Fixed points and maximal independent sets in AND–OR networks

Aracena, Julio; Demongeot, Jacques; Goles, Éric

doi:10.1016/s0166-218x(03)00461-x

Cited by 48 publications

(48 citation statements)

References 6 publications

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“…• For Boolean cellular automata with the majority rule, the number of fixed points was determined in Agur et al (1988). • In Aracena et al (2004), the authors studied AND-OR networks (Boolean networks where each local function is either an OR or AND function) with directed dependency graphs. Formulae for the maximum number of fixed points are obtained.…”

Section: Figmentioning

confidence: 99%

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

2010

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For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

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Section: Figmentioning

confidence: 99%

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

2010

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“…Close to the 16th Hilbert problem concerning the number of limit cycles of dynamical systems [16], this question has already been considered in some works [17,18,19]. It is of particular relevance in the neighbouring field of formal neural networks where attractors represent memorisation capacity [9,10].…”

Section: Introductionmentioning

confidence: 99%

On the Number of Attractors of Positive and Negative Boolean Automata Circuits

Demongeot

Noual

Sené

2010

2010 IEEE 24th International Conference on Advanced Information Networking and Applications Workshops

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Abstract. In line with fields of theoretical computer science and biology that study Boolean automata networks often seen as models of regulation networks, we present some results concerning the dynamics of networks whose underlying interaction graphs are circuits, 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. We prove how to obtain formally 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. Keywords: Discrete dynamical system, formal neural network, positive and negative circuit, asymptotic behaviour, attractor. IntroductionThe theme of this article is set in the general framework of complex dynamical systems and, more precisely, that of regulation or neural networks modeled by means of discrete mathematical tools. Since McCulloch and Pitts [1] proposed threshold Boolean automata networks to represent neural networks formally in 1943 and later, at the end of the 1960's, Kauffman [2] and Thomas [3] introduced the first models of genetic regulation networks, many other studies based on the same or different formalisms were carried out on the theoretical properties of such networks. One of the main motivations of many of them was to better understand those emergent dynamical behaviours that networks display and that cannot be explained or predicted by a simple analysis of the local interactions existing between the components of the networks. Amongst the studies published in this context since the end of the 1990's, one may cite [4,5,6,7,8]. Earlier on, Hopfield [9,10] emphasised the notions of memory and learning and Goles et al. revealed in [11,12,13] interesting dynamical properties of some particular networks. Further, later works by Thomas and Kauffman [14,15] yielded conjectures and gave rise to problematics that are still relevant in the field of regulation networks beyond the particular definition of the models one may choose to use. For instance, Thomas highlighted the importance of specific patterns on the dynamics of discrete regulation networks and Kauffman gave an approximation of the number of different possible asymptotic behaviours of Boolean networks.From the point of view of theoretical biology as well as that of theoretical computer science, it seems to be of great interest to address the question of the number of attractors in the dynamics of a network. Close to the 16th Hilbert problem concerning the number of limit cycles of dynamical systems [16], this question has already been considered in some works [17,1...

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“…Note that the interaction graph contains only one connected component having at least one (here two) positive circuit of interactions (a circuit is positive if its number of inhibiting edges is even). Hence, from [16,17,18,19,20,21,22], we can expect only 2 1 = 2 fixed configurations for the network dynamics and an upper bound for this number of 2 2 . On Table 3, we see that, if the state of p27 and miRNA 159 are not fixed to particular values, then this number is in reality 2, plus one (resp.…”

Section: Cell Cyclementioning

confidence: 97%

Structural Sensitivity of Neural and Genetic Networks

Amor

Demongeot

Sené

2008

MICAI 2008: Advances in Artificial Intelligence

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Abstract. This paper aims at giving new results on the structural sensitivity of biological networks represented by threshold Boolean networks and ruled by Hopfield-like evolution laws classically used in the context of neural and genetic networks. Indeed, the objective is to present how certain changes and/or perturbations in such networks can modify significantly their asymptotic behaviour. More precisely, this work has been focused on three different kinds of what we think to be relevant in the biological area of robustness (in both theoretical and applied frameworks): the boundary sensitivity (external fields, hormone flows, ...), the state sensitivity (axonal or somatic modulations, microRNAs actions, ...) and the updating sensitivity.

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Fixed points and maximal independent sets in AND–OR networks

Cited by 48 publications

References 6 publications

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

On the Number of Attractors of Positive and Negative Boolean Automata Circuits

Structural Sensitivity of Neural and Genetic Networks

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