Spiral and Scroll Waves in Excitable Media: Cardiological Applications A Calculus III Honors Project by David Hausner And Katrina McAlpin

What is Excitable Media? “In excitable media, such as the B-Z( Belousov- Zhabotinsky) reaction or the heart, constant amplitude waves propagate, but the collision of two waves leads to the annihilation of both.

What are Scroll and Spiral Waves in Excitable Media? In excitable media, there are 2 different types of waves. Concentric spherical waves are the normal waves or patterns that the heart should produce. Scroll waves are irregular patterns that the heart may produce alluding to a type of arrhythmia.

2D and 3D Waves

Arrhythmia Researchers study heart arrhythmia in hopes to understand rhythm disorders to better treat patients. More than 400,000 Americans die each year from rhythm disorders.

For years mathematicians and biophysicists have been trying to understand flutter and fibrillation of the heart by studying spiral waves of excitation in mathematically idealized excitable media. How do researchers study excitable media?

The mathematical model for excitation waves in excitable media is based on a map that assigns at a fixed time a reaction phase to spatial locations, and computes the obstruction to extending waves observed on the boundary of a manifold to waves on the entire manifold.

Researchers study the formation and existence of spiral excitation wave patterns in orientable 2D and 3D excitable media.

Excitable media are most naturally represented as partial differential equations. A generic excitable media model can be represented in a simplified form by the interaction of two variables.

An excitation variable (u) and a recovery variable (v) interact locally according to the ordinary differential equations du/dt=g(u,v). The excitation variable u increases rapidly until the trajectory approaches the rightmost arm of f(u,v)=0. At this point the trajectory moves slowly up the nullcline as the recovery variable increases.

When the trajectory reaches the top of the left arm of f(u,v)=0, the trajectory moves to the leftmost arm as variable u rapidly decreases, which is followed by a slow decrease of v back to the rest state.

A simple mathematical model is FitzHugh-Nagumo model of excitation in nerve and muscle tissue: du/dt = f(u,v)=u-u^3/3-v dv/dt = g(u,v)=E(u+a-bg) where u is membrane potential and v approximates a slow current (when E<<1.)

Modeling wave propgation in 2D with a model similar to the 1 st involves having several equations coupled to each other by diffusion: du/dt = f(u,v) + Dxd^2/dx^2 + Dyd^2/dx^2, dv/dt = g(u,v)tDxd^2(v)/dx^2 + Dyd^2(v)/dy^2

…..where Dx and Dy are diffusion constants in the x and y directions. This equation approximates continuos diffusion if the integration time step is small.

Other Mathematical Models Plane Waves in linear flows.

Angles The propagation angle at time t is the angle between the normal to the wave front and the x axis (or the y axis).

Researchers hope that this work will help diagnose rhythm disorders and identify patients at risk for sudden death. This work could assist surgeons in planning anti-arrhythmic heart surgery, steer them to affected sites, or help in guiding catheters to remove the foci of the arrhythmia without surgery.