A local measure of symmetry and orientation for individual spikes of grid cells

Grid cells have attracted broad attention because of their highly symmetric hexagonal firing patterns. Recently, research has shifted its focus from the global symmetry of grid cell activity to local distortions both in space and time, such as drifts in orientation, local defects of the hexagonal symmetry, and the decay and reappearance of grid patterns after changes in lighting condition. Here, we introduce a method that allows to visualize and quantify such local distortions, by assigning both a local grid score and a local orientation to each individual spike of a neuronal recording. The score is inspired by a standard measure from crystallography, which has been introduced to quantify local order in crystals. By averaging over spikes recorded within arbitrary regions or time periods, we can quantify local variations in symmetry and orientation of firing patterns in both space and time.Author summary: Grid cells are neurons in mammals whose activity depends on the location of the animal in a striking way: grid cells fire spikes at multiple locations that form a symmetric lattice of repeating hexagons. Recent experiments have shown that the symmetry of these patterns is not as stable as initially thought. For example, patterns show local distortions or drifts in orientation. Here, we propose a method to visualize and quantify such local distortions directly from the recorded spike locations. Our method is inspired by an approach used in crystallography and we show that it reliably detects distortions in grid patterns. Moreover, we demonstrate that it can be used to study the stability of grid patterns not only in space, but also in time. Our method enables researchers to analyze the spatial symmetry of neuronal activity in more detail and could thus contribute to the understanding of spatial representations in mammals.