Before revision:   We identify an isotropic condition  so that the local change of this vector is a conformal embedding of the  self-motion.
After revision:  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 integration model. We identify an isotropic condition so that the local change of this vector is a conformal embedding of the self-motion.