A unified theory for the origin of grid cells through the lens of pattern formation

Ben Sorscher, Gabriel Mel, Surya Ganguli, Samuel A Ocko

Abstract: Grid cells in the medial entorhinal cortex fire in strikingly regular hexagonal patterns. There are currently two seemingly unrelated frameworks for understanding these patterns. A mechanistic framework accounts for hexagonal firing fields as the result of pattern-forming dynamics in a neural network with hand-tuned center-surround connectivity. A normative framework accounts for grid maps as optimal solutions for decoding position. Consistent with this, neural networks trained to decode position by path integrating learn grid-like responses in their hidden units. Here we provide an analytic theory that unifies the two perspectives by casting the decoding problem as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the spatial decoding problem and correctly predicts whether neural networks trained to decode position under different circumstances develop square or hexagonal grids. We extend this theory to the case of learning multiple grid maps and demonstrate that optimal solutions consist of a hierarchy of maps with increasing length scales. These results unify previous accounts of grid cell firing and provide a novel framework for predicting the learned representations of recurrent neural networks.

Code Link: https://ganguli-lab.github.io/grid-pattern-formation/

CMT Num: 5290