On the number of attractors of Boolean automata circuits

Jacques Demongeot (1) , Mathilde Noual (2) , Sylvain Sené (3)

1 TIMB
2 LIP - Laboratoire de l'Informatique du Parallélisme
3 IBISC - Informatique, Biologie Intégrative et Systèmes Complexes

In line with fields of theoretical computer science and biology that study Boolean automata networks often seen as models of regulation networks, we present some results concerning the dynamics of networks whose underlying interaction graphs are circuits, that is Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work focuses on the number of attractors of Boolean automata circuits. We prove how to obtain formally the exact value of the total number of attractors of a circuit of arbitrary size n as well as, for every positive integer p, the number of its attractors of period p depending on whether the circuit has an even or an odd number of inhibitions. As a consequence, we obtain that both numbers depend only on the parity of the number of inhibitions and not on their distribution along the circuit.

Keywords: asymptotic behaviour; attractor.; formal neural network; Discrete dynamical system; attractor; positive and negative circuit

Domains: Discrete Mathematics [cs.DM]; Combinatorics [math.CO]; Bioinformatics [q-bio.QM]; Quantitative Methods [q-bio.QM]; Neural and Evolutionary Computing [cs.NE]

Jacques Demongeot, Mathilde Noual, Sylvain Sené. On the number of attractors of Boolean automata circuits. 2009. ⟨hal-00424722⟩