Fractals and Chaos in Biology (Presentation)

7/31/2019 Fractals and Chaos in Biology (Presentation)

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Fractals & Chaos In BiologyFractals & Chaos In Biology

By: DanielBy: Daniel


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IntroductionIntroduction

Have you ever awed at the fact that the heartbeats more than 30 million times a year? Perhaps you

have wondered how part of our lungs can have a surface

area the size of a tennis court. Youre in luck! Thanks to

fractals and chaos we can accurately describe and

understand various parts of the human body.Researchers are working around the clock to try to solve

biological problems such as cancer and cystic fibrosis.

Instead of some new dangerous method, wouldnt it be

wonderful if we could cure diseases by the knowledge we

have gained in fractals and chaos theory? Stem cellresearch, radiation treatment, and any other uncertain

cures may all be ruled out if fractals and chaos can take a

large role in the field of biology. Come, lets find out

more!Picture from:

http://web.ukonline.co.uk/members/jill.lawson/


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What exactly is Fractals and Chaos andWhat exactly is Fractals and Chaos and

why is it so important in Biology?why is it so important in Biology?

Chaos is traditionally thought of as being confusion, hysteria, andturmoil. Chaos, however, in the sense of chaos theory is the idea thatthe final outcome of something can be extremely sensitive upon initialconditions. Amazingly, you can actually find brief organized patternswithin chaotic systems. Chaotic systems have the three mainproperties of sensitivity, mixing, and periodicity. As chaos theory isstill a fairly new field of research, the properties may be apt to changein the near future. Fractals are the models generated by mathequations resulting in chaotic systems. Fractals are very artistic,complex, and intricate. They also have properties which includehaving a fine structure, being defined by a recursive process, being tooirregular to be described by traditional geometry, having self-similarity, and having fractal dimension. In Biology, Chaotic systemscan be used to show the rhythms of heartbeats, walking strides, andeven the biological changes of aging. Fractals can be used to modelthe structures of nerve networks, circulatory systems, lungs, and evenDNA.

(Click to see full view of background)


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When was fractals and chaos first researched?

When was chaos theory relating to biology firstresearched?

Henri Poincar, a notable French mathematician, was doing researchon celestial mechanics in 1887 when he stumbled upon chaos. Adiscrete error in one of his solutions later gained fame as thefoundation of all chaos theory. Since then numerous notable scientistssuch as Robert Shaw and Edward Lorenz have researched the subject

of chaos. (Lorenz had the butterfly-effect idea)

Picture from:

www.emsf.rai.it/interviste/interviste.asp?d=502 Henri Poincar

The butterfly effect- A butterflyflapping its wings in China can cause

tornadoes in Texas.


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The father of fractals is often considered to be a man by the name of

Gaston Julia. In the early 1900s, Julia did much research on iteratedfunctions, and even drew some of his famous Julia sets by hand. True,there were some other works out there, such as Sierpinskis triangleand Kochs curve, but Julias work was a major breakthrough. Until the1960s much of the work with fractals was abandoned due to lack of

technology. That changed in the 1970s when Mandelbrot usedcomputers to create what we now know today as the Mandelbrot Set.

Gaston Julia,who sadly,lost his nosein WWI

Pictures from:http://www.fractalus.ch/index_pic/Julia.jpg

http://www.spsu.edu/math/edwards/mandel/bigman.jpg

The Mandelbrot Set


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Chaos theory relating to biology was first researched in the early 70s.

Researchers were looking at how chaos theory could be used to modelpopulation trends. Several researchers, such as George Oster, RobertMay, and Jim Yorke, looked at equations such as this one in their effortto model population:xt+l = lxt(1 xt). As for human biology, shortlyafter the Mandelbrot Set was discovered this also took off. Dynamical

diseases, a term coined in this era, described diseases that showchaotic systems. Researchers such as Leon Glass and Michael Mackeydid research in this field. Now, there are organizations dedicated toresearch with chaos theory and fractals in the field of biology.

