Geometric properties of a class of piecewise affine biological network models
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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.
Key words or phrasesGene and neural networks Piecewise-affine dynamical systems Symbolic dynamics
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- 7.de Jong, H., Gouzé, J.-L., Hernandez, C., Page, M., Sari, T., Geiselmann, J.: Hybrid modeling and simulation of genetic regulatory networks: A qualitative approach. HSCC'2003, A. Pnueli, O. Maler (eds.), LNCS 2623, Springer-Verlag, Berlin, pp. 267–282, (2003)Google Scholar
- 8.Demongeot, J., Aracena, J., Thuderoz, F., Baum, T.-P., Cohen, O.: Genetic regulation networks: circuits, regulons and attractors. C. R. Biologies 326, (2003)Google Scholar
- 14.Farcot, E.: Transitions d'états dans un réseau génétique affine par morceaux. technical report available at http://www-lmc.imag.fr/lmc-cf/Etienne.Farcot/Text.html (in French), 2003
- 19.Glass, L.: Classification of biological networks by their qualitative dynamics. J. Theor. Biol. 54, 85–107 (1975)Google Scholar
- 24.Henk, M., Richter-Gebert, J., Ziegler, G.M.: Basic properties of convex polytopes. In: CRC Hanbook of discrete and computational geometry, J.E. Goodman, J.O'Rourke, (eds.), Boca Raton, New York, CRC Press, 1997Google Scholar
- 28.Kauffman, S.A.: The origins of order. Oxford University Press, 1993Google Scholar
- 30.Lewis, J.E., Glass, L.: Nonlinear and symbolic dynamics of neural networks. Neural Computation 4, 621–642 (1992)Google Scholar
- 31.Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, 1995Google Scholar
- 38.Snoussi, E.H.: Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn. Stab. Syst. 4 (3–4), 189–207 (1989)Google Scholar
- 39.Thom, R.: Modèles mathématiques de la morphogenèse. Bourgois, 10-18, 1974, réed., 1980Google Scholar
- 40.Thomas, R., D'Ari, R.: Biological Feedback. CRC-Press, Boca Raton, Florida, 1990Google Scholar
- 41.Ziegler, G.M.: Lectures on polytopes. Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995Google Scholar