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- Gilles Bernot, Jean-Paul Comet, Adrien Richard, Janine Guespin
- Journal of theoretical biology
- 2004

Based on the discrete definition of biological regulatory networks developed by René Thomas, we provide a computer science formal approach to treat temporal properties of biological regulatory networks, expressed in computational tree logic. It is then possible to build all the models satisfying a set of given temporal properties. Our approach is… (More)

- Adrien Richard
- ArXiv
- 2009

The biologist René Thomas conjectured, twenty years ago, that the presence of a negative feedback circuit in the interaction graph of a dynamical system is a necessary condition for this system to produce sustained oscillations. In this paper, we state and prove this conjecture for asynchronous automata networks, a class of discrete dynamical systems… (More)

- Adrien Richard
- Discrete Applied Mathematics
- 2011

Given a Boolean function F : {0, 1} n → {0, 1} n , and a point x in {0, 1} n , we represent the discrete Jacobian matrix of F at point x by a signed directed graph G F (x). We then focus on the following open problem: Is the absence of a negative circuit in G F (x) for every x in {0, 1} n a sufficient condition for F to have at least one fixed point? As… (More)

- Loïc Paulevé, Adrien Richard
- Electr. Notes Theor. Comput. Sci.
- 2012

Boolean networks are discrete dynamical systems extensively used to model biological regulatory networks. The dynamical analysis of these networks suffers from the combinatorial explosion of the state space, which grows exponentially with the number n of components. To face this problem, a classical approach consists in deducing from the interaction graph… (More)

- Adrien Richard, Jean-Paul Comet
- Discrete Applied Mathematics
- 2007

R. Thomas conjectured, twenty years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical… (More)

- Adrien Richard
- Discrete Applied Mathematics
- 2009

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components.… (More)

- Jamil Ahmad, Olivier F. Roux, Gilles Bernot, Jean-Paul Comet, Adrien Richard
- IJBRA
- 2008

In this paper, we propose a refinement of the modelling of biological regulatory networks based on the discrete approach of Rene Thomas. We refine and automatise the use of delays of activation/inhibition in order to specify which variable is more quickly affected by a change of its regulators. The formalism of linear hybrid automata is well suited to allow… (More)

- Jean-Paul Comet, Mathilde Noual, +7 authors Denis Thieffry
- Bulletin of mathematical biology
- 2013

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it "generates"… (More)

- Adrien Richard, Jean-Paul Comet, Gilles Bernot
- Fundam. Inform.
- 2005

In the field of biological regulation, models extracted from experimental works are usually complex networks comprising intertwined feedback circuits. The overall behavior is difficult to grasp and the development of formal methods is needed in order to model and simulate biological regulatory networks. To model the behavior of such systems, R. Thomas and… (More)

- Maximilien Gadouleau, Adrien Richard, Søren Riis
- SIAM J. Discrete Math.
- 2015

In this paper, we are interested in the number of fixed points of functions f : A n → A n over a finite alphabet A defined on a given signed digraph D. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on D. We then discover relationships between the number of fixed points of f and… (More)