Asynchronous Random Boolean Network Model With Variable Number of Parents Based on Elementary Cellular Automata Rule 126

Abstract

A Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which can vary from one node to another, is considered. This is a generalization of previous results obtained for a constant number of parent nodes, by Matache and Heidel in "Asynchronous Random Boolean Network Model Based on Elementary Cellular Automata Rule 126", Phys. Rev. E71, 026 232, 2005. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. The Boolean rule is a generalization of rule 126 of elementary cellular automata, and is assumed to be the same for all the nodes. We provide a model for the probability of finding a node in state 1 at a time t for the class of generalized asynchronous random Boolean networks (GARBN) in which a random number of nodes can be updated at each time point. We generate consecutive states of the network for both the real system and the models under the various schemes, and use simulation algorithms to show that the results match well. We use the model to study the dynamics of the system through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the GARBN's dynamics range from order to chaos, depending on the type of random variable generating the asynchrony and the parameter combinations.

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

“…If c n and each of its parents have the same value at time t (that is they are all either 0 or 1), then c n (t + 1) = 0, otherwise c n (t + 1) = 1. This generalizes rule 126 of cellular automata (Matache andHeidel, 2004, 2005;Wolfram, 2002). The parents of a node are chosen randomly from the remaining N − 1 nodes and do not change thereafter.…”

“…In these formulae k is the number of parents of each node (considered fixed), and x t is the number of nodes to be updated at time t (randomly generated). In Matache (2006) the formula above is generalized for an arbitrary number of parents for each node. It is shown that the probability that a node is in state 1 at time t + 1 is given by…”

“…Given the variety of stochastic processes that one could consider, we narrow our study to several examples of processes that generalize the distributions used in Matache and Heidel (2005) and Matache (2006) or represent natural choices for applications to biology, genetics, physics, chemistry or sociology. We focus first on integer-valued processes, and later apply also some non-integer-valued processes for studying the dynamics of the system.…”

“…A large class of biological networks, cellular automata, or artificial neural networks have been modelled as Boolean networks in recent years Andrecut and Ali, 2001;Anthony, in press;De Garis et al, 2002;Fox and Hill, 2001;Heidel et al, 2003;Huang, 2001;Huepe and AldanaGonzález, 2002;Kürten, 1988;Matache andHeidel, 2004, 2005;Matache, 2006;Silvescu and Honavar, The present work is an extension of previous work by Matache and Heidel (2005) and Matache (2006). Those papers generalize rule 126 of elementary cellular automata (ECA) (Wolfram, 2002) and provide models for the probability of finding a node in state 1 (or ON) at time t. Rule 126 can be very simply described in terms of cell evolution as follows: complete crowding of live, ON, cells causes death, OFF, in the next generation, while complete isolation of a cell prevents birth in the next generation (Matache and Heidel, 2004).…”

“…In the paper Matache and Heidel (2005) the authors consider an asynchronous Boolean network with N nodes, each node having a number of parents that is fixed for all nodes. In Matache (2006) the author allows for a varying number of parents, generalizing results in Matache and Heidel (2005); in both these papers the asynchrony is generated using some classical random variables, including the uniform, binomial, Poisson, power law, and hypergeometric distributions. These distributions represent a more general updating scheme than the ones used so far in the literature, e.g.…”