Mathematical models of biological phenomena come in two distinct forms. High-dimensional, biophysically detailed models give us realism; they talk about biophysical quantities that can be experimentally and clinically altered.

But there is another critical kind of modeling: Low-dimensional modeling tries to isolate the essential dynamical phenomena responsible for a qualitative type of solution. It gives us deeper insights into causal mechanisms.

The most important technique of low-dimensional modeling is bifurcation theory. A bifurcation is a qualitative change in the solution to a differential equation (ODE or PDE), as key parameters are varied. Identification of these key parameters then becomes the central task, because it is by acting through them that we can produce or prevent the qualitative phenomenon.

Simple bifurcations, especially the saddle/node, are now being used as models of “cell decisions” such as stem cell commitment and the turning on of “biological switches”. Hopf bifurcations are being used to model the onset of biological oscillations. We will discuss the cardiac arrhythmias that arise from the pathological voltage oscillations that are known as early after depolarizations (EADs). We will show that the onset of these pathological oscillations is a Hopf bifurcation at the cellular level, and that manipulation of its key parameters can produce and prevent EADs.

We will also discuss the onset of Ventricular Fibrillation, the leading cause of Sudden Cardiac Death. We will show that VF arises as a series of bifurcations from simple reentry to complex spatio- temporally chaotic behavior, and that manipulations of key parameters can prevent its onset.

Finally, we will discuss some bifurcations of PDEs, especially the Turing bifurcation explaining the appearance of spot patterns in morphogenesis.