Guerra interpolation for place cells

Pyramidal cells that emit spikes when the animal is at specific locations of the environment are known as "place cells": these neurons are thought to provide an internal representation of space via "cognitive maps". Here, we consider the Battaglia-Treves neural network model for cognitive map storage and reconstruction, instantiated with McCulloch & Pitts binary neurons. To quantify the information processing capabilities of these networks, we exploit spin-glass techniques based on Guerra's interpolation: in the low-storage regime (i.e., when the number of stored maps scales sub-linearly with the network size and the order parameters self-average around their means) we obtain an exact phase diagram in the noise vs inhibition strength plane (in agreement with previous findings) by adapting the Hamilton-Jacobi PDE-approach. Conversely, in the high-storage regime, we find that -- for mild inhibition and not too high noise -- memorization and retrieval of an extensive number of spatial maps is indeed possible, since the maximal storage capacity is shown to be strictly positive. These results, holding under the replica-symmetry assumption, are obtained by adapting the standard interpolation based on stochastic stability and are further corroborated by Monte Carlo simulations (and replica-trick outcomes for the sake of completeness). Finally, by relying upon an interpretation in terms of hidden units, in the last part of the work, we adapt the Battaglia-Treves model to cope with more general frameworks, such as bats flying in long tunnels.