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。该项目将从理论上理解非光滑分支作为分段光滑问题的展开，这将使我们能够更全面地理解分段光滑分支问题与原始光滑分支问题之间的关系。本项目中发展的理论将应用于现有的和新的心脏模型，以检验交替的混合光滑/非光滑分叉。理论和数值预测将通过使用分离的心肌细胞和整个分离的兔心进行实验验证，以探索心脏动力学中杂交现象的生理机制。心脏交替是室颤的标志，室颤是一种致命的心律，在西方世界是导致死亡的主要原因之一。结合本研究的数学分析和实验研究，可能会对交替的生理机制有新的认识，从而影响心律失常的诊断和治疗。发展的分叉技术将为解决工程和生物学中的其他非光滑问题提供新的途径，如冲击振动系统、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训研究生和本科生，使他们能够使用非线性分析来理解工程和生物系统中的复杂性。外展活动将让高中生接触到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.