Discretization of Continuous Dynamical Systems Using UPPAAL

  • Stefano Schivo
  • Rom Langerak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10500)


We want to enable the analysis of continuous dynamical systems (where the evolution of a vector of continuous state variables is described by differential equations) by model checking. We do this by showing how such a dynamical system can be translated into a discrete model of communicating timed automata that can be analyzed by the UPPAAL tool. The basis of the translation is the well-known Euler approach for solving differential equations where we use fixed discrete value steps instead of fixed time steps. Each state variable is represented by a timed automaton in which the delay for taking the next value is calculated on the fly using the differential equations. The state variable automata proceed independently but may notify each other when a value step has been completed; this leads to a recalculation of delays. The approach has been implemented in the tool ANIMO for analyzing biological kinase networks in cells. This tool has been used in actual biological research on osteoarthritis dealing with systems where the dimension of the state vector (the number of nodes in the network) is in the order of one hundred.


Discretization Euler method Model checking Timed automata Systems biology 



We thank Arend Rensink for an important comment on an earlier version of this work. We thank Wim Bos, Liesbeth Geris, Marcel Karperien, Johan Kerkhofs, Jaco van de Pol, Janine Post, Jetse Scholma, Ricardo Urquidi Camacho, Paul van der Vet, and Brend Wanders for the fruitful and pleasant collaboration leading to ANIMO.


