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- Patterns are organized arrangements of objects in space and time, such as schools of fish, columns of ants, stripes in the visual cortex, flashing fireflies, and pigmentation on shells. - Patterns can form through simple rules and interactions between units, even without natural selection, in phenomena like self-organization and morphogenesis. - Alan Turing's 1952 paper was a classic that studied how patterns form through reaction-diffusion processes.

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11 Artificial Life & Complex Systems Lecture 3Lecture 3 PatternsPatterns May 25, 2007May 25, 2007 MaxMax LungarellaLungarella
2 Contents •• PatternsPatterns •• Dynamical SystemsDynamical Systems •• FractalsFractals •• LL--SystemsSystems •• Pattern formationPattern formation •• MorphogenesisMorphogenesis [Paper:[Paper: A.M.TuringA.M.Turing (1952) CLASSIC!](1952) CLASSIC!]
3 What is a Pattern? Pattern = organized arrangement of objects in space andPattern = organized arrangement of objects in space and timetime Examples of biological patterns:Examples of biological patterns: -- Schools of fishSchools of fish -- Raiding column of army antsRaiding column of army ants -- Ocular dominance stripes in visual cortexOcular dominance stripes in visual cortex -- Synchronous flashing of firefliesSynchronous flashing of fireflies -- Pigmentation patterns on shellsPigmentation patterns on shells inputs from left eye (black) andinputs from left eye (black) and right eye (white)right eye (white)firefliesfireflies school of fishschool of fish
4 More Patterns
5 More Patterns
6 Pattern Formation If Nature is at all economical (probably, yes), weIf Nature is at all economical (probably, yes), we can expect that she will choose to createcan expect that she will choose to create complex form not by piececomplex form not by piece--toto--piece constructionpiece construction by organizational andby organizational and patternpattern--forming phenomenaforming phenomena we see also in the nonwe see also in the non--living world.living world. If this is true, then there must be similarities in theIf this is true, then there must be similarities in the form and patterns of living and purely inorganicform and patterns of living and purely inorganic or physical systems.or physical systems.
7 How Does Nature Build Patterns? Buildings blocks are:Buildings blocks are: -- Living units (fish, ants, nerve cells,Living units (fish, ants, nerve cells, ……)) biological systemsbiological systems -- Inanimate objects (bits of dirt, sand,Inanimate objects (bits of dirt, sand, ……)) physical/physical/nonbiologicalnonbiological systemssystems Differences between physical and biologicalDifferences between physical and biological systems:systems: -- In biological systems the subunits are moreIn biological systems the subunits are more complexcomplex -- The interactions among the subunits areThe interactions among the subunits are influenced also by genetically controlledinfluenced also by genetically controlled properties (in physical systems interactions areproperties (in physical systems interactions are purely physical)purely physical)
8 How Are Patterns Formed? Any Ideas?
9 How Are Patterns Formed? Simple answer:Simple answer: ““Form follows functionForm follows function”” That is, the shape and structure of aThat is, the shape and structure of a biological/technological systembiological/technological system –– a proteina protein molecule, a limb, an organism, a colony, themolecule, a limb, an organism, a colony, the InternetInternet –– is that which best equips the system foris that which best equips the system for survivalsurvival A form which gives organism (system) anA form which gives organism (system) an evolutionary advantage tends to stickevolutionary advantage tends to stick Power of cumulative selectionPower of cumulative selection
10 Natural selection is not entirely satisfyingNatural selection is not entirely satisfying …… Why?Why? How Are Patterns Formed?
11 How Are Patterns Formed? 1)1) It says nothing about theIt says nothing about the underlyingunderlying mechanismsmechanisms!! 2)2) Why should the pattern on a shellWhy should the pattern on a shell be of advantage?be of advantage? Once you start to ask theOnce you start to ask the ‘‘how?how?’’ ofof mechanism, you are up against themechanism, you are up against the rules of chemistry, physics, andrules of chemistry, physics, and mechanism, and the questionmechanism, and the question becomes not justbecomes not just ‘‘is the formis the form successful?successful?’’ butbut ‘‘is it physicallyis it physically possible?possible?’’ (1917)(1917)
12 Another simple answer: Emergence!Another simple answer: Emergence! Patterns emerge from large populations ofPatterns emerge from large populations of interactinginteracting ‘‘unitsunits’’ (living organisms and non(living organisms and non--livingliving entities)entities) Emergence implies that the largeEmergence implies that the large--scale organizationscale organization could never be deduced by close inspection of thecould never be deduced by close inspection of the individual units (holistic perspective, e.g. Hindividual units (holistic perspective, e.g. H220)0) How Are Patterns Formed?
13 How Are Patterns Formed? Can simple rules give rise to patterns and forms?Can simple rules give rise to patterns and forms?
14 Mechanisms -- IterationIteration -- RecursionRecursion -- FeedbackFeedback Can give rise to selfCan give rise to self--similar structures such as trees,similar structures such as trees, plants, clouds, mountains,plants, clouds, mountains, ……
15 Characterization:Characterization: •• AA systemsystem thatthat evolves in time according to a set of rulesevolves in time according to a set of rules •• What changes in time is a variable (e.g. position,What changes in time is a variable (e.g. position, concentration, temperature, viscosity,concentration, temperature, viscosity, ……)) •• PPresent conditions determine the future.resent conditions determine the future. •• The rules are usually nonlinearThe rules are usually nonlinear (e.g. set of diff.(e.g. set of diff. eqeq. defining. defining rates of change)rates of change) •• There may be many interacting variablesThere may be many interacting variables What is a Dynamical System?
