# A Combinatorial Exploration of Boolean Dynamics Generated by Isolated and Chorded Circuits

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## Abstract

Most studies of motifs of biological regulatory networks focus on the analysis of asymptotical behaviours (attractors, and even often only stable states), but transient properties are rarely addressed. In the line of our previous study devoted to isolated circuits (Remy et al. in Bioinformatics (Oxford, England) 19(Suppl. 2):172–178, 2003), we consider chorded circuits, that are motifs made of an elementary positive or negative circuit with a chord, possibly a self-loop. We provide detailed descriptions of the boolean dynamics of chorded circuits versus isolated circuits, under the synchronous and asynchronous updating schemes within the logical formalism. To this end, we address the description of the trajectories in the dynamics of isolated circuits with coding techniques and adapt them for chorded circuits. The use of the logical modeling gives access to mathematical tools (group actions, analysis of recurrent sequences, coding of trajectories, specific abacus...) allowing complete analytical analysis of basic yet important motifs. In particular, we show that whatever the chosen updating rule, the dynamics depends on a small number of parameters.

## Keywords

Boolean modeling Regulatory network Motifs Circuits Chorded circuits Discrete dynamical systems## Notes

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