Collaborative Research: Hybrid Bifurcation With Applications In Cardiac Dynamics

这项合作研究的目的是探索边界碰撞模型在描述心脏交替的全部潜力和局限性。该项目将发展非光滑分叉的理论理解为分段光滑问题的展开，这将使分段光滑分叉问题和原始光滑问题之间的关系得到更全面的理解。在本项目中开发的理论将被应用到现有的和新的心脏模型，检查混合光滑/非光滑分叉交替。理论和数值预测将通过实验验证，使用离体心肌细胞以及整个离体兔心脏来探索心脏动力学混合现象的生理机制。心脏电交替是心室颤动的标志，心室颤动是一种致命的心律，是西方世界死亡率的主要原因之一。本研究的数学分析与实验研究相结合，可能会对交替的生理机制产生新的认识，从而影响心律失常的诊断和治疗。分叉技术的发展将为工程和生物学中的其他非光滑问题提供新的方法，如振动碰撞系统，摩擦系统和电路中的开关。该研究计划将在多学科环境中培养研究生和本科生，使他们能够使用非线性分析来理解工程和生物系统的复杂性。外展活动将使高中生接触数学生物学和非线性思维方式。

The objective of this collaborative research is to explore the full potential and limitations of border-collision models in describing cardiac alternans. The project will develop a theoretical understanding of nonsmooth bifurcations as unfoldings of piecewise smooth problems, which will allow a more complete comprehension of the relation between a piecewise smooth bifurcation problem and the original smooth one. The theory developed in the present project will be applied to existing and new cardiac models to examine hybrid smooth/nonsmooth bifurcation to alternans. Theoretical and numerical predictions will be validated experimentally using isolated cardiac myocytes as well as whole isolated rabbit hearts to explore physiological mechanisms accounting for hybrid phenomena in cardiac dynamics.Cardiac alternans is a marker of ventricular fibrillation, a fatal heart rhythm that is one of the leading causes of mortality in the Western world. Mathematical analyses combined with experimental studies of this research may lead to new insights on the physiological mechanism of alternans and therefore will impact the diagnosis and treatment of cardiac arrhythmias. The developed bifurcation techniques will provide new approaches to other nonsmooth problems in engineering and biology such as vibro-impacting systems, systems with friction, and switching in electrical circuits. The research program will train graduate and undergraduate students in a multidisciplinary environment giving them the ability to use nonlinear analyses to understand complexities in engineering and biological systems. The outreach activities will expose high school students to mathematical biology and nonlinear ways of thinking.