8,532 research outputs found

    Positive circuits and maximal number of fixed points in discrete dynamical systems

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    We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive circuits of the interaction graph associated with F. The proof uses and strongly generalizes a theorem of Richard and Comet which corresponds to a discrete version of the Thomas' conjecture: if the interaction graph associated with F has no positive circuit, then F has at most one fixed point. The obtained upper bound on the number of fixed points also strongly generalizes the one established by Aracena et al for a particular class of Boolean networks.Comment: 13 page

    Homogeneous and Scalable Gene Expression Regulatory Networks with Random Layouts of Switching Parameters

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    We consider a model of large regulatory gene expression networks where the thresholds activating the sigmoidal interactions between genes and the signs of these interactions are shuffled randomly. Such an approach allows for a qualitative understanding of network dynamics in a lack of empirical data concerning the large genomes of living organisms. Local dynamics of network nodes exhibits the multistationarity and oscillations and depends crucially upon the global topology of a "maximal" graph (comprising of all possible interactions between genes in the network). The long time behavior observed in the network defined on the homogeneous "maximal" graphs is featured by the fraction of positive interactions (0≤η≤10\leq \eta\leq 1) allowed between genes. There exists a critical value ηc<1\eta_c<1 such that if η<ηc\eta<\eta_c, the oscillations persist in the system, otherwise, when η>ηc,\eta>\eta_c, it tends to a fixed point (which position in the phase space is determined by the initial conditions and the certain layout of switching parameters). In networks defined on the inhomogeneous directed graphs depleted in cycles, no oscillations arise in the system even if the negative interactions in between genes present therein in abundance (ηc=0\eta_c=0). For such networks, the bidirectional edges (if occur) influence on the dynamics essentially. In particular, if a number of edges in the "maximal" graph is bidirectional, oscillations can arise and persist in the system at any low rate of negative interactions between genes (ηc=1\eta_c=1). Local dynamics observed in the inhomogeneous scalable regulatory networks is less sensitive to the choice of initial conditions. The scale free networks demonstrate their high error tolerance.Comment: LaTeX, 30 pages, 20 picture

    Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

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    A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

    General Iteration graphs and Boolean automata circuits

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    This article is set in the field of regulation networks modeled by discrete dynamical systems. It focuses on Boolean automata networks. In such networks, there are many ways to update the states of every element. When this is done deterministically, at each time step of a discretised time flow and according to a predefined order, we say that the network is updated according to block-sequential update schedule (blocks of elements are updated sequentially while, within each block, the elements are updated synchronously). Many studies, for the sake of simplicity and with some biologically motivated reasons, have concentrated on networks updated with one particular block-sequential update schedule (more often the synchronous/parallel update schedule or the sequential update schedules). The aim of this paper is to give an argument formally proven and inspired by biological considerations in favour of the fact that the choice of a particular update schedule does not matter so much in terms of the possible and likely dynamical behaviours that networks may display

    Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

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    We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

    Dynamical complexity of discrete time regulatory networks

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    Genetic regulatory networks are usually modeled by systems of coupled differential equations and by finite state models, better known as logical networks, are also used. In this paper we consider a class of models of regulatory networks which present both discrete and continuous aspects. Our models consist of a network of units, whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations, according to a rule which depends on the state of the neighboring units. As a first approximation to the complete description of the dynamics of this networks we focus on a global characteristic, the dynamical complexity, related to the proliferation of distinguishable temporal behaviors. In this work we give explicit conditions under which explicit relations between the topological structure of the regulatory network, and the growth rate of the dynamical complexity can be established. We illustrate our results by means of some biologically motivated examples.Comment: 28 pages, 4 figure

    Bifurcations and Chaos in Time Delayed Piecewise Linear Dynamical Systems

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    We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay which is evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.Comment: 34 Pages, 14 Figure

    AND-NOT logic framework for steady state analysis of Boolean network models

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    Finite dynamical systems (e.g. Boolean networks and logical models) have been used in modeling biological systems to focus attention on the qualitative features of the system, such as the wiring diagram. Since the analysis of such systems is hard, it is necessary to focus on subclasses that have the properties of being general enough for modeling and simple enough for theoretical analysis. In this paper we propose the class of AND-NOT networks for modeling biological systems and show that it provides several advantages. Some of the advantages include: Any finite dynamical system can be written as an AND-NOT network with similar dynamical properties. There is a one-to-one correspondence between AND-NOT networks, their wiring diagrams, and their dynamics. Results about AND-NOT networks can be stated at the wiring diagram level without losing any information. Results about AND-NOT networks are applicable to any Boolean network. We apply our results to a Boolean model of Th-cell differentiation
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