On neural circuits of working memory sequence permutation: optimizing circuit architectures via Cayley graphs

The brain's ability to store and manipulate working memory (WM) sequences is pivotal for higher cognitive reasoning. Although the neural circuit mechanisms for storing WM sequences have been extensively studied, those for manipulating WM sequences remain largely unknown. Inspired by a recent WM sequence manipulation experiment in monkeys, we design a functional, biologically plausible neural circuit model that realizes WM sequence permutations using guidance from permutation groups and their Cayley graph representations. The circuit consists of two interconnected modules: a memory module composed of continuous attractor-based memory motifs that store and interchange items in WM sequences, and a control module that sends gain modulations to guide permutation operations within the memory module. The control module features a hierarchical tree structure that decomposes complex permutations into a sequence of basic two-item swaps, simplifying circuit implementations. We demonstrate that permutation circuit architectures have one-to-one correspondence with Cayley graphs representing permutation group structure, where the group generating set directly determines the connectivity between memory motifs. Since each permutation group may have multiple generating sets, there are multiple circuit architectures implementing the same permutation. We therefore utilize Cayley graph analysis to determine trade-offs between computational efficiency, circuit complexity, and circuit robustness. Our study establishes connections between abstract group theory, Cayley graphs, and biologically plausible circuit architectures, providing insights into principled circuit design via algebraic frameworks.