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. 2025 Aug 22;15(3):20240059. doi: 10.1098/rsfs.2024.0059

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© 2025 The Author(s).

Published by the Royal Society. All rights reserved.

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(a) Regulatory network RN consisting of a negative cycle. — (a) Regulatory network RN consisting of a negative cycle. (b) The expanded network that admits an oscillatory motif consisting of all nodes. The two sets of token distributions represent two distinct 8-periodic sets $C_{0} = {v_{1}^{0}, v_{2}^{0}, v_{3}^{0}, v_{4}^{0}}$ (blue/upper) and $C_{0}^{'} = {v_{1}^{0}, v_{2}^{0}, v_{3}^{1}, v_{4}^{0}}$ (green/lower). $C_{0}$ corresponds to a stable FC in the STG, while $C_{0}^{'}$ corresponds to an unstable FC. (c) Depiction of the two FCs in STG, stable (left) and unstable (right). Each node label a, b, c, d corresponds to domain in STG and a collection of virtual nodes $v_{1}^{a}, v_{2}^{b}, v_{3}^{c}, v_{4}^{d}$ . The coloured nodes indicate the domain represented by the 8-periodic sets $C_{0}$ and $C_{0}^{'}$ in (b) for each cycle.