16 Dynamical Systems at a Glance
17 Other Examples of (Nonlinear) Dynamical Systems •• The Solar SystemThe Solar System •• The atmosphere (the weather)The atmosphere (the weather) •• The economy (stock market)The economy (stock market) •• The human body (heart, brain, lungs, ...)The human body (heart, brain, lungs, ...) •• Ecology (plant and animal populations)Ecology (plant and animal populations) •• Cancer growthCancer growth •• Spread of epidemicsSpread of epidemics •• Chemical reactionsChemical reactions •• The electrical power gridThe electrical power grid •• The InternetThe Internet
18 Def.: Map of variables in timeDef.: Map of variables in time -- Time is parameterTime is parameter -- Trajectory (orbit) in stateTrajectory (orbit) in state spacespace E.g.E.g. Z(tZ(t) = [x(t),x(t+1),x(t+2)]) = [x(t),x(t+1),x(t+2)] Interestingly, forInterestingly, for continuouscontinuous (reversible) systems(reversible) systems: Only: Only one trajectory passesone trajectory passes through each point of athrough each point of a statestate--space (not true inspace (not true in discrete systems)discrete systems) Phase or State-Space
19 Vector Fields DEMO:DEMO:
20 Attractors PhasePhase--space volume to where dynamical system convergesspace volume to where dynamical system converges asymptotically over timeasymptotically over time
21 Why Attractor Behavior? Energy dissipation (thermodynamic systems)Energy dissipation (thermodynamic systems) -- Friction, thermodynamic loss, loss of material, etc.Friction, thermodynamic loss, loss of material, etc. -- Volume contraction in phaseVolume contraction in phase--space (system tends tospace (system tends to restrict itself to small basins of attraction)restrict itself to small basins of attraction) -- SelfSelf--organization (dissipative systems)organization (dissipative systems) Compare withCompare with …… Hamiltonian systemsHamiltonian systems -- frictionless, no attractorsfrictionless, no attractors -- Conservation of energyConservation of energy -- ergodicityergodicity
22 Types of Attractors Fixed pointFixed point Limit cycleLimit cycle QuasiQuasi--periodicperiodic StrangeStrange
23 Types of Attractors
24 Fix-Point Behavior •• 00--dimensional attractordimensional attractor •• Volume of phase space to which the systemVolume of phase space to which the system converges after a long enough time = basin ofconverges after a long enough time = basin of attraction (defined by all trajectories leading intoattraction (defined by all trajectories leading into the attractor)the attractor) •• Example: Pendulum when friction and gravityExample: Pendulum when friction and gravity bring its motion to a haltbring its motion to a halt
25 Limit Cycle Behavior •• Periodic motionPeriodic motion •• Repetitive oscillation among a number of statesRepetitive oscillation among a number of states (e.g. loop)(e.g. loop) •• Example: lone planet orbiting around a star in anExample: lone planet orbiting around a star in an elliptical orbitelliptical orbit
26 Quasi-Periodic Attractor •• Several independent cyclic motionsSeveral independent cyclic motions •• Never quite repeat themselvesNever quite repeat themselves •• ToroidalToroidal attractorsattractors •• Example: the moon orbits Earth, which orbits theExample: the moon orbits Earth, which orbits the sun, which, in turn, orbits the galactic centersun, which, in turn, orbits the galactic center
27 Strange or Chaotic Attractors -- Pervasive (chaos is everywhere: weather patterns, humanPervasive (chaos is everywhere: weather patterns, human brainbrain’’s electrochemical activity,s electrochemical activity, ……)) -- Sensitivity to initial conditions (Butterfly effect)Sensitivity to initial conditions (Butterfly effect) -- Deterministic chaos (if we could know the exact initialDeterministic chaos (if we could know the exact initial conditions, trajectories would be determined)conditions, trajectories would be determined) -- LowLow--dimensional chaos (strange attractors are restricteddimensional chaos (strange attractors are restricted to small volumes of phaseto small volumes of phase--space)space) -- Weak causalityWeak causality - 3-body problem (any slight measurement difference results in very different predictions) - Butterfly effect (Lorenz attractor)
28 Simple Chaotic Systems in Physics
29 Strange Attractors Limit set asLimit set as tt →→ ∞∞ Set of measure zeroSet of measure zero Basin of attractionBasin of attraction Fractal structureFractal structure • non-integer dimension • self-similarity • infinite detail Chaotic dynamicsChaotic dynamics • sensitivity to initial conditions • topological transitivity • dense periodic orbits Aesthetic appealAesthetic appeal
30 Characteristics of Chaos •• Never repeatsNever repeats •• Depends sensitively on initial conditions (Butterfly effect)Depends sensitively on initial conditions (Butterfly effect) •• Allows shortAllows short--term prediction but not longterm prediction but not long--term predictionterm prediction •• Comes and goes with a small change in some control knobComes and goes with a small change in some control knob •• Usually produces a fractal patternUsually produces a fractal pattern •• Not random (or stochastic); unpredictable in the shortNot random (or stochastic); unpredictable in the short-- term butterm but ““predictablepredictable”” in the longin the long--termterm ……
31 Example 1: A Planet Orbiting a Star Elliptical OrbitElliptical Orbit Chaotic OrbitChaotic Orbit
32 Example 2: The Logistic Map Also known as the quadratic map or the Feigenbaum map Simple population growth model defined by the iterative map: x(t+1) = rx(t)*(1-x(t)), 0 < r < 4, 0 < x < 1 (r = reproduction rate or bifurcation parameter) if x = 0: population is extinct if x = 1: population will be extinct at the next time step rx(t) positive feedback term 1-x(t) negative feedback term (decrease to overpopulation)
33 Example 2: The Logistic Map Question: What happens to the longQuestion: What happens to the long--term behaviorterm behavior ofof x(tx(t) for different values of) for different values of ‘‘rr’’?? Special cases:Special cases: 0<r<1: all terms <10<r<1: all terms <1 extinctionextinction 1<r<3: fix point behavior1<r<3: fix point behavior r>3: period doubling (bifurcation)r>3: period doubling (bifurcation)
34 Logistic Map – Fix Point
35 Logistic Map – Chaos Cobweb plot at:Cobweb plot at: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/CobwebPlot.htmcs/yanikjoe/Chaos/CobwebPlot.htm
36 The Logistic Map – Four Types of Attractors
37 The Logistic Map – Bifurcation Diagram DEMO:DEMO: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/Bifurcation.htmcs/yanikjoe/Chaos/Bifurcation.htm
38 Logistic Map – Bifurcations and Self-Similarity
39 Discovered sensitivity to initial conditions inDiscovered sensitivity to initial conditions in a simple 3a simple 3--variable dynamical system;variable dynamical system; simplified model of weather; convectionsimplified model of weather; convection flows in the atmosphere (x: convection; y:flows in the atmosphere (x: convection; y: temperature difference; z: distortion of thetemperature difference; z: distortion of the temperature profile)temperature profile) DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.htmlhttp://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.html DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.htmlhttp://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html Example 3: Lorenz Attractor
40 Example 3: Lorenz Attractor
41 Example 3: The Hénon Attractor xn+1 = a - xn 2 + bxn-1 Example ofExample of infinite detailinfinite detail ……
42 Example 3: The Hénon Attractor
43 Example 4: General 2-D Quadratic Map xn+1 = a1 + a2xn + a3xn 2 + a4xnyn + a5yn + a6yn 2 yn+1 = a7 + a8xn + a9xn 2 + a10xnyn + a11yn + a12yn 2 xn+1 = xn 2 - yn 2 + a yn+1 = 2xnyn + b a b
44 Take-Home-Message •• Simple equations can give rise to complexSimple equations can give rise to complex behaviorbehavior •• Chaotic systems are deterministic and not randomChaotic systems are deterministic and not random (but also unpredictable in the long(but also unpredictable in the long--term;term; prediction in the shortprediction in the short--term fairly accurate)term fairly accurate) •• Chaotic systems are extremely sensitive to theChaotic systems are extremely sensitive to the initial conditionsinitial conditions •• Presence of iterationsPresence of iterations •• The devil is in the detailsThe devil is in the details
45 Fractals •• Termed coined by Benoit Mandelbrot toTermed coined by Benoit Mandelbrot to differentiate pure geometric figuresdifferentiate pure geometric figures (circle, square,(circle, square, ……) from other types of) from other types of figures that defy such simplefigures that defy such simple classificationclassification •• Structure on all scales (detail persistsStructure on all scales (detail persists when zoomed arbitrarily)when zoomed arbitrarily) •• Geometrical objects generally with nonGeometrical objects generally with non-- integer dimensioninteger dimension •• SelfSelf--similarity (contains infinite copies ofsimilarity (contains infinite copies of itself)itself)
