  In this paper we study the dynamical behavior of Boolean networks with firing
memory, namely Boolean networks whose vertices are updated synchronously
depending on their proper Boolean local transition functions so that each
vertex remains at its firing state a finite number of steps. We prove in
particular that these networks have the same computational power than the
classical ones, ie any Boolean network with firing memory composed of $m$
vertices can be simulated by a Boolean network by adding vertices. We also
prove general results on specific classes of networks. For instance, we show
that the existence of at least one delay greater than 1 in disjunctive networks
makes such networks have only fixed points as attractors. Moreover, for
arbitrary networks composed of two vertices, we characterize the delay phase
space, \ie the delay values such that networks admits limit cycles or fixed
points. Finally, we analyze two classical biological models by introducing
delays: the model of the immune control of the $\lambda$-phage and that of the
genetic control of the floral morphogenesis of the plant \emph{Arabidopsis
thaliana}.
