Bifurcations in biological systems

为了更好地理解控制心脏跳动的机制，我们必须更好地理解心脏组织中的电流（神经冲动）。 重要的是要了解为什么以及如何可能电脉冲被心脏组织中的不均匀性阻断（波阻）。 我们还必须了解螺旋波是如何出现的，以及它们对心脏的影响。波传播的数学模型也可以用来理解传染病的传播。 人口可以被看作是一个兴奋的媒介和被动的差距作为一个区域的非传染性的个人。 波的阻断与阻止疾病的传播有关。T.刘易斯和J.P.Keener研究了一个模拟波在可激发介质中传播的反应扩散方程。 他们把波阻塞与可激发介质中的被动隙的宽度联系起来。 特别是，他们已经表明，波阻塞的存在与折叠分叉的被动间隙的宽度增加。 然而，数值模拟表明，通过扩散函数的导数表示的波的形状，也应该在波阻塞的出现中发挥作用。一个目标是找到一个“通用”的标准形式来描述可能的分叉（神经冲动行为），当参数变化时，神经反应扩散模型可以支持。 使用分叉理论的方法具有额外的优点，可以潜在地扩展到更复杂的模型中的电流在可激发介质。 对更复杂模型的分析也是拟议研究项目的一部分。

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.