Picture from:www.physionet.org/

Leon Glass

OnwardOnward


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Select a topicSelect a topic

Brain, DNABrain, DNABrain, DNABrain, DNA LungsLungsLungsLungs

The human heartThe human heartThe human heartThe human heart

Click whenClick whenClick whenClick whenfinishedfinishedfinishedfinished


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The Human HeartThe Human Heart

The rate of the beating of the human

heart can be described as chaotic. Thetime between heart beats fluctuatesdue to a variety of factors including

exercising, stress, and physical

activities.

A good way to listen to your heart iswith a heart rate monitor or just placeyour hand on your chest or anotherplace where you can measure your

pulse. Then try a variety of exercises

that cause your heart rate to changeand see how rapidly and slowly yourheart goes.

Order and Chaos:

Healthy hearts vs.Diseased hearts

The Sound of the Heart

Fractal Geometry of the

Heart and Circulatory

Structures

Click to go

to the mainmenu


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Order and Chaos:Healthy hearts vs. Diseased hearts

Healthy heartHealthy heart Has slight variations inHas slight variations in

the time between onethe time between one

beat to the next.beat to the next.

Has a heartHas a heart--rate that is arate that is achaotic pattern that ischaotic pattern that is

selfself--similar.similar.

Diseased heartDiseased heart DoesnDoesnt exhibit slightt exhibit slight

variations in timevariations in time

between each beatbetween each beat

Has a heartHas a heart--rate that israte that issteady, constant, andsteady, constant, and

predictable or eitherpredictable or either

extremely randomextremely random

Evidence for this

Interesting Fact:

This concept also applies to human

walking. When humans walk with

variation in their step they are normal.

With the onset of a disease, such as

Parkinsons disease, the human strideis more constant.


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Evidence for chaotic healthy heartsEvidence for chaotic healthy hearts

Dr. Ary L. Goldberger is director ofDr. Ary L. Goldberger is director of

electrocardiography at a hospital in Boston, MA.electrocardiography at a hospital in Boston, MA.

In the years gone past, Mr. Goldberger has doneIn the years gone past, Mr. Goldberger has done

extensive research on heart rates and chaoticextensive research on heart rates and chaotic

patterns.patterns.

As Dr. Goldberger was in medical school he was learning the

traditional taught methods of how hearts should be in equilibrium andshould be constant. Yet, after listening to countless heart rates of hispatients, he began to notice variations in completely healthy hearts.

By researching with his colleagues, Goldberger was able to discoverthat heart rates show fractal patterns. This is not because of physical

reasons, as many might believe, but because of physiological reasons.

As for why the fractal patterns break down in diseased hearts is stillinconclusive. Goldberger has helped establish a resource center athttp://www.physionet.org/ to help in the discovery process throughthe share of ideas and data.

Next page

Dr. Goldberger

Picture

from:http://focus.hms.harvard.edu/2002/March8_2002/cardiology.html


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Evidence for chaotic healthy heartsEvidence for chaotic healthy hearts

As you can see the top graph shows anormal healthy heart. The graph of thehealthy heart has more complexity thenthe bottom graph. Complexity = healthy

in many physiological aspects. Thebottom time scale graph shows a heartwith CHF (congestive heart failure). CHFis just one of the many diseases thatcauses the heart rate to lose its chaoticproperty. Click on the image for a further

view of the normal heart-rate.Picture from:http://www.physionet.org/tutorials/ndc/http://www.physionet.org/tutorials/ndc/


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As you can see the heart rate is very complexand does show signs of self-similarity. Whatis more surprising is how the heart-rateseems to lose its long range correlation as

the heart becomes diseased or break down.(Click to continue)

Picture from:http://www.physionet.org/tutorials/ndchttp://www.physionet.org/tutorials/ndc


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Both of these images showwhat happens as the heartgoes out of its normal state.The bottom left graph shows a

subject with heart failure.This graph has highly periodicvalues with little variation.The bottom right graph showsa subject with atrial

fibrillation. This heart rate isvery erratic jumping from thehigh end of heart rate to thelow end, with no particularpattern.

