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In this paper we study the dynamical behavior of Boolean networks with firingmemory, namely Boolean networks whose vertices are updated synchronouslydepending on their proper Boolean local transition functions so that eachvertex remains at its firing state a finite number of steps. We prove inparticular that these networks have the same computational power than theclassical ones, ie any Boolean network with firing memory composed of $m$vertices can be simulated by a Boolean network by adding vertices. We alsoprove general results on specific classes of networks. For instance, we showthat the existence of at least one delay greater than 1 in disjunctive networksmakes such networks have only fixed points as attractors. Moreover, forarbitrary networks composed of two vertices, we characterize the delay phasespace, \ie the delay values such that networks admits limit cycles or fixedpoints. Finally, we analyze two classical biological models by introducingdelays: the model of the immune control of the $\lambda$-phage and that of thegenetic control of the floral morphogenesis of the plant \emph{Arabidopsisthaliana}.

Attractor landscapes in Boolean networks with firing memory