New tools are available for reconstructing the attractor basins of discrete dynamical networks where state-space is linked according to the network's dynamics. In this thesis the computer software 'Discrete Dynamics Lab' is applied to examine simple networks ranging from cellular automata (CA) to random Boolean networks (RBN), that have been widely applied as idealised models of physical and biological systems, to search for general principles underlying their dynamics. The algorithms and methods for generating pre-images for both CA and RBN, and reconstructing and representing attractor basins are described, and also considered in the mathematical context of random directed graphs. RBN and CA provide contrasting notions of self-organisation. RBN provide models of hierarchical categorisation in biology, for example memory in neural and genomic networks. CA provide models at the lower level of emergent complex pattern. New measures and results are presented on CA attractor basins and how they relate to measures on local dynamics and the Z parameter, characterising ordered to 'complex' to chaotic behaviour. A method is described for classifying CA rules by an entropy-variance measure which allows glider rules and related complex rules to be found automatically giving a virtually unlimited sample for further study. The dynamics of RBN and intermediate network architectures are examined in the context of memory, where categorisation occurs at the roots of subtrees as well as at attractors. Learning algorithms are proposed for 'sculpting' the basin of attraction field. RBN are proposed as a possible neural network model, and also discussed as a model of genomic regulatory networks, where cell types have been explained as attractors.
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