Combinatorics of Boolean automata circuits dynamics

In line with fields of theoretical computer science and biology that study Boolean automata networks to model regulation networks, we present some results concerning the dynamics of networks whose underlying structures are oriented cycles, that is, Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work focuses on the number of attractors of Boolean automata circuits whose elements are updated in parallel. In particular, we give the exact value of the total number of attractors of a circuit of arbitrary size n as well as, for every positive integer p, the number of its attractors of period p depending on whether the circuit has an even or an odd number of inhibitions. As a consequence, we obtain that both numbers depend only on the parity of the number of inhibitions and not on their distribution along the circuit. We also relate the counting of attractors of Boolean automata circuits to other known combinatorial problems and give intuition about how circuits interact by studying their dynamics when they intersect one another in one point.

[1]  Jacques Demongeot,et al.  Boundary conditions and phase transitions in neural networks. Theoretical results , 2008, Neural Networks.

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

[3]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[4]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

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

[6]  Harold Fredricksen,et al.  Lexicographic Compositions and deBruijn Sequences , 1977, J. Comb. Theory, Ser. A.

[7]  Jacques Demongeot,et al.  Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic , 2009, International journal of molecular sciences.

[8]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[9]  Eric Goles Ch.,et al.  Block-sequential update schedules and Boolean automata circuits , 2010, Automata.

[10]  Françoise Fogelman-Soulié,et al.  Specific roles of the different Boolean mappings in random networks , 1982 .

[11]  Adrien Elena,et al.  Robustesse des réseaux d'automates booléens à seuil aux modes d'itération. Application à la modélisation des réseaux de régulation génétique. (Robustness of threshold boolean automata networks to iteration modes. Application to genetic regulation network modelling) , 2009 .

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

[13]  Stuart A. Kauffman,et al.  The origins of order , 1993 .

[14]  Adrien Richard,et al.  Positive circuits and maximal number of fixed points in discrete dynamical systems , 2008, Discret. Appl. Math..

[15]  N.J.A. Sloane,et al.  On Single-Deletion-Correcting Codes , 2002, math/0207197.

[16]  Jeffrey Shallit,et al.  On the iteration of certain quadratic maps over GF(p) , 2004, Discret. Math..

[17]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

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

[19]  El Houssine Snoussi.  Necessary Conditions for Multistationarity and Stable Periodicity , 1998 .

[20]  Eric Goles Ch.,et al.  Transient length in sequential iteration of threshold functions , 1983, Discret. Appl. Math..

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

[22]  S. Grillner.  The motor infrastructure: from ion channels to neuronal networks , 2003, Nature Reviews Neuroscience.

[23]  Denis Thieffry,et al.  A description of dynamical graphs associated to elementary regulatory circuits , 2003, ECCB.

[24]  E. Goles.  Fixed Point Behavior of Threshold Functions on a Finite Set , 1982 .

[25]  Lorenzo Cangiano,et al.  Mechanisms of rhythm generation in the lamprey locomotor network , 2004 .

[26]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Christophe Soulé,et al.  Mathematical approaches to differentiation and gene regulation. , 2005, Comptes rendus biologies.

[28]  Hu Chuan-Gan,et al.  On The Shift Register Sequences , 2004 .

[29]  R. Thomas,et al.  A new necessary condition on interaction graphs for multistationarity. , 2007, Journal of Theoretical Biology.

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

[31]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[32]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

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

[34]  D. Hilbert.  Mathematical Problems , 2019, Mathematics: People · Problems · Results.

[35]  G. Frobenius,et al.  Ueber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul. , 1887 .

[36]  D Swandulla,et al.  Synaptic connectivity in cultured hypothalamic neuronal networks. , 1997, Journal of neurophysiology.

[37]  W S McCulloch,et al.  A logical calculus of the ideas immanent in nervous activity , 1990, The Philosophy of Artificial Intelligence.

[38]  Eric Goles Ch.,et al.  Fixed points and maximal independent sets in AND-OR networks , 2004, Discret. Appl. Math..

[39]  Eric Goles Ch.,et al.  On limit cycles of monotone functions with symmetric connection graph , 2004, Theor. Comput. Sci..

[40]  Jacques Demongeot,et al.  Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits , 2003, IEEE Trans. Syst. Man Cybern. Part B.

[41]  Eric Goles Ch.,et al.  Comportement periodique des fonctions a seuil binaires et applications , 1981, Discret. Appl. Math..