When genetic circuits are described as Boolean networks, the gene state is quantized to only two levels, true and false, but this is obviously an approximation. Even if this two-valued status were precisely correct, since the heredity is groupwise, a group of genes bounded by a chromosome may also be considered as a multivalued logical variable. Multivalued networks also appear in some other complex systems, for instance, in chemical reactions and cognitive science (Adamatzky, Chaos Solitons Fractals 18:917–936, 2003; Volkert and Conrad, J. Theor. Biol. 193(2):287–306, 1998). When the gene state is not limited to true and false, such as when the effect of one gene on another is not strong, we should modify the model.
In this chapter we consider k-valued networks. As with a Boolean network, the dynamics of a k-valued network can be converted into an conventional discrete-time dynamical system. Many results obtained for Boolean networks can then be transferred in a parallel way to the k-valued case. This chapter is mainly devoted to the transfer of some major results from Boolean networks to k-valued networks. In the last section, a more general network, called a mix-valued network, is introduced. This chapter is based on Li and Cheng (Int. J. Bifurc. Chaos 20(3):561–582, 2010).
KeywordsLogical Operator Structure Matrix Structure Matrice Boolean Network Logical Network
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