A spike-based grid score.

a) Illustration of the ψ measure for 6-fold symmetry. Left: A reference ‘atom’ (black disc) with four neighbors (colored discs). Arcs show the angles, ϕl, to a reference axis (dashed black line) for all neighbors l ∈ {1, 2, 3, 4}. The blueish neighbors lie on the corners of a hexagon. The red neighbor is a defect to this hexagonal structure. Right: Sum of four unit vectors with directions given by 6ϕl (colored arcs). Vectors associated with neighbors that lie on the corners of the same hexagon (blueish colors) point in the same direction. The vector associated with the defect (red) points in a different direction. The length |ψ| of the resulting vector, normalized by the number of neighbors (black arrow), quantifies how much the reference atom is embedded in a hexagonal structure. The direction arg(ψ) of the vector—divided by 6 to reverse the previous rotation—indicates the orientation of the hexagon. b) Detection of the neighborhood shell. Locations of spikes of a grid cell and smoothed histogram of the pairwise distances between all spikes. Dashed lines indicate the automatically detected neighborhood shell. Top: Generated data. Bottom: Experimental recording kindly provided by [10]. c) Preventing false positives by discarding other symmetries. Example |ψ(M)| and values for a spike (black) with few neighbors (blue) in six different configurations. d) Same spike maps as in b, color-coded with the spike-based grid score (left) and the spike-based orientation (right).