  Grid cells in the entorhinal cortex, together with head direction, place,
speed and border cells, are major contributors to the organization of spatial
representations in the brain. In this work we introduce a novel theoretical and
algorithmic framework able to explain the emergence of hexagonal grid-like
response patterns from head direction cells' responses. We show that this
pattern is a result of minimal variance encoding of neurons. The novelty lies
into the formulation of the encoding problem through the modern Frame Theory
language, specifically that of equiangular Frames, providing new insights about
the optimality of hexagonal grid receptive fields. The model proposed overcomes
some crucial limitations of the current attractor and oscillatory models. It is
based on the well-accepted and tested hypothesis of Hebbian learning, providing
a simplified cortical-based framework that does not require the presence of
theta velocity-driven oscillations (oscillatory model) or translational
symmetries in the synaptic connections (attractor model). We moreover
demonstrate that the proposed encoding mechanism naturally explains axis
alignment of neighbor grid cells and maps shifts, rotations and scaling of the
stimuli onto the shape of grid cells' receptive fields, giving a
straightforward explanation of the experimental evidence of grid cells
remapping under transformations of environmental cues.
