A boolean network is a map $ F:\{0,1\}^n \to \{0,1\}^n $ that defines a discrete dynamical system by the subsequent iterations of $ F $. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.

中文翻译：

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激发记忆在联合网络动力学中的作用

布尔网络是映射$ F：\ {0,1 \} ^ n \到\ {0,1 \} ^ n $，它由$ F $的后续迭代定义了离散的动力系统。尽管如此，据认为该定义在应用中，尤其是在生物学中并不总是可靠的。关于这个问题，提出了基于为动力学增加异步性的概念的模型。特别地，我们对基于延迟概念的方法感兴趣。我们关注一种称为触发记忆的特定类型的延迟，它会影响对称（非定向）联合网络的动力学。我们发现，在延迟执行不均匀的情况下，所有动力学复杂性都以某种方式封装在延迟起作用的组件中。因此，我们表明，在齐次的情况下，有可能表现出非多项式周期的吸引子。此外，我们研究了预测问题，其中包括在给定初始条件的情况下确定固定坐标最终是否会改变其状态的问题。我们再次发现，在非齐次情况下，所有复杂度均由受延迟影响的组件确定，并且在齐次情况下，我们得出结论，此问题是PSPACE完全的。