Dynamics of Circuits and Intersecting Circuits

  • Mathilde Noual
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7183)

Abstract

This paper presents a combinatorial study to characterise the dynamics of intersecting Boolean automata circuits and more specifically that of double Boolean automata circuits. Explicit formulae are given to count the number of periodic configurations and attractors of these networks and a conjecture proposes a comparison between the number of attractors of isolated circuits and that of double circuits. The aim of this study is to give intuition on the way circuits interact and how a circuits intersection modifies the “degrees of freedom” of the overall network.

Keywords

Positive and Negative Circuits Boolean Automata Network Regulation Network Dynamical Behaviour Attractor 

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References

  1. 1.
    Adams, W., Shanks, D.: Strong primality tests that are not sufficient. Mathematics of Computation (American Mathematical Society) 39(159), 255–300 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Apostol, T.M.: Introduction to analytic number theory. Springer, Heidelberg (1976)MATHGoogle Scholar
  3. 3.
    Aracena, J., Ben Lamine, S., Mermet, O., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Transactions on Systems, Man, and Cybernetics 33, 825–834 (2003)CrossRefGoogle Scholar
  4. 4.
    Demongeot, J., Noual, M., Sené, S.: Combinatorics of boolean automata circuits dynamics (2011) (submitted)Google Scholar
  5. 5.
    Elena, A.: Algorithme pour la simulation dynamique des réseaux de régulation génétique. Master’s thesis, University J. Fourier (2004)Google Scholar
  6. 6.
    Goles, E.: Comportement oscillatoire d’une famille d’automates cellulaires non uniformes. Ph.D. thesis, Université scientifique et médicale de Grenoble, France (1980), http://tel.archives-ouvertes.fr/tel-00293368/fr/
  7. 7.
    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences of the USA 79, 2554–2558 (1982)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology 22, 437–467 (1969)MathSciNetCrossRefGoogle Scholar
  9. 9.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Puri, Y., Ward, T.: Arithmetic and growth of periodic orbits. Journal of Integer Sequences 4(2) (2001)Google Scholar
  11. 11.
    Puri, Y., Ward, T.: A dynamical property unique to the Lucas sequence. The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 39(5), 398–402 (2001)MathSciNetMATHGoogle Scholar
  12. 12.
    Remy, É., Ruet, P., Thieffry, D.: Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Advances in Applied Mathematics 41, 335–350 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ribenboim, P.: The New Book of Prime Number Records. Springer, Heidelberg (1996)CrossRefMATHGoogle Scholar
  14. 14.
    Richard, A.: Positive circuits and maximal number of fixed points in discrete dynamical systems. Discrete Applied Mathematics 157, 3281–3288 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Riordan, J.: An Introduction to Combinatorial Analysis. Wiley, New York (1980)CrossRefMATHGoogle Scholar
  16. 16.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences, OEIS (2008)Google Scholar
  17. 17.
    Thomas, R.: Boolean formalisation of genetic control circuits. Journal of Theoretical Biology 42, 563–585 (1973)CrossRefGoogle Scholar
  18. 18.
    Thomas, R.: On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Series in Synergetics, vol. 9, pp. 180–193 (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mathilde Noual
    • 1
    • 2
  1. 1.Université de Lyon, ÉNS-Lyon, LIP, CNRS UMR5668LyonFrance
  2. 2.IXXI, Institut Rhône-alpin des systèmes complexesLyonFrance

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