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