Before revision:  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.
After revision:  We identify an isotropic condition on the norm of the directional derivative of the recurrent network , 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 representation of 2D self-motion, as well as explicit geometric structure where the self-motion is represented by the rotation of the vector.