The Self-Organization of Grid Cells in 3D

TL;DR

This paper models the self-organization of grid cells in three-dimensional environments, revealing complex dynamics and potential crystal-like structures, with some features emerging quickly and others taking longer to coordinate.

Contribution

It introduces a self-organization model based on firing-rate adaptation to predict 3D grid cell arrangements and analyzes their emergence and structure.

Findings

FCC and HCP-like asymptotic states are mathematically close in cost.

Initial grid-like patterns form rapidly in large 3D spaces.

Full network coordination and proper orientation take longer, with no convergence to a single structure.

Abstract

What sort of grid cells do we expect to see in bats exploring a three-dimensional environment? How long will it take for them to emerge? We address these questions within our self-organization model based on firing-rate adaptation. The model indicates that the answer to the first question may be simple, and to the second one rather complex. The mathematical analysis of the simplified version of the model points at asymptotic states resembling FCC and HCP crystal structures, which are calculated to be very close to each other in terms of cost function. The simulation of the full model, however, shows that the approach to such asymptotic states involves several sub-processes over distinct time scales. The smoothing of the initially irregular multiple fields of individual units and their arrangement into hexagonal grids over certain best planes are observed to occur relatively fast, even in large 3D volumes. The correct mutual orientation of the planes, however, and the coordinated arrangement of different units, take a longer time, with the network showing no sign of convergence towards either a pure FCC or HCP ordering.