Interesting that you naturally evoke eigenvalues for formalizing the grid cell “thing”.
Did you know that a theory explains grid cell patterns by eigenvalues?
Place cells can be modeled with Successor Representations (a technique from Reinforcement Learning). And surprisingly, grid cell patterns look accurately similar to the eigendecomposition of those Successor Representations.
A good and not too technical blog post on Successor Representations: The present in terms of the future: Successor representations in Reinforcement learning | by Arthur Juliani | Medium?
Experimental grid cell fields:
Eigenvector from modeled Successor Representations:
More in Stachenfeld, 2017: http://gershmanlab.webfactional.com/pubs/Stachenfeld17.pdf
Compared to other models, this one reproduces the pattern irregualirities in non-squared environments with boundaries. So it seems more robust. But it can just be a coincidence…