Advertisement

Invariant Verification of Nonlinear Hybrid Automata Networks of Cardiac Cells

  • Zhenqi Huang
  • Chuchu Fan
  • Alexandru Mereacre
  • Sayan Mitra
  • Marta Kwiatkowska
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8559)

Abstract

Verification algorithms for networks of nonlinear hybrid automata (HA) can aid us understand and control biological processes such as cardiac arrhythmia, formation of memory, and genetic regulation. We present an algorithm for over-approximating reach sets of networks of nonlinear HA which can be used for sound and relatively complete invariant checking. First, it uses automatically computed input-to-state discrepancy functions for the individual automata modules in the network \(\mathcal{A}\) for constructing a low-dimensional model \(\mathcal{M}\). Simulations of both \(\mathcal{A}\) and \(\mathcal{M}\) are then used to compute the reach tubes for \(\mathcal{A}\). These techniques enable us to handle a challenging verification problem involving a network of cardiac cells, where each cell has four continuous variables and 29 locations. Our prototype tool can check bounded-time invariants for networks with 5 cells (20 continuous variables, 295 locations) typically in less than 15 minutes for up to reasonable time horizons. From the computed reach tubes we can infer biologically relevant properties of the network from a set of initial states.

Keywords

Biological networks hybrid systems invariants verification 
Download to read the full conference paper text

