Before revision:  Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for path integral, which models the change of the vector representation given the self-motion, and we discern two matrix Lie algebras and their Lie groups that are naturally coupled together.
After revision:  Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras and their Lie groups that are naturally coupled together.