Before revision:  We connect the isotropic condition under the linear model to the hexagon grid patterns of the response maps of grid cells . Our numerical experiments demonstrate that our model learns hexagon grid patterns which share various observed properties of the grid cells in the rodent brain.
After revision:  We connect the isotropic condition under the linear model to the hexagon grid patterns of grid cells under the linear transformation. Finally, with our optimization-based approach, we manage to learn hexagon grid patterns which share various observed properties of the grid cells in the rodent brain.