Onaral, B., Cammarota, J. P. Complexity, Scaling, and Fractals in Biomedical Signals.The Biomedical Engineering Handbook: Second Edition.Ed. Joseph D. BronzinoBoca Raton: CRC Press LLC, 2000

2000 by CRC Pr

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59Complexity, Scaling,

and Fractals inBiomedical Signals

59.1 Complex Dynamics.Overcoming the Limits of Newtonian Mathematics Critical Phenomena: Phase Transitions An Illustration of Critical Phenomena: Magnetism A Model for Phase Transitions: Percolation Self-Organized Criticality Dynamics at the Edge of Chaos

59.2 Introduction to Scaling TheoriesFractal Preliminaries Mathematical and Natural Fractals Fractal Measures Power Law and 1/f Processes Distributed Relaxation Processes Multifractals

59.3 An Example of the Use of Complexity Theory in the Development of a Model of the Central Nervous System

a contemporary theme embraced by physical as well as social sciences, is concerned with thehavior observed in composite systems in which long-range order is induced by short-range of the constituent parts. Complex forms and functions abound in nature. Particularly in physiology, branched, nested, granular, or otherwise richly packed, irregular, or disorderedhe rule rather than the exception. Similarly ubiquitous are distributed, broad-band phenom-pear to fluctuate randomly. The rising science of complexity holds the promise to lead tols to analyze, model, process, and control the global behavior of complex biomedical systems. tenets of the complexity theory rest on the revelation that large classes of complex systemsof a multitude of richly interacting components) are reducible to simple rules. In particular,e and dynamics of complex systems invariably exist or evolve over a multitude of spatial andales. Moreover, they exhibit a systematic relationship between scales. From the biomedical standpoint, the worthwhile outcome is the ability to characterize these intricate objects and terms of straightforward scaling and fractal concepts and measures that often can beto simple iterative rules. In this sense, the set of concepts and tools, emerging under themplexity, complements the prediction made by the chaos theory that simple (low-orderc) systems may generate complex behavior.any incarnations, the concepts of complexity and scaling are playing a refreshingly unifying

diverse scientific pursuits; therein lie compelling opportunities for scientific discoveries andovations. Since these advances span a host of disciplines, hence different scientific languages,

d dissemination media, finding ones path has become confusing. One of the aims of this

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Cammarotarfare Center, ion

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is to serve as a resource for key literature. We hope to guide the reader toward substantials and away from figments of fascination in the popular press that have tended to stretchncepts ahead of the rigorous examination of evidence and the scientific verification of facts.ter is organized in three mains parts. The first part is intended to serve as a primer for thel aspects of the complexity theory. An overview of the attendant notions of scaling theoriesthe core of the second part. In the third part, we illustrate the potential of the complexity presenting an application to predict acceleration-induced loss of consciousness in pilots.

omplex Dynamics

a class of systems in which very complex spatial and temporal behavior is produced throughractions among a large number of local subsystems. Complexity theory is concerned with

t have many degrees of freedom (composite systems), are spatially extended (systems with and temporal degrees of freedom), and are dissipative as well as nonlinear due to the interplay components (agents). In general, such systems exhibit emergent global behavior. This meanscopic characteristics cannot be deduced from the microscopic characteristics of the elemen-ents considered in isolation. The global behavior emerges from the interactions of the local

ity theories draw their power from recognition that the behavior of a complex dynamic system general, depend on the physical particulars of the local elements but rather on how theyollectively (cooperatively or competitively) produce the globally observable behavior. The

of a complex dynamic system interact with their neighbors through a set of usually (very).gent global organization that occurs through the interplay of the local agents arises withouttion of a central controller. That is, there is self-organization, a spontaneous emergence of. Long-range correlations between local elements are not explicitly defined in such models,e induced through local interactions. The global organization also may exert a top-down the local elements, providing feedback between the macroscopic and microscopic structures

90] (Fig. 59.1).

