Fractional Calculus In Physiologic Networks
US Army Research Office
Last modified: May 24, 2006
Complex adaptive systems have become popular in the last few years, particularly in the field of medicine, but the modeling and understanding of complex phenomena has always been a major concern of the physical scientist. In my presentation I will adopt a historical perspective and review ways in which complexity has been described by mathematical models over the past hundred years or so. There have been two main strategies for modeling the time development of complex physical phenomena over this period; dynamic equations and phase space equations. The dynamic equations start with Newton’s law, or their equivalent, and incorporate complexity through random forces, for example, Langevin’s development of ordinary stochastic differential equations for particle trajectories (e.g., Brownian motion). The phase space equations are obtained by averaging over the single-particle trajectories to obtain equations of motion for the probability density, usually partial differential equations that are first-order in time and second-order in space, such as the Fokker-Planck equation (e.g., Einstein diffusion).
However these approaches are no longer adequate when phenomena contain long-range memory in time and/or nonlocal interactions in space. Such mechanisms are manifest in inverse power-law correlation functions and/or inverse power-law probability densities, which introduce the notion of fractal stochastic processes. Physiology is found to be replete with such fractal time series, for example, heart beats, respiration rate, walking and cerebral blood flow. The evolution of such fractal stochastic phenomena is found to be well described by fractional partial differential equations for the evolution of the probability density and fractional Langevin equations for the evolution of particle trajectories. These ‘modern’ descriptions of complexity require the application of the fractional calculus, which we interpret in the context of a number of biomedical phenomena. These fractional operators are necessary for a proper description of the calculus of medicine.