46 Fractals in Nature
47 Which One is Digital?
48 Fractals in the Human Body
49 Properties: Anyone?
50 Properties •• They are fractal!They are fractal! ☺☺ •• Display selfDisplay self--similaritysimilarity •• Display scaleDisplay scale--invarianceinvariance
51 Non-Fractal Size of features 1 cm 1 characteristic scale
52 Fractal 2 cm 1 cm 1/2 cm 1/4 cm manymany characteristic scalecharacteristic scaless Size ofSize of ffeatureseatures Structure on all scales (detail persists when zoomedStructure on all scales (detail persists when zoomed arbitrarily)arbitrarily)
53 Self-Similarity and Scale-Invariance •• ““When each piece of a shape is geometricallyWhen each piece of a shape is geometrically similar to the whole, both the shape and thesimilar to the whole, both the shape and the cascade that generate it are called selfcascade that generate it are called self-- similar.similar.”” (B. Mandelbrot)(B. Mandelbrot) •• CContains infinite copies of itselfontains infinite copies of itself •• Scaling/Scale =Scaling/Scale = The value measured for aThe value measured for a propertyproperty does notdoes not depend on the resolution atdepend on the resolution at which it is measuredwhich it is measured •• Two types of invariance:Two types of invariance: - Geometrical - Statistical ImportantImportant ConceptConcept
54 Self-Similarity and Scale-Invariance
55 Self-Similarity and Scale Invariance
56 Scale-Invariance
57 Scaling one measurement:one measurement: not sonot so interestinginteresting slope Logarithmof themeasurement Logarithmof themeasurement one value Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement scaling relationship:scaling relationship: muchmuch more interestingmore interesting Fractal?
58 Why Fractals? -- Packing efficiency: fractals are very good atPacking efficiency: fractals are very good at squeezing a great deal length (or surface)squeezing a great deal length (or surface) –– perhaps infiniteperhaps infinite –– into a small finite areainto a small finite area -- Reason: minimize material requirements whileReason: minimize material requirements while maximizing functionalitymaximizing functionality
59 Branching Patterns nerve cells in the retina, and in culture
60 Branching Patterns blood vessels Family, Masters, and Platt 1989 Physica D38:98-103 Mainster 1990 Eye 4:235-241 in the retina air ways in the lungs West and Goldberger 1987 Am. Sci. 75:354-365
61 Branching Patterns
62 Currents Through Ion Channels ATP sensitive potassium channel in cell from the pancreas Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391) 5 sec 5 msec 5 pA FC = 10 Hz FC = 1k Hz
63 How Are Fractal Patterns Generated? How are fractals grown?How are fractals grown? Three known mechanismsThree known mechanisms -- IterationIteration -- DiffusionDiffusion -- RecursionRecursion
64 Example: The Cantor Set (Mathematical Monster) Requires: iteration and recursionRequires: iteration and recursion Note: points in Cantor set areNote: points in Cantor set are uncountablyuncountably infinite but width of allinfinite but width of all the points is (2/3)^nthe points is (2/3)^n 0, for n0, for n 00
65 Example: The Koch Curve (Math. Monster) Question: what is the length ofQuestion: what is the length of the Koch curve after n=100the Koch curve after n=100 iterations if the length of oneiterations if the length of one segment is 1m?segment is 1m?
66 Example: Sierpinski Triangle
67 Example: The Mandelbrot Set
68 Example: The Mandelbrot Set Technically speaking:Technically speaking: ““iterative dynamical systemiterative dynamical system”” PseudoPseudo--code:code: For each number, c, in a subset of the complex planeFor each number, c, in a subset of the complex plane SetSet xtxt=0=0 For t=1 toFor t=1 to tmaxtmax ComputeCompute xtxt = xt^2 + c= xt^2 + c If |If |xtxt|>2, then break out of loop|>2, then break out of loop EndForEndFor If t<If t<tmaxtmax, then color point c white, then color point c white If t=If t=tmaxtmax, then color point c black, then color point c black EndForEndFor Definition: The Mandelbrot set is defined as the set of pointsDefinition: The Mandelbrot set is defined as the set of points thatthat never divergesnever diverges!!
69 Example: The Mandelbrot Set Special cases:Special cases: 1.1. c = 0c = 0 (origin of complex plane):(origin of complex plane): xtxt = xt^2 = 0= xt^2 = 0 (stays bounded)(stays bounded) 2.2. c = ic = i: x[0] = 0; x[1] = i; x[2] =: x[0] = 0; x[1] = i; x[2] = --1 + i;1 + i; …… 3.3. c = 1+ic = 1+i:: x[tx[t]] unboudedunbouded (sequence diverges before(sequence diverges before tmaxtmax iterations)iterations) 4.4. c = 0.01+ic = 0.01+i (compare with 2): sequence diverges(compare with 2): sequence diverges beforebefore tmaxtmax iterationsiterations Butterfly effect!Butterfly effect!
70 Example: The Mandelbrot Set For each pixel on the screen do:For each pixel on the screen do: {{ x =x = aa = x co= x co--ordinate of pixelordinate of pixel y =y = bb = y co= y co--ordinate of pixelordinate of pixel x2 = x*xx2 = x*x y2 = y*yy2 = y*y iteration = 0iteration = 0 maxitermaxiter = 1000= 1000 while ( x2 + y2 < (2*2) AND iteration <while ( x2 + y2 < (2*2) AND iteration < maxitermaxiter )) {{ x = x2x = x2 -- y2 +y2 + aa y = 2*x*y +y = 2*x*y + bb x2 = x*xx2 = x*x y2 = y*yy2 = y*y iteration =iteration = iterationiteration + 1+ 1 }} if ( iteration ==if ( iteration == maxitermaxiter )) color = blackcolor = black elseelse color = iterationcolor = iteration }} xn+1 = xn 2 - yn 2 + a yn+1 = 2xnyn + b
71 On Compressibility The Mandelbrot set despite itThe Mandelbrot set despite it’’s complex nature iss complex nature is actually extremely compressible in the sense thatactually extremely compressible in the sense that an algorithm for the image is far simpler to storean algorithm for the image is far simpler to store than the image itself!than the image itself!
