HEART-MUSCLE CONTRACTION OSCILLATION

van der Pol developed a mathematical model for self-sustained radio oscillations described by his non-linear differential equation D(2)X + epsilon(X(2) - 1)DX + X = 0 in which X is a function of time T and D/DT the differential operator to T. For epsilon = 0, this is the differential equation for the harmonic oscillator which has sinusoidal solutions. For epsilon not equal 0 the equation is non-linear, If epsilon > 1 van der Pol coined the name relaxation oscillations for its solutions, These are nonlinear and quite different from simple sinusoidal oscillations. They are mathematical models for many physical and biological phenomena. van der Pol suggested that his equation is also a model for the heartbeat, However, biomedical oscillations, including the heartbeat, have a threshold which the mathematical model described by van der Pol's equation does not possess. It has, in addition to an unstable origin, only a stable limit cycle of Poincare. In this paper, van der Pol's equation is extended in such a way that it has in addition to a stable origin and a stable limit cycle, an unstable limit cycle. Because it possesses such an unstable limit cycle, the extension obtained is a mathematical model for a threshold oscillation, It is also shown that an asymmetric analogy of the extended equation is a mathematical model for an isometric contraction of the mammalian cardiac muscle.