Hierarchical neural networks perform both serial and parallel processing

TL;DR

This paper investigates hierarchical neural networks that can perform both serial and parallel processing, revealing a richer phase diagram and a trade-off between multitasking ability and capacity, with implications for biological systems.

Contribution

It introduces a detailed analysis of hierarchical neural networks, demonstrating their ability to perform both serial and parallel processing, and characterizes the phase diagram and capacity trade-offs.

Findings

Networks perform both serial and parallel processing.

Wider multitasking reduces network capacity.

Phase diagram is richer than classical models.

Abstract

In this work we study a Hebbian neural network, where neurons are arranged according to a hierarchical architecture such that their couplings scale with their reciprocal distance. As a full statistical mechanics solution is not yet available, after a streamlined introduction to the state of the art via that route, the problem is consistently approached through signal- to-noise technique and extensive numerical simulations. Focusing on the low-storage regime, where the amount of stored patterns grows at most logarithmical with the system size, we prove that these non-mean-field Hopfield-like networks display a richer phase diagram than their classical counterparts. In particular, these networks are able to perform serial processing (i.e. retrieve one pattern at a time through a complete rearrangement of the whole ensemble of neurons) as well as parallel processing (i.e. retrieve several patterns simultaneously, delegating the management of diff erent patterns to diverse communities that build network). The tune between the two regimes is given by the rate of the coupling decay and by the level of noise affecting the system. The price to pay for those remarkable capabilities lies in a network's capacity smaller than the mean field counterpart, thus yielding a new budget principle: the wider the multitasking capabilities, the lower the network load and viceversa. This may have important implications in our understanding of biological complexity.