72 The Mandelbrot Set
73 The Mandelbrot Set
74 Self-Similarity and Scale-Invariance
75 Diffusion-Limited Aggregation Is this a living organism?Is this a living organism?
76 Diffusion-Limited Aggregation What about this one?What about this one?
77 Diffusion-Limited Aggregation How was this pattern generated?How was this pattern generated?
78 Diffusion-Limited Aggregation Two processes:Two processes: -- DiffusionDiffusion -- AggregationAggregation
79 Diffusion (Random Walk) •• Diffusion is a random motionDiffusion is a random motion •• Although the motion of individual particles is totallyAlthough the motion of individual particles is totally random with respect to the direction, it may happen thatrandom with respect to the direction, it may happen that particles walk somewhat far relative to a starting pointparticles walk somewhat far relative to a starting point •• But, in contrary to a normal flow, where all particlesBut, in contrary to a normal flow, where all particles under investigation move more or less into the sameunder investigation move more or less into the same direction,direction, the average of walked distance of all particlesthe average of walked distance of all particles within a random walk is zerowithin a random walk is zero
80 •• When particles have the possibility to attract each otherWhen particles have the possibility to attract each other and stick together, they may formand stick together, they may form aggregatesaggregates •• The forces between the particles may be weak or strongThe forces between the particles may be weak or strong •• For particles which carry some electrical charge (ions) theFor particles which carry some electrical charge (ions) the forces are typically very strong and thus there is a gain inforces are typically very strong and thus there is a gain in energy when they build aggregatesenergy when they build aggregates Aggregation
81 Diffusion-Limited Aggregation In diffusionIn diffusion--limited aggregateslimited aggregates the shape of thethe shape of the structurestructure is controlled by the possibility of particles to reach theis controlled by the possibility of particles to reach the aggregateaggregate Starting with aStarting with a uniform distribution,uniform distribution, and someand some ‘‘seedseed’’ particles which start the growth processparticles which start the growth process The rate of growth is governed/limited by the diffusionThe rate of growth is governed/limited by the diffusion SSome particles might meetome particles might meet, and t, and the aggregate may growhe aggregate may grow as long there are particles moving aroundas long there are particles moving around Preferential attachment at tip of branchesPreferential attachment at tip of branches The volume is not filled in its entirety, but there areThe volume is not filled in its entirety, but there are many gapsmany gaps FractalityFractality!! Found in crystal growth, coral reefs, bacterial colonies,Found in crystal growth, coral reefs, bacterial colonies, and other natural systemsand other natural systems
82 Diffusion-Limited Aggregation http://apricot.ap.polyu.edu.hk/~lam/dla/dla.htmlhttp://apricot.ap.polyu.edu.hk/~lam/dla/dla.html
83 How Do We Measure Self-Similarity and Scaling?
84 Dimension •• A quantitative measure of selfA quantitative measure of self--similarity andsimilarity and scalingscaling •• Intuitively: The dimension tells us how manyIntuitively: The dimension tells us how many pieces we see when we look at a finer resolution.pieces we see when we look at a finer resolution. •• Definitions:Definitions: - Topological dimension (always integer) - Self-similarity dimension (fractal) - Capacity dimension (fractal) - Hausdorff dimension - Embedding dimension
85 Topological Dimension -- Always an integer (not fractal!)Always an integer (not fractal!) - in a minimal covering, each point of the object is covered by no more than G sets - Topological dimension d = G-1 - For a plane: G = 3 d = 3-1 = 2
86 Self-Similarity Dimension Simplest fractals are selfSimplest fractals are self--similar!similar! How many times does scaledHow many times does scaled--downdown copy fit into the whole object?copy fit into the whole object? N = number of copiesN = number of copies rr = scale factor= scale factor SelfSelf--ssimilarity dimension =imilarity dimension = ln(N)/lnln(N)/ln(r(r)) E.g. square: N = 9, r = 3E.g. square: N = 9, r = 3 SsdSsd == ln(9)/ln(3) = 2ln(9)/ln(3) = 2 What is the selfWhat is the self--similarity dimension ofsimilarity dimension of the Koch curve?the Koch curve?
87 Fractal Dimension: Koch Curve Log (number of new pieces) Log (factor of finer resolution) D = Log 4 = 1.2619= Log 3Log 3
88 Box Dimension •• Used for fractals that are not selfUsed for fractals that are not self--similarsimilar •• NN (r)(r) balls or squaresballs or squares of radiusof radius 1/1/r needed to cover ther needed to cover the objectobject •• Box dimensionBox dimension •• D = 1.65 (?)D = 1.65 (?) r rN d r ln )(ln lim 0→ =
89 Definition of a Fractal Mathematical:Mathematical: d(fractald(fractal) >) > d(topologicald(topological)) Origin:Origin: Fractal = fragmented, many pieces, one alsoFractal = fragmented, many pieces, one also speaks of fractal dimensionspeaks of fractal dimension
90 Example: Koch Curve Perimeter:Perimeter: d(fractald(fractal) = 1.2619) = 1.2619 d(topologicald(topological) = 1.) = 1. In other words: Perimeter covers more space than aIn other words: Perimeter covers more space than a 11--D line; covers less space than a 2D line; covers less space than a 2--D area.D area.
91 Fractal Dimensions in Biology: Measured •• surfaces of proteinssurfaces of proteins •• surface of cell membranessurface of cell membranes •• growth of bacterial coloniesgrowth of bacterial colonies •• islands of types of lipids inislands of types of lipids in cell membranescell membranes •• dendrites of neuronsdendrites of neurons •• blood flow in the heartblood flow in the heart •• blood vessels in the eye,blood vessels in the eye, heart, and lungheart, and lung •• shape of herpes simplexshape of herpes simplex •• ulcers in the corneaulcers in the cornea •• textures of Xtextures of X--rays of bonerays of bone and teethand teeth •• texture of radioisotopetexture of radioisotope tracer in the livertracer in the liver •• MANY MORE!MANY MORE!
92 Pulmonary Hypertension HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100 D = 1.65 normal 20% 02
93 D = 1.53 hypoxic 10% 02 HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100 Pulmonary Hypertension
94 Pulmonary Hypertension D = 1.43 hyperoxic 90% 02 HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100
95 Statistical Indicators of Fractal StructureMean More Data Mean More Data 0 More Data Mean ∞ areaarea 00 lengthlength infinf
96 Statistical Indicators of Fractal Structure Log avg density within radius r Log radius r r rN d r ln )(ln lim 0→ = Remember!