ing the Limits of Newtonian Mathematics

well as the inherent predictive ability, was an important factor in the success of NewtonianIf a linear system is perturbed by a small amount, then the system response will change bynally small amount. In nonlinear systems, however, if the system is perturbed by a small response could be no change, a small change, a large change, oscillations (limit cycle), or

FIGURE 59.1 A complex dynamic system.ess LLC

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tion), there measuremenincreased, i.vior. The response depends on the state of the system at the time it was perturbed. Sincere is nonlinear, the key to success in understanding nature lies in embracing this nonlinearity.eature found in linear systems is the property of superposition. Superposition means that theal to the sum of the parts. All the properties of a linear system can be understood through

of each of its parts. This is not the case for complex systems, where the interaction among elements can produce complex emergent global behavior.ity theory stands in stark contrast to a purely reductionist approach that would seek to explainvior by breaking down the system into its most elementary components. The reductionist not guaranteed to generate knowledge about the behavior of a complex system, since it is

e information about the local interactions (which determine the global behavior) will notin such an analysis. For example, knowing everything there is to know about a single ant willng about why an ant colony is capable of such complex behaviors as waging war, farming,and the ability to quickly adapt to changing environmental conditions. The approach thatheory proposes is to look at the system as a whole and not merely as a collection of irreducible

ity research depends on digital computers for simulation of interactions. Cellular automataprincipal tools of complexity) have been constructed to model sand piles, earthquakes, trafficellite communication networks, evolution, molecular autocatalysis, forest fires, and species (among others) [Toffoli & Margoulis, 1987]. We note here that complexity is building on, cases unifying, developments made in the fields of chaotic dynamics [Devaney, 1992], critical, phase transitions, renormalization [Wilson, 1983], percolation [Stauffer & Aharony, 1992],orks [Harvey, 1994; Simpson, 1990], genetic algorithms [Goldberg, 1989] and artificial life989; Langton et al., 1992].

henomena: Phase Transitions

pose of this discussion, a phase transition can be defined as any abrupt change between the/or dynamic states of a system. The most familiar examples of phase transitions are between

ental stages of matter: solid, liquid, gas, and plasma. Phase transitions are also used to definees in matter, such as changes in the crystalline structure or state of magnetism. There areransitions in the dynamics of systems from ordered (fixed-point and limit-cycle stability) tochaos). Determining the state of matter is not always straightforward. Sometimes the apparentter changes when the scale of the observation (macroscopic versus microscopic) is changed.int is a special case of phase transitions where order and disorder are intermixed at all scales83]. At criticality, all spatial and temporal features become scale invariant or self-similar.is a good example of this phenomenon.

ration of Critical Phenomena: Magnetism

of a ferromagnetic substance have more electrons with spins in one direction than in theing in a net magnetic field for the atom as a whole. The individual magnetic fields of theto line up in one direction, with the result that there is a measurable level of magnetism inl. At a temperature of absolute zero, all the atomic dipoles are perfectly aligned. At normalrature, however, some of the atoms are not aligned to the global magnetic field due to thermal. This creates small regions that are nonmagnetic, although the substance is still magnetic.rmalization, or coarse graining, is the process of averaging the microscopic properties of theer a specified range in order to replace the multiple elements with a single equivalent element.ements of the magnetic property were taken at a very fine resolution (without renormaliza-would be some measurements that detect small pockets of nonmagnetism, although mostts would indicate that the substance was magnetic. As the scale of the measurements is

e., spatially renormalized, the small pockets of nonmagnetism would be averaged out andess LLC

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would not bis magnetic,dent on the That is, spamagnetism

At the Cregions. Theization tranapparent temsimilarity) oCurie tempgoverns the

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1992]. For spercolation,populated-bprobability

