Note: Again, these are notes for myself to understand the topic. It is not a finished paper and this piece is mainly quotes from various papers.
Truncated Fractals in Organic Structure
The word ‘fractal’ was coined by Benoit Mandelbrot. Mandelbrot is an intuitive genius who is able to intuit patterns that others have not perceived. His mathematics allows measurement of forms that Euclidian geometry is unable to model. Briggs and Peat write this about the fractal:
In general, fractals are characterized by infinite detail, infinite length, no slope or derivative, fractional dimension, self similarity, and they can be generated by iteration.
We can now understand why fractals and strange attractors are so intimately connected. Remember, in a phase space diagram, a strange attractor is traced by the point which represents the system. In its movement the system point folds and refolds in the phase space with infinite complexity. Thus, a strange attractor is a fractal curve. Fractal shapes have self-similarity at descending scales. For systems under the folding and stretching influence of the strange attractor, any single folding motion of the system represents (though in a unique instance) a mirror of the entire folding operation.
Briggs John, F. David Peat. Turbulent Mirror; an illustrated guide to chaos theory and the science of wholeness. Harper & Row, Publishers, Inc. New York, NY 1989. p. 95.
Mandelbrot found similarities in the roughness or brokenness at widely varying levels of scale and in very different fields. He found that the graphic pattern formed by daily and monthly price variations in the stock market, looked very much like the graphic representation of major recessions. Using this idea, he mathematically modeled a distribution for the galaxies in the universe that was later confirmed by astrophysicists.
Briggs, J., Peat, D., Turbulent Mirror. An illustrated guide to chaos theory and the science of wholeness. Harper and Row, NY 1989. P.83+
There are two major aspects to the fractal theories. This first is self-similarity. The basic idea is that any part of the object is quite similar to the entire object except for size. This can be seen in a fern or a cauliflower. In the case of a cauliflower, the entire cauliflower is shaped identically to a segment of the plant. Undersegments that comprise the segment also show this same self-similarity. The other major idea is that of dimensionality. In Euclidean geometry, a point possesses a dimension of zero. A straight line is one-dimensional, a plane has two dimensions, space has three. Fractal geometry permits the use of fractional dimensions. This allows a quantitative approach to parameters which until now have been fairly abstract such as heterogeneity, irregularity and complexity.
Heymans O, Fissette P., Vico S. et al. Is fractal geometry useful in medicine and biomedical science? Medical Hypotheses 2000. 54(3):360-366.
[Discuss truncated fractals and oscillatory behavior]
Many biological phenomena appear to be fractal, for example the structure of the bronchial tree (Schlesinger & West, 1991), heartbeat dynamics (Goldberger, et al., 1985; Peng et al., 1996; Ivanov et al., 1999; Makikallio et al., 1999), protein surfaces (Goetze & Brickman, 1992), chromatin microscopic images of breast epithelial cell nuclei (Einstein et al., 1998), fetal breathing dynamics (Szeto et al., 1992), microbial growth pattern (Obert et al., 1990), reduction law of metabolism (Sernetz et al., 1985), fetal heart rate (Gough, 1993), convoluted surface of mammalian brain (hofman, 1991), neural networks (Goldberger & West 1987), long-range power-law correlation in DNA (Peng et al., 1992; Voss, 1992; Buldyrev et al., 1993), neuronal shape (Castera et al., 1990), pattern in human retinal vessels (Family et al., 1989), structure of biomembranes (Nonnenmacher, 1989), blood vessel system (Kalda, 1993; Zamir, 1999), etc. In some fractal biological phenomena it is the spatial shape of a biological object itself that exhibits obvious fractal features, while in other cases the fractal properties are more hidden and can only be perceived if data are studied as a function of time or of some other variable, or mapped in some particular way (Buldyrev et al., 1995). The latter cases of fractality are referred to as hidden fractal properties.
