  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 conformally 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 explore the connection between the isotropic
scaling condition and a special class of hexagon grid patterns. 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. Code is available
at https://github.com/ruiqigao/grid-cell-path.
