Skip to main content

On Boolean Automata Isolated Cycles and Tangential Double-Cycles Dynamics

  • 180 Accesses

Part of the Emergence, Complexity and Computation book series (ECC,volume 42)

Abstract

Our daily social and political life is more and more impacted by social networks. The functioning of our living bodies is deeply dependent on biological regulation networks such as neural, genetic, and protein networks. And the physical world in which we evolve, is also structured by systems of interacting particles. Interaction networks can be seen in all spheres of existence that concern us, and yet, our understanding of interaction networks remains severely limited by our present lack of both theoretical and applied insight into their clockworks. In the past, efforts at understanding interaction networks have mostly been directed towards applications. This has happened at the expense of developing understanding of the generic and fundamental aspects of interaction networks (properties and behaviours due primarily to the fact that a system is an interaction network, as opposed to properties and behaviours rather due to the fact a system is a genetic interaction network for instance). Intrinsic properties of interaction networks (e.g., the ways in which they transmit information along entities, their ability to produce this or that kind of global dynamical behaviour depending on local interactions) are thus still not well understood. Lack of fundamental knowledge tends to limit the innovating power of applications. Without more theoretical fundamental knowledge, applications cannot evolve deeply and become more impacting. Hence, it is necessary to better apprehend and comprehend the intrinsic properties of interaction networks, notably the relations between their architecture and their dynamics and how they are affected by and set in time. In this chapter, we use the elementary mathematical model of Boolean automata networks as a formal archetype of interaction networks. We survey results concerning the role of feedback cycles and the role of intersections between feedback cycles, in shaping the asymptotic dynamical behaviours of interaction networks. We pay special attention to the impact of the automata updating modes.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-92551-2_11
  • Chapter length: 34 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   149.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-92551-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   199.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Albert R (2009) Discrete dynamic modeling of cellular signaling networks. Methods Enzym 467:281–306

    CrossRef  Google Scholar 

  2. Aldana M (2003) Boolean dynamics of networks with scale-free topology. Phys D 185:45–66

    MathSciNet  CrossRef  Google Scholar 

  3. Apostol TM (1976) Introduction to analytic number theory. Springer

    Google Scholar 

  4. Aracena J, González M, Zuñiga A, Mendez MA, Cambiazo V (2006) Regulatory network for cell shape changes during drosophila ventral furrow formation. J Theor Biol 239:49–62

    MathSciNet  CrossRef  Google Scholar 

  5. Berlekamp ER, Conway JH, Guy RK (1982) Winning ways for your mathematical plays. Academic Press

    Google Scholar 

  6. Berstel J, Perrin D (2007) The origins of combinatorics on words. Eur J Comb 28:996–1022

    MathSciNet  CrossRef  Google Scholar 

  7. Church A (1932) A set of postulates for the foundation of logic. Ann Math 33:346–366

    MathSciNet  CrossRef  Google Scholar 

  8. Cook M (2004) Universality in elementary cellular automata. Complex Syst 15:1–40

    MathSciNet  MATH  Google Scholar 

  9. Cosnard M, Demongeot J (1985) On the definitions of attractors. In: Iteration theory and its functional equations, vol 1163 of Lecture notes in mathematics. Springer, pp 23–31

    Google Scholar 

  10. Cosnard M, Demongeot J, Le Breton A (eds) (1983) Rhythms in biology and other fields of application, vol 49 of Lecture notes in biomathematics. Springer

    Google Scholar 

  11. Cull P (1971) Linear analysis of switching nets. Biol Cybern 8:31–39

    Google Scholar 

  12. Delbrück M (1949) Génétique du bactériophage. In: Unités biologiques douées de continuité génétique

    Google Scholar 

  13. Demongeot J (1975) Au sujet de quelques modèles stochastiques appliqués à la biologie. PhD thesis, Université scientifique et médicale de Grenoble

    Google Scholar 

  14. Demongeot J, Goles E, Morvan M, Noual M, Sené S (2010) Attraction basins as gauges of the robustness against boundary conditions in biological complex systems. PLoS One 5:e11793

    Google Scholar 

  15. Demongeot J, Goles E, Tchuente M (eds) Dynamical systems and cellular automata. Academic Press

    Google Scholar 

  16. Demongeot J, Noual M, Sené S (2012) Combinatorics of boolean automata circuits dynamics. Discret Appl Math 160:398–415

    MathSciNet  CrossRef  Google Scholar 

  17. Demongeot J, Sené S (2020) About block-parallel Boolean networks: a position paper. Nat Comput 19:5–13

    MathSciNet  CrossRef  Google Scholar 

  18. Elspas B (1959) The theory of autonomous linear sequential networks. IRE Trans Circuit Theory 6:45–60

    CrossRef  Google Scholar 

  19. Gershenson C (2004) Updating schemes in random Boolean networks: do they really matter? In: Proceedings of artificial life. MIT Press, pp 238–243

    Google Scholar 

  20. Gödel KF (1986) Kurt Gödel collected works, volume I—Publications 1929–1936, chapter on undecidable propositions of formal mathematical systems. Oxford University Press, pp 346–372

