## Preprint 2012-10

## Julio Aracena, Laurence Calzone, Jean - Paul Comet, Jacques Demongeot, Marcelle Kaufman, Aurélien Naldi, Adrien Richard, El Houssine Snoussi, Denis Thieffry:

### On circuit functionality in Boolean networks

### Abstract:

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 then said to be functional when it "generates" several stable states (resp. a cyclic attractor). However, there are no mathematical frameworks translating the underlying meaning of "generates". Focusing on Boolean networks, we recall and propose some definitions around the notion of functionality and state associated mathematical results.

This preprint gave rise to the following definitive publication(s):

**Julio ARACENA, Laurence CALZONE, Jean - Paul COMET, Jacques DEMONGEOT, Marcelle KAUFMAN, Aurélien NALDI, Adrien RICHARD, El Houssine SNOUSSI, Denis THIEFFRY: ***On circuit functionality in Boolean networks*. Bulletin of Mathematical Biology, vol. 75, 6, pp. 906-919, (2013).