97 Statistical Indicators of Fractal Structure Log avg density within radius r Log radius r .5 -1.0 -2.0 -1.5 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00 -2.5 0 Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135
98 Natural Design Principles
99 Mathematics: the Language of Nature
100 On Growth and Form On Growth and Form (1917) laid the foundations ofOn Growth and Form (1917) laid the foundations of biobio--mathematicsmathematics ““Organic form itself is found, mathematicallyOrganic form itself is found, mathematically speaking, to be a function of time [speaking, to be a function of time [……] We might call] We might call the form of an organism an event in spacethe form of an organism an event in space--time, andtime, and not merely a configuration in space.not merely a configuration in space.”” EqsEqs. to describe static patterns of living organisms. to describe static patterns of living organisms Transforming form changing a few parametersTransforming form changing a few parameters ““In many growth processes of living organism,In many growth processes of living organism, especially of plants, regularly repeated appearancesespecially of plants, regularly repeated appearances of certainof certain multicellularmulticellular structures are readilystructures are readily noticeable [noticeable [……] In the case of a compound lead, for] In the case of a compound lead, for instance, some of the leaflets which are parts of ainstance, some of the leaflets which are parts of a leaf at an advanced stage, have the same shape asleaf at an advanced stage, have the same shape as the whole leave has at an earlier stage.the whole leave has at an earlier stage.”” (Herman et(Herman et al.)al.)
101 On Growth and Form
102 Affine Transformations Theory of transformation:Theory of transformation: Comparison of relatedComparison of related morphological forms (Thompson,morphological forms (Thompson, 1917)1917)
103 Homology of Structure
104 On Growth Three mechanisms:Three mechanisms: -- IterationIteration -- RecursionRecursion -- DiffusionDiffusion
105 On Growth and Form: L-Systems •• The selfThe self--similarity of plants is the result ofsimilarity of plants is the result of developmental processesdevelopmental processes •• Mathematical formalism introduced in 1968Mathematical formalism introduced in 1968 byby AristidAristid LindenmeyerLindenmeyer (1925(1925--1989)1989) •• Essentially, production systemEssentially, production system •• Productions are rewriting rules which stateProductions are rewriting rules which state how new symbols (or cells) can be producedhow new symbols (or cells) can be produced from old symbols (or cells)from old symbols (or cells)
106 Example •• Axiom = seedAxiom = seed ‘‘cellcell’’ •• Rule = description of how to grow newRule = description of how to grow new ‘‘cellcell’’ F = draw forwardF = draw forward G = go forward (but do not draw)G = go forward (but do not draw) + = right turn+ = right turn -- = left turn= left turn [ = save turtle[ = save turtle’’s current poss current pos ] = remove last saved pos] = remove last saved pos
107 L-Systems
108 More sophisticated L-Systems
109 Three-Dimensional L-Systems
110 Movies ……
111 Book Available for free online!Available for free online!
112 Project Idea http://http://algorithmicbotany.org/virtual_laboratoryalgorithmicbotany.org/virtual_laboratory//
113 L-Systems Needless to sayNeedless to say …… LL--Systems display fractalSystems display fractal structurestructure ☺☺
114 On Growth and Form: Morphogenesis Morphogenesis = morphology (shape, form) + genesisMorphogenesis = morphology (shape, form) + genesis (development)(development) Technically speaking: stage in embryonic developmentTechnically speaking: stage in embryonic development Cells begin to cluster and form patterns:Cells begin to cluster and form patterns: -- Cell differentiationCell differentiation -- Cell growthCell growth -- Cell divisionCell division -- Cell migrationCell migration -- Chemical secretion/diffusionChemical secretion/diffusion ……
115 Cell Differentiation: Vertebrates •• Three stagesThree stages •• Four cell typesFour cell types
116 On Growth and Form: Morphogenesis •• How does body structure arise?How does body structure arise? •• If arm cells had an extra cycle of cell division, our armsIf arm cells had an extra cycle of cell division, our arms would be 1.8m longwould be 1.8m long …… how is growth controlled?how is growth controlled? •• How is cell division turned on or off? The early embryoHow is cell division turned on or off? The early embryo is like a cancerous tumoris like a cancerous tumor –– growing very rapidlygrowing very rapidly –– butbut the embryothe embryo ““knowsknows”” when to stop growing and diving.when to stop growing and diving. •• What are theWhat are the mechanisms that determine themechanisms that determine the anatomical structure of an organismanatomical structure of an organism??
117 Striped Patterns and Spotted Bodies
118 Morphogenetic Model [Classic][Classic] ““The Chemical Basis of Morphogenesis,The Chemical Basis of Morphogenesis,”” 1952,1952, Phil. Trans.Phil. Trans. Roy. Soc. of LondonRoy. Soc. of London, Series B: Biological Sciences,, Series B: Biological Sciences, 237237:37:37——72.72.
119 Reaction-Diffusion Systems Key idea: Morphology is the result of two substancesKey idea: Morphology is the result of two substances ‘‘aa’’ andand ‘‘bb’’ ((morphogensmorphogens) diffusing at different rates) diffusing at different rates MorphogenMorphogen:: ““is simply the kind of substance concerned in thisis simply the kind of substance concerned in this theorytheory…”…” in fact, anything thatin fact, anything that diffusesdiffuses into the tissue andinto the tissue and ““somehow persuades it to develop along different lines fromsomehow persuades it to develop along different lines from those which would have been followed in its absencethose which would have been followed in its absence”” qualifies.qualifies. (Generator of form)(Generator of form) Investigation of pattern due to breakdown of symmetry andInvestigation of pattern due to breakdown of symmetry and homogeneity in initially homogeneous continuous mediahomogeneity in initially homogeneous continuous media Breakdown captured by set of differential equationsBreakdown captured by set of differential equations Equations capture dynamics of autocatalytic andEquations capture dynamics of autocatalytic and antagonistic chemical reactions with diffusionantagonistic chemical reactions with diffusion Result = Turing patternsResult = Turing patterns
120 Pattern Formation from Homogenous I.C. ActivatorActivator--Inhibitor ModelInhibitor Model Requirements:Requirements: -- Activator:Activator: local selflocal self--enhancement (positive feedback)enhancement (positive feedback) -- Inhibitor:Inhibitor: LongLong--range antagonistic effect (negative feedback)range antagonistic effect (negative feedback) -- Diffusion:Diffusion: Diffusion of inhibitor >> diffusion of activator (>7 timesDiffusion of inhibitor >> diffusion of activator (>7 times faster)faster) inhibitor must act rapidly (rapid adaptation of inhibitorinhibitor must act rapidly (rapid adaptation of inhibitor concentration)concentration) –– otherwise oscillations will occurotherwise oscillations will occur -- Decay (removal):Decay (removal): Decay inhibitor > decay activatorDecay inhibitor > decay activator Pattern formation:Pattern formation: -- Small deviations from a homogeneous distribution create a strongSmall deviations from a homogeneous distribution create a strong positive feedbackpositive feedback which causes deviations to grow even more.which causes deviations to grow even more. -- AA longlong--range antagonistic effectrange antagonistic effect restricts the selfrestricts the self--enhancing reactionenhancing reaction and causes a localization.and causes a localization.
