Related papers: Negative circuits and sustained oscillations in as…
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…
We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. Once such a graph has been…
We discuss properties which must be satisfied by a genetic network in order for it to allow differentiation. These conditions are expressed as follows in mathematical terms. Let $F$ be a differentiable mapping from a finite dimensional real…
We are interested in the relationships between the number fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ and its interaction graph, which is the arc-signed digraph $G$ on $\{1,\dots,n\}$ that describes the positive and negative…
In the context of discrete dynamical systems and their applications, fixed points often have a clear interpretation. This is indeed a central topic of gene regulatory mechanisms modeled by Boolean automata networks (BANs), where a…
We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables…
In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph…
In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown…
Automata networks, and in particular Boolean networks, are used to model diverse networks of interacting entities. The interaction graph of an automata network is its most important parameter, as it represents the overall architecture of…
The asynchronous systems are the non-deterministic models of the asynchronous circuits from the digital electrical engineering. In the autonomous version, such a system is a set of functions x:R{\to}{0,1}^{n} called states (R is the time…
We show that a continuous map or a continuous flow on $\R^{n}$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in $\R^{n}$…
The aim of this paper is to analyze a class of consensus algorithms with finite-time or fixed-time convergence for dynamic networks formed by agents with first-order dynamics. In particular, in the analyzed class a single evaluation of a…
A finite dynamical system with $n$ components is a function $f:X\to X$ where $X=X_1\times\dots\times X_n$ is a product of $n$ finite intervals of integers. The structure of such a system $f$ is represented by a signed digraph $G$, called…
This paper considers synchronous discrete-time dynamical systems on graphs based on the threshold model. It is well known that after a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of…
We analyse the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling the stability of fixed points in large systems defined…
Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical models of assemblies of interacting self-repressing genes. For mean-field interactions, which most mathematical studies have assumed so far, every trajectory must…
The structure of the graph defined by the interactions in a Boolean network can determine properties of the asymptotic dynamics. For instance, considering the asynchronous dynamics, the absence of positive cycles guarantees the existence of…
We study nonautonomous discrete dynamical systems with randomly perturbed trajectories. We suppose that such a system is generated by a sequence of continuous maps which converges uniformly to a map $f$. We give conditions, under which a…
Networks of degrade-and-fire oscillators are elementary models of populations of synthetic gene circuits with negative feedback, which show elaborate phenomenology while being amenable to mathematical analysis. In addition to thorough…
Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous…