If my understanding is correct, the triangular lattice pattern of neuron activations in the grid cell modules exist because of local inhibition effects. The set of winning/activated cells suppress the firing of their neighbors. The overlapping spheres of inhibition are what give rise to the lattice pattern. As the sensor is shifted (translated/rotated), there is a subtle change in the proximal inputs that result in a shift of the firing pattern that is proportional to the sensor shift. These same inhibitory effects then generate a similar (but shifted) lattice pattern. If after some number of these shifts, a sufficient number of the original nodes become active again, then the original lattice pattern will reestablish itself. This periodic behavior (amongst individual neurons) is perhaps what is being observed in the experiments with mice moving about their environment. (I go into a little more detail on these ideas here.)
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