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, blocksequential, 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.

Goles, E., & Moreira, A. (2012). NumberConserving Cellular Automata and Communication Complexity: A Numerical Exploration Beyond Elementary CAs. J. Cell. Autom., 7(2), 151–165.
Abstract: We perform a numerical exploration of numberconserving cellular automata (NCCA) beyond the class of elementary CAs, in search of examples with high communication complexity. We consider some possible generalizations of the elementary rule 184 (a minimal model of traffic, which is the only nontrivial elementary NCCA). as well as the classes of NCCAs which minimally extend either the radius or the state set (with respect to the 2 states and radius 1 of the elementary case). Both for 3 states and radius 1, and for 2 stales and radius 2, NCCA appear that are conjectured to have maximal (exponential) communication complexity. Examples are given also for (conjectured) linear and quadratic behaviour.

Goles, E., Moreira, A., & Rapaport, I. (2011). Communication complexity in numberconserving and monotone cellular automata. Theor. Comput. Sci., 412(29), 3616–3628.
Abstract: One third of the elementary cellular automata (CAs) are either numberconserving (NCCAs) or monotone (increasing or decreasing). In this paper we prove that, for all of them, we can find linear or constant communication protocols for the prediction problem. In other words, we are able to give a succinct description for their dynamics. This is not necessarily true for general NCCAs. In fact, we also show how to explicitly construct, from any CA, a new NCCA which preserves the original communication complexity. (C) 2011 Elsevier B.V. All rights reserved.
