  We propose a normative model for spatial representation in the hippocampal
formation that combines optimality principles, such as maximizing coding range
and spatial information per neuron, with an algebraic framework for computing
in distributed representation. Spatial position is encoded in a residue number
system, with individual residues represented by high-dimensional,
complex-valued vectors. These are composed into a single vector representing
position by a similarity-preserving, conjunctive vector-binding operation.
Self-consistency between the representations of the overall position and of the
individual residues is enforced by a modular attractor network whose modules
correspond to the grid cell modules in entorhinal cortex. The vector binding
operation can also associate different contexts to spatial representations,
yielding a model for entorhinal cortex and hippocampus. We show that the model
achieves normative desiderata including superlinear scaling of patterns with
dimension, robust error correction, and hexagonal, carry-free encoding of
spatial position. These properties in turn enable robust path integration and
association with sensory inputs. More generally, the model formalizes how
compositional computations could occur in the hippocampal formation and leads
to testable experimental predictions.