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e measurable. Therefore, measurements at the larger scale would indicate that the substance thereby decreasing its apparent temperature and making the apparent magnetic state depen-resolution of the measurements. The situation is similar (but reversed) at high temperatures.tial renormalization results in apparently higher temperatures, since microscopic islands ofare missed because of the large areas of disorder in the material.urie temperature there is long-range correlation in both the magnetic and nonmagnetic distribution of magnetic and nonmagnetic regions is invariant under the spatial renormal-sform. These results are independent of the scale at which the measure is taken, and the

perature does not change under the renormalization transform. This scale invariance (self-ccurs at only three temperatures: absolute zero, infinity, and the Curie temperature. The

erature represents a critical point (criticality) in the tuning parameter (temperature) that phase transition from a magnetic to a nonmagnetic state [Pietgen & Richter, 1986].

for Phase Transitions: Percolation

n model is created by using a simple regular geometric framework and by establishing simplerules among the elements on the grid. Yet these models give rise to very complex structuresships that can be described by using scaling concepts such as fractals and power laws. Amodel can be constructed on any regular infinite n-dimensional lattice [Stauffer & Aharony,implicity, the example discussed here will use a two-dimensional finite square grid. In site each node in the grid has only two states, occupied or vacant. The nodes in the lattice areased on a uniform probability distribution, independent of the sate of any other node. Theof a node being occupied is p (and thus the probability of a node being vacant is 1 p).are neighbors on the grid link together to form clusters (Fig. 59.2).

FIGURE 59.2 A percolation network.ess LLC

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(fixed-pointadapt and echaotic (higlearning, antransients inAs the dynamgrows. The epresent connections between nodes in the lattice. Anything associated with the cluster canavel (flow) to any node that belongs to the cluster. Percolation can describe the ability ofw through a porous medium such as igneous rock, oil fields, or finely ground Colombiane occupation probability increases, the clusters of the percolation network grow from local

ess to global connectedness [Feder, 1988]. At the critical occupation probability, a cluster thattire lattice emerges. It is easy to see how percolation could be used to describe such phenomenaansitions by viewing occupied nodes as ordered matter, with vacant nodes representingmatter. Percolation networks have been used to model magnetism, forest fires, and the of ion channels in cell membranes.

nized Criticality

t of self-organized criticality has been introduced as a possible underlying principle of[Bak et al., 1988; Bak & Chen, 1991]. The class of self-organized critical systems is spatiallymposite, and dissipative with many locally interacting degrees of freedom. These systemsability to naturally evolve (i.e., there is no explicit tuning parameter such as temperature orward a critical state.nized criticality is best illustrated by a sand pile. Start with a flat plate. Begin to add sand one

e. The mound will continue to grow until criticality is reached. This criticality is dependent local interactions among the grains of sand. The local slope determines what will happen ifin of sand is added. If the local slope is below the criticality (i.e., flat) the new grain of sand and increase the local slope. If the local slope is at the criticality, then adding the new grain increase the slope beyond the criticality, causing it to collapse. The collapsing grains of sandjoining areas. If those areas are at the criticality, then the avalanche will continue until local

lopes below the criticality are reached. Long-range correlations (up to the length of the sanderge from the interactions of the local elements. Small avalanches are very common, while

ches are rare. The size (and duration) of the avalanche plotted against the frequency ofof the avalanche can be described by a power law [Bak et al., 1988]. The sand pile seeks then its own. The slope in the sand pile will remain constant regardless of even the largestThese same power laws are observed in traffic patterns, earthquakes, and many other complex.

