English abstract
This project is devoted to a study of two proposed models of the heartbeat. One model is provided by the Van der Pol equation * +B(x2 - 1)x +x = 0) where x is the length of the muscle fibre of the heart, and the 'dot' signifies differentiation with time, t. The other model is the Zeeman model defined by the dynamical system x =-&1(x3 + ax +b), b=x -x,, in which b is a measure of the electrochemical wave controlling the contraction of the muscle fibre, ε is a small parameter, a is a parameter representing the tension in the muscle fibre and x, is the length of the muscle fibre at the equilibrium state of diastole.
The two systems are studied and solved numerically by the use of singular perturbation techniques. The Van der Pol equation possesses only one parameter while the Zeeman model has three parameters. This makes the Zeeman model more general than the Van der Pol equation.
The Van der Pol equation has a simple harmonic motion for B=0. Its solution as represented in the phase plane initially starts as a spiral but eventually reaches a limit cycle of a closed orbit representing an oscillatory solution of the type known as relaxation oscillations which are similar to the heartbeat. This behaviour of the solution is independent of the initial conditions. This indicates that the Van der Pol equation gives a reasonable general representation of the dynamics of the heartbeat.
The Zeeman model solution is discussed in the full parameter space €,a,x,. It provides a rich variety of solutions of different types. Restricting ourselves to the case e«<l, the solutions of the Zeeman model are found to fall into two different categories: one category has properties similar to those of the Van der Pol equation which possesses a limit cycle giving a solution of relaxation oscillations, whatever the initial conditions. This category is present provided that the length of the muscle fibre at the diastole state, xo, does not exceed a certain value, *(a), which depends on a. This result has not been identified before. The other category corresponds to x, >xm(a). Here the Zeeman model does not possess a closed orbit in the phase plane and the solution for x (t) tends to x, as time increases while b(t) approaches a constant value dependent on a. Thus the solution approaches the diastole state whatever the initial condition used. This result, that the heart will not beat if the muscle fibre exceeds a certain value dependent on the muscle fibre tension, is new and cannot be obtained from the Van der Pol equation.
