Wave instabilities in models of excitable heart tissue – from alternans to ventricular fibrillation

Marcus Bär*

Physikalische-Technische Bundesanstalt, Berlin, Germany

* Markus.Baer@ptb.de

 

Cardiac arrhythmias like alternans or ventricular fibrillation stem from dynamic instabilities of electrical propagation waves in excitable cardiac tissue. The talk presents mathematical background and numerical tools for a bifurcation analysis of regular periodic waves and classification of several instabilities relevant to the heart. The first example provided by alternans in the modified Beeler-Reuter model for cardiac tissue [1]. A stability analysis of alternans and its interaction with supernormal conduction velocity is presented [2]. The resulting bifurcation diagrams explain the behaviour found in simulations in simple 1D and 2D geometries as well as in realistic 3D rabbit heart geometries. Bifurcation and related continuations methods are also used to study systematically changes of individual ionic channels and their impact on the onset of alternans. They allow also a quantitative test of the widespread theory of steep action potential duration (APD) restitution.  In the second part of the talk, we will compare different scenarios of spiral breakup in two-dimensional excitable media and discuss their relevance to the understanding of ventricular fibrillation. In particular, scenarios based on wave breaks generated from dynamical instabilities will be compared to those resulting from heterogeneities in the medium [3, 4].

References:

 

1.      S. Bauer, G. Röder, and M. Bär, Alternans and the influence of ionic channel modifications: Cardiac 3D simulations and 1D numerical bifurcation analysis, Chaos 17, 015104 (2007).

2.      G. Röder, B. Echebarria, H. Engel, J. Davidsen, and M. Bär, Cardiac alternans and supernormal conduction, Physical Review E, in preparation (2009).

3.      M. Bär and L. Brusch, Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wavenumber instabilities, New J.  Phys. 6, 5 (2004).

4.      S. Alonso, R. Kapral, and M. Bär, Effective medium theory for heterogeneous reaction-diffusion systems, Phys. Rev. Lett. 102, 238302 (2009).