s at the Edge of Chaos

ics of systems can be divided into several categories. Dynamic systems that exhibit a fixed-ty will return to their initial state after being perturbed. A periodic evolution of states willa system that exhibits a limit-cycle stability. Either of these systems may display a transientf states before the stable regions are reached. Dynamic systems also may exhibit chaotice evolution of states associated with chaotic behavior is aperiodic, well-bounded, and very

nitial conditions and resembles noise but is completely deterministic [Tsonis & Tsonis, 1989].ality that lies between highly ordered and highly disordered dynamics has been referred toof chaos [Langton, 1990] and is analogous to a phase transition between states of matter,ighly ordered system can be thought of as a solid and the highly disordered system a liquid.of chaos is the critical boundary between order and chaos. If the system dynamics are stagnant stability, highly ordered system), then there is no mechanism for change. The system cannotvolve because new states cannot be encoded into the system. If the system dynamics arehly disordered), then the system is in a constant state of flux, and there is no memory, nod no adaptation (some of the main qualities associated with life). Systems may exhibit the evolution of states before settling down into either fixed-point or limit-cycle behavior.ics of a complex system enter the edge of chaos region, the length of these transients quickly

chaotic region is where the length of the transient is infinite. At the edge of chaos (theess LLC

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dynamic phcorrelation Per Bak is acharacteristiconsists of tsand pile.

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Despite econcepts, it ase transition) there is no characteristic scale due to the emergence of arbitrarily longlengths in space and time [Langton, 1990]. The self-organized criticality in the sand piles ofn example of a system that exists at the edge of chaos. It is in this region that there is noc space or time scale. A single grain of sand added to the pile could cause an avalanche thatwo grains of sand, or it could cause an avalanche that spreads over the entire surface of the

troduction to Scaling Theories

rise of complexity theories, the existence of a systematic relationship between scales eludedeam sciences. As a consequence, natural structures and dynamics have been commonlys too irregular and complex and often rejected as monstrous formations, intractable noise,The advent of scaling concepts [Mandelbrot, 1983] has uncovered a remarkable hierarchicalersists over a significant number of spatial or temporal scales.eories capitalize on scale-invariant symmetries exhibited by many natural broadband (i.e.,henomena. According to the theory of self-organized criticality (see Section 59.1), this scalinganifestation of dilation (compression) symmetries that define the organization inherent totems which naturally evolve toward a critical state while dissipating energies on broad ranges time scales. Long overlooked, this symmetry is now added to the repertoire of mathematicalncepts, which had included approaches based largely on displacement invariances under

and/or rotation.ural forms and functions maintain some form of exact or statistical invariance under trans-of scale and thus belong in the scaling category. Objects and processes that remain invariantary geometric similarity constitute the self-similar subset in this class.to capture scaling information in the form of simple rules that relate features on differenttively developed in many scientific fields [Barnsley, 1993]. Engineers are coping with scalingrms and functions by investigating multiscale system theory [Basseville et al., 1992], multi-nd multirate signal processing [Akansu & Hadad, 1992; Vaidyanathan, 1993], subband coding,d filter banks [Meyer, 1993], and fractal compression [Barnsley & Hurd, 1993].erging tools empower engineers to reexamine old data and to re-formulate the question atmany unresolved inverse problemsWhat can small patterns say about large patterns, andhey also offer the possibility to establish cause-effect relationships between a given physicaldium and the monitored dynamic (temporal) behavior that constitutes the primary preoc- diagnostic scientists.

reliminaries

est sense, the noun or adjective fractal refers to physical objects or dynamic processes thatetails on space or time magnification. A staple of a truly fractal object or process is therefore

characteristic scale in time or space. Most structures in nature are broadband over a finiteing at least a number of frequency decades in space or time. Scaling fractals often consist ofr heterarchy of spatial or temporal structures in cascade and are often accomplished throughlication of patterns at finer scales. If the replication rule preserves scale invariance throughout

uch fractals are recognized as self-similar in either an exact or a statistical sense.ent feature of fractals is their ability to pack structure with economy of resources, whethere, or whatever other real estate. Fitting nearly infinite networks into finite spaces is just onement. These types of fractals are pervasive in physiology, i.e., the branching patterns of the cardiovascular tree, and the nervous tissue [West and Goldberger, 1987], which have the

eature of being fault tolerant [West, 1990].xpectations heightened by the colorful publicity campaign mounted by promoters of fractalis advisable to view fractals only as a starting approximation in analyzing scaling shapes andess LLC

