Pedro Arroyo: Mathematical model with discretization-dependend parameters: Aplication to the study of phenomena of propagation discrete in excitable media

Pedro Andre Arroyo successfully defended his pHD thesis in the Universidade Federal de Juiz de Fora (Brazil). He spent during his thesis one year in our group.

Patterns formations are observed in extended chemical and biological processes. Although
biochemical systems are highly heterogeneous, homogenized continuum approaches formed
by partial dierential equations have been employed frequently. These approaches are usually
justied by the dierence scales between the characteristic spatial size of the patterns.
Under dierent conditions, for example under weak coupling, discrete models are more
adequate. On the other hand discrete models may be less manageable, for instance, in
terms of numerical implementation and mesh generation, compared to the continuum
models. Here we study a model to approach the discreteness which permits the computer
implementation on general unstructured meshes. The model is cast as a partial dierential
equation but with a parameter that depends on the discretization mesh. Therefor
we refer to it as a discretization-dependent model. We validate the approach in a generic
excitable media that simulates three dierent phenomena: the propagation of action potential
in cardiac tissue, the propation of the action potentialin laments of axons wrapped
by myelin sheaths, and the propagation of the activator/inhibitor in chemical microemulsions.
For the 2D case we develop a version to our approach in microemulsions where it
was possible to reproduce spiral waves with weak coupling of the medium.