Picture from:http://www.physionet.org/tutorials/ndchttp://www.physionet.org/tutorials/ndc

BACK TO THE

HUMAN HEART


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Fractal Heart MusicFractal Heart Music The son of Dr. Goldberger, aThe son of Dr. Goldberger, a

man by the name of Zachman by the name of ZachDavids, actually recorded theDavids, actually recorded the

melodies of cardiograms onmelodies of cardiograms on

piano. He used much of hispiano. He used much of his

fatherfathers research to add thes research to add theright theme of music to theright theme of music to the

cardiograms and give themcardiograms and give them

the tempo they needed.the tempo they needed.

BACK TO THEHUMAN HEART

This may take a momentto load. It should openup a web page and load.(must have realplayer toview)


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Fractal Geometry of the Heart and

Circulatory Structures

First, the main areas where fractal geometry canFirst, the main areas where fractal geometry canbe seen in the circulatory system are:be seen in the circulatory system are:

Arteries and veinsArteries and veins

--Their cells and organization display the properties ofTheir cells and organization display the properties offractals, such as the powerfractals, such as the power--law distribution in thelaw distribution in thediameter distribution of arteries and veins.diameter distribution of arteries and veins.

Organization of heart muscle groupsOrganization of heart muscle groups

-- Show properties of selfShow properties of self--similarity, fine structure, etc.similarity, fine structure, etc.

Branching of certain muscles inside the heartBranching of certain muscles inside the heart

-- resemble the bifurcations seen in fractals such asresemble the bifurcations seen in fractals such asthe Feigenbaum plotthe Feigenbaum plot

HisHis--Purkinje networkPurkinje network

-- The branches and bifurcation of this electricalThe branches and bifurcation of this electricalsystem are essential to human biology andsystem are essential to human biology andresilience.resilience.

The tendons that connect the tricuspid valve toThe tendons that connect the tricuspid valve tothe papillary muscles.the papillary muscles.

--These again show bifurcation along with other fractalThese again show bifurcation along with other fractalproperties.properties.

The aortic valve leafletsThe aortic valve leaflets

-- These are layered providing a huge surface area,These are layered providing a huge surface area,while keeping a small volumewhile keeping a small volume

How does the fractal structurehelp?

-The fractal structure of the veins,arteries, and heart muscles help

protect the circulatory system fromthe strong, violent pumping of thehuman heart.

-The fractal structure, which isusually unnecessary, can come into

play when the His-Purkinje networkis damaged. This helps the heart beresilient and resistant to damage.

- The fractal geometry of the heart

could possibly save us everyday.

Fractals are very useful in modeling the heart and circulatory sFractals are very useful in modeling the heart and circulatory structures,tructures,and they play an important role in maintaining homeostasisand they play an important role in maintaining homeostasis..


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Fractal Geometry of the Heart andCirculatory Structures

Pictures from:http://webvision.med.utah.edu/imageswv/ARTERIES.jpg,

www.synecor.com/images/

www.medtronic.com/cardsurgery/images/fix4.jpg

Arteries and

veins

Aortic valve

leaflets


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The LungsThe Lungs

What you are hearing is a ventilator,which hopefully you will never have touse.

Thanks to the fractal geometry of ourlungs, most of us dont have to useventilators or other breathing devices.

Continue

As Benoit Mandelbrot was doing hisresearch on Fractals and Chaos, he

proposed that the lung shows signs offractal geometry. Since then severalstudies have been done to prove this

as being true. Another notablescientist to research fractals in thelungs was Sergey V. Buldyrev.

