Serial vs parallel recall in the Blume-Every-Griffiths neural networks

TL;DR

This paper rigorously analyzes Blume-Emery-Griffiths neural networks, revealing how pattern dilution enables a transition from serial to parallel recall, with different regimes exhibiting hierarchical or equal-strength multi-pattern retrieval.

Contribution

It provides a rigorous analysis of pattern recall dynamics in diluted Blume-Emery-Griffiths networks, highlighting the transition from serial to parallel recall and exploring graded neural responses.

Findings

Dilution enables switching from serial to parallel recall.

Mild dilution leads to hierarchical recall with different amplitudes.

Extreme dilution results in equal-strength multi-pattern recall.

Abstract

Fully connected Blume-Emery-Griffiths neural networks performing pattern recognition and associative memory have been heuristically studied in the past (mainly via the replica trick and under the replica symmetric assumption) as generalization of the standard Hopfield reference. In these notes, at first, by relying upon Guerra interpolation, we re-obtain the existing picture rigorously. Next we show that, due to dilution in the patterns, these networks are able to switch from serial recall (where one pattern is retrieved per time) to parallel recall (where several patterns are retrieved at once) and the larger the dilution, the stronger this emerging multi-tasking capability. In particular, we inspect the regimes of mild dilution (where solely a low storage of pattern can be enabled) and extreme dilution (where a medium storage of patterns can be sustained) separately as they give rise to different outcomes: the former displays hierarchical recall (distributing the amplitudes of the retrieved signals with different amplitudes), the latter -- instead -- performs a equal-strength recall (where a O(1) fraction of all the patterns is simultaneously retrieved with the same amplitude per pattern). Finally, in order to implement graded responses in the neurons, variations on theme obtained by enlarging the possible values of neural activity these neurons may sustain are also discussed generalizing the Ghatak-Sherrington model for inverse freezing in Hebbian terms.