Logo PTB

Instabilities and homogenization in discrete and heterogeneous non-linear reaction-diffusion equations

31.05.2010

Many important equations of classical and modern physics are linear partial differential equations which describe quantities which continuously depend on space and time. Since 1990, mathematic-theoretic research has increasingly turned its attention to non-linear evolution equations. An important class here are reaction-diffusion equations which have important applications in biological and chemical systems. Hence, many processes in cell biology and, in particular, the propagation of electrical signals in the human (and animal) heart are, for example, modelled with reaction-diffusion equations. This opens up the possibility of realistically simulating biosignals such as the electrocardiogram (ECG) or the magnetocardiogram (MCG). On the other hand it is, however, known that cardiac tissue is composed of relatively large muscle cells (length: 100 micrometers) which are coupled to each other by so-called "gap junctions". Electrical signals in the heart thus propagate in a coarse-grain discrete medium. Scientists in the "Mathematical Modelling and Data Analysis" Department have gained new knowledge related to the influence of the discrete structure and to the heterogeneity of medium on the propagation and stability of non-linear waves. Both aspects (heterogeneity and discreteness) can destabilize three-dimensional waves (Figures 1 and 2, [1]) what, in the case of the heart, may lead to life-threatening arrhythmia (ventricular fibrillation). In a further work, the conditions could be found for which a continuum model represents a sufficient approximation: if the length scale of the wavefront is clearly larger than the length scale of the heterogeneity or the discreteness in the medium, a continuum model - which can be obtained by a so-called homogenization method - describes the dynamics in a quantitatively correct way [2].

 

[1] S. Alonso, M. Bär, A. Panfilov, eingereicht (2010).
[2] S. Alonso, R. Kapral, M. Bär, Phys. Rev. Lett. 102, 238302 (2009).




Dynamik von dreidimensionalen rotierenden Wellen aus einer Anfangsbedingung für Medien mit zunehmer Diskretheit (a - c) bei identischern Anfangsbedingungen und gleicher lokaler Dynamik.

Figure 1: Dynamics of three-dimensionally rotating waves from an initial condition for media with increasing discreteness (a – c) at identical initial conditions and identical local dynamics.


Dynamik von dreidimensionalen rotierenden Wellen aus einer Anfangsbedingung für Medien mit zunehmer Diskretheit (a – c) bei identischern Anfangsbedingungen und gleicher lokaler Dynamik.

Figure 2: Dynamics of three-dimensionally rotating waves from an initial condition for media with increasing heterogeneities (a – c) at identical initial conditions.

 

Contact:

Dr. S. Alonso, WG 8.41,
PD Dr. M. Bär, Dept. 8.4