Fractal PhysiologyThis volume delineates the use of fractal patterns and measures of fractal dimensions in describing and understanding general aspects of biology, particularly human physiology. After describing the ubiquitous nature of fractal phenomena, the authors give examples of the properties of fractals in space and time. Proceeding from mathematical definitions, they develop detailed practical methods for assessing the fractal characteristics of wave forms varying with time, tissue density variation, and surface irregularities. Most importantly, the authors show how fractal variation defines internal spatial or temporal correlations within the fractal system or object. Simple, recursively applied rules can give rise to complex biological structures by a variety of methods. This suggests that genetic rules govern the general structuring of an organism, while rules implied by interactions at the biochemical, cellular, and tissue levels govern ontogenic development and therefore play the major role in the growth of an organism. Chaos, or non-linear dynamics, is introduced as a stimulating way to examine biological behavior at the cellular and whole animal levels, even though proof of the chaotic nature of normal physiologic events is as yet meager. The later chapters give sets of examples of structural and behavioral fractal phenomena in nerve and muscle, in the cardiovascular and respiratory systems and in growth processes. Why molecular interactions and complex systems give rise to fractals is explored and related to the ideas of emergent properties of systems operating at high levels of complexity. |
Contents
Fractals Really Are Everywhere | 3 |
Properties of Fractal Phenomena in Space and Time | 11 |
SelfSimilar and SelfAffine Scaling | 45 |
Fractal Measures of Heterogeneity and Correlation | 63 |
Generating Fractals | 108 |
Properties of Chaotic Phenomena | 136 |
Is a Process Driven by Chance or Necessity? | 147 |
A Fractal Time Sequence | 177 |
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action potentials aggregation algorithm analysis analyzed attractor average Bassingthwaighte behavior branching cells channel protein chaos coefficient complex correlation correlation dimension curve data set described determine deterministic diameter diffusion-limited aggregation distribution effective kinetic rate embedding energy barriers equation estimates example exponent exponential Fano factor Figure fluctuations fractal dimension fractal model fractal scaling frequency function growth heart heterogeneity histograms Hurst intervals inverse power law ion channel iteration Julia sets kinetic rate constant left panel length log-log logarithm Lyapunov exponents Mandelbrot Mandelbrot set Markov model mathematical mean measured membrane method neurons noise nonlinear observed open and closed parameters patterns phase plane phase space set physiological pieces plot power law probability density function processes properties r₁ random walk range recursion regions relative dispersion rescaled range Right panel self-similarity shown in Fig signal slope spatial statistical structure teff tissue two-dimensional values variables variance variation vascular versus