Geometric properties of a class of piecewise affine biological network models |
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Authors: | Etienne Farcot |
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Institution: | (1) COMORE INRIA, U.R. Sophia Antipolis, 2004 route des Lucioles, BP93, 06902 Sophia Antipolis, France |
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Abstract: | The purpose of this report is to investigate some dynamical properties common to several biological systems. A model is chosen,
which consists of a system of piecewise affine differential equations. Such a model has been previously studied in the context
of gene regulation and neural networks, as well as biochemical kinetics. Unlike most of these studies, nonuniform decay rates
and several thresholds per variable are assumed, thus considering a more realistic model. This model is investigated with
the aid of a geometric formalism. We first provide an analysis of a continuous-space, discrete-time dynamical system equivalent
to the initial one, by the way of a transition map. This is similar to former studies. Especially, the analysis of periodic
trajectories is carried out in the case of multiple thresholds, thus extending previous results, which all concerned the restricted
case of binary systems.
The piecewise affine structure of such models is then used to provide a partition of the phase space, in terms of explicit
cells. Allowed transitions between these cells define a language on a finite alphabet. Some words are proved to be forbidden
in this language, thus improving the knowledge on such systems in terms of symbolic dynamics. More precisely, we show that
taking these forbidden words into account leads to a dynamical system with strictly lower topological entropy. This holds
for a class of systems, characterized by the presence of a splitting box, with additional conditions. We conclude after an
illustrative three-dimensional example. |
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Keywords: | Gene and neural networks Piecewise-affine dynamical systems Symbolic dynamics |
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