Abstract
The basis of an analytical description of the behaviour of large random nets of binary elements of the type first investigated in detail by S. A. Kauffman is presented. It is shown that information about the network dynamics can be deduced from quite general considerations of the properties of the state transition graph and matrix. An expression for the matrix elements of the state transition matrix in terms of the Boolean function specification of the net is derived. Using these ideas the distribution of limit cycle lengthsl for a completely random net is calculated and shown to bex 1/l, a result which agrees well with experimental data.
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Literature
Abramowitz, M. and I. A. Segun. 1968.Handbook of Mathematical Functions. New York: Dover.
Aleksander, I. 1973. “Random Logic Nets: Stability and Adaptation.”Int. J. Man-Mach. Stud.,5, 115–131.
— and P. Atlas. 1973. “Cyclic Activity in Nature.”Int. J. Neurosci.,6, 45–50.
Andrews, G. E. 1976.The Theory of Partitions, Vol. 2 ofEncyclopedia of Mathematics and its Applications. Reading, Mass: Addison-Wesley.
Cavender, J. A. 1977. “Kauffman's Square-root Law: Possible Correlatives.”J. Theor. Biol.,65, 791–793.
Glass, L. and S. A. Kauffman. 1973. “The Logical Analysis of Continuous Non-linear Biochemical Control Networks.”J. Theor. Biol.,39, 103–129.
Goodwin, B. C. 1976.Analytical Physiology of Cells and Developing Organisms pp. 53–56. London: Academic Press.
Jacob, F. and J. Monod. 1961. “Genetic Regulatory Mechanisms in the Synthesis of Proteins.”J. Mol. Biol.,3, 318–356.
Kauffman, S. A. 1969a. “Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets.”J. Theor. Biol.,22, 437–467.
— 1969b. “Homeostasis and Differentiation in Random Genetic Networks.”Nature,224, 177–178.
— 1970a. “Behaviour of Randomly Constructed Genetic Nets: Binary Element Nets.” InTowards a Theoretical Biology 3: Drafts, Ed. C. H. Waddington, pp. 18–37. Edinburgh University Press.
— 1970b. “Behaviour of Randomly Constructed Genetic Nets: Continuous Element Nets.” InTowards a Theoretical Biology 3: Drafts. Ed. C. H. Waddington, pp. 38–46. Edinburgh: Edinburgh University Press.
— 1971. “Gene Regulation Networks: A Theory for their Global Structure and Behaviour.”Curr. Topics Dev. Biol.,6, 145–182.
— 1973. “Control Circuits for Determination and Transdetermination.”Science,181, 310–318.
Pattee, H. H. 1969. “How Does a Molecule Become a Message?”Dev. Biol. Suppl.,3, 1–16.
Torng, H. C. 1972.Switching Circuits: Theory and Logic Design. Reading, Mass: Addison Wesley.
Wilson, R. J. 1972.Introduction to Graph Theory. Glasgow: Bell & Bain.
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Sherlock, R.A. Analysis of the behaviour of Kauffman binary networks—I. State space description and the distribution of limit cycle lengths. Bltn Mathcal Biology 41, 687–705 (1979). https://doi.org/10.1007/BF02462422
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DOI: https://doi.org/10.1007/BF02462422