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Aracena, J., Goles, E., Moreira, A., & Salinas, L. (2009). On the robustness of update schedules in Boolean networks. Biosystems, 97(1), 1–8.
Abstract: Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems. (C) 2009 Elsevier Ireland Ltd. All rights reserved.
Keywords: Boolean network; Update schedule; Robustness; Attractor; Dynamical cycle
Goles, E., & Salinas, L. (2008). Comparison between parallel and serial dynamics of Boolean networks. Theor. Comput. Sci., 396(1-3), 247–253.
Abstract: In this article we study some aspects about the graph associated with parallel and serial behavior of a Boolean network. We conclude that the structure of the associated graph can give some information about the attractors of the network. We show that the length of the attractors of Boolean networks with a graph by layers is a power of two and under certain conditions the only attractors are fixed points. Also, we show that, under certain conditions, dynamical cycles are not the same for parallel and serial updates of the same Boolean network. (C) 2007 Elsevier B.V. All rights reserved.
Keywords: Boolean network; synchronous update; asynchronous update; attractor; dynamical cycle; fixed point