Title: A geometric attractor mechanism for self-organization of entorhinal grid modules,
Abstract: Grid cells in the medial entorhinal cortex (mEC) respond when an animal
occupies a periodic lattice of "grid fields" in the environment. The grids are
organized in modules with spatial periods clustered around discrete values
separated by constant ratios reported in the range 1.3-1.8. We propose a
mechanism for dynamical self-organization in the mEC that can produce this
modular structure. In attractor network models of grid formation, the period of
a single module is set by the length scale of recurrent inhibition between
neurons. We show that grid cells will instead form a hierarchy of discrete
modules if a continuous increase in inhibition distance along the dorso-ventral
axis of the mEC is accompanied by excitatory interactions along this axis.
Moreover, constant scale ratios between successive modules arise through
geometric relationships between triangular grids, whose lattice constants are
separated by $\sqrt{3} \approx 1.7$, $\sqrt{7}/2 \approx 1.3$, or other ratios.
We discuss how the interactions required by our model might be tested
experimentally and realized by circuits in the mEC.