On the supra-linear storage in dense networks of grid and place cells

Place-cell networks, typically forced to pairwise synaptic interactions, are widely studied as models of cognitive maps: such models, however, share a severely limited storage capacity, scaling linearly with network size and with a very small critical storage. This limitation is a challenge for navigation in three-dimensional space because, oversimplifying, if encoding motion along a one-dimensional trajectory embedded in two dimensions requires 𝑂(𝐾) patterns (interpreted as bins), extending this to a two-dimensional manifold embedded in a three dimensional space – yet preserving the same resolution – requires roughly 𝑂(𝐾2) patterns, namely a supra-linear amount of patterns. In these regards, dense Hebbian architectures, where higher-order neural assemblies mediate memory retrieval, display much larger capacities and are increasingly recognized as biologically plausible, but have never linked to place cells so far. Here we propose a minimal two-layer model, with place cells building a layer and leaving the other layer populated by neural units that account for the internal representations (so to qualitatively resemble grid cells in the medial entorhinal cortex of mammals): crucially, by assuming that each place cell interacts with pairs of grid cells, we show how such a model is formally equivalent to a dense Battaglia–Treves-like Hebbian network of grid cells only endowed with four-body interactions. By studying its emergent computational properties by means of statistical mechanics of disordered systems, we prove – analytically – that such effective higher-order assemblies (constructed under the guise of biological plausibility) can support supra-linear storage of continuous attractors; furthermore, we prove – numerically – that the present neural network (namely the simplest dense generalization of the interplay between grid and cells) is, thus, already capable of recognition and navigation on general surfaces embedded in a three-dimensional space and, in particular, we deepen the cases of toroidal and spherical topologies. We remark two emergent properties of this type of networks: the former being that – despite keeping the tessellation performed by grid cells a random distributed process – their signaling allow place cells to behave in a quasi grand-mother fashion by spontaneously acquiring selectivity for firing, the latter being that these networks can sacrifice memory storage to lower their pattern detection threshold, a phenomenon that can be particularly relevant in spatial map’s recognition.