Invariant Cycles in the Random Mapping of N Integers Onto Themselves. Comparison with Kauffman Binary Network

  • Jean Coste
  • Michel Henon
Conference paper
Part of the NATO ASI Series book series (volume 20)


According to Kauffman’s idea [Kauffman 1970a,b, 1979], one considers an ensemble of P genes which may be found in two possible states s., labelled as 0 and 1. An overall state S of the ensemble is the set {s1, s2,…, sp}, which is an element of {0,1}p. Given a mapping of {0,l}p→{0,1}p, the iteration of this mapping defines the dynamics of any initial S. In Kauffman model si at time (t+1) is determined by the states of k genes at time t, -possibly including si itself. Therefore the dynamics is defined by the set of all gene connections and, for each gene, by the data of a Boolean function, that is by an array of 2k elements whose values are either 0 or 1 (there are \( {2^{{{2^k}}}} \) possible Boolean functions). The dynamics drives any S towards a cycle of period m (1 ⩽ m ⩽ 2p), and the problem is to find out the number and the periods of those cycles when S is varied over the various possible states. A numerical study has been performed by Kauffman for k=2 and choosing at random the set of gene connections and the P Boolean functions. It appeared that:
  1. i)

    The average number of cycles is of the order of \( \sqrt {P} \)

  2. ii)

    The average period of the cycles is also of the order of \( \sqrt {P} \).

This remarkable result shows up some amazing simplicity in the dynamics of a large system and, in particular, helps one to understand how a so large number of interacting genes can produce only few cellular types.


Boolean Function Random Mapping Average Period Successive Image Cellular Type 
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  1. Kauffman, S. 1970 a. “Behaviour of randomly constructed nets”. In Towards a theoretical biology Ed. C.H. Waddington, vol. 3, Edimburg University Press.Google Scholar
  2. Kauffman, S. 1970 b. “The organization of cellular genetic cintrol systems”. Math. Life Sci. 3, 63–116.Google Scholar
  3. Kauffman, S. 1979 “Assessing the probable regulatory structures and dynamics of the metazoan genome. Kinetic logic”. In Lecture notes for Biomathematics Ed. R. Thomas, 29, 30–61. Berlin Springer Verlag.Google Scholar
  4. Sherlock, R.A. 1979 “Analysis of Kauffman binary networks” Bull. Math. Biol. 41, 687–724.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Jean Coste
    • 1
  • Michel Henon
    • 2
  1. 1.Laboratoire de physique de la matière condenséeNiceFrance
  2. 2.Observatoire de NICENice CedexFrance

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