Advertisement

Journal of Systems Science and Complexity

, Volume 26, Issue 6, pp 871–885 | Cite as

Solvability and control design for synchronization of Boolean networks

  • Xiangru Xu
  • Yiguang HongEmail author
Article

Abstract

This paper addresses the synchronization problem of Boolean networks. Based on the matrix expression of logic, solvability conditions and design procedures of the synchronization of Boolean networks with outputs are given for both open-loop and feedback control. Necessary and sufficient conditions on open-loop control are proposed first with a constructive design procedure. Then sufficient condition for the feedback control case is obtained, and corresponding design procedure is proposed with the help of algorithms to solve logic matrix equations. Numerical examples are also provided to illustrate the proposed control design.

Key words

Boolean networks open-loop and feedback control semi-tensor product synchronization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Kauffman S, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theoretical Biology, 1969, 22: 437–467.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Kunz W and Stoffel D, Reasoning in Boolean Networks: Logic Synthesis and Verification Using Testing Techniques, Kluwer Academic Publishers, Boston, 1997.CrossRefzbMATHGoogle Scholar
  3. [3]
    Kauffman S, The Origins of Order: Self-organization and Selection in Evolution, Oxford University Press, New York, 1993.Google Scholar
  4. [4]
    Farrow C, Heidel J, Maloney J, and Roger J, Scale equations for synchronous Boolean networks with biological applications, IEEE Trans. Neural Networks, 2004, 15(2): 348–354.CrossRefGoogle Scholar
  5. [5]
    Cheng D, Qi H, Li Z Q, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach, Springer, London, 2010.Google Scholar
  6. [6]
    Ma J, Cheng D, Hong Y, and Sun Y, On complexity of power systems, Journal of Systems Science and Complexity, 2003, 16(3): 391–403.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Zhao Y, Li Z, and Cheng D, Optimal control of logical control networks, IEEE Trans. Automatical Control, 2011, 56(8): 1766–1776.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Cheng D and Qi H, Controllability and observability of Boolean control networks, Automatica, 2009, 45(7): 1659–1667.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Cheng D, Qi H, Hu X, and Liu J, Stability and stabilization of Boolean networks, Int. J. Rubost and Nonlinear Control, 2010, 21(2): 134–156.CrossRefGoogle Scholar
  10. [10]
    Laschov D and Margaliot M, A maximum principle for single-input of Boolean control network, IEEE Trans. Automatic Control, 2011, 56(4): 913–917.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Li F and Sun J, Controllability of Boolean control networks with time delays in states, Automatica, 2011, 47(3): 603–607.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Danino T, Mondragon-Palomino O, Tsimiring L, et al., A synchronized quorum of genetic clocks, Nature, 2010, 463: 326–330.CrossRefGoogle Scholar
  13. [13]
    Hong Y, Qin H, and Chen G, Adaptive synchronization of chaotic systems via state and output feedback, Int. J. of Bifurcation and Chaos, 2001, 11(5): 1149–1158.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Morelli L G and Zanette D H, Synchronization of Kauffman networks, Phys. Rev. E, 2001, 63(3): 036204-1–036204-10.Google Scholar
  15. [15]
    Millérioux G and Guillot P, Self-synchronizing stream ciphers and dynamical systems: State of the art and open issues, Int. Journal of Bifurcation and Chaos, 2010, 20(9): 2979–2991.CrossRefzbMATHGoogle Scholar
  16. [16]
    Glass L, Synchronization and rhythmic processes in physiology, Nature, 2001, 410: 277–284.CrossRefGoogle Scholar
  17. [17]
    Ballesteros F and Luque B, Random Boolean networks response to external periodic signals, Physica A, 2002, 313: 289–300.CrossRefzbMATHGoogle Scholar
  18. [18]
    Hung Y C, Ho M C, Lih J S, and Jiang I M, Chaos synchronization of two stochastically coupled random Boolean networks, Phys. Lett. A, 2006, 356(1): 35–43.CrossRefzbMATHGoogle Scholar
  19. [19]
    Hong Y and Xu X, Solvability and control design for dynamic synchronization of Boolean networks, 29th Chinese Control Conference, Beijing, China, 2010, 805–810.Google Scholar
  20. [20]
    Akutsu T, Hayashida M, Ching W, and Ng M K, Control of Boolean networks: Hardness results and algorithms for tree structured networks, J. Theoretical Biology, 2007, 244: 670–679.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Ho M C, Hung Y C, and Jiang I M, Stochastic coupling of two random Boolean networks, Phys. Lett. A, 2005, 344(1): 36–42.CrossRefzbMATHGoogle Scholar
  22. [22]
    Robert F, Discrete Iterations: A Metric Study, Translated by Rokne J, Springer-Verlag, Berlin, 1986.CrossRefzbMATHGoogle Scholar
  23. [23]
    Tutte W T, Graph Theory, Cambridge University Press, UK, 2001.zbMATHGoogle Scholar
  24. [24]
    Huepe-Minoletti C and Aldana-Gonzalez M, Dynamical phase transition in a neural network model with noise: An exact solution, Journal of Statistical Physics, 2002, 108: 527–540.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Kitano H, Cancer as a robust system: Implications for anticancer therapy, Nature Reviews Cancer, 2004, 4(3): 227–235.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Huang J, Nonlinear Output Regulation: Theory & Applications, SIAM, Phildelphia, 2004.CrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations