# Zeeman's Model For The Heartbeat

In this lab you will be introduced to a model motivated by the concept of catastrophes. It originally appeared in an article "Differential Equations for the Heartbeat and Nerve Impulse," by E.C.Zeeman. The model is also described in "Differential Equations and Mathematical Biology," by D.S.Jones and B.D.Sleeman.

## Introduction:

We begin by establishing the three fundamental features of the heartbeat cycle upon which we can develop a mathematical model. These features are the following:
• The model must exhibit an equilibrium state corresponding to the diastole, or relaxed state of the heart;
• It should contain a threshold for triggering the electrochemical wave eminating from a pacemaker, causing the heart to contract into its systole, or fully contracted state;
• It should reflect the property of a limit cycle which includes the rapid return to the original equilibrium state.
Our model will include two dependent variables, x and b. x will refer to muscle fiber length, and b to electrochemical activity. Thus, the values for diastole will be b0,x0, and the values for systole will be b1,x1. The main parameter shall be T, which will refer to the overall tension of the system. You will now construct a variety of phase portraits, using a mathematical model for a heartbeat.