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Electrical waves in a onedimensional 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 onedimensional model of cardiac tissue. SIAM Journal on Applied Dynamical Systems, Vol.7 (No.4). pp. 15581581. ISSN 15360040
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Official URL: http://dx.doi.org/10.1137/070709980
Abstract
The electrical dynamics in the heart is modeled by a twocomponent 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 blowup 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 unstablein this case marginally stablefront 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:  15360040  
Official Date:  2008  
Dates: 


Volume:  Vol.7  
Number:  No.4  
Number of Pages:  24  
Page Range:  pp. 15581581  
Identifier:  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:  PHY0549259 (NSF), 1R01HL7283 (NIH)  
References:  [AGJ90] J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability 

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