Random Boolean network

Now consider a latticized version of this model. Populate a square lattice -which may represent a tissue sample in which the modeled immune reactions are assumed to occur - with each of the four cell types C, H, M and V and initialize the system so that a fraction po of each cell type is in its high (i.e. = 1) concentration state. Assign the value 1 to each site i,j) if the sum of the concentrations of its nearest neighbors that are of the same cell type as site (i, j) is nonzero. After all sites have been assigned new values in this manner, update the system according to equations 8.92. 

Stauffer [stauff92] reports that the concentration of cells in this system quickly saturates. For low initial concentrations po, the viral population wins over the population of cells of the immune system and the system can be said to develop Aids. For larger po, the population of helper T cells becomes greater than that of the virus and the immune system wins. 

More complex models must carefully consider additional factors such as the receptor structure of helper T cells and allow for, what in reality, is a less than perfect lock and key match between antibody and antigen. For the latter case, Stauffer [staufF92] describes two schemes in which more than one type of antibody fits a given antigen and more than one type of antigen corresponds to a given antibody. 

Consider a directed graph of size N with adjacency matrix Aij,V We will say that Qn,k represents a size N directed graph of connectivity k if each site of the graph 

Boolean Network with connectivity k- or N, )-net - generalizes the basic binary k = 2) CA model by evolving each site variable Xi 0,1 of according to a randomly selected Boolean function of k inputs  

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Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

We might mention here in passing that while class-1, class-3 and class-4 (but not class-2) can all be obtained from one another with these two simple rules, class-2 behavior can only be obtained if the system is first quenched [vich86b]. That is, if the lattice is initially randomly populated with AND and XOR rules according to the prescribed value of p and is then frozen for all later times. Such quenched random rules harbor some interesting properties of their own in dimensions d > 1, and are the basis of much of Kauffman s findings on random Boolean networks (see section 8.6). 

Fig. 8.16 Typical random Boolean network with connectivity k == 1. Arrows indicate inputs to given sites and represent the Boolean functions at the sites toward which the arrows are pointing.

DDLab is an interactive graphics program for studying many different kinds of discrete dynamical systems. Arbitrary architectures can be defined, ranging from Id, 2d or 3d CA to random Boolean networks. 

The random Boolean network of a seeker cell showing internal and external connections... 

In this section, two models of development were presented, a complex model consisting of a multioperon genome and a cytoplasm, and a simple model based on random Boolean networks. The simpler model was explained in more detail, as it is the basis for the extended example described here. This model utilizes both development and evolution to get to a cell that can develop into a multicellular organism able to seek a chemical trace. 

Abstract. In this paper we show that a well-known model of genetic regulatory networks, namely that of Random Boolean Networks (RBNs), allows one to study in depth the relationship between two important properties of complex systems, i.e. dynamical criticality and power-law distributions. The study is based upon an analysis of the response of a RBN to permanent perturbations, that may lead to avalanches of changes in activation levels, whose statistical properties are determined by the same parameter that characterizes the dynamical state of the network (ordered, critical or disordered). Under suitable approximations, in the case of large sparse random networks an analytical expression for the probability density of avalanches of different sizes is proposed, and it is shown that for not-too-smaU avalanches of critical systems it may be approximated by a power law. In the case of small networks the above-mentioned formula does not maintain its validity, because of the phenomenon of self-interference of avalanches, which is also explored by numerical simulations. 

It is also often assumed that the presence of power-law distributions is the hallmark of criticality. Indeed, slightly different (although overlapping) notions of criticality have been used [7]. In this paper we show that a well-known model of genetic regulatory networks, introduced hy one of us several years ago [8], i.e. that of Random Boolean Networks (RBNs), can he used to study the relationships between power-law distributions and criticality issues. 

Villani, M., Serra, R., Ingrami, P., Kauffman, S.A. Coupled random boolean network forming an artificial tissue. In El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI2006. LNCS, vol. 4173, pp. 548-556. Springer, Heidelberg (2006)... 

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