Extracting grid characteristics from spatially distributed place cell inputs using non-negative PCA

Many recent models study the downstream projection from grid cells to place cells, while recent data has pointed out the importance of the feedback projection. We thus asked how grid cells are affected by the nature of the input from the place cells. We propose a two-layered neural network with feedforward weights connecting place-like input cells to grid cell outputs. Place-to-grid weights were learned via a generalized Hebbian rule. The architecture of this network highly resembles neural networks used to perform Principal Component Analysis (PCA). Our results indicate that if the components of the feedforward neural network were nonnegative, the output converged to a hexagonal lattice. Without the non-negativity constraint the output converged to a square lattice. Consistent with experiments, grid alignment to walls was ~7° and grid spacing ratio between consecutive modules was ~1.4. Our results express a possible linkage between place cell to grid cell interactions and PCA.