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Bifurcation Analysis of Cardiac Alternans Using \(\delta \)-Decidability

Part of the Lecture Notes in Computer Science book series (LNBI,volume 9859)

Abstract

We present a bifurcation analysis of electrical alternans in the two-current Mitchell-Schaeffer (MS) cardiac-cell model using the theory of \(\delta \)-decidability over the reals. Electrical alternans is a phenomenon characterized by a variation in the successive Action Potential Durations (APDs) generated by a single cardiac cell or tissue. Alternans are known to initiate re-entrant waves and are an important physiological indicator of an impending life-threatening arrhythmia such as ventricular fibrillation. The bifurcation analysis we perform determines, for each control parameter \(\tau \) of the MS model, the bifurcation point in the range of \(\tau \) such that a small perturbation to this value results in a transition from alternans to non-alternans behavior. To the best of our knowledge, our analysis represents the first formal verification of non-trivial dynamics in a numerical cardiac-cell model.

Our approach to this problem rests on encoding alternans-like behavior in the MS model as a 11-mode, multinomial hybrid automaton (HA). For each model parameter, we then apply a sophisticated, guided-search-based reachability analysis to this HA to estimate parameter ranges for both alternans and non-alternans behavior. The bifurcation point separates these two ranges, but with an uncertainty region due to the underlying \(\delta \)-decision procedure. This uncertainty region, however, can be reduced by decreasing \(\delta \) at the expense of increasing the model exploration time. Experimental results are provided that highlight the effectiveness of this method.

Keywords

  • Bifurcation Point
  • Bifurcation Analysis
  • Hybrid Automaton
  • Reachability Analysis
  • Reachability Property

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    A third current \(I_{s}\), which is not intrinsic to the MS model, is used to stimulate the cell to produce an action potential.

References

  1. Shrier, A., Dubarsky, H., Rosengarten, M., Guevara, M.R., Nattel, S., Glass, L.: Prediction of complex atrioventricular conduction rhythms in humans with use of the atrioventricular nodal recovery curve. Circulation 76(6), 1196–1205 (1987)

    CrossRef  Google Scholar 

  2. Alpern, B., Schneider, F.B.: Recognizing safety and liveness. Distrib. Comput. 2(3), 117–126 (1987)

    CrossRef  MATH  Google Scholar 

  3. Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Hybrid Systems, pp. 209–229 (1992)

    Google Scholar 

  4. Alur, R., Henzinger, T.A., Ho, P.-H.: Automatic symbolic verification of embedded systems. IEEE Trans. Softw. Eng. 22(3), 181–201 (1996)

    CrossRef  Google Scholar 

  5. Bae, K., Ölveczky, P.C., Kong, S., Gao, S., Clarke, E.M.: SMT-based analysis of virtually synchronous distributed hybrid systems. In: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control, pp. 145–154. ACM (2016)

    Google Scholar 

  6. Batt, G., De Jong, H., Page, M., Geiselmann, J.: Symbolic reachability analysis of genetic regulatory networks using discrete abstractions. Automatica 44(4), 982–989 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Brim, L., Demko, M., Pastva, S., Šafránek, D.: High-performance discrete bifurcation analysis for piecewise-affine dynamical systems. In: Abate, A., et al. (eds.) HSB 2015. LNCS, vol. 9271, pp. 58–74. Springer, Heidelberg (2015). doi:10.1007/978-3-319-26916-0_4

    CrossRef  Google Scholar 

  8. Bryce, D., Gao, S., Musliner, D.J., Goldman, R.P.: SMT-based nonlinear PDDL+ planning. In: 29th AAAI Conference on Artificial Intelligence, p. 3247

    Google Scholar 

  9. Fenton, F.H., Cherry, E.M., Hastings, H.M., Harold, M., Evans, S.J.: Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. chaos: an interdisciplinary. J. Nonlinear Sci. 12(3), 852 (2002)

    Google Scholar 

  10. Fenton, F.H., Cherry, E.M.: Models of cardiac cell. Scholarpedia 3(8), 1868 (2008)

    CrossRef  Google Scholar 

  11. Feret, J.: Reachability analysis of biological signalling pathways by abstract interpretation. In: Computation in Modern Science and Engineering, vol. 2, Part A (AIP Conference Proceedings vol. 963), pp. 619–622 (2007)

    Google Scholar 

  12. Gao, S., Avigad, J., Clarke, E.M.: \(\delta \)-complete decision procedures for satisfiability over the reals. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 286–300. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  13. Gao, S., Kong, S., Chen, W., Clarke, E.M.: Delta-complete analysis for bounded reachability of hybrid systems. CoRR, abs/1404.7171 (2014)

    Google Scholar 

  14. Gao, S., Kong, S., Clarke, E.M.: dReal: an smt solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 208–214. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  15. Gizzi, A., Cherry, E.M., Gilmour, R.F., Luther, S., Filippi, S., Fenton, F.H.: Effects of pacing site and stimulation history on alternans dynamics and the development of complex spatiotemporal patterns in cardiac tissue. Front. Physiol. 4, 71 (2013)