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fluctuationsto reveal thconclusive ecreated the

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in nature. Fractal concepts are usually descriptive at a phenomenologic level without pretensee exact nature of the underlying elementary processes. They do not offer, for that matter,vidence of whatever particular collective, coupled, or isolated repetitive mechanism that

fractal object.situations, the power of invoking fractal concepts resides in the fact that they bring the logicts, whether in the form of asymmetry of motion caused by defects, traps, energy barriers,mories, irreversibility, or any other appropriate interaction or coupling mechanisms thatandom behavior. As discussed earlier, the spontaneous or forced organization and the ensuingn correlations and coherences that emerges out of random behavior are presumably respon- irregular structures pervasive throughout the physical world.portant, the versatility of fractal concepts as a magnifying tool is rooted in the facility toscale hierarchies and/or scale invariances in an exact or statistical sense. In the role of a scale they suggest a fresh look, with due respect to all scales of significance, at many structuralc problems deemed thus far anomalous or insoluble.

tical and Natural Fractals

of mathematics is rife with pathologic constructions of the iterated kind which defy themension concepts. The collection once included an assortment of anomalous dust sets, lines,umes, and other mathematical miscellenia mostly born out of the continuous yet nondiffer-gory of functions such as the Weierstrass series.re unifying these mathematical creations with natural fractals is the fractional or integerdistinct from the euclidian definition. Simply stated, a fractional dimension positions anween two integer dimensions in the euclidian sense, best articulated by the critical dimensiondorff-Besicovith derivation [Feder, 1988]. When this notion of dimension is pursued to the the dimension reaches an integer value, one is confronted with the counterintuitive realitying curves, volume-filling planes, etc. These objects can be seen readily to share intrinsicerties with the nearly infinite networks accomplished by the branching patterns of bronchiessels and the intricate folding of the cortex.ing outcome afforded by the advent of scaling concepts is the ability to characterize such terms of straightforward scaling or dimension measures. From these, simple iterative

e deduced to yield models with maximum economy (or minimum number) or parameters993]. This principle is suspected to underlie the succinct, coding adopted by nature in ordernsive information needed to create complex shapes and forms.

easures

e most often used in the diagnosis of a fractal is the basic fractal dimension, which, in thef fractals, has eluded a rigorous definition embracing the entire family of fractal objects. Theor in the choice of the appropriate measures is the recognition that most fractal objects scaley; in other words, they can be characterized by a measure expressed in the form of a poweraling exponent , that links the change in the observed dependent quantity V to the inde-iable x as V(x) x [Falconer, 1990, p 36]. Clearly, is proportional to the ratio of thef V(x) and x, i.e., = log V(x)/log x. In the case of fractal objects, is the scaling exponentl sense and may have a fractional value. In the final analysis, most scaling relationships cansome form of a logarithmic dependence on the independent variable with respect to whichperty is analyzed, the latter also expressed on the logarithmic scale. A number of dimensionve been developed based on this observation, and comprehensive compilations are nowlconer, 1990; Feder, 1988].oach to formalize the concept of scale invariance utilizes the homogeneity or the renormal-

ciple given by f() = f(a)/b, where a and b are constants and is the independent variableess LLC

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ldberger, 1987]. The function f that satisfies this relationship is referred as a scaling function.law function f() is a prominent example in this category provided = log b/log a. Thef this particular scaling function has been proven many times over in many areas of science,e thermodynamics of phase transitions and the threshold behavior of percolation networks1991; Stauffer & Aharony, 1992; Wilson, 1983].

w and 1/f Processes

interest in power-law behavior largely stems from the recognition that a large class of noisybits spectra that attenuate with a fractional power dependence on frequency [West &989; Wornell, 1993]. Such behavior is often viewed as a manifestation of the interplay of af local processes evolving on a spectrum of time scales that collectively give rise to the so-r, more generally, the 1/f-type behavior. As in the case of spatial fractals that lack a charac-h scale, 1/f processes such as the fractional brownian motion cannot be described adequatelyonfines of a characteristic time scale and hence exhibit the fractal time property [Man-7].

