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

  • Md. Ariful IslamEmail author
  • Greg Byrne
  • Soonho Kong
  • Edmund M. Clarke
  • Rance Cleaveland
  • Flavio H. Fenton
  • Radu Grosu
  • Paul L. Jones
  • Scott A. Smolka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Notes

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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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Md. Ariful Islam
    • 1
    Email author
  • Greg Byrne
    • 2
  • Soonho Kong
    • 1
  • Edmund M. Clarke
    • 1
  • Rance Cleaveland
    • 3
  • Flavio H. Fenton
    • 4
  • Radu Grosu
    • 5
    • 6
  • Paul L. Jones
    • 2
  • Scott A. Smolka
    • 5
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.US Food and Drug AdministrationSilver SpringUSA
  3. 3.University of MarylandCollege ParkUSA
  4. 4.Georgia Institute of TechnologyAtlantaUSA
  5. 5.Vienna University of TechnologyViennaAustria
  6. 6.Stony Brook UniversityStony BrookUSA

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