Before revision:  We study a general representational model of path integration 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. 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 integrationmodel.
After revision:  We study a general representational model of path integration 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 a general transformation 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 integrationmodel.