    CrossRef  Google Scholar 

  16. Huang, Z., Fan, C., Mereacre, A., Mitra, S., Kwiatkowska, M.: Invariant verification of nonlinear hybrid automata networks of cardiac cells. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 373–390. Springer, Heidelberg (2014)

    Google Scholar 

  17. Islam, M.A., De Francisco, R., Fan, C., Grosu, R., Mitra, S., Smolka, S.A.: Model checking tap withdrawal in C. Elegans. In: Hybrid Systems Biology, p. 195

    Google Scholar 

  18. Islam, M.A., Murthy, A., Girard, A., Smolka, S.A., Grosu, R.: Compositionality results for cardiac cell dynamics. In: Proceedings of the 17th international conference on Hybrid systems: computation and control, pp. 243–252. ACM (2014)

    Google Scholar 

  19. Weiss, J.N., Alain, S., Shiferaw, Y., Chen, P., Garfinkel, A., Qu, Z.: From pulsus to pulseless the saga of cardiac alternans. Circ. Res. 98(10), 1244–1253 (2006). WOS: 000237812200006

    CrossRef  Google Scholar 

  20. Kapinski, J., Deshmukh, J.V., Sankaranarayanan, S., Arechiga, N.: Simulation-guided Lyapunov analysis for hybrid dynamical systems. In: Hybrid Systems: Computation and Control (HSCC), pp. 133–142. ACM Press (2014)

    Google Scholar 

  21. Kong, S., Gao, S., Chen, W., Clarke, E.M.: dReach: \(\delta \)-reachability analysis for hybrid systems. In: Proceedings of Tools and Algorithms for the Construction and Analysis of Systems - 21st International Conference, TACAS, London, UK, April 11–18, 2015, pp. 200–205 (2015)

    Google Scholar 

  22. Liu, B., Kong, S., Gao, S., Zuliani, P., Clarke, E.M.: Towards personalized prostate cancer therapy using delta-reachability analysis. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 227–232. ACM (2015)

    Google Scholar 

  23. Guevara, M., Glass, L., Shrier, A.: Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science 214(4527), 1350–1353 (1981)

    CrossRef  Google Scholar 

  24. Mitchell, C.C., Schaeffer, D.G.: A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol. 65(5), 767–793 (2003)

    CrossRef  MATH  Google Scholar 

  25. Murthy, A., Islam, M.A., Smolka, S.A., Grosu, R.: Computing bisimulation functions using SOS optimization and \(\delta \)-decidability over the reals. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 78–87. ACM (2015)

    Google Scholar 

  26. Murthy, A., Islam, M.A., Smolka, S.A., Grosu, R.: Computing compositional proofs of input-to-output stability using SOS optimization and \(\delta \)-decidability. Hybrid Systems, Nonlinear Analysis (2016)

    Google Scholar 

  27. Gilmour, R.F., Chialvo, D.R.: Electrical restitution, critical mass, and the riddle of fibrillation. J. Cardiovas. Electrophysiol. 10(8), 1087–1089 (1999)

    CrossRef  Google Scholar 

  28. Shiferaw, Y., Sato, D., Karma, A.: Coupled dynamics of voltage and calcium in paced cardiac cells. Phys. Rev. E 71(2), 021903 (2005)

    CrossRef  Google Scholar 

  29. Quail, T., Shrier, A., Glass, L.: Predicting the onset of period-doubling bifurcations in noisy cardiac systems. Proc. Nat. Acad. Sci. 112(30), 9358–9363 (2015)

    CrossRef  Google Scholar 

  30. Watanabe, M.A., Fenton, F.H., Evans, S.J., Hastings, H.M., Karma, A.: Mechanisms for discordant alternans. J. Cardiovasc. Electrophysiol. 12(2), 196–206 (2001)

    CrossRef  Google Scholar 

  31. Yang, Y., Lin, H.: Reachability analysis based model validation in systems biology. In: 2010 IEEE Conference on Cybernetics and Intelligent Systems (CIS), pp. 14–19. IEEE (2010)

    Google Scholar 

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Acknowledgments

We would like to thank the anonymous reviewers for their helpful comments. Research supported in part by the following grants: NSF IIS-1447549, NSF CPS-1446832, NSF CPS-1446725, NSF CNS-1446665, NSF CPS 1446365, NSF CAR 1054247, AFOSR FA9550-14-1-0261, AFOSR YIP FA9550-12-1-0336, CCF-0926190, ONR N00014-13-1-0090, and NASA NNX12AN15H.

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Correspondence to Md. Ariful Islam .

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Islam, M.A. et al. (2016). Bifurcation Analysis of Cardiac Alternans Using \(\delta \)-Decidability. In: Bartocci, E., Lio, P., Paoletti, N. (eds) Computational Methods in Systems Biology. CMSB 2016. Lecture Notes in Computer Science(), vol 9859. Springer, Cham. https://doi.org/10.1007/978-3-319-45177-0_9

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  • DOI: https://doi.org/10.1007/978-3-319-45177-0_9

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