    Google Scholar 

  21. Goles E (1982) Fixed point behavior of threshold functions on a finite set. SIAM J Algebr Discret Methods 3:529–531

    MathSciNet  CrossRef  Google Scholar 

  22. Goles E, Fogelman-Soulié F, Pellegrin D (1985) Decreasing energy functions as a tool for studying threshold networks. Discret Appl Math 12:261–277

    MathSciNet  CrossRef  Google Scholar 

  23. Goles E, Martínez S (1990) Neural and automata networks: dynamical behavior and applications, vol 58 of Mathematics and its applications. Kluwer Academic Publishers

    Google Scholar 

  24. Goles E, Noual M (2010) Block-sequential update schedules and Boolean automata circuits. In: Proceedings of AUTOMATA. Discrete mathematics and theoretical computer science, pp 41–50

    Google Scholar 

  25. Golomb SW (1967) Shift register sequences. Holden-Day Inc

    Google Scholar 

  26. Graham RL, Knuth DE, Patashnik O (1989) Concrete mathematics: a foundation for computer science. Addison-Wesley

    Google Scholar 

  27. Guet CC, Elowitz MB, Hsing W, Leibler S (2002) Combinatorial synthesis of genetic networks. Science 296:1466–1470

    CrossRef  Google Scholar 

  28. Gupta S, Bisht SS, Kukreti R, Jain S, Brahmachari SK (2007) Boolean network analysis of a neurotransmitter signaling pathway. J Theor Biol 244:463–469

    MathSciNet  CrossRef  Google Scholar 

  29. Hesper B, Hogeweg P (1970) Bioinformatica: EEN werkconcept. Kameleon 1:18–29

    Google Scholar 

  30. Hogeweg P, Hesper B (1978) Interactive instruction on population interactions. Comput Biol Med 8:319–327

    CrossRef  Google Scholar 

  31. Huffman DA (1959) Canonical forms for information-lossless finite-state logical machines. IRE Trans Inf Theory 5:41–59

    CrossRef  Google Scholar 

  32. Jacob F, Monod J (1961) Genetic regulatory mechanisms in the synthesis of proteins. J Mol Biol 3:318–356

    CrossRef  Google Scholar 

  33. Kauffman SA (1969) Homeostasis and differentiation in random genetic control networks. Nature 224:177–178

    CrossRef  Google Scholar 

  34. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467

    MathSciNet  CrossRef  Google Scholar 

  35. Kauffman SA (1971) Current topics in developmental biology, vol 6, chapter Gene regulation networks: a theory for their global structures and behaviors. Elsevier, pp 145–181

    Google Scholar 

  36. Kauffman SA, Peterson C, Samuelsson B, Troein C (2003) Random boolean network models and the yeast transcriptional network. Proc Natl Acad Sci USA 100:14796–14799

    CrossRef  Google Scholar 

  37. Kaufman M, Thomas R (1985) Towards a logical analysis of the immune response. J Theor Biol 114:527–561

    MathSciNet  CrossRef  Google Scholar 

  38. Kleene SC (1956) Automata studies, vol 34 of Annals of mathematics studies, chapter Representation of events in nerve nets and finite automata. Princeton Universtity Press, pp 3–41

    Google Scholar 

  39. Kung HT, Leiserson CE (1980) Introduction to VLSI systems, chapter Algorithms for VLSI processor arrays. Addison-Wesley, pp 271–292

    Google Scholar 

  40. Kurka P (1997) Languages, equicontinuity and attractors in cellular automata. Ergod Theory Dyn Syst 17:417–433

    MathSciNet  CrossRef  Google Scholar 

  41. McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. J Math Biophys 5:115–133

    MathSciNet  CrossRef  Google Scholar 

  42. Melliti T, Noual M, Regnault D, Sené S, Sobieraj J (2015) Asynchronous dynamics of Boolean automata double-cycles. In: Proceedings of UCNC, vol 9252 of Lecture notes in computer science. Springer, pp 250–262

    Google Scholar 

  43. Mendoza L (2006) A network model for the control of the differentiation process in Th cells. Biosystems 84:101–114

    CrossRef  Google Scholar 

  44. Mendoza L, Alvarez-Buylla ER (1998) Dynamics of the genetic regulatory network for arabidopsis thaliana flower morphogenesis. J Theor Biol 193:307–319

    CrossRef  Google Scholar 

  45. Monod J, Changeux J-P, Jacob F (1963) Allosteric proteins and cellular control systems. J Mol Biol 6:306–329

    CrossRef  Google Scholar 

  46. Noual M (2012) Dynamics of circuits and intersection circuits. In: Proceedings of LATA, vol 7183 of Lecture notes in computer science. Springer, pp 433–444

    Google Scholar 

  47. Noual M (2012) Updating automata networks. PhD thesis, École normale supérieure de Lyon. http://tel.archives-ouvertes.fr/tel-00726560

