Grid-like structure is optimal for path integration.

Grid cells in medial entorhinal cortex are believed to play a key role in path integration. However, the relation between path integration and the grid-like arrangement of their firing field remains unclear. We provide theoretical evidence that grid-like structure and path integration are closely related. In one dimension, the grid-like structure provides the optimal solution for path integration assuming that the noise correlation structure is Gaussian. In two dimensions, assuming that the noise is Gaussian, rectangular grid-like structure is the optimal solution provided that 1- both noise correlation and receptive field structures of the neurons can be multiplicatively decomposed into orthogonal components and 2- the eigenvalues of the decomposed correlation matrices decrease faster than the square of the frequency of the corresponding eigenvectors. We will also address the decoding mechanism and show that the problem of decoding reduces to the problem of extracting task relevant information in the presence of task irrelevant information. Change-based Population Coding provides the optimal solution for this problem.