I have to ask some incredibly basic questions.
As I understand grid cells, the thing that makes them “griddy” is that there is a global (in the sense of a single region or map in the entorhinal cortex) pattern that extends beyond the reach of the processes of any cell in any single mini-column. There is some overall coordinating mechanism that distributes whatever input is being represented (motion, location, …) so that small local clusters of cells respond in a phased and spatially repeating pattern.
This focus on the possible local phase mechanism seems to be ignoring the larger coordination that seems to be an important part of the grid cell story.
This cluster of questions are all looking at the same underlying question from different directions:
Q: Put plainly, why is that group of cells “way over there” responding in lockstep to what is happening locally? “Way over there” is so far away that no dendrite from any cell in this mini or macro-column is able to receive any direct input from that remote area.
Q: Though most of the hierarchy topology is maintained - any given part of a brain region/map projects to roughly the same part of the topologically related area in the next map. How do we get from a local part of a region/map to some region/map-wide global representation pattern that we are calling a grid? I am not aware of any “dispersion” mechanism in the hierarchy.
Q: When a critter enters a room and some grid pattern is established, what is the input to that grid, and is that part of the coordinating mechanism?
Q: If you are talking about the location signal in a single macro-column is it really correct to label that as having anything to do with grid cells if that area of the cortex does not display the signature grid pattern?