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}.
翻译:在本文中,我们研究了布林网络与燃烧记忆的动态行为,即布林网络的脊椎根据适当的布林地方过渡功能同步更新,以使每个顶端保持一定数量的步骤。我们特别证明,这些网络的计算能力与古典网络相同,即任何布林网络与由1百万美元的脊椎组成的燃烧内存可以通过布林网络添加脊椎来模拟。我们还证明了特定网络类别的一般结果。例如,我们表明,至少有一个超过1个断交网络的延迟使这种网络只有固定点作为吸引器。此外,对于由两个脊椎组成的任意网络,我们描述延迟阶段空间的特性,用网络允许限制循环或固定点的延迟值。最后,我们通过引入延迟来分析两种典型的生物模型:美元/升巴达元的免疫控制模型和植物致癌的基因控制模型。