Heart muscle contraction oscillation.

Abstract

van der Pol developed a mathematical model for self-sustained radio oscillations described by his non-linear differential equation D2X + epsilon(X2-1)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 to 0 the equation is non-linear. If epsilon > 1 van der Pol coined the name relaxation oscillations for its solutions. These are non-linear 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 Poincaré. 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.