121 Activator-Inhibitor Scheme
122 Diffusion: 1-D Case x a D t a txa a 2 2 ),( ∂ ∂ = ∂ ∂ •• Constant concentration difference betweenConstant concentration difference between neighboring cellsneighboring cells a(x,ta(x,t) and a(x+1,t): net) and a(x+1,t): net exchange of molecules by diffusion is 0exchange of molecules by diffusion is 0 •• Linear concentration gradient: net exchangeLinear concentration gradient: net exchange of molecules by diffusion is also 0 (whatof molecules by diffusion is also 0 (what goes out come in)goes out come in) •• Hence, second derivate!Hence, second derivate! first temporal derivative: rate second spatial derivative: flux
123 Activator-Inhibitor System: Sea Shells
124 Activator-Inhibitor System: Sea Shells 2 2 2 2 22 x b Dbbrsa t b x a Darb b a s t a bba aaa ∂ ∂ ++−= ∂ ∂ ∂ ∂ +−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ += ∂ ∂ a(x,ta(x,t),), b(x,tb(x,t): concentration of activator and inhibitor): concentration of activator and inhibitor s: ability of cells to perform autos: ability of cells to perform auto--catalysiscatalysis DaDa, Db: diffusion constants, Db: diffusion constants rara,, rbrb: decay rates: decay rates sasa22 /b : non/b : non--linear autocatalytic influencelinear autocatalytic influence --rraaaa : rate of removal (proportional to concentration): rate of removal (proportional to concentration) bbaa, b, bbb: basic activator/inhibitor production: basic activator/inhibitor production
125 Turing Patterns in a Chemical Medium Patterns arise spontaneously through competition between localizPatterns arise spontaneously through competition between localizeded autocatalytic chemical reaction and longautocatalytic chemical reaction and long--ranged diffusion of aranged diffusion of a substance inhibiting the reactionsubstance inhibiting the reaction
126 Results: Often Surprising!
127 Activator-Inhibitor System •• ShortShort--range autocatalysis / longrange autocatalysis / long--range inhibitionrange inhibition •• Local instability but global stabilizationLocal instability but global stabilization •• Behavior of system is often difficult to predictBehavior of system is often difficult to predict (fluctuations in initial conditions render accurate(fluctuations in initial conditions render accurate predictions impossible)predictions impossible)

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Internet Basica

  • 1. 11 Artificial Life & Complex Systems Lecture 3Lecture 3 PatternsPatterns May 25, 2007May 25, 2007 MaxMax LungarellaLungarella
  • 2. 2 Contents •• PatternsPatterns •• Dynamical SystemsDynamical Systems •• FractalsFractals •• LL--SystemsSystems •• Pattern formationPattern formation •• MorphogenesisMorphogenesis [Paper:[Paper: A.M.TuringA.M.Turing (1952) CLASSIC!](1952) CLASSIC!]
  • 3. 3 What is a Pattern? Pattern = organized arrangement of objects in space andPattern = organized arrangement of objects in space and timetime Examples of biological patterns:Examples of biological patterns: -- Schools of fishSchools of fish -- Raiding column of army antsRaiding column of army ants -- Ocular dominance stripes in visual cortexOcular dominance stripes in visual cortex -- Synchronous flashing of firefliesSynchronous flashing of fireflies -- Pigmentation patterns on shellsPigmentation patterns on shells inputs from left eye (black) andinputs from left eye (black) and right eye (white)right eye (white)firefliesfireflies school of fishschool of fish
  • 6. 6 Pattern Formation If Nature is at all economical (probably, yes), weIf Nature is at all economical (probably, yes), we can expect that she will choose to createcan expect that she will choose to create complex form not by piececomplex form not by piece--toto--piece constructionpiece construction by organizational andby organizational and patternpattern--forming phenomenaforming phenomena we see also in the nonwe see also in the non--living world.living world. If this is true, then there must be similarities in theIf this is true, then there must be similarities in the form and patterns of living and purely inorganicform and patterns of living and purely inorganic or physical systems.or physical systems.
  • 7. 7 How Does Nature Build Patterns? Buildings blocks are:Buildings blocks are: -- Living units (fish, ants, nerve cells,Living units (fish, ants, nerve cells, ……)) biological systemsbiological systems -- Inanimate objects (bits of dirt, sand,Inanimate objects (bits of dirt, sand, ……)) physical/physical/nonbiologicalnonbiological systemssystems Differences between physical and biologicalDifferences between physical and biological systems:systems: -- In biological systems the subunits are moreIn biological systems the subunits are more complexcomplex -- The interactions among the subunits areThe interactions among the subunits are influenced also by genetically controlledinfluenced also by genetically controlled properties (in physical systems interactions areproperties (in physical systems interactions are purely physical)purely physical)
  • 8. 8 How Are Patterns Formed? Any Ideas?
  • 9. 9 How Are Patterns Formed? Simple answer:Simple answer: ““Form follows functionForm follows function”” That is, the shape and structure of aThat is, the shape and structure of a biological/technological systembiological/technological system –– a proteina protein molecule, a limb, an organism, a colony, themolecule, a limb, an organism, a colony, the InternetInternet –– is that which best equips the system foris that which best equips the system for survivalsurvival A form which gives organism (system) anA form which gives organism (system) an evolutionary advantage tends to stickevolutionary advantage tends to stick Power of cumulative selectionPower of cumulative selection
  • 10. 10 Natural selection is not entirely satisfyingNatural selection is not entirely satisfying …… Why?Why? How Are Patterns Formed?
  • 11. 11 How Are Patterns Formed? 1)1) It says nothing about theIt says nothing about the underlyingunderlying mechanismsmechanisms!! 2)2) Why should the pattern on a shellWhy should the pattern on a shell be of advantage?be of advantage? Once you start to ask theOnce you start to ask the ‘‘how?how?’’ ofof mechanism, you are up against themechanism, you are up against the rules of chemistry, physics, andrules of chemistry, physics, and mechanism, and the questionmechanism, and the question becomes not justbecomes not just ‘‘is the formis the form successful?successful?’’ butbut ‘‘is it physicallyis it physically possible?possible?’’ (1917)(1917)
  • 12. 12 Another simple answer: Emergence!Another simple answer: Emergence! Patterns emerge from large populations ofPatterns emerge from large populations of interactinginteracting ‘‘unitsunits’’ (living organisms and non(living organisms and non--livingliving entities)entities) Emergence implies that the largeEmergence implies that the large--scale organizationscale organization could never be deduced by close inspection of thecould never be deduced by close inspection of the individual units (holistic perspective, e.g. Hindividual units (holistic perspective, e.g. H220)0) How Are Patterns Formed?
  • 13. 13 How Are Patterns Formed? Can simple rules give rise to patterns and forms?Can simple rules give rise to patterns and forms?
