General Iteration graphs and Boolean automata circuits

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

Negative circuits and sustained oscillations in asynchronous automata networks

The biologist Ren\'e 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 extensively used to model the behaviors of gene networks. As a corollary, we obtain the following fixed point theorem: given a product $X$ of $n$ finite intervals of integers, and a map $F$ from $X$ to itself, if the interaction graph associated with $F$ has no negative circuit, then $F$ has at least one fixed point

Block-sequential update schedules and Boolean automata circuits

International audienceOur work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period $p$, $\forall p\in \mathbb{N}$ and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule