Natural Computing

, Volume 17, Issue 2, pp 393–402 | Cite as

Synchronism versus asynchronism in monotonic Boolean automata networks

  • Mathilde Noual
  • Sylvain SenéEmail author


This paper focuses on Boolean automata networks and the updatings of automata states in these networks. More specifically, we study how synchronous updates impact on the global behaviour of a network. On this basis, we define different types of network sensitivity to synchronism, which are effectively satisfied by some networks. We also relate this synchronism-sensitivity to some properties of the structure of networks and to their underlying mechanisms.


Boolean automata networks Sensitivity to synchronism Asynchronous transition graph Elementary transition graph Time in interaction systems 



This work has been partially supported by the project PACA APEX FRI.


  1. Aldana M (2003) Boolean dynamics of networks with scale-free topology. Phys D 185:45–66MathSciNetCrossRefzbMATHGoogle Scholar
  2. Aracena J, Demongeot J, Goles E (2004a) On limit cycles of monotone functions with symmetric connection graph. Theor Comput Sci 322:237–244MathSciNetCrossRefzbMATHGoogle Scholar
  3. Aracena J, Demongeot J, Goles E (2004b) Positive and negative circuits in discrete neural networks. IEEE Trans Neural Netw 15:77–83CrossRefzbMATHGoogle Scholar
  4. Chandesris J, Dennunzio A, Formenti E, Manzoni L (2011) Computational aspects of asynchronous cellular automata. In: Proceedings of DLT, lecture notes in computer science, vol 6795, pp 466–468. Springer, BerlinGoogle Scholar
  5. Choffrut C (ed) (1988) Automata networks, lecture notes in computer science, vol 316. Springer, BerlinGoogle Scholar
  6. Combe P, Nencka H (1997) Frustration and overblocking on graphs. Math Comput Model 26:307–309MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cosnard M, Demongeot J (1985) Iteration theory and its functional equations, lecture notes in mathematics, vol 1163, chap. On the definition of attractors, Springer, Berlin, pp 23–31Google Scholar
  8. Demongeot J, Goles E, Morvan M, Noual M, Sené S (2010) Attraction basins as gauges of robustness against boundary conditions in biological complex systems. PLoS ONE 5:e11,793CrossRefGoogle Scholar
  9. Derrida B, Pomeau Y (1986) Random networks of automata: a simple annealed approximation. Europhys Lett 1:45–49CrossRefGoogle Scholar
  10. Fatès N (2003) Experimental study of elementary cellular automata dynamics using the density parameter. Discret Math Theor Comput Sci AB:155–166MathSciNetzbMATHGoogle Scholar
  11. Floréen P (1992) Computational complexity problems in neural associative memories. PhD thesis, University of HelsinkiGoogle Scholar
  12. Goles E, Olivos J (1981) Comportement périodique des fonctions à seuil binaires et applications. Discret Appl Math 3:93–105CrossRefzbMATHGoogle Scholar
  13. Goles E, Martínez S (1990) Neural and automata networks: dynamical behaviour and applications. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  14. Goles E, Salinas L (2008) Comparison between parallel and serial dynamics of Boolean networks. Theor Comput Sci 396:247–253MathSciNetCrossRefzbMATHGoogle Scholar
  15. Harvey I, Bossomaier T (1997) Time out of joint: attractors in asynchronous random Boolean networks. In: Proceedings of ECAL, MIT Press, Cambridge, pp 67–75Google Scholar
  16. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467MathSciNetCrossRefGoogle Scholar
  17. Kleene SC (1956) Automata studies, annals of mathematics studies, vol 34, chap. Representation of events in nerve nets and finite automata, Princeton Universtity Press, Princeton, pp 3–41Google Scholar
  18. Manzoni L (2012) Asynchronous cellular automata and dynamical properties. Natural Comput 11:269–276MathSciNetCrossRefzbMATHGoogle Scholar
  19. McCulloch WS, Pitts WH (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5:115–133MathSciNetCrossRefzbMATHGoogle Scholar
  20. Noual M, Regnault D, Sené S (2012) Boolean networks synchronism sensitivity and XOR circulant networks convergence time. In: Full papers proceedings of AUTOMATA & JAC’2012, electronic proceedings in theoretical computer science, vol 90, Open Publishing Association, pp 37–52Google Scholar
  21. Noual M, Regnault D, Sené S (2013) About non-monotony in Boolean automata networks. Theor Comput Sci 504:12–25MathSciNetCrossRefzbMATHGoogle Scholar
  22. Remy É, Ruet P (2008) From minimal signed circuits to the dynamics of Boolean regulatory networks. Bioinformatics 24:i220–i226CrossRefGoogle Scholar
  23. Richard A (2010) Negative circuits and sustained oscillations in asynchronous automata networks. Adv Appl Math 44:378–392MathSciNetCrossRefzbMATHGoogle Scholar
  24. Robert F (1986) Discret iterations: a metric study. Springer, BerlinCrossRefGoogle Scholar
  25. Saint Savage N (2005) The effects of state dependent and state independent probabilistic updating on Boolean network dynamics. PhD thesis, University of ManchesterGoogle Scholar
  26. Schabanel N, Regnault D, Thierry É (2009) Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority. Theor Comput Sci 410:4844–4855MathSciNetCrossRefzbMATHGoogle Scholar
  27. Thieffry D, Thomas R (1995) Dynamical behaviour of biological regulatory networks-II. Immunity control in bacteriophage lambda. Bull Math Biol 57:277–297zbMATHGoogle Scholar
  28. Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585CrossRefGoogle Scholar
  29. Thomas R (1991) Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 153:1–23CrossRefGoogle Scholar
  30. Toulouse G (1977) Theory of the frustration effect in spin glasses. I. Commun Phys 2:115–119Google Scholar
  31. Vannimenus J, Toulouse G (1977) Theory of the frustration effect in spin glasses. II. Ising spins on a square lattice. J Phys C 10:L537–L542CrossRefGoogle Scholar
  32. von Neumann J (1966) Theory of self-reproducing automata. University of Illinois Press, ChampaignGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Freie Universität BerlinFB Mathematik und InformatikBerlinGermany
  2. 2.Aix-Marseille UniversityCNRS, LIFMarseilleFrance

Personalised recommendations