  • 14. 14 Mechanisms -- IterationIteration -- RecursionRecursion -- FeedbackFeedback Can give rise to selfCan give rise to self--similar structures such as trees,similar structures such as trees, plants, clouds, mountains,plants, clouds, mountains, ……
  • 15. 15 Characterization:Characterization: •• AA systemsystem thatthat evolves in time according to a set of rulesevolves in time according to a set of rules •• What changes in time is a variable (e.g. position,What changes in time is a variable (e.g. position, concentration, temperature, viscosity,concentration, temperature, viscosity, ……)) •• PPresent conditions determine the future.resent conditions determine the future. •• The rules are usually nonlinearThe rules are usually nonlinear (e.g. set of diff.(e.g. set of diff. eqeq. defining. defining rates of change)rates of change) •• There may be many interacting variablesThere may be many interacting variables What is a Dynamical System?
  • 17. 17 Other Examples of (Nonlinear) Dynamical Systems •• The Solar SystemThe Solar System •• The atmosphere (the weather)The atmosphere (the weather) •• The economy (stock market)The economy (stock market) •• The human body (heart, brain, lungs, ...)The human body (heart, brain, lungs, ...) •• Ecology (plant and animal populations)Ecology (plant and animal populations) •• Cancer growthCancer growth •• Spread of epidemicsSpread of epidemics •• Chemical reactionsChemical reactions •• The electrical power gridThe electrical power grid •• The InternetThe Internet
  • 18. 18 Def.: Map of variables in timeDef.: Map of variables in time -- Time is parameterTime is parameter -- Trajectory (orbit) in stateTrajectory (orbit) in state spacespace E.g.E.g. Z(tZ(t) = [x(t),x(t+1),x(t+2)]) = [x(t),x(t+1),x(t+2)] Interestingly, forInterestingly, for continuouscontinuous (reversible) systems(reversible) systems: Only: Only one trajectory passesone trajectory passes through each point of athrough each point of a statestate--space (not true inspace (not true in discrete systems)discrete systems) Phase or State-Space
  • 20. 20 Attractors PhasePhase--space volume to where dynamical system convergesspace volume to where dynamical system converges asymptotically over timeasymptotically over time
  • 21. 21 Why Attractor Behavior? Energy dissipation (thermodynamic systems)Energy dissipation (thermodynamic systems) -- Friction, thermodynamic loss, loss of material, etc.Friction, thermodynamic loss, loss of material, etc. -- Volume contraction in phaseVolume contraction in phase--space (system tends tospace (system tends to restrict itself to small basins of attraction)restrict itself to small basins of attraction) -- SelfSelf--organization (dissipative systems)organization (dissipative systems) Compare withCompare with …… Hamiltonian systemsHamiltonian systems -- frictionless, no attractorsfrictionless, no attractors -- Conservation of energyConservation of energy -- ergodicityergodicity
  • 22. 22 Types of Attractors Fixed pointFixed point Limit cycleLimit cycle QuasiQuasi--periodicperiodic StrangeStrange
  • 24. 24 Fix-Point Behavior •• 00--dimensional attractordimensional attractor •• Volume of phase space to which the systemVolume of phase space to which the system converges after a long enough time = basin ofconverges after a long enough time = basin of attraction (defined by all trajectories leading intoattraction (defined by all trajectories leading into the attractor)the attractor) •• Example: Pendulum when friction and gravityExample: Pendulum when friction and gravity bring its motion to a haltbring its motion to a halt
  • 25. 25 Limit Cycle Behavior •• Periodic motionPeriodic motion •• Repetitive oscillation among a number of statesRepetitive oscillation among a number of states (e.g. loop)(e.g. loop) •• Example: lone planet orbiting around a star in anExample: lone planet orbiting around a star in an elliptical orbitelliptical orbit
  • 26. 26 Quasi-Periodic Attractor •• Several independent cyclic motionsSeveral independent cyclic motions •• Never quite repeat themselvesNever quite repeat themselves •• ToroidalToroidal attractorsattractors •• Example: the moon orbits Earth, which orbits theExample: the moon orbits Earth, which orbits the sun, which, in turn, orbits the galactic centersun, which, in turn, orbits the galactic center
  • 27. 27 Strange or Chaotic Attractors -- Pervasive (chaos is everywhere: weather patterns, humanPervasive (chaos is everywhere: weather patterns, human brainbrain’’s electrochemical activity,s electrochemical activity, ……)) -- Sensitivity to initial conditions (Butterfly effect)Sensitivity to initial conditions (Butterfly effect) -- Deterministic chaos (if we could know the exact initialDeterministic chaos (if we could know the exact initial conditions, trajectories would be determined)conditions, trajectories would be determined) -- LowLow--dimensional chaos (strange attractors are restricteddimensional chaos (strange attractors are restricted to small volumes of phaseto small volumes of phase--space)space) -- Weak causalityWeak causality - 3-body problem (any slight measurement difference results in very different predictions) - Butterfly effect (Lorenz attractor)
  • 29. 29 Strange Attractors Limit set asLimit set as tt →→ ∞∞ Set of measure zeroSet of measure zero Basin of attractionBasin of attraction Fractal structureFractal structure • non-integer dimension • self-similarity • infinite detail Chaotic dynamicsChaotic dynamics • sensitivity to initial conditions • topological transitivity • dense periodic orbits Aesthetic appealAesthetic appeal
  • 30. 30 Characteristics of Chaos •• Never repeatsNever repeats •• Depends sensitively on initial conditions (Butterfly effect)Depends sensitively on initial conditions (Butterfly effect) •• Allows shortAllows short--term prediction but not longterm prediction but not long--term predictionterm prediction •• Comes and goes with a small change in some control knobComes and goes with a small change in some control knob •• Usually produces a fractal patternUsually produces a fractal pattern •• Not random (or stochastic); unpredictable in the shortNot random (or stochastic); unpredictable in the short-- term butterm but ““predictablepredictable”” in the longin the long--termterm ……
  • 31. 31 Example 1: A Planet Orbiting a Star Elliptical OrbitElliptical Orbit Chaotic OrbitChaotic Orbit
  • 32. 32 Example 2: The Logistic Map Also known as the quadratic map or the Feigenbaum map Simple population growth model defined by the iterative map: x(t+1) = rx(t)*(1-x(t)), 0 < r < 4, 0 < x < 1 (r = reproduction rate or bifurcation parameter) if x = 0: population is extinct if x = 1: population will be extinct at the next time step rx(t) positive feedback term 1-x(t) negative feedback term (decrease to overpopulation)
  • 33. 33 Example 2: The Logistic Map Question: What happens to the longQuestion: What happens to the long--term behaviorterm behavior ofof x(tx(t) for different values of) for different values of ‘‘rr’’?? Special cases:Special cases: 0<r<1: all terms <10<r<1: all terms <1 extinctionextinction 1<r<3: fix point behavior1<r<3: fix point behavior r>3: period doubling (bifurcation)r>3: period doubling (bifurcation)
