Self-organization of entorhinal grid modules through commensurate lattice relationships

A grid cell is a neuron that only fires when an animal reaches certain locations in its enclosure. These locations form a triangular grid of a certain spatial scale. The medial entorhinal cortex contains many grid cells spanning a wide range of scales. These scales are not distributed smoothly but instead cluster around discrete values separated by constant ratios reported in the range 1.3–1.8. Although this modular organization has been shown to be an efficient encoding of spatial location, its origin is unknown. We propose an extension of the standard continuous attractor model for generating grid responses that naturally produces geometric sequences of scales. By introducing excitatory connections between attractor networks, an otherwise smooth distribution of grid scales becomes modular with discrete transitions between preferred values. Moreover, constant scale ratios between successive modules arise through robust geometric relationships between commensurate triangular grids, whose lattice constants are separated by √3 ≈ 1.7, √7/2 ≈ 1.3, or other ratios. We suggest analyses and experiments that test our model and describe its connection to the Frenkel-Kontorova model of condensed matter physics.