ed Relaxation Processes

er part of the nineteenth century, the fractional power function dependence of the frequencyso has been recognized as a macroscopic dynamic property manifested by strongly interactingiscoelastic, and magnetic materials and interfaces between different conducting materials7]. More recently, the 1/f-type dynamic behavior has been observed in percolating networksf random mixtures of conductors and insulators and layered wave propagation in heteroge- [Orbach, 1986]. In immittance (impedance or admittance) studies, this frequency dispersion

alyzed conventionally to distinguish a broad class of the so-called anomalous, i.e., nonexpo-xation/dispersion systems from those which can be described by the ideal single exponential Debye [Daniel, 1967].l time or the multiplicity of times scales prevalent in distributed relaxation systems necessarilyto fractional constitutive models amenable to analysis by fractional calculus [Ross, 1977] andate-space methods [Bagley & Calico, 1991]. This corresponds to logarithmic distributionnging in symmetry from the log-normal with even center symmetry at one extreme to single-bolic distributions with diverging moments at the other. The realization that systems that doa characteristic time can be described in terms of distributions renewed the interest in theersion/relaxation analysis. Logarithmic distribution functions have been used conventionally characterize such complexity [West, 1994].

tals

cts and processes in nature are rarely strictly homogeneous in their scaling properties andy a distribution of scaling exponents that echos the structural heterogeneities occurring at angth or time scales. In systems with spectra that attenuate following a pure power law overquency scales, as in the case of Davidson-Cole dispersion [Daniel, 1967], the corresponding of relaxation times is logarithmic and single-tailed. In many natural relaxation systems, spectral dimension exhibits a gradual dependence on frequency, as in phenomena conven-eled by the Cole-Cole type dispersion. The equivalent distribution functions exhibit double-

etries on the logarithmic relaxation time scale ranging from the even symmetry of the log-ugh intermediate symmetries down to strictly one-sided functions.ept that a fractal structure can be composed of fractal subsets with uniform scaling propertyubset has gained popularity in recent years [Feder, 1988]. From this perspective, one mayplicated fractal object, say, the strange attractor of a chaotic process, as a superposition ofal subsystems. The idea has been formalized under the term multifractal. It follows thatess LLC

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n Example of the Use of Complexity Theory in the evelopment of a Model of the Central Nervous System

ess can be viewed as an emergent behavior arising from the interactions among a very largelocal agents, which, in this case, range from electrons through neurons and glial cells to neurons. The hierarchical organization of the brain [Churchland & Sejnowski, 1992; Newell,h exists and evolves on a multitude of spatial and temporal scales, is a good example of theacteristics found in many complex dynamic systems. There is no master controller for thisehavior, which results from the intricate interactions among a very large number of local

that duplicates the global dynamics of the induction of unconsciousness in humans due toemia produced by linear acceleration stress (G-LOC) was constructed using some of theplexity [Cammarota, 1994]. It was an attempt to provide a theory that could both replicate

man acceleration tolerance data and present a possible underlying mechanism. The model realization that an abrupt loss of consciousness could be thought of as a phase transitionousness to unconsciousness with the proposed neurophysiologic theory of G-LOC [Whinnery,phase transition was modeled using a percolation network to evaluate the connectivity ofays within the central nervous system.

