Modeled grid cells aligned by a flexible attractor.

Entorhinal grid cells implement a spatial code with hexagonal periodicity, signaling the position of the animal within an environment. Grid maps of cells belonging to the same module share spacing and orientation, only differing in relative two-dimensional spatial phase, which could result from being part of a two-dimensional attractor guided by path integration. However, this architecture has the drawbacks of being complex to construct and rigid, path integration allowing for no deviations from the hexagonal pattern such as the ones observed under a variety of experimental manipulations. Here, we show that a simpler one-dimensional attractor is enough to align grid cells equally well. Using topological data analysis, we show that the resulting population activity is a sample of a torus, while the ensemble of maps preserves features of the network architecture. The flexibility of this low dimensional attractor allows it to negotiate the geometry of the representation manifold with the feedforward inputs, rather than imposing it. More generally, our results represent a proof of principle against the intuition that the architecture and the representation manifold of an attractor are topological objects of the same dimensionality, with implications to the study of attractor networks across the brain.