# Links between topology of the transition graph and limit cycles in a two-dimensional piecewise affine biological model

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## Abstract

A class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103–129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles.

## Keywords

Biological regulatory networks Piecewise affine differential models Limit cycles Structural principles Transition graph## Mathematics Subject Classification (2000)

34K13 92B05## Notes

### Acknowledgments

We would like to thank Denis Thieffry for critical reading of the manuscript, and also on of the reviewers for his careful reading and many useful suggestions. W. Abou-Jaoudé was supported in part by the LabEx MemoLife (http://www.memolife.biologie.ens.fr). M. Chaves and J. L. Gouzé were supported in part by the projects GeMCo (ANR 2010 BLANC020101), ColAge (Inria-Inserm Large Scale Initiative Action), RESET (Investissements dAvenir, Bioinformatique), and also by the LABEX SIGNALIFE (ANR-11-LABX-0028-01).

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