to construct the model, several hypotheses had to be formulated to account for the unob-erplay among the local elements of the central nervous system. The inspiration for thecs of the locally interacting elements (the nodes of the percolation lattice) was provided bygic mechanism of arousal (the all-or-nothing aspect of consciousness), the utilization ofeural tissue during ischemia, and the response of neural cells to metabolic threats. Thelogic theory of acceleration tolerance views unconsciousness as an active protective mecha- triggered by a metabolic threat which in this case is acceleration-induced ischemia. Theong the local systems is determined by using a percolation network that models the con-

the arousal mechanism (the reticular activating system). When normal neuronal function isdue to local cerebral ischemia, the corresponding node is removed from the percolatione configuration of the percolation network varies as a function of time. When the network able to support arousal, unconsciousness results.el simulated a wide range of human data with a high degree of fidelity. It duplicated theresponse (measured as the time it took to lose consciousness) over a range of stresses that a simulation of the acute arrest of cerebral circulation to a gradual application of accelerationover, the model was able to offer a possible unified explanation for apparently contradictoryata. An analysis of the parameters responsible for the determination of the time of LOCat there is a phase transition in the dynamics that was not explicitly incorporated into the of the model. The model spontaneously captured an interplay of the cardiovascular andystems that could not have been predicted based on existing data.

to the models success are the reasonable assumptions that were made about the characteristicsion of the local dynamic subsystems through the integration of a wide range of human andiologic data in the design of the model. None of the local parameters was explicitly tuned to global (input-output) behavior. By successfully duplicating the observed global behavior ofer acceleration stress, however, this model provided insight into some (currently) unobserv-ynamics of the central nervous system. Furthermore, the model suggests new experimentalecifically aimed at exploring further the microscopic interplay responsible for the macroscopic) behavior.ess LLC

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Signals or systems that exhibit spectra which attenuate following a fractional power depen- on frequency.omata: Composite discrete-time and discrete space dynamic systems defined on a regular. Neighborhood rules determine the state transitions of the individual local elements (cells).state the produces a signal that resembles noise and is aperiodic, well-bounded, and veryive to initial conditions but is governed by a low-order deterministic differential or differenceion.: Complexity theory is concerned with systems that have many degrees of freedom (com- systems), are spatially extended (systems with both spatial and temporal degrees of freedom),re dissipative as well as nonlinear due to the rich interactions among the local componentsts). Some of the terms associated with such systems are emergent global behavior, collectiveior, cooperative behavior, self-organization, critical phenomena, and scale invariance.

A state of a system where spatial and/or temporal characteristics are scale invariant.lobal behavior: The observable behavior of a system that cannot be deduced from therties of constituent components considered in isolation and results from the collective (coop-e or competitive) evolution of local events.efers to physical objects or dynamic processes that reveal new details on space or timeification. Fractals lack a characteristic scale.rownian motion: A generalization of the random function created by the record of then of a brownian particle executing random walk. Brownian motion is commonly used tol diffusion in constraint-free media. Fractional brownian motion is often used to modelion of particles in constrained environments or anomalous diffusion.: A simple mathematical construct commonly used to measure the extent of connectednessartially occupied (site percolation) or connected (bond percolation) lattice structure.ition: Any abrupt change between the physical and/or the dynamic states of a system,

ly between ordered and disordered organization or behavior.ation: Changing the characteristic scale of a measurement though a process of systematicing applied to the microscopic elements of a system (also referred to as coarse graining).

tructures or dynamics that maintain some form of exact or statistical invariance underormations of scale.ation: The spontaneous emergence of order. This occurs without the direction of a global

oller.ity: A subset of objects and processes in the scaling category that remain invariant underary geometric similarity.

es

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The Biomedical Engineering Handbook: Second Edition.ContentsComplexity, Scaling, and Fractals in Biomedical Signals59.1 Complex DynamicsOvercoming the Limits of Newtonian MathematicsCritical Phenomena: Phase TransitionsAll Illustration of Critical Phenomena: MagnetismA Model for Phase Transitions: PercolationSelf-Organized CriticalityDynamics at the Edge of Chaos

59.2 Introduction to Scaling TheoriesFractal PreliminariesMathematical and Natural FractalsFractal MeasuresPower Law and 1/f ProcessesDistributed Relaxation ProcessesMultifractals

59.3 An Example of the Use of Complexity Theory in the Development of a Model of the Central Nerv...Defining TermsReferences