On Circuit Functionality in Boolean Networks
It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it “generates” several stable states (resp. a cyclic attractor). However, there are no definite mathematical frameworks translating the underlying meaning of “generates.” Focusing on Boolean networks, we recall and propose some definitions concerning the notion of functionality along with associated mathematical results.
KeywordsBoolean network Interaction graph Feedback circuit Fixed point Multistability Cyclic attractor
This paper results from a collective discussion that took place during the workshop Logical formalism, gains, and challenges for the modeling of regulatory networks, held at Rabat, Morocco, from 12th to 15th of April 2011. We are grateful to Mohamed Hedi Ben Amor, Claudine Chaouiya, Madalena Chaves, Eric Fanchon, Pedro Monteiro, Elisabeth Remy, Paul Ruet, Sylvain Sené, Christophe Soulé, Laurent Tournier, and Laurent Trilling for taking part to this discussion. This work is partially supported by FONDECYT project 1090549 and BASAL project CMM, Universidad de Chile, and also by the French National Agency for Research (ANR-10-BLANC-0218 BioTempo project).
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