Numerical Solutions of the scroll wave type on Reaction-Diffusion systems: Applications to Cardiac Dynamic

Abstract

Mathematical models have played a very important role throughout the history of science. With the theory of differential equations developed by Newton, an infinite amount of possibilities arose to describe phenomena that appears in nature. In molecular biology, the use of mathematical models had its great breakthrough with the work done in 1952 by professors Alan Lloyd Hodgkin and Andrew Huxley where they developed a mathematical model to describe and explain the ionic mechanisms that underlie in the initiation and the propagation of action potentials in nerve cells. In 1963 they were given the Nobel prize in physiology-medicine due to this remarkable achievement. The work done by Hodgkin and Huxley not only has been used to study the nervous system. Together, Arturo Rosenblueth and Norbert Wiener, on their research paper “The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle”, in 1946, was the starting point of theoretical research in this field. Their paper seemingly deals with cardiac arrythmia and its mathematical formulation. The model of Wiener and Rosenblueth describes the propagation of an excitable wave. It considers the motion of curves with free ends representing the wave front. The attractive feature of this kinematic model is that it perfectly mimics biophysical reaction-diffusion equations of waves in excitable media in the parameter window of weak excitability (Brazhnik et al. (1988); Mikhailov et al. (1994)). The kinematic theory of wave propagation attempts to follow the spatial and temporal aspects based only on the fundamental underlying biophysical processes. It can predict differences between the spatio-temporal aura pattern caused by a neural phenomena and those caused by a vascular phenomena.