References

  1. 1.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theoretical Computer Science 138(1), 3–34 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Angeli, D.: Further results on incremental input-to-state stability. IEEE Transactions on Automatic Control 54(6), 1386–1391 (2009)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Angeli, D.: A lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control 47(3), 410–421 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Angeli, D., Sontag, E.D., Wang, Y.: A characterization of integral input-to-state stability. IEEE Transactions on Automatic Control 45(6), 1082–1097 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Annpureddy, Y., Liu, C., Fainekos, G., Sankaranarayanan, S.: S-taLiRo: A tool for temporal logic falsification for hybrid systems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 254–257. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Axmacher, N., Mormann, F., Fernández, G., Elger, C.E., Fell, J.: Memory formation by neuronal synchronization. Brain Research Reviews 52(1), 170–182 (2006)CrossRefGoogle Scholar
  7. 7.
    Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical processes on complex networks, vol. 1. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bartocci, E., Corradini, F., Berardini, M.R.D., Entcheva, E., Smolka, S.A., Grosu, R.: Modeling and simulation of cardiac tissue using hybrid I/O automata. Theor. Comput. Sci. 410(33-34), 3149–3165 (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: Structure and dynamics. Physics Reports 424(4), 175–308 (2006)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bouissou, O., Martel, M.: Grklib: A guaranteed runge kutta library. In: 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN 2006, p. 8. IEEE (2006)Google Scholar
  11. 11.
    Bueno-Orovio, A., Cherry, E.M., Fenton, F.H.: Minimal model for human ventricular action potentials in tissue. Journal of Theoretical Biology 253(3), 544–560 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    CAPD. Computer assisted proofs in dynamics (2002)Google Scholar
  13. 13.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: An analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 167–170. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Duggirala, P.S., Mitra, S., Viswanathan, M.: Verification of annotated models from executions. In: EMSOFT (2013)Google Scholar
  16. 16.
    Fenton, F., Karma, A.: Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos: An Interdisciplinary Journal of Nonlinear Science 8(1), 20–47 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Garzón, A., Grigoriev, R.O., Fenton, F.H.: Model-based control of cardiac alternans on a ring. Physical Review E 80 (2009)Google Scholar
  18. 18.
    Greenhut, S., Jenkins, J., MacDonald, R.: A stochastic network model of the interaction between cardiac rhythm and artificial pacemaker. IEEE Transactions on Biomedical Engineering 40(9), 845–858 (1993)CrossRefGoogle Scholar
  19. 19.
    Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From cardiac cells to genetic regulatory networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From cardiac cells to genetic regulatory networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Grosu, R., Smolka, S.A., Corradini, F., Wasilewska, A., Entcheva, E., Bartocci, E.: Learning and detecting emergent behavior in networks of cardiac myocytes. Commun. ACM 52(3), 97–105 (2009)CrossRefGoogle Scholar
  22. 22.
    Guevara, M.R., Ward, G., Shrier, A., Glass, L.: Electrical alternans and period-doubling bifurcations. In: Computers in Cardiology, pp. 167–170 (1984)Google Scholar
  23. 23.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In. In: ACM Symposium on Theory of Computing, pp. 373–382 (1995)Google Scholar
  24. 24.
    Hespanha, J.P., Morse, A.: Stability of switched systems with average dwell-time. In: Proceedings of 38th IEEE Conference on Decision and Control, pp. 2655–2660 (1999)Google Scholar
  25. 25.
    Huang, Z., Fan, C., Mitra, S., Mereacre, A., Kwiatkowska, M.: Invariant verification of nonlinear hybrid automata networks of cardiac cells, Online supporting material: Models, code, and data (2014), https://wiki.cites.illinois.edu/wiki/display/MitraResearch/Complex+Networks+of+Nonlinear+Modules
  26. 26.
    Huang, Z., Mitra, S.: Proofs from simulations and modular annotations. In: In 17th International Conference on Hybrid Systems: Computation and Control. ACM Press, BerlinGoogle Scholar
  27. 27.
    Ideker, R.E., Rogers, J.M.: Human ventricular fibrillation: Wandering wavelets, mother rotors, or both? Circulation 114(6), 530–532 (2006)CrossRefGoogle Scholar
  28. 28.
    Jee, E., Wang, S., Kim, J.-K., Lee, J., Sokolsky, O., Lee, I.: A safety-assured development approach for real-time software. In: RTCSA, pp. 133–142 (2010)Google Scholar
  29. 29.
    Jiang, Z., Pajic, M., Connolly, A., Dixit, S., Mangharam, R.: Real-Time Heart model for implantable cardiac device validation and verification. In: ECRTS, pp. 239–248. IEEE Computer Society (2010)Google Scholar
  30. 30.
    Jiang, Z., Pajic, M., Moarref, S., Alur, R., Mangharam, R.: Modeling and verification of a dual chamber implantable pacemaker. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 188–203. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  31. 31.
    Kaynar, D.K., Lynch, N., Segala, R., Vaandrager, F.: The Theory of Timed I/O Automata. Synthesis Lectures on Computer Science. Morgan Claypool, Also available as Technical Report MIT-LCS-TR-917 (November 2005)Google Scholar
  32. 32.
    Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 137–151. Springer, Heidelberg (1999)Google Scholar
  33. 33.
    Lian, J., Krätschmer, H., Müssig, D.: Open source modeling of heart rhythm and cardiac pacing. The Open Pacing, Electrophysiology & Therapy Journal, 28–44 (2010)Google Scholar
  34. 34.
    Lynch, N., Segala, R., Vaandrager, F.: Hybrid I/O automata. Information and Computation 185(1), 105–157 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Mitra, S.: A Verification Framework for Hybrid Systems. PhD thesis. Massachusetts Institute of Technology, Cambridge, MA 02139 (September 2007)Google Scholar
  36. 36.
    Mitra, S., Liberzon, D., Lynch, N.: Verifying average dwell time of hybrid systems. ACM Trans. Embed. Comput. Syst. 8(1), 1–37 (2008)CrossRefGoogle Scholar
  37. 37.
    Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation 105(1), 21–68 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  39. 39.
    Pajic, M., Jiang, Z., Lee, I., Sokolsky, O., Mangharam, R.: From verification to implementation: A model translation tool and a pacemaker case study. In: IEEE Real-Time and Embedded Technology and Applications Symposium, pp. 173–184 (2012)Google Scholar
  40. 40.
    Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic dynamics and endemic states in complex networks. Physical Review E 63(6), 066117 (2001)Google Scholar
  41. 41.
    Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Introducing sostools: A general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, vol. 1, pp. 741–746. IEEE (2002)Google Scholar
  42. 42.
    Sontag, E.D.: Comments on integral variants of iss. Systems & Control Letters 34(1-2), 93–100 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268–276 (2001)CrossRefGoogle Scholar
  44. 44.
    Vladimerou, V., Prabhakar, P., Viswanathan, M., Dullerud, G.: Stormed hybrid systems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 136–147. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  45. 45.
    Ye, P., Entcheva, E., Grosu, R., Smolka, S.A.: Efficient Modeling of Excitable Cells Using Hybrid Automata. In: CMSB, pp. 216–227 (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Zhenqi Huang
    • 1
  • Chuchu Fan
    • 1
  • Alexandru Mereacre
    • 2
  • Sayan Mitra
    • 1
  • Marta Kwiatkowska
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignUSA
  2. 2.Department of Computer ScienceUniversity of OxfordUnited Kingdom

Personalised recommendations

Cite paper