Full characterization of attractors for two intersected asynchronous Boolean automata cycles

The understanding of Boolean automata networks dynamics takes an important place in various domains of computer science such as computability, complexity and discrete dynamical systems. In this paper, we make a step further in this understanding by focusing on their cycles, whose necessity in networks is known as the brick of their complexity. We present new results that provide a characterisation of the transient and asymptotic dynamics, i.e. of the computational abilities, of asynchronous Boolean automata networks composed of two cycles that intersect at one automaton, the so-called double-cycles. To do so, we introduce an efficient formalism inspired by algorithms to define long sequences of updates, that allows a better description of their dynamics than previous works in this area.

[1]  J. Schwartz,et al.  Theory of Self-Reproducing Automata , 1967 .

[2]  James Bonner,et al.  Current Topics in Developmental Biology, Vol. 2 , 1968 .

[3]  Denis Thieffry,et al.  Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework , 2008, Adv. Appl. Math..

[4]  Nazim Fatès,et al.  Asynchronous Behavior of Double-Quiescent Elementary Cellular Automata , 2006, LATIN.

[5]  René Thomas.  On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations , 1981 .

[6]  Mathilde Noual,et al.  Updating Automata Networks , 2012 .

[7]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[8]  François Robert,et al.  Discrete iterations - a metric study , 1986, Springer series in computational mathematics.

[9]  W. Pitts,et al.  A Logical Calculus of the Ideas Immanent in Nervous Activity (1943) , 2021, Ideas That Created the Future.

[10]  Adrien Richard,et al.  Negative circuits and sustained oscillations in asynchronous automata networks , 2009, Adv. Appl. Math..

[11]  S. Kauffman.  Gene regulation networks: a theory for their global structure and behaviors. , 1971, Current topics in developmental biology.

[12]  J. Monod,et al.  Genetic regulatory mechanisms in the synthesis of proteins. , 1961, Journal of Molecular Biology.

[13]  Adrien Richard,et al.  Necessary conditions for multistationarity in discrete dynamical systems , 2007, Discret. Appl. Math..

[14]  Elisabeth Remy,et al.  Relations between gene regulatory networks and cell dynamics in Boolean models , 2012, Discret. Appl. Math..

[15]  Damien Regnault,et al.  About non-monotony in Boolean automata networks , 2011, Theor. Comput. Sci..

[16]  Thomas C. Schelling,et al.  Dynamic models of segregation , 1971 .

[17]  Mathilde Noual,et al.  Dynamics of Circuits and Intersecting Circuits , 2012, LATA.

[18]  Eric Goles,et al.  Neural and automata networks , 1990 .

[19]  R. Thomas,et al.  Boolean formalization of genetic control circuits. , 1973, Journal of theoretical biology.

[20]  F. Jacob,et al.  L'opéron : groupe de gènes à expression coordonnée par un opérateur [C. R. Acad. Sci. Paris 250 (1960) 1727–1729] , 2005 .

[21]  Jacques Demongeot,et al.  Combinatorics of Boolean automata circuits dynamics , 2012, Discret. Appl. Math..

[22]  Adrien Richard,et al.  On the Convergence of Boolean Automata Networks without Negative Cycles , 2013, Automata.

[23]  E. Ising.  Beitrag zur Theorie des Ferromagnetismus , 1925 .