So, I’ve been trying to develop an intuition for the behavior of the grid cells. My thoughts have clarified a bit since watching the video shared by @bdsaglam. In the talk, the presenter described the grid like structure in the cells arising from local inhibition fields around the currently active neurons in the cortical layer. The resulting field of overlapping inhibition spheres naturally give rise to a triangular lattice pattern of cell activations. When the sensor moves, the pattern of active cells in the layer all shift together in roughly the same direction. I imagine this shift occurring in a manner similar to how a flock of birds or school of fish move en masse in response to subtle variations in the movement of its constituent parts. The tell-tale repetitive grid pattern observed in the lab rats are a result of these cells periodically firing as the shifting pattern realigns with the original pattern, which will occur at regularly spaced intervals as the sensor is shifted (or rotated). Different grid cell modules will have different responses (in phase, period, and orientation), which then gives us multiple populations of cells that can be correlated to obtain unique location/orientation representations.
So, this is where I am at the moment. I’m currently trying to work up a visualization of this inhibition generated pattern, but I’m also very interested in the necessary input requirements for shifting the grid cell representation. The presenter in the video seemed to think that the network was something like a self-organizing map (EDIT: actually it’s a Hopfield network.) (see around here in video), but instead of having a finite set of stable fixed point attractor states, there could exist continuous manifolds of stable states with similar attraction strengths (Lyapunov function with a flat valey in K dimensions). One could potentially move along these states like walking along the floor of a canyon rather than having to climb up out of a local minimum valley before descending into another. These manifolds would basically represent all of the known transitions from one state to another. (e.g. Teleporting from one location to another is currently not possible, so we don’t have a convenient way to represent how such a transition would be able to properly update the internal representation.)