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are discontinuous, inhomogeneous and irregular. The variable, complicated structure
and behavior of living webs do not seem to converge on some regular pattern, but
instead maintain or increase their variability as data collection is increased. Conse-
quently, new models are required to characterize these kinds of complex webs. Herein
we explore how the concepts of fractals, non-analytic mathematical functions, the
fractional calculus, non-ergodic stochastic processes and renormalization-group trans-
formations provide novel approaches to the study of physiologic form and function, as
well as a broad-based understanding of complex webs. Do not be discouraged if you do
not recognize these technical terms; most experts do not. They will become familiar to
you in due course.
This is one of those forks in the road that ought to be flagged but is usually part
of a smooth transition from what we know and understand into what we do not know
and do not understand; a stepping from the sunlight into the shade and eventually into
total darkness. It is a decision point for most scientists who historically formed a career
involving the dynamics of differential equations, continuous differentiable functions,
predictability and a sense of certainty. The scientists who made this choice developed
the disciplines of biophysics, sociophysics, psychophysics and so on. These disciplines
were birthed to make the complexity in biology, sociology, psychology and so on
compatible with the scientific paradigm of physics. The other choice led to the road
less-traveled, where scientists have grappled with complexity directly; first in the phys-
ical sciences, producing such successes as the renormalization and scaling models of
phase transitions. More recently there has been success in understanding complexity
outside the physical domain. One area having some success is the physiology briefly
discussed in this section, which emphasizes the deviation from the traditional modeling
of living webs. G. Werner commented that this fork is a rupture from classical to fractal
scaling and deserves signaling with alarm bells.
Complexity is perhaps the most obvious characteristic of physiologic networks and
consequently living webs provide a rich experimental data base with which to test
mathematical models of complexity. Modern biology is challenged by the richness of
physiologic structure and function and has not as yet been able to capture this variability
in a single law or set of laws. For the sake of this preliminary discussion, consider the
bronchial tree as a paradigm for anatomic complexity. Subsequently we investigate the
dynamics underlying physiologic time series, but let's focus on the static situation for
the time being.
West and Goldberger [ 93 ] point out that two paradoxical features of the bronchial tree
are immediately evident to the untrained eye, see Figure 2.11 . The first is the extreme
variability of tube lengths and diameters; the second is the high level of organization
limiting this variability. One labeling convention in the literature is to number successive
bifurcations of the bronchial tree as “generations.” The first generation of tubes in the
counting scheme adopted in this context includes just two members, the left and right
mainstem bronchi, which branch off from the trachea. The second generation consists of
four tubes, and so forth. One observes that from one generation to the next the tube sizes
vary, tending to get shorter and narrower with each generation. However, this variability
is not restricted to comparisons between generations; tubes vary markedly also within
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