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, i.e. 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, i.e. 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 Arabidopsis thaliana.

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具有激发记忆的布尔网络中的吸引子景观：应用于遗传网络的理论研究

在本文中，我们研究具有触发记忆的布尔网络的动力学行为，即布尔网络的顶点根据其适当的布尔局部转移函数而同步更新，以便每个顶点在其触发状态下保持有限的步数。我们特别证明了这些网络与经典网络具有相同的计算能力，即任何具有由m组成的触发内存的布尔网络。布尔网络可以通过添加顶点来模拟顶点。我们还证明了特定网络类别的一般结果。例如，我们表明析取网络中至少存在一个大于1的延迟，这使得此类网络仅具有固定点作为吸引子。此外，对于由两个顶点组成的任意网络，我们表征了延迟相空间，即延迟值，使得网络允许存在极限环或固定点。最后，我们通过引入延迟来分析两个经典的生物学模型：\（\ lambda \）-噬菌体的免疫控制模型和植物拟南芥花形态发生的遗传控制模型。