Guerra interpolation for place cells

TL;DR

This paper applies Guerra's interpolation and spin-glass techniques to analyze the capacity of neural network models for cognitive maps, demonstrating conditions for storage and retrieval of multiple spatial representations.

Contribution

It introduces an exact phase diagram for the Battaglia-Treves model using Guerra's interpolation, extending understanding of neural network capacity for spatial memory.

Findings

Exact phase diagram in noise vs inhibition plane

Positive storage capacity in high-storage regime under certain conditions

Validation through Monte Carlo simulations and replica analysis

Abstract

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.