Sergey V. Buldryev

Picture from: polymer.bu.edu/ ~sergey/home.html


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The LungsThe Lungs

Continue

The fractal geometry of

the lungs helps the lungsresist and overcomeproblems and physicalstress during their growth.

By the lungs having fractal geometrythey are more efficient. But how? Therate at which the diffusion of air throughthe alveoli occurs is directly proportionalto the surface area of the alveoli and

lungs.

All of the alveoli in a human adult, havea total surface area of about 750 sq ft.!

Yet, they have a tiny volume.

Also, In the lungs, there aresmall air sacs, called alveoli,that are responsible for thediffusion of oxygen into the

blood.

Thus, the only way to model thesealveoli is through fractals. As imagined,the fractal geometry of the alveoli is

very high, usually around 2.9 or so.


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So what other parts of the lungs show fractalSo what other parts of the lungs show fractal

geometry?geometry?

As a whole, the lungs

show much fractalgeometry. Also, in thebronchi and bronchioletubes, there can begreatly seen the property

of bifurcation. If youlook back to theFeigenbaum plot, youwill find similar suchbifurcations.

A model of the lungs as awhole. The branching bronchiole tubes

(top). The bronchiole tubes

and the branching arteries

(bottom).

Main MenuPictures from:

http://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.html

A digital representation oflung tissue.


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Fractals and Chaos in the Brain,

DNAOut of the three topics given on the previous page, this one is the most new and

waiting to be researched. As research is in its early stages, there have beenfew discoveries made in how fractals and chaos relates to the brain and DNA.

Some discoveries that have been made: By modeling electrical signals in the brain and nerves, scientists have been

able to model the *alpha rhythm using chaos. Due to this discovery, theteaching of what areas in the brain control which neurological functions maybe in need of a change. If wave analysis in the brain is indeed modeled by

time series graphs it will indeed show that chaotic dynamics is important inneurology.

The Large surface area where neurons are packed in layers in the brain canonly be modeled by using fractal geometry.

Brain oscillations and waves is where most of the chaos theory and timeseries graphs come into play. The Brain itself may very well be organizedstrictly by the laws of chaos.

DNA sequences can be very similar to fractals in that they display qualities ofBrownian self-similarity, which involve small random lines making up linesinstead of small, more patterned lines making up lines.

DNA and the Brain both can be modeled by fractal geometry.

*Alpha rhythm- frequency of brainwaves between 8 -12 Hz.

*Image from:http://www.geocities.com/Omegaman_UK/fractal.html


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DNA By modeling electrical signals in the brain and nerves, scientists have been able to

model the *alpha rhythm using chaos. Alpha rhythm, along with other wave

frequencies, can often be modeled using time series graphs.

*Alpha rhythm- frequency of brainwaves between 8 -12 Hz.

Alpha rhythm Does this lookfamiliar to any time plots seen inthe study of fractals and chaos?

Match the graph to the type

of rhythm (frequency band).

Delta: .1Delta: .1Delta: .1Delta: .1 3 Hz3 Hz3 Hz3 Hz

Theta: 4Theta: 4Theta: 4Theta: 4 8 Hz8 Hz8 Hz8 Hz

Beta: above 12 HzBeta: above 12 HzBeta: above 12 HzBeta: above 12 Hz

If alpha rhythm is

between 8 -12 Hz and

has the graph shown

try to guess the

following.

Click to see answer

Images

from:http://www.crossroadsinstitute.org/eeg.html


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DNA

Delta: .1Delta: .1Delta: .1Delta: .1 3 Hz3 Hz3 Hz3 Hz

Beta: above 12 HzBeta: above 12 HzBeta: above 12 HzBeta: above 12 Hz

Theta: 4Theta: 4Theta: 4Theta: 4 8 Hz8 Hz8 Hz8 Hz

Images

from:http://www.crossroadsinstitute.org/eeg.html


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Applications of Fractals and

Chaos in the Brain

The brain is one of the most intricateparts of the human body. It is very

unlikely that science will ever be able to

grasp every part of its perplex design,

however chaos theory and dynamical

equations may very well be a startingground. In addition to being able to

model electrical signals of nerves and

the brain, chaos theory may help solve

neurological diseases and progress the

invention of artificial intelligence.

