# 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.