ConchaVega, P., Goles, E., Montealegre, P., RiosWilson, M., & Santivanez, J. (2022). Introducing the activity parameter for elementary cellular automata. Int. J. Mod Phys. C, 33(09), 2250121.
Abstract: Given an elementary cellular automaton (ECA) with local transition rule R, two different types of local transitions are identified: the ones in which a cell remains in its current state, called inactive transitions, and the ones in which the cell changes its current state, which are called active transitions. The number of active transitions of a rule is called its activity value. Based on latter identification, a rule R1 is called a subrule of R2 if the set of active transitions of R1 is a subset of the active transitions of R2.
In this paper, the notion of subrule for elementary cellular automata is introduced and explored: first, we consider a lattice that illustrates relations of nonequivalent elementary cellular automata according to nearby subrules. Then, we introduce statistical measures that allow us to compare rules and subrules. Finally, we explore the possible similarities in the dynamics of a rule with respect to its subrules, obtaining both empirical and theoretical results.

Formenti, E., Goles, E., & Martin, B. (2012). Computational Complexity of Avalanches in the Kadanoff Sandpile Model. Fundam. Inform., 115(1), 107–124.
Abstract: This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is Pcomplete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for onedimensional sandpiles and Pcomplete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak's model of twodimensional sandpiles.

Goles, E., Lobos, F., Ruz, G. A., & Sene, S. (2020). Attractor landscapes in Boolean networks with firing memory: a theoretical study applied to genetic networks. Nat. Comput., 19(2), 295–319.
Abstract: 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$$\lambda $$\end{document}phage and that of the genetic control of the floral morphogenesis of the plant Arabidopsis thaliana.

Goles, E., Montealegre, P., & RiosWilson, M. (2020). On The Effects Of Firing Memory In The Dynamics Of Conjunctive Networks. Discret. Contin. Dyn. Syst., 40(10), 5765–5793.
Abstract: A boolean network is a map F : {0, 1}(n) > {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 (nondirected) 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 nonpolynomial 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 nonhomogeneous 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 PSPACEcomplete.
