There’s so much to unpack here. I really need to sit down with @mrcslws for a serious discussion about the ideas he’s expressing here. My own thoughts on grid cell modules have been running pretty much in parallel to what he’s been presenting over the past year.
At about T+8:00, @subutai asks if the input needed to be continuous in time. @mrcslws replies that there are not specific assumptions about discontinuities in the input, but that in this work he was focused on continuous inputs. @jhawkins then commented that some seemingly discontinuous events, such as saccades could be interpreted as really fast motions since path integration is occurring during saccades.
I’ve given some thought to continuous input (e.g. persistent sensory input) vs. discrete input (e.g. sudden shift in a significant portion of the sensory input). It occurs to me that while continuous path integration of physical (or conceptual) space is likely to be enabled by the grid cell modules described here, the discrete jumps in input (like tapping an icon on your phone, or walking into a different room) would require another mechanism to reanchor the grid cell modules to the new context. Whether these are place cells, or a specific population of cells which respond to the input (e.g. by filter matching), would probably depend on the specific context.
The 2D polar coordinate frame is a good place to start. By training the input filters to recognize shifted periodic features, it would seem like you are training a sort of Fourier Transform filter. This filter would essentially extract the real and imaginary components of spatial (or temporal) frequency responses to the input.
Now, as @mrcslws points out, restricting these filters to two cells anchored to two specific phases (90 degrees apart) is a problem for biological plausibility since it would require both positive and negative response values in order to represent a complete ring. @subutai is correct in proposing that adding more cells to the module and forming an over-complete basis set would be the way to address this issue. In the diagram below, you can see that a minimum of three cells would be needed to represent an entire ring wile avoiding negative response amplitudes from the grid cells.
With more grid cells in a module, each representing a finer grained (higher resolution) phase shift for a particular feature, it may be necessary to include inhibition to reduce the response of the filters that are close to, but not the best match to the phase orientation of the input.