  • 34. 34 Logistic Map – Fix Point
  • 35. 35 Logistic Map – Chaos Cobweb plot at:Cobweb plot at: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/CobwebPlot.htmcs/yanikjoe/Chaos/CobwebPlot.htm
  • 36. 36 The Logistic Map – Four Types of Attractors
  • 37. 37 The Logistic Map – Bifurcation Diagram DEMO:DEMO: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/Bifurcation.htmcs/yanikjoe/Chaos/Bifurcation.htm
  • 38. 38 Logistic Map – Bifurcations and Self-Similarity
  • 39. 39 Discovered sensitivity to initial conditions inDiscovered sensitivity to initial conditions in a simple 3a simple 3--variable dynamical system;variable dynamical system; simplified model of weather; convectionsimplified model of weather; convection flows in the atmosphere (x: convection; y:flows in the atmosphere (x: convection; y: temperature difference; z: distortion of thetemperature difference; z: distortion of the temperature profile)temperature profile) DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.htmlhttp://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.html DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.htmlhttp://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html Example 3: Lorenz Attractor
  • 40. 40 Example 3: Lorenz Attractor
  • 41. 41 Example 3: The Hénon Attractor xn+1 = a - xn 2 + bxn-1 Example ofExample of infinite detailinfinite detail ……
  • 42. 42 Example 3: The Hénon Attractor
  • 43. 43 Example 4: General 2-D Quadratic Map xn+1 = a1 + a2xn + a3xn 2 + a4xnyn + a5yn + a6yn 2 yn+1 = a7 + a8xn + a9xn 2 + a10xnyn + a11yn + a12yn 2 xn+1 = xn 2 - yn 2 + a yn+1 = 2xnyn + b a b
  • 44. 44 Take-Home-Message •• Simple equations can give rise to complexSimple equations can give rise to complex behaviorbehavior •• Chaotic systems are deterministic and not randomChaotic systems are deterministic and not random (but also unpredictable in the long(but also unpredictable in the long--term;term; prediction in the shortprediction in the short--term fairly accurate)term fairly accurate) •• Chaotic systems are extremely sensitive to theChaotic systems are extremely sensitive to the initial conditionsinitial conditions •• Presence of iterationsPresence of iterations •• The devil is in the detailsThe devil is in the details
  • 45. 45 Fractals •• Termed coined by Benoit Mandelbrot toTermed coined by Benoit Mandelbrot to differentiate pure geometric figuresdifferentiate pure geometric figures (circle, square,(circle, square, ……) from other types of) from other types of figures that defy such simplefigures that defy such simple classificationclassification •• Structure on all scales (detail persistsStructure on all scales (detail persists when zoomed arbitrarily)when zoomed arbitrarily) •• Geometrical objects generally with nonGeometrical objects generally with non-- integer dimensioninteger dimension •• SelfSelf--similarity (contains infinite copies ofsimilarity (contains infinite copies of itself)itself)
  • 47. 47 Which One is Digital?
  • 48. 48 Fractals in the Human Body
  • 50. 50 Properties •• They are fractal!They are fractal! ☺☺ •• Display selfDisplay self--similaritysimilarity •• Display scaleDisplay scale--invarianceinvariance
  • 51. 51 Non-Fractal Size of features 1 cm 1 characteristic scale
  • 52. 52 Fractal 2 cm 1 cm 1/2 cm 1/4 cm manymany characteristic scalecharacteristic scaless Size ofSize of ffeatureseatures Structure on all scales (detail persists when zoomedStructure on all scales (detail persists when zoomed arbitrarily)arbitrarily)
  • 53. 53 Self-Similarity and Scale-Invariance •• ““When each piece of a shape is geometricallyWhen each piece of a shape is geometrically similar to the whole, both the shape and thesimilar to the whole, both the shape and the cascade that generate it are called selfcascade that generate it are called self-- similar.similar.”” (B. Mandelbrot)(B. Mandelbrot) •• CContains infinite copies of itselfontains infinite copies of itself •• Scaling/Scale =Scaling/Scale = The value measured for aThe value measured for a propertyproperty does notdoes not depend on the resolution atdepend on the resolution at which it is measuredwhich it is measured •• Two types of invariance:Two types of invariance: - Geometrical - Statistical ImportantImportant ConceptConcept
  • 57. 57 Scaling one measurement:one measurement: not sonot so interestinginteresting slope Logarithmof themeasurement Logarithmof themeasurement one value Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement scaling relationship:scaling relationship: muchmuch more interestingmore interesting Fractal?
  • 58. 58 Why Fractals? -- Packing efficiency: fractals are very good atPacking efficiency: fractals are very good at squeezing a great deal length (or surface)squeezing a great deal length (or surface) –– perhaps infiniteperhaps infinite –– into a small finite areainto a small finite area -- Reason: minimize material requirements whileReason: minimize material requirements while maximizing functionalitymaximizing functionality
  • 59. 59 Branching Patterns nerve cells in the retina, and in culture
  • 60. 60 Branching Patterns blood vessels Family, Masters, and Platt 1989 Physica D38:98-103 Mainster 1990 Eye 4:235-241 in the retina air ways in the lungs West and Goldberger 1987 Am. Sci. 75:354-365
  • 62. 62 Currents Through Ion Channels ATP sensitive potassium channel in cell from the pancreas Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391) 5 sec 5 msec 5 pA FC = 10 Hz FC = 1k Hz
  • 63. 63 How Are Fractal Patterns Generated? How are fractals grown?How are fractals grown? Three known mechanismsThree known mechanisms -- IterationIteration -- DiffusionDiffusion -- RecursionRecursion
  • 64. 64 Example: The Cantor Set (Mathematical Monster) Requires: iteration and recursionRequires: iteration and recursion Note: points in Cantor set areNote: points in Cantor set are uncountablyuncountably infinite but width of allinfinite but width of all the points is (2/3)^nthe points is (2/3)^n 0, for n0, for n 00
  • 65. 65 Example: The Koch Curve (Math. Monster) Question: what is the length ofQuestion: what is the length of the Koch curve after n=100the Koch curve after n=100 iterations if the length of oneiterations if the length of one segment is 1m?segment is 1m?
  • 68. 68 Example: The Mandelbrot Set Technically speaking:Technically speaking: ““iterative dynamical systemiterative dynamical system”” PseudoPseudo--code:code: For each number, c, in a subset of the complex planeFor each number, c, in a subset of the complex plane SetSet xtxt=0=0 For t=1 toFor t=1 to tmaxtmax ComputeCompute xtxt = xt^2 + c= xt^2 + c If |If |xtxt|>2, then break out of loop|>2, then break out of loop EndForEndFor If t<If t<tmaxtmax, then color point c white, then color point c white If t=If t=tmaxtmax, then color point c black, then color point c black EndForEndFor Definition: The Mandelbrot set is defined as the