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 integrationmodel. 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 transformationof 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.
Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation model of path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector , and the 2D self-motion is represented by a general transformation of the vector . We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding , so that the error in the vector representation translates proportionally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation, uncover its explicit algebraic and geometric structure as matrix Lie group of rotation, and establish the connection between the isotropic scaling condition and hexagon grid patterns of grid cells under the linear transformation. Finally, with our optimization-based approach, we manage to learn hexagon grid patterns that share similar properties of the grid cells in the rodent brain. The learned model is capable of accurate long distance path integration.
R
The purpose of this paper is to understand how the grid cells may
Understanding how grid cells
R
. We study a general representational
remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation
R
in which the
by grid cells, where the 2D
R
represented by a vector formed by the activities of a population of grid cells
encoded as a higher dimensional vector
A
2D
R
the change of this vector which is transformed by a general recurrent network for path integration. For local infinitesimal self-motion, the change
a general transformation
R
is determined by the directional derivative of the recurrent network, and the norm of the directional derivative captures the metric of the path integrationmodel. We identify an isotropic condition on the norm of the directional derivative of the recurrent network
. We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding
R
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 transformationof the vector.This linear model gives rise to explicit algebraic structure in terms of
error in the vector representation translates proportionally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation, uncover its explicit algebraic and geometric structure as
R
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
of rotation, and establish the connection between the isotropic scaling condition and
R
the response maps of grid cells . Our numerical experiments demonstrate that our model learns
grid cells under the linear transformation. Finally, with our optimization-based approach, we manage to learn
R
which share various observed
that share similar
R
Furthermore, the
The
R
near exact
accurate long distance