  1. 1.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126, 183–235 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. In: Proceedings of the IEEE, pp. 971–984 (2000)Google Scholar
  3. 3.
    Bos, W.: Interactive signaling network analysis tool. Master’s thesis, University of Twente (2009)Google Scholar
  4. 4.
    Bouyer, P., Markey, N., Perrin, N., Schlehuber-Caissier, P.: Timed-automata abstraction of switched dynamical systems using control funnels. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 60–75. Springer, Cham (2015). doi: 10.1007/978-3-319-22975-1_5 CrossRefGoogle Scholar
  5. 5.
    Brim, L., Češka, M., Šafránek, D.: Model checking of biological systems. In: Bernardo, M., de Vink, E., Di Pierro, A., Wiklicky, H. (eds.) SFM 2013. LNCS, vol. 7938, pp. 63–112. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38874-3_3 CrossRefGoogle Scholar
  6. 6.
    Brim, L., Fabriková, J., Drazan, S., Safránek, D.: Reachability in biochemical dynamical systems by quantitative discrete approximation. CoRR, abs/1107.5924 (2011)Google Scholar
  7. 7.
    Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Cardelli, L., Tribastone, M., Tschaikowski, M., Vandin, A.: ERODE: a tool for the evaluation and reduction of ordinary differential equations. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 310–328. Springer, Heidelberg (2017). doi: 10.1007/978-3-662-54580-5_19 CrossRefGoogle Scholar
  9. 9.
    Carter, R., Navarro-López, E.M.: Dynamically-driven timed automaton abstractions for proving liveness of continuous systems. In: Jurdziński, M., Ničković, D. (eds.) FORMATS 2012. LNCS, vol. 7595, pp. 59–74. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33365-1_6 CrossRefGoogle Scholar
  10. 10.
    Clarke, E.M.: Model checking. In: Ramesh, S., Sivakumar, G. (eds.) FSTTCS 1997. LNCS, vol. 1346, pp. 54–56. Springer, Heidelberg (1997). doi: 10.1007/BFb0058022 CrossRefGoogle Scholar
  11. 11.
    Cytoscape 3 ANIMO app.
  12. 12.
    David, A., Grunnet, J.D., Jessen, J.J., Larsen, K.G., Rasmussen, J.I.: Application of model-checking technology to controller synthesis. In: Aichernig, B.K., de Boer, F.S., Bonsangue, M.M. (eds.) FMCO 2010. LNCS, vol. 6957, pp. 336–351. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-25271-6_18 CrossRefGoogle Scholar
  13. 13.
    David, A., Larsen, K.G., Legay, A., Mikučionis, M., Poulsen, D.B., Sedwards, S.: Statistical model checking for biological systems. Int. J. Softw. Tools Technol. Transfer 17(3), 351–367 (2015)CrossRefGoogle Scholar
  14. 14.
    Donzé, A., Krogh, B., Rajhans, A.: Parameter synthesis for hybrid systems with an application to simulink models. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 165–179. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-00602-9_12 CrossRefGoogle Scholar
  15. 15.
    Emerson, E.A., Clarke, E.M.: Characterizing correctness properties of parallel programs using fixpoints. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 169–181. Springer, Heidelberg (1980). doi: 10.1007/3-540-10003-2_69 CrossRefGoogle Scholar
  16. 16.
    Fehnker, A.: Scheduling a steel plant with timed automata. In: Proceedings of the Sixth International Conference on Real-Time Computing Systems and Applications, RTCSA 1999, p. 280. IEEE Computer Society, Washington, DC (1999)Google Scholar
  17. 17.
    Goethem, S.V., Jacquet, J.-M., Brim, L., Šafránek, D.: Timed modelling of gene networks with arbitrarily precise expression discretization. Electron. Notes Theoret. Comput. Sci. 293, 67–81 (2013). Proceedings of the Third International Workshop on Interactions Between Computer Science and Biology (CS2Bio 2012)CrossRefGoogle Scholar
  18. 18.
    Habets, L.C.G.J.M., van Schuppen, J.H.: Control of piecewise-linear hybrid systems on simplices and rectangles. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 261–274. Springer, Heidelberg (2001). doi: 10.1007/3-540-45351-2_23 CrossRefGoogle Scholar
  19. 19.
    Habets, L.C.G.J.M., van Schuppen, J.H.: Control to facet problems for affine systems on simplices and polytopes - with applications to control of hybrid systems. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 4175–4180, December 2005Google Scholar
  20. 20.
    Jongerden, M., Haverkort, B., Bohnenkamp, H., Katoen, J.: Maximizing system lifetime by battery scheduling. In: 39th Annual IEEE/IFIP International Conference on Dependable Systems and Networks, DSN 2009, Los Alamitos. IEEE Computer Society Press, June 2009Google Scholar
  21. 21.
    Killcoyne, S., Carter, G.W., Smith, J., Boyle, J.: Cytoscape: a community-based framework for network modeling. Methods Mol. Biol. (Clifton, N.J.) 563, 219–239 (2009)CrossRefGoogle Scholar
