The purpose of this paper is to understand how the grid cells may perform path integration calculations. We study a general representational model  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.  We identify an isotropic condition  so that the local change of this vector is a conformal embedding of the  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 algebra and matrix Lie group , 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.
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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of path integration
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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.
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on the norm of the directional derivative of the recurrent network,
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local
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algebra and matrix Lie group
group representation of 2D self-motion