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

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Book cover Computational Methods in Systems Biology (CMSB 2016)

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

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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.

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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.

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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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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, USA

    Md. Ariful Islam, Soonho Kong & Edmund M. Clarke

  2. US Food and Drug Administration, Silver Spring, MD, USA

    Greg Byrne & Paul L. Jones

  3. University of Maryland, College Park, MD, USA

    Rance Cleaveland

  4. Georgia Institute of Technology, Atlanta, GA, USA

    Flavio H. Fenton

  5. Vienna University of Technology, Vienna, Austria

    Radu Grosu & Scott A. Smolka

  6. Stony Brook University, Stony Brook, NY, USA

    Radu Grosu

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  1. Md. Ariful Islam
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Correspondence to Md. Ariful Islam .

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Editors and Affiliations

  1. TU Wien, Vienna, Austria

    Ezio Bartocci

  2. Computer Laboratory, University of Cambridge, Cambridge, United Kingdom

    Pietro Lio

  3. Department of Computer Science, University of Oxford, Oxford, United Kingdom

    Nicola Paoletti

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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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