2,692 research outputs found
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
Discrete time piecewise affine models of genetic regulatory networks
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 represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.Comment: 34 page
Boolean networks synchronism sensitivity and XOR circulant networks convergence time
In this paper are presented first results of a theoretical study on the role
of non-monotone interactions in Boolean automata networks. We propose to
analyse the contribution of non-monotony to the diversity and complexity in
their dynamical behaviours according to two axes. The first one consists in
supporting the idea that non-monotony has a peculiar influence on the
sensitivity to synchronism of such networks. It leads us to the second axis
that presents preliminary results and builds an understanding of the dynamical
behaviours, in particular concerning convergence times, of specific
non-monotone Boolean automata networks called XOR circulant networks.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Positive circuits and maximal number of fixed points in discrete dynamical systems
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 circuits of the interaction graph associated with F. The proof uses
and strongly generalizes a theorem of Richard and Comet which corresponds to a
discrete version of the Thomas' conjecture: if the interaction graph associated
with F has no positive circuit, then F has at most one fixed point. The
obtained upper bound on the number of fixed points also strongly generalizes
the one established by Aracena et al for a particular class of Boolean
networks.Comment: 13 page
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