Probabilistic approach for predicting periodic orbits in piecewise affine differential models

Madalena Chaves 1 Etienne Farcot 2, 3 Jean-Luc Gouzé 1
1 BIOCORE - Biological control of artificial ecosystems
INRA - Institut National de la Recherche Agronomique, CRISAM - Inria Sophia Antipolis - Méditerranée , LOV - Laboratoire d'océanographie de Villefranche
2 VIRTUAL PLANTS - Modeling plant morphogenesis at different scales, from genes to phenotype
UMR AGAP - Amélioration génétique et adaptation des plantes méditerranéennes et tropicales, INRA - Institut National de la Recherche Agronomique, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles, which can be represented as nodes in a directed graph, so that the system's trajectories follow a path in a transition graph.This paper proposes and compares two definitions of probability of transition between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.
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Madalena Chaves, Etienne Farcot, Jean-Luc Gouzé. Probabilistic approach for predicting periodic orbits in piecewise affine differential models. Bulletin of Mathematical Biology, Springer Verlag, 2013, 75 (6), pp.967-987. ⟨10.1007/s11538-012-9773-6⟩. ⟨hal-00828847⟩

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