Analytical Methods for Continuous Attractor Neural Networks

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, namely the interpolation method and the replica trick. In particular, 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), by adapting the Hamilton-Jacobi PDE-approach, we obtain an exact phase diagram in the noise vs inhibition strength plane. In the high-storage regime, by adapting the standard interpolation based on stochastic stability, we find that—for mild inhibition and not too high noise—memorization and retrieval of an extensive number of spatial maps is possible. These results, holding under the replica-symmetry assumption, are recovered, for completeness, also by the replica method and they are corroborated by Monte Carlo simulations. Finally, by leveraging the integral representation of the model (in terms of a bipartite network equipped with highly-selective hidden units), we successfully test its robustness versus various distributions of place fields, including the log-normal distribution observed in recent experiments on bats navigating long tunnels. Additionally, we demonstrate that, by appropriately coupling these hidden units, the network can effectively orient itself even in dynamic environments.