Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells

Dehong Xu, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, Ying Nian Wu
Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations, PMLR 197:370-387, 2023.

Abstract

The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact and connected abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at \url{https://github.com/DehongXu/grid-cell-rnn}.

Cite this Paper

BibTeX
@InProceedings{pmlr-v197-xu23a, title = {Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells}, author = {Xu, Dehong and Gao, Ruiqi and Zhang, Wen-Hao and Wei, Xue-Xin and Wu, Ying Nian}, booktitle = {Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations}, pages = {370--387}, year = {2023}, editor = {Sanborn, Sophia and Shewmake, Christian and Azeglio, Simone and Di Bernardo, Arianna and Miolane, Nina}, volume = {197}, series = {Proceedings of Machine Learning Research}, month = {03 Dec}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v197/xu23a/xu23a.pdf}, url = {https://proceedings.mlr.press/v197/xu23a.html}, abstract = {The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact and connected abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at \url{https://github.com/DehongXu/grid-cell-rnn}.} }
Endnote
%0 Conference Paper %T Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells %A Dehong Xu %A Ruiqi Gao %A Wen-Hao Zhang %A Xue-Xin Wei %A Ying Nian Wu %B Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2023 %E Sophia Sanborn %E Christian Shewmake %E Simone Azeglio %E Arianna Di Bernardo %E Nina Miolane %F pmlr-v197-xu23a %I PMLR %P 370--387 %U https://proceedings.mlr.press/v197/xu23a.html %V 197 %X The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact and connected abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at \url{https://github.com/DehongXu/grid-cell-rnn}.
APA
Xu, D., Gao, R., Zhang, W., Wei, X. & Wu, Y.N.. (2023). Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells. Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 197:370-387 Available from https://proceedings.mlr.press/v197/xu23a.html.

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