The reason why nature prefers fractal structures to those generated by classical scaling is that more effective function is achieved, but it may also be related to higher tolerance that fractal structures and processes possess over those of classical structures and processes (West, 1990). It was argued that fractal geometry may not only be a design principle for living organisms, but may also underlie an evolutionary advantage of biological systems having fractal dimension (West & Deering, 1994).
the dimension of a naturally occurring fractal is associated with self-similarity over some region of space or interval of time. Therefore, such fractals are referred to as truncated fractals.
On the basis of an analysis of 96 reports on fractality of a wide range of natural systems, the narrow range of appropriate scaling properties for declared fractal objects was pointed out, centered around 1.3 orders of magnitude.
the key biological systems, such as, for example, cardiac, neural, respiratory, neuromuscular, and hormonal, display intrinsic oscillatory behavior (Othmer, 1980; Glass et al., 1984; Sporns et al., 1987; Glass 1988; Ermentrout, 1980; Murray, 1993; Han et al., 1995; Barrio et al., 1997; Leloup & Goldbeter, 1999; McLeod et al., 1998; Kaern & Hunding, 1999). Biological oscillators interact with one another and with the environment. Moreover, there are innumerable feedback loops acting on physiological variables. Instigated by this observation, it is tempting to look for a possible origin of truncated biological fractals at the level of a simple mathematical model of coupled oscillators.
Paar V, Pavin N, Rosandic M. Link between truncated fractals and coupled oscillators in biological systems. J. theor. Biol. (201) 212, 47-8.
In this paper we show that a nonlinear system of coupled oscillators can generate truncated fractal basin boundaries. This can shed new light on the coexistence of coupled oscillators and truncated fractals in biological systems.
Paar et al., p.49.
One should note an essential difference between a dissipative system with external forcing and an autonomous system with dissipation. In a system with external forcing energy is brought into the system, while simultaneously dissipation causes a loss of energy and thus an interplay between external force and dissipation takes place. Therefore, it is energetically allowed for the system to return to its initial state. On the other hand, in an autonomous system dissipation leads to permanent loss of energy and the energy of the system gradually decreases. Therefore, the autonomous system with dissipation cannot return to its initial state.
Paar et al., p. 50.
a simple mechanism of coupled oscillators can lead to the complex coexistence of various modes involving a truncated fractal pattern, as truncated fractal basin boundaries and consequently the fractal boundaries in the parameter space. This fractality may play a role in generating some basic features of biological systems. On the one hand, the appearance of fractality at certain ranges of scale can be associated with a higher tolerance in physiological functions which is important for the adaptability of biological systems (West & Deering, 1994). On the other hand, the appearance of truncation in the fractal pattern enables the appearance of a predictable long-term behavior of the system in conjunction with fractality once a certain level of precision in investigating and/or treating a biological system has been achieved. Consequently, a possible erratic nature of the systems behavior due to truncated fractality may disappear once the experimental errors in the measurement and/or treatment of biological systems reaches a certain level of precision.
Paar, et al., p. 54.
We argue that an integrated control of gait and posture is made possible because these two motor functions share some common principles of spatial organization.
The issue of the relationship between posture and locomotion is of great theoretical and experimental relevance (see Burleigh et al. 1994; Lacquaniti et al. 1997; Massion 1992; Mori 1987; Winter 1991; Zernicke and Smith 1996). Neurophysiological studies indicate that the control of posture and locomotion are interdependent at many different levels of the CNS, from the motor cortex to the basal ganglia, the brain stem, and the spinal cord.
Grasso R, Zago M, Lacquaniti F. Interactions between posture and locomotion: motor patterns in humans walking with bent posture versus erect posture. J Neurophysiol. 83:288.
Biological coordination patterns may thus emerge naturally as properties of appropriately coupled oscillators. P.89
Kay BA, Warren WH Jr. Coupling of posture and gait: mode locking and parametric excitation. Biol Cybern. 2001 Aug; 85(2)89-106.