The hit movie, AI displayed many

types of neurosurgery and inspired

feelings about how artificial intelligence

would be viewed in the future.


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Examples of Fractal Geometry of

DNA and of the Brain

A fractal of DNA.

Images from: www.sgeier.net/ fractals/flam3/, www. sprott.physics.wisc.edu , .

A Brain Fractal.

Main Menu


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See you later!See you later!

Dr. Goldberger, with

his dog, Willy.

Pictures from: www.elnidodelescorpion.com/N23/ilustraciones/..., reylab.bidmc.harvard.edu,http://sv.wikipedia.org/wiki/Helge_von_Koch

www.colorcube.com/ illusions/dither.htm

http://math.youngzones.org/Fractal%20webpages/history_fractals.html

Benoit

Mandelbrot.

Helge Von Koch, witha comical friend.

Gaston Julia,displaying one of hismany talents.


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CreditsCredits

Slayden, Cameron. American Association for the Advancement of ScSlayden, Cameron. American Association for the Advancement of Science.ience.

Welcome to Modern BiologyWelcome to Modern Biology ScienceScience291291, 1177 (2001)., 1177 (2001).http://web.jjay.cuny.edu/~mwallace/ModernBiology/http://web.jjay.cuny.edu/~mwallace/ModernBiology/(image)(image)

May, Robert.May, Robert. The chaotic rhythms of lifeThe chaotic rhythms of life 10/2/200210/2/2002http://members.fortunecity.com/templarser/rhythm.htmlhttp://members.fortunecity.com/templarser/rhythm.html

Microsoft PowerPoint 2002 SP3, Microsoft Clipart GalleryMicrosoft PowerPoint 2002 SP3, Microsoft Clipart Gallery

Rae, Greg.Rae, Greg.Chaos Theory: A Brief IntroductionChaos Theory: A Brief Introduction Jan. 31st, 2003Jan. 31st, 2003http://www.imho.com/grae/chaos/chaos.htmlhttp://www.imho.com/grae/chaos/chaos.html

With sources of:With sources of: "Bach to Chaos: Chaotic Variations on a Classical Theme", Scienc"Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994, pg. 428.e News, Dec. 24, 1994, pg. 428.

Gleick, James, ChaosGleick, James, Chaos -- Making a New Science, Penguin Books Ltd, Harmondsworth, MiddlesMaking a New Science, Penguin Books Ltd, Harmondsworth, Middlesex,ex,1987.1987.

Lowrie, Peter, personal interview over the Internet, May 17, 199Lowrie, Peter, personal interview over the Internet, May 17, 1995.5.

Rae, Kevin, "Chaos", unpublished paper, submitted to Professor GRae, Kevin, "Chaos", unpublished paper, submitted to Professor Gould, Modern Physics class,ould, Modern Physics class,Claremont McKenna College, December 5, 1994.Claremont McKenna College, December 5, 1994.

Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, PengStewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin Books Ltd, Harmondsworth,uin Books Ltd, Harmondsworth,

Middlesex, 1989.Middlesex, 1989.

Browne, Malcolm.Browne, Malcolm. The Fractal Heart.The Fractal Heart. New York Times, unknown dateNew York Times, unknown datehttp://www.nyu.edu/classes/neimark/FRACT1.HTMhttp://www.nyu.edu/classes/neimark/FRACT1.HTM

Landau, Misia.Landau, Misia. Healthy Heart Keeps Polyrhythmic BeatHealthy Heart Keeps Polyrhythmic Beat, March 8th, 2002., March 8th, 2002.Harvard Medical Dental and Public Health Schools

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Fractals and Chaos in Biology (Presentation)