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A Brief look at Complexity and Non-linear Dynamical Systems

Note:  These topics are all connected together for me and it is difficult to separate them out.  This seems to be much the way they work in life too.

 

Complexity

 

“…the idea of treating complexity as a coherent scientific concept potentially amenable to explicit definition is quite new: indeed this became popular only in the late 1980’s…”

 

“That what one would usually call complexity can be present in mathematical systems was for example already noted in the 1890’s by Henri Poincare in connection with the three-body problem.”

 

“Most often complexity seems to have been thought of as associated with the presence of large numbers of components with different types of behavior, and typically also with the presence of extensive interconnections or interdependencies.”

 

pp.1068-9 Wolfram, Stephen “A New Kind of Science” Wolfram Media, Inc. 2002. 

 

 

Non-Linear Dynamical Systems

 

Isaac Newton’s physics provided a way to predict outcomes of many physical events and the power of these equations came to be seen over centuries of time as ‘laws of nature’.  If all the variables were known, the outcome of an equation could be known in a precise way.  A change in one variable caused a proportional change in the outcome.  As computers came onto the world scene in the early 1960’s, it became possible to study much more complicated systems.  Some systems had long stymied modeling; they seemed ‘chaotic’.  As Gleick noted:

 

“Where chaos begins, classical science stops.  For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and the brain.  The irregular side of nature, the discontinuous and erratic side – these have been puzzles to science, or worse, monstrosities.”

 

Gleick, James. Chaos: making a new science.  Penguin Books 1987. New York, N.Y. p. 3.

 

Gleick, in his popular book, “Chaos”, largely introduced the concept of chaos to the general population.  He recounted how Edward Lorenz, a research meteorologist began in 1960, to use computer programs to model the weather.  He had developed twelve equations that could create a primitive version of global weather patterns.  He could run the program and, as Gleick wrote:

 

“If you knew how to read the printouts, you would see a prevailing westerly wind swing now to the north, now to the south, now back to the north.  Digitized cyclones spun slowly around an idealized globe.”

 

Gleick ibid. p. 11.

 

In 1961, Lorenz decided to re-examine one specific sequence in more detail and re-entered the values into his computer.  To his surprise, the results were distinctly different than his first run.  He realized that the initial values in the computer memory had been stored to six digits.  His read-out, from which he had re-entered the numbers, only gave the first three.  This was only a difference of one part in one thousand and Lorenz had assumed the difference would be inconsequential.  Instead, he found that it created an entirely different outcome. 

 

This came to be known as “The Butterfly Effect”.  It is known in science as ‘sensitive dependence on initial conditions’.  In the decades since this initial insight, we have come to understand that measurement is always approximate.  It is not possible to precisely account for every variable in complex systems and therefore outcomes are not ever going to be completely predictable. As more complex models were studied, it was found that even though the models would never exactly repeat themselves, there is often an irregular regularity, a pattern, which recurs.   Some factor appeared to cause this near-periodic behavior and this factor became known as an ‘attractor’.  Gleick explained:

 

“The strange attractor lives in phase space, one of the most powerful inventions of modern science.  Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible road map to all its possibilities.  Physicists already worked with two simpler kinds of “attractors”: fixed points and limit cycles, representing behavior that reached a steady state or repeated itself continuously.

 

In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point.  That point is the dynamical system-at that instant.  At the next instant, though, the system will have changed, ever so slightly, and so the point moves.  The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time.”

 

Gleick ibid, p. 134.

 

The above explanation constitutes a brief non-mathematical description of the topic of non-linear complexity, but was necessary to provide a minimum perspective that can enable us to discuss the particular strange attractor which is the topic of this paper: gravity.

Truncated Fractals in Organic Structure

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.

 

They continue:

 

“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.

Gravity, Strange Attractors, Hausdorff Dimensions and Collagen

Gravity, Strange Attractors, Hausdorff Dimensions and Collagen

 

Note:  I really understand that upper cervical doctors may not be looking at many of these topics that I am putting on my blog.  I have come to think that they are however, important topics for some of us to look at as we try to understand exactly what it is that we are doing in our clinical work.  Again, these are just my own notes to myself as I try to educate myself regarding various topics that present themselves as potentially important to understand the vast landscape that this work constitutes.   

 

 

 

The one constant for any creature who ventures beyond the water’s edge is gravity.  Structure and function in any land-based organism is closely related to the effects of the Earth’s gravitational field. 

 

Gravity is a constant force of attraction between two bodies.  Unless you are one of the intrepid travelers aboard the space station, every action you make must take gravity into account (even at 300 km above the earth, gravitational forces remain substantial (approximately 0.9 G).   The reason for the apparent weightlessness in the space station is that the centrifugal and centripetal forces are almost balanced, canceling each other –and the effects of gravity- out.  (resultant gravitational forces range from 10^-3 to 10^-6 G).  [ from: Klaus DM. (2001) Clinostats and bioreactors. Gravitational and Space Biology Bulletin 14:55-64].  Microgravity (defined as 10^-6 G) requires approximately 1000 earth radii or 6.37 x 10⁶ km.)  This is because the universal gravitational constant (6.67 x 10^-8 cm ³/g.s² is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. [Morey-Holton ER. “The Impact of Gravity on Life”, in Evolution on Planet Earth 2003 Elsevier. Edit. Rothschild. P. 144]

 

Physics has identified four fundamental forces.  Gravity is by far the weakest.  For example, if gravitational force is given an arbitrary value of 1, the relative value of the weak nuclear force is 10²⁶, electromagnetic force is 10³⁸, and strong nuclear force is 10⁴⁰. [http://learn.lincoln.ac.nz/phsc103/lectures/intro/4_forces_of_physics.htm] However, the great mass of the earth (approx. 6.0 x 10²⁴ kg) and the relatively miniscule distances involved, creates continual loading, pressing any object against the surface. And so, humans, with their bipedal stance, must contend with gravitational force every time they move about. 

