0
Research Papers

Cardiac Alternans Arising From an Unfolded Border-Collision Bifurcation

[+] Author and Article Information
Xiaopeng Zhao1

Department of Biomedical Engineering, Duke University, Durham, NC 27708; Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708xzhao9@utk.edu

David G. Schaeffer

Department of Mathematics, Duke University, Durham, NC 27708; Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708

Carolyn M. Berger

Department of Physics, Duke University, Durham, NC 27708; Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708

Wanda Krassowska

Department of Biomedical Engineering, Duke University, Durham, NC 27708; Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708

Daniel J. Gauthier

Department of Biomedical Engineering, and Department of Physics, Duke University, Durham, NC 27708; Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708

1

Corresponding author. Present address: Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee, Knoxville TN 37996.

J. Comput. Nonlinear Dynam 3(4), 041004 (Aug 19, 2008) (7 pages) doi:10.1115/1.2960467 History: Received April 16, 2007; Revised January 10, 2008; Published August 19, 2008

Following an electrical stimulus, the transmembrane voltage of cardiac tissue rises rapidly and remains at a constant value before returning to the resting value, a phenomenon known as an action potential. When the pacing rate of a periodic train of stimuli is increased above a critical value, the action potential undergoes a period-doubling bifurcation, where the resulting alternation of the action potential duration is known as alternans in medical literature. Existing cardiac models treat alternans either as a smooth or as a border-collision bifurcation. However, recent experiments in paced cardiac tissue reveal that the bifurcation to alternans exhibits hybrid smooth∕nonsmooth behaviors, which can be qualitatively described by a model of so-called unfolded border-collision bifurcation. In this paper, we obtain analytical solutions of the unfolded border-collision model and use it to explore the crossover between smooth and nonsmooth behaviors. Our analysis shows that the hybrid smooth∕nonsmooth behavior is due to large variations in the system’s properties over a small interval of the bifurcation parameter, providing guidance for the development of future models.

FIGURES IN THIS ARTICLE
<>
Copyright © 2008 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Schematic action potential showing the response of the transmembrane voltage to periodic electrical stimuli

Grahic Jump Location
Figure 2

Schematic bifurcation diagrams of period-doubling bifurcation: (a) a smooth type and (b) a border-collision type. Here, B represents a bifurcation parameter and A represents fixed-point solutions.

Grahic Jump Location
Figure 3

Prebifurcation gain Γ of a classical period-doubling bifurcation ((a) and (b)) and of a border-collision period-doubling bifurcation ((c) and (d)). In Panels (a) and (c), δ stays constant; in Panels (b) and (d), Bbif<B stays constant. Comparison between Panels (b) and (d) provides the most revealing difference between the two bifurcations.

Grahic Jump Location
Figure 4

Schematic showing a border-collision bifurcation (solid) and the unfolded bifurcation (dashed)

Grahic Jump Location
Figure 5

Comparison between the model of Guevara (solid) and the unfolded border-collision model (dashed) in fitting the experimental data in Ref. 31 (points). Although both models fit the dynamic restitution curve well (top panel), the unfolded border-collision model fits alternans data much better (bottom panel).

Grahic Jump Location
Figure 6

Prebifurcation amplification predicted by the unfolded border-collision model (12): δ=5ms (solid), δ=10ms (dashed), and δ=15ms (dotted)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In