76 research outputs found
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
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
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
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