On Path Integration Model of Grid Cells: Locally Conformal Embedding and Matrix Lie Algebra

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.