title: 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.
authors.affiliation: Department of Animal Biology, School of Veterinary Medicine, University of Pennsylvania, Philadelphia 19104, USA.
authors.forename: G
authors.initials: G
authors.lastname: Karreman
authors.forename: C
authors.initials: C
authors.lastname: Prood
mesh_terms: D000465:Algorithms; D000818:Animals; D006321:Heart / Q000502:physiology*; D006339:Heart Rate / Q000502:physiology; D007537:Isometric Contraction / Q000502:physiology; D008322:Mammals; D008955:Models, Cardiovascular; D009119:Muscle Contraction / Q000502:physiology*; D010210:Papillary Muscles / Q000502:physiology; D051381:Rats