
2Total citation count in OpenAlex
- Citations by year (recent)
-
2021: 2Per-year citation counts (last 5 years)
- Related works (count)
-
20Other works algorithmically related by OpenAlex
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