 

As noted above, dynamic, non-linear systems tend toward certain states over time.  A pendulum is often used as an example of an attractor.  Regardless of where it is released, the pendulum will end at the same point, perpendicular to the ground.  Gravity is the unseen force that gradually brings the pendulum to its final position of equilibrium.  Dissipative dynamic systems tend to give rise to strange attractors.  Again, strange attractors are not completely defined, but are often described as having non-integral Hausdorff Dimensions (or else dependant on initial conditions). [ Note:  Hausdorff Dimension represents a way to measure the dimensions of a mathematical object.  A point has a dimension of ‘0’.  A line occurs in ‘1’ dimension, a plane has ‘2’, three-dimensional space obviously has ‘3’.]   Such (non-integral) attractors are non-periodic (never exactly repeat themselves) and are fractal in nature. 

 

Because gravity is a constant and biologically significant force, it is a critical component in the ongoing calculations of the central nervous system as it continually effects ‘new’ postural strategies.  Gracovetsky writes about the constraints of the fascial envelope and the strategies of the central nervous system in dealing with varying loads:

 

“The continuously loading and unloading of collagen requires the CNS to rapidly re-direct forces.  It is known for quite some time that there exist many combinations of muscles and ligaments that correspond to a single posture.  A “steady” erect stance can be kept because the musculo-skeletal system oscillates from one combination to the next in such a way that no structure ends up being continuously loaded.”…

 

He continues:

 

“It could be argued that the need to switch from one muscular combination to the next is determined by the properties of collagen.  In a series of studies in Sweden, Kazarian (1968) realized that collagen has a complex time dependent response to loading.  The most important factor for the purpose of this discussion is the fact that collagen has at least two time constants.  One of them is about 20 minutes and another one of about one third of a second.”

 

Gracovetsky, S. (2005).Personal communication –data from his latest paper accepted for publication by Elsevier LTD, UK.

[ Note:  Kazarian L; 1968.  Acta orthopedic Scandinavia supplemental.  ]

 

This means that even standing erect posture is an inherently dynamic state which develops from constantly updated integration of afferent input and which is then made possible by the constantly shifting postural strategies which load and unload the joints and fascial envelope, in order to minimize and equalize stress throughout the organism in a tensegral manner.  As Gracovetsky further notes:

 

“Stability at minimum energy cost is essentially a collagen issue, and the forces that shaped us cannot be left out of the equation.  I believe that sooner or later we will end up being confronted with the need to understand the relation between the visco-elastic properties of collagen and gravity.”

 

Gracovetsky  2005. ibid.

 

The effect of gravity on the human body is profound.  In the standing neutral position, the alignment of the human frame is most biomechanically efficient when consistent with the vertical axis.  The vertical axis represents the greatest stability with respect to gravity in the upright position.  Obviously, a horizontal posture (lying flat on the ground) would constitute the most stable position (equilibrium) but it also represents the least potential for kinesis.  Standing upright on two legs with our center of mass some distance above our feet is an inherently unstable configuration.  This instability may however, be the point.  Gracovetsky writes:

 

“Interestingly, engineers purposefully construct unstable machines.  The modern jet fighter is one of them.  This machine can be flown because there are dozens of computers forcing that instable machine into stable flight.  Why not design a stable fighter in the first place?  Survival is the short answer: a stable fighter may not be agile enough to escape an incoming missile.  It takes less time to execute a maneuver by letting go an unstable machine than force a stable fighter into an evasive maneuver at a considerable energy cost.”   

 

Gracovetsky, 2005 ibid.

 

Gracovetsky reminds us that even standing posture is not static but extremely dynamic as the central nervous system constantly recalculates muscle strategies to deal with the time constants of the fascia.  Our agility too, is greatly enhanced by the inherent instability of the bipedal stance, allowing rapid shifts in posture and weight balance with movement. 

 

An Introductory Note from Dr. Thomas

I am begining to get happy about this new website.  I have always wanted a place to put my writings and I have always wanted a place I could just say what is on my mind too.  This blog will perform several tasks: I’ll let you know about changes in our hours at the office or other special news.  In addition, I will continue to write and publish papers; some for my patient’s uses and some of a more professional nature.  I am frequently in the literature reading and studying so I can understand this work better and perhaps be able to communicate it’s benefits more clearly. 

Some of the pieces, like the two below will be a bit thick for most people but some of my professional friends do enjoy reading my musings too.  If you have any comments please email me from this site.  For now I am going to keep the comments section closed on the website. 

I hope to put in some of the testimonials that patients have written regarding the results they have obtained under NUCCA care.  We are going to start work on that project this weekend. 

Thank you for your interest. I am so pleased to have this new way to communicate.