Bifurcations in biological systems

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To better understand the mechanism governing heart beat, we have to better understand the electrical flow (nervous impulses) in cardiac tissues.  It is important to understand why and how may electrical impulses be blocked (wave-block) by inhomogeneities in the cardiac tissue.  We also have to understand how do spiral waves appear and what are their effects on the heart. The mathematical models of wave propagation can also be used to understand the spread of contagious diseases.  The population can be seen as an excitable medium and the passive gap as a zone of non-contagious individuals.  Wave blocking is then associated to stopping the spread of a disease. T. Lewis and J. P. Keener have studied a bistable reaction-diffusion equation modeling wave propagation in an excitable medium.  They have related wave blocking to the width of a passive gap in an excitable medium.  In particular, they have shown that the existence of wave blocking is associated to a fold bifurcation as the width of the passive gap increases.  However, numerical simulations indicate that the shape of the wave, expressed through the derivative of the diffusion function, should also play a role in the appearance of wave blocking. One goal is to find a ``generic'' normal form to describe the possible bifurcations (nervous impulse behaviors) that the bistable reaction-diffusion model can support when the parameters vary.  The approach using bifurcation theory has the additional advantage to be potentially expandable to more complex models of electrical flow in excitable media.  This analysis of more complex models is also part of the proposed research project.