It may be that the individual grid cells are only responding to locally available information, but that information is likely strongly correlated with information local to other cells “way over there” due to the fundamental spatial and temporal structure of the sensed environment and the spatially correlated layout of the cells in the cortex. My current working theory for grid cells is that the patterns of activation from the input are parsed in such a way that features are filtered in both space and time.
Specific combinations of features are lighting up something like place cell SDRs. These place cell representations are encoding the presence of specific features and, through apical connections, their locations and/or orientations on the sensor or relative to other features on the sensor.
Features that are changing rapidly in time** would then activate cells that are part of modules with smaller length scales. This corresponds to grids that are responsive to objects that are physically closer to the subject. Similarly, features that are further away are changing more slowly** on the sensor would activate cells in grid modules with larger length scales.
While I’ve not worked out the specific dynamics yet, my intuition tells me that something like this is what gives rise to the grid cell like behavior.
** There may also be an additional mechanism that is able to distinguish between sensor updates due to self-movement and those due to independent external phenomena. I’m still working on that one too.