A necessary condition for boundary sensitivity of attractive non-linear stochastic cellular automata in ZxZ

International audienceThis paper tackles the question of the environmental robustness of a particular class of two-dimensional finite threshold Boolean cellular automata when they are subjected to distinct fixed boundary instances. More precisely, focusing on a non-linear stochastic version of the classical threshold function governing the evolution of formal neural networks, we show the existence of a necessary condition under which attractive cellular automata of this form become boundary sensitive, i.e., we highlight a condition without which a cellular automaton hits the same asymptotic dynamical behaviour whatever its boundary conditions are. To go further, we give an explicit formula for this necessary condition

Turning block-sequential automata networks into smaller parallel networks with isomorphic limit dynamics

We state an algorithm that, given an automata network and a block-sequential update schedule, produces an automata network of the same size or smaller with the same limit dynamics under the parallel update schedule. Then, we focus on the family of automata cycles which share a unique path of automata, called tangential cycles, and show that a restriction of our algorithm allows to reduce any instance of these networks under a block-sequential update schedule into a smaller parallel network of the family and to characterize the number of reductions operated while conserving their limit dynamics. We also show that any tangential cycles reduced by our main algorithm are transformed into a network whose size is that of the largest cycle of the initial network. We end by showing that the restricted algorithm allows the direct characterization of block-sequential double cycles as parallel ones.Comment: Accepted at CIE 202

Modular organisation of interaction networks based on asymptotic dynamics

This paper investigates questions related to the modularity in discrete models of biological interaction networks. We develop a theoretical framework based on the analysis of their asymptotic dynamics. More precisely, we exhibit formal conditions under which agents of interaction networks can be grouped into modules. As a main result, we show that the usual decomposition in strongly connected components fulfils the conditions of being a modular organisation. Furthermore, we point out that our framework enables a finer analysis providing a decomposition in elementary modules

On the number of attractors of Boolean automata circuits

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

About non-monotony in Boolean automata networks

International audienceThis paper aims at presenting motivations and rst results of a prospective theoretical study on the role of non-monotone interactions in the modelling process of biological regulation networks. Focusing on discrete models of these networks, namely, Boolean automata networks, we propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours. More precisely, in this paper, we start by detail- ing some motivations, both mathematical and biological, for our interest in non-monotony, and we discuss how it may account for phenomena that cannot be produced by monotony only. Then, to build some understanding in this direction, we show some preliminary results on the dynamical be- haviours of some speci c non-monotone Boolean automata networks called xor circulant networks

On countings and enumerations of block-parallel automata networks

When we focus on finite dynamical systems from both the computability/complexity and the modelling standpoints, automata networks seem to be a particularly appropriate mathematical model on which theory shall be developed. In this paper, automata networks are finite collections of entities (the automata), each automaton having its own set of possible states, which interact with each other over discrete time, interactions being defined as local functions allowing the automata to change their state according to the states of their neighbourhoods. The studies on this model of computation have underlined the very importance of the way (i.e. the schedule) according to which the automata update their states, namely the update modes which can be deterministic, periodic, fair, or not. Indeed, a given network may admit numerous underlying dynamics, these latter depending highly on the update modes under which we let the former evolve. In this paper, we pay attention to a new kind of deterministic, periodic and fair update mode family introduced recently in a modelling framework, called the block-parallel update modes by duality with the well-known and studied block-sequential update modes. More precisely, in the general context of automata networks, this work aims at presenting what distinguish block-parallel update modes from block-sequential ones, and at counting and enumerating them: in absolute terms, by keeping only representatives leading to distinct dynamics, and by keeping only representatives giving rise to distinct isomorphic limit dynamics. Put together, this paper constitutes a first theoretical analysis of these update modes and their impact on automata networks dynamics