  To solve the spatial problems of mapping, localization and navigation, the
mammalian lineage has developed striking spatial representations. One important
spatial representation is the Nobel-prize winning grid cells: neurons that
represent self-location, a local and aperiodic quantity, with seemingly bizarre
non-local and spatially periodic activity patterns of a few discrete periods.
Why has the mammalian lineage learnt this peculiar grid representation?
Mathematical analysis suggests that this multi-periodic representation has
excellent properties as an algebraic code with high capacity and intrinsic
error-correction, but to date, there is no satisfactory synthesis of core
principles that lead to multi-modular grid cells in deep recurrent neural
networks. In this work, we begin by identifying key insights from four families
of approaches to answering the grid cell question: coding theory, dynamical
systems, function optimization and supervised deep learning. We then leverage
our insights to propose a new approach that combines the strengths of all four
approaches. Our approach is a self-supervised learning (SSL) framework -
including data, data augmentations, loss functions and a network architecture -
motivated from a normative perspective, without access to supervised position
information or engineering of particular readout representations as needed in
previous approaches. We show that multiple grid cell modules can emerge in
networks trained on our SSL framework and that the networks and emergent
representations generalize well outside their training distribution. This work
contains insights for neuroscientists interested in the origins of grid cells
as well as machine learning researchers interested in novel SSL frameworks.
