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Local cycles and dynamical properties of Boolean networks

Published online by Cambridge University Press:  20 November 2014

PAUL RUET
PAUL RUET*
Affiliation:
CNRS, Laboratoire Preuves, Programmes et Systèmes, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France Email: [email protected]
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Abstract

We investigate the relationships between the dynamical properties of Boolean networks and properties of their Jacobian matrices, in particular the existence of local cycles in the associated interaction graphs. We define the notion of hereditarily bijective maps, and we use it to strengthen the property of unicity of a fixed point and to provide simplified proofs and generalizations of theorems relating attractors to the existence of local cycles, in particular local positive cycles. We then argue that this notion may not suffice to prove, under a suitable hypothesis such as the existence of a cyclic attractor or a stronger hypothesis, the existence of local negative cycles. We then consider a class of Boolean networks called and-or-nets, and for this class, we prove that the hypothesis of an antipodal attractive cycle implies the existence of a local negative cycle.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 26 , Issue 4 , May 2016 , pp. 702 - 718
Copyright
Copyright © Cambridge University Press 2014 

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References

Bass, H., Connell, E. H. and Wright, D. (1982) The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bulletin of the American Mathematical Society 7 (2) 287–330.CrossRefGoogle Scholar
Bazsó, F. (2000) Derivations of vector-valued Boolean functions. Acta Mathematica Hungarica 87 (3) 197–203.Google Scholar
Cima, A., Gasull, A. and Mañosas, F. (1999) The discrete Markus–Yamabe problem. Nonlinear Analysis 35 343–354.CrossRefGoogle Scholar
Kauffman, S. A. (1969) Metabolic stability and epigenesis in randomly connected nets. Journal of Theoretical Biology 22 437–467.CrossRefGoogle Scholar
Kauffman, S. A. (1993) The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press.Google Scholar
Kim, K. H. (1982) Boolean Matrix Theory and Applications, Marcel Dekker.Google Scholar
Remy, E. and Ruet, P. (2007) On differentiation and homeostatic behaviours of Boolean dynamical systems. In: Transactions on Computational Systems Biology. Springer Lecture Notes in Computer Science 4780 92–101.Google Scholar
Remy, E., Ruet, P. and Thieffry, D. (2008) Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Advances in Applied Mathematics 41 (3) 335–350.CrossRefGoogle Scholar
Richard, A. (2010) Negative circuits and sustained oscillations in asynchronous automata networks. Advances in Applied Mathematics 44 (4) 378–392.Google Scholar
Richard, A. and Ruet, P. (2013) From kernels in directed graphs to negative cycles in Boolean networks. Discrete Applied Mathematics 161 1106–1117.Google Scholar
Robert, F. (1995) Les Systèmes Dynamiques Discrets, Mathématiques et Applications, volume 19, Springer.Google Scholar
Rudeanu, S. (1974) Boolean Functions and Equations, North-Holland.Google Scholar
Shih, M.-H. and Dong, J.-L. (2005) A combinatorial analogue of the Jacobian problem in automata networks. Advances in Applied Mathematics 34 (1) 30–46.Google Scholar
Thomas, R. (1973) Boolean formalization of genetic control circuits. Journal of Theoretical Biology 42 563–585.Google Scholar
Thomas, R. (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states and sustained oscillations. In: Series in Synergetics, volume 9, Springer 180–193.Google Scholar
Thomas, R. and Kaufman, M. (2001) Multistationarity, the basis of cell differentiation and memory. i. Structural conditions of multistationarity and other non-trivial behaviour. Chaos 11 170–179.Google Scholar
von Neumann, J. (1966) Theory of Self-Reproducing Automata, University of Illinois Press.Google Scholar