  22. 22.
    Larsen, K.G., Pettersson, P., Yi, W.: UPPAAL in a nutshell. Int. J. Softw. Tools Technol. Transf. (STTT) 1, 134–152 (1997)CrossRefMATHGoogle Scholar
  23. 23.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000). doi: 10.1007/10722167_15 CrossRefGoogle Scholar
  24. 24.
    Maler, O., Batt, G.: Approximating continuous systems by timed automata. In: Fisher, J. (ed.) FMSB 2008. LNCS, vol. 5054, pp. 77–89. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-68413-8_6 CrossRefGoogle Scholar
  25. 25.
    Monteiro, P.T., Ropers, D., Mateescu, R., Freitas, A.T., de Jong, H.: Temporal logic patterns for querying dynamic models of cellular interaction networks. Bioinformatics 24(16), i227–i233 (2008)CrossRefGoogle Scholar
  26. 26.
  27. 27.
    Schivo, S., Scholma, J., Karperien, H.B.J., Post, J.N., van de Pol, J.C., Langerak, R.: Setting parameters for biological models with ANIMO. In: André, E., Frehse, G. (eds.) Proceedings 1st International Workshop on Synthesis of Continuous Parameters, Grenoble, France. Electronic Proceedings in Theoretical Computer Science, vol. 145, pp. 35–47. Open Publishing Association, April 2014Google Scholar
  28. 28.
    Schivo, S., Scholma, J., van der Vet, P.E., Karperien, M., Post, J.N., van de Pol, J., Langerak, R.: Modelling with ANIMO: between fuzzy logic and differential equations. BMC Syst. Biol. 10(1), 56 (2016)CrossRefGoogle Scholar
  29. 29.
    Schivo, S., Scholma, J., Wanders, B., Urquidi Camacho, R., van der Vet, P., Karperien, M., Langerak, R., van de Pol, J., Post, J.: Modelling biological pathway dynamics with Timed Automata. IEEE J. Biomed. Health Inform. 18(3), 832–839 (2013)CrossRefGoogle Scholar
  30. 30.
    Schivo, S., Scholma, J., Wanders, B., Urquidi Camacho, R.A., van der Vet, P.E., Karperien, H.B.J., Langerak, R., van de Pol, J.C., Post, J.N.: Modelling biological pathway dynamics with timed automata. In: 12th IC on Bioinformatics and Bioengineering (BIBE 2012), pp. 447–453. IEEE Computer Society (2012)Google Scholar
  31. 31.
    Scholma, J., Kerkhofs, J., Schivo, S., Langerak, R., van der Vet, P.E., Karperien, H.B.J., van de Pol, J.C., Geris, L., Post, J.N.: Mathematical modeling of signalingpathways in osteoarthritis. In: Lohmander, S. (ed.) 2013 Osteoarthritis Research Society International (OARSI) World Congress, Philadelphia, USA, vol. 21, Supplement, p. S123. Elsevier, Amsterdam, April 2013Google Scholar
  32. 32.
    Scholma, J., Schivo, S., Kerkhofs, J., Langerak, R., Karperien, H.B.J., van de Pol, J.C., Geris, L., Post, J.N.: ECHO: the executable chondrocyte. In: Tissue Engineering & Regenerative Medicine International Society, European Chapter Meeting, Genova, Italy, vol. 8, p. 54. Wiley, Malden, June 2014Google Scholar
  33. 33.
    Scholma, J., Schivo, S., Urquidi Camacho, R., van de Pol, J., Karperien, M., Post, J.: Biological networks 101: computational modeling for molecular biologists. Gene 533(1), 379–384 (2014)CrossRefGoogle Scholar
  34. 34.
    Siebert, H., Bockmayr, A.: Temporal constraints in the logical analysis of regulatory networks. Theoret. Comput. Sci. 391(3), 258–275 (2008). Converging Sciences: Informatics and BiologyMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sloth, C., Wisniewski, R.: Verification of continuous dynamical systems by timed automata. Formal Methods Syst. Des. 39(1), 47–82 (2011)CrossRefMATHGoogle Scholar
  36. 36.
    Sloth, C., Wisniewski, R.: Complete abstractions of dynamical systems by timed automata. Nonlinear Anal.: Hybrid Syst. 7(1), 80–100 (2013). (IFAC) World Congress 2011MathSciNetMATHGoogle Scholar
  37. 37.
    Stursberg, O., Kowalewski, S., Engell, S.: On the generation of timed discrete approximations for continuous systems. Math. Comput. Modell. Dyn. Syst. 6(1), 51–70 (2000)CrossRefMATHGoogle Scholar
  38. 38.
  39. 39.
    Urquidi Camacho, R.: Modeling osteoarthritic cartilage with ANIMO: an executable biology approach to osteoarthritic signaling and gene expression. Master’s thesis, University of Twente, The Netherlands, July 2013Google Scholar
  40. 40.
    Wisniewski, R., Sloth, C.: Abstraction of dynamical systems by timed automata. Model. Ident. Control 32(2), 79 (2011)CrossRefMATHGoogle Scholar
  41. 41.
    Xing, J., Theelen, B.D., Langerak, R., van de Pol, J., Tretmans, J., Voeten, J.P.M.: UPPAAL in practice: quantitative verification of a RapidIO network. In: Margaria, T., Steffen, B. (eds.) ISoLA 2010. LNCS, vol. 6416, pp. 160–174. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-16561-0_20 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Formal Methods and Tools Group, Faculty of EEMCSUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations