Journal of Mathematical Biology

, Volume 52, Issue 1, pp 27–56 | Cite as

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

  • Richard Casey
  • Hidde de Jong
  • Jean-Luc Gouzé


A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.

Key words or phrases

Piecewise-linear systems Filippov solutions stability genetic regulatory networks 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.COMORE INRIAUnité de recherche Sophia AntipolisSophia AntipolisFrance
  2. 2.HELIX INRIAUnité de recherche Rhône-AlpesSaint Ismier CedexFrance

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