  48. Puri Y, Ward T (2001) Arithmetic and growth of periodic orbits. J Integer Seq 4:01.2.1

    Google Scholar 

  49. Remy É, Mossé B, Chaouiya C, Thieffry D (2003) A description of dynamical graphs associated to elementary regulatory circuits. Bioinformatics 19:ii172–ii178

    Google Scholar 

  50. Remy É, Ruet P, Thieffry D (2008) Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Adv Appl Math 41:335–350

    MathSciNet  CrossRef  Google Scholar 

  51. Ribenboim P (1996) The new book of prime number records. Springer

    Google Scholar 

  52. Richard A (2010) Negative circuits and sustained oscillations in asynchronous automata networks. Adv Appl Math 44:378–392

    MathSciNet  CrossRef  Google Scholar 

  53. Richard A, Comet J-P (2007) Necessary conditions for multistationarity in discrete dynamical systems. Discret Appl Math 155:2403–2413

    MathSciNet  CrossRef  Google Scholar 

  54. Richard A, Comet J-P, Bernot G, Thomas R, dynamics (2004) Modeling of biological regulatory networks introduction of singular states in the qualitative. Fundam Inform 65:373–392

    MathSciNet  MATH  Google Scholar 

  55. Robert F (1969) Blocs-H-matrices et convergence des méthodes itératives classiques par blocs. Linear Algebr Its Appl 2:223–265

    CrossRef  Google Scholar 

  56. Robert F (1976) Contraction en norme vectorielle: convergence d’itérations chaotiques pour des équations non linéaires de point fixe à plusieurs variables. Linear Algebr Its Appl 13:19–35

    CrossRef  Google Scholar 

  57. Robert F (1980) Itérations sur des ensembles finis et automates cellulaires contractants. Linear Algebr Its Appl 29:393–412

    CrossRef  Google Scholar 

  58. Robert F (1986) Discrete iterations: a metric study, vol 6 of Springer series in computational mathematics. Springer

    Google Scholar 

  59. Robert F (1995) Les systèmes dynamiques discrets, vo 19 of Mathématiques & applications. Springer

    Google Scholar 

  60. Ruskey F (2003) Combinatorial generation. Book preliminary working draft

    Google Scholar 

  61. Ruz GA, Goles E, Sené S (2018) Reconstruction of Boolean regulatory models of flower development exploiting an evolution strategy. In: Proceedings of CEC. IEEE Press, pp 1–8

    Google Scholar 

  62. Saez-Rodriguez J, Simeoni L, Lindquist JA, Hemenway R, Bommhardt U, Arndt B, Haus U-U, Weismantel R, Gilles ED, Klamt S, Schraven B (2007) A logical model provides insights into T cell receptor signaling. PLoS Comput Biol 3:e163

    Google Scholar 

  63. Sené S (2012) Sur la bio-informatique des réseaux d’automates. Habilitation thesis, Université d’Évry – Val d’Essonne http://tel.archives-ouvertes.fr/tel-00759287

  64. Smith AR (1971) Simple computation-universal cellular spaces. J ACM 18:339–353

    MathSciNet  CrossRef  Google Scholar 

  65. Thieffry D, Thomas R (1995) Dynamical behaviour of biological regulatory networks—II. Immunity control in bacteriophage lambda. Bull Math Biol 57:277–297

    Google Scholar 

  66. Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585

    CrossRef  Google Scholar 

  67. Thomas R (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Numerical methods in the study of critical phenomena, vol 9 of Springer series in synergetics. Springer, pp 180–193

    Google Scholar 

  68. Thomas R (1991) Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 153:1–23

    CrossRef  Google Scholar 

  69. Thomas R, Thieffry D, Kaufman M (1995) Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull Math Biol 57:247–276

    Google Scholar 

  70. Turing AM (1936) On computable numbers, with an application to the entscheidungsproblem. Proc Lond Math Soc 2:230–265

    MathSciNet  MATH  Google Scholar 

  71. van der Laan H (1960) Le nombre plastique, quinze leÃğons sur l’ordonnance architectonique. E J Brill

    Google Scholar 

  72. von Neumann J (1966) Theory of self-reproducing automata. University of Illinois Press

    Google Scholar 

  73. Wolfram S (1984) Universality and complexity in cellular automata. Phys D 10:1–35

    MathSciNet  CrossRef  Google Scholar 

Download references

Acknowledgements

This work was funded mainly by our salaries as French or German State agents or pensioner (affiliated to Université Grenoble-Alpes (JD), Université d’Évry (TM and DR), Freie Universität Berlin (MN), and Université d’Aix-Marseille (SS)), and secondarily by the ANR-18-CE40-0002 FANs project (SS).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sylvain Sené .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Demongeot, J., Melliti, T., Noual, M., Regnault, D., Sené, S. (2022). On Boolean Automata Isolated Cycles and Tangential Double-Cycles Dynamics. In: Adamatzky, A. (eds) Automata and Complexity. Emergence, Complexity and Computation, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-92551-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-92551-2_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92550-5

  • Online ISBN: 978-3-030-92551-2

  • eBook Packages: EngineeringEngineering (R0)