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### Electrical waves in a one-dimensional model of cardiac tissue

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Beck, Margaret, Jones, C. K. R. T. (Christopher K. R. T.), Schaeffer, David G. and Wechselberger, Martin.
(2008)
*Electrical waves in a one-dimensional model of cardiac tissue.*
SIAM Journal on Applied Dynamical Systems, Vol.7
(No.4).
pp. 1558-1581.
ISSN 1536-0040

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1137/070709980

## Abstract

The electrical dynamics in the heart is modeled by a two-component PDE. Using geometric singular perturbation theory, it is shown that a traveling pulse solution, which corresponds to a single heartbeat, exists. One key aspect of the proof involves tracking the solution near a point on the slow manifold that is not normally hyperbolic. This is achieved by desingularizing the vector field using a blow-up technique. This feature is relevant because it distinguishes cardiac impulses from, for example, nerve impulses. Stability of the pulse is also shown, by computing the zeros of the Evans function. Although the spectrum of one of the fast components is only marginally stable, due to essential spectrum that accumulates at the origin, it is shown that the spectrum of the full pulse consists of an isolated eigenvalue at zero and essential spectrum that is bounded away from the imaginary axis. Thus, this model provides an example in a biological application reminiscent of a previously observed mathematical phenomenon: that connecting an unstable-in this case marginally stable-front and back can produce a stable pulse. Finally, remarks are made regarding the existence and stability of spatially periodic pulses, corresponding to successive heartbeats, and their relationship with alternans, irregular action potentials that have been linked with arrhythmia.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QP Physiology |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Heart -- Electric properties -- Mathematical models, Singular perturbations (Mathematics), Blowing up (Algebraic geometry), Arrhythmia |

Journal or Publication Title: | SIAM Journal on Applied Dynamical Systems |

Publisher: | Society for Industrial and Applied Mathematics |

ISSN: | 1536-0040 |

Date: | 2008 |

Volume: | Vol.7 |

Number: | No.4 |

Number of Pages: | 24 |

Page Range: | pp. 1558-1581 |

Identification Number: | 10.1137/070709980 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | National Science Foundation (U.S.) (NSF), National Institutes of Health (U.S.) (NIH) |

Grant number: | PHY-0549259 (NSF), 1R01-HL-7283 (NIH) |

References: | [AGJ90] J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math., 410 (1990), pp. 167–212. [BJ89] P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), pp. 1–38. [BR77] G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol., 268 (1977), pp. 177–210. [Car77] G. A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differential Equations, 23 (1977), pp. 335–367. [CS06] J. W. Cain and D. G. Schaeffer, Two-term asymptotic approximation of a cardiac restitution curve, SIAM Rev., 48 (2006), pp. 537–546. [EK02] B. Echebarria and A. Karma, Instability and spatiotemporal dynamics of alternans in paced cardiac tissue, Phys. Rev. Lett., 88 (2002), paper 208101. [EK06] B. Echebarria and A. Karma, Amplitude equation approach to spatiotemporal dynamics of cardiac alternans, Phys. Rev. E, 76 (2007), paper 051911. [Eva73] J. W. Evans, Nerve axon equations. III. Stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972/73), pp. 577–593. [Fen71] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), pp. 193–226.[Fen79] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53–98. [Gar97] R. A. Gardner, Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math., 491 (1997), pp. 149–181. [GZ98] R. A. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), pp. 797–855. [Hen81] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. [HN60] O. F. Hutter and D. Noble, Rectifying properties of heart muscle, Nature, 188 (1960), pp. 495–497. [JK94] C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), pp. 64–88. [JKL91] C. K. R. T. Jones, N. Kopell, and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and Dynamics in Reactive Media (Minneapolis, MN, 1989), IMA Vol. Math. Appl. 37, Springer, New York, 1991, pp. 101–115. [Jon84] C. K. R. T. Jones, Stability of the traveling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), pp. 431–469. [Jon94] C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Lecture Notes in Math. 1609, R. Johnson, ed., Springer-Verlag, Berlin, 1994, pp. 44–118. [Kap99] T. J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, in Proc. Sympos. Appl. Math. 56, AMS, Providence, RI, 1999, pp. 85–131. [Kar93] A. Karma, Spiral breakup in model equations of action potential propagation in cardiac tissue, Phys. Rev. Lett., 71 (1993), pp. 1103–1106. [Kar94] A. Karma, Electrical alternans and spiral wave breakup in cardiac tissue, Chaos, 4 (1994), pp. 461–472. [KS98] T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrodinger equations, Phys. D, 124 (1998), pp. 58–103. [KS01] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—Fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), pp. 286– 314. [LR91] C. H. Luo and Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction, Circ. Res., 68 (1991), pp. 1501–1526. [LR94] C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes, Circ. Res., 74 (1994), pp. 1071–1096. [MS03] C. C. Mitchell and D. G. Schaeffer, A two-current model for the dynamics of cardiac membrane, Bull. Math. Biol., 65 (2003), pp. 767–793. [Nob62] D. Noble, A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials, J. Physiol., 160 (1962), pp. 317–352. [PK06] N. Popovi´c and T. J. Kaper, Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), pp. 103–139. [San02] B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems, Vol. 2, North- Holland, Amsterdam, 2002, pp. 983–1055. [SS01] B. Sandstede and A. Scheel, On the stability of periodic traveling waves with large spatial period, J. Differential Equations, 172 (2001), pp. 134–188. [WXY06] Y. Wu, X. Xing, and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dynam. Systems, 16 (2006), pp. 47–66. [Xin00] J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), pp. 161–230. |

URI: | http://wrap.warwick.ac.uk/id/eprint/28556 |

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