Lecture: The Mathematics of the Heart Beat
Myocites, a class of heart cells, when put into a petri dish will oscillate independently, but after some time, nearby cells will have similar oscillatory behavior. Our goal is to give a mathematical model of synchronization to help understand this phenomenon.
Important work on this subject goes back to Huygens more than 350 years ago (pre-Newton). More recently, Turing (morphogenesis), Winfree, and Strogatz,
(a one-time post-doc of my student Nancy Kopell) made contributions to these studies.
Here we focus especially on the work of Kuramoto on the collective behavior of phases. We will give a geometric analysis of Kuramoto's ordinary differential equations. Starting from the graph Laplacian of the cellular architecture of the heart, and a "hard-wiring" hypothesis of the associated genome dynamics, we obtain a phase setting of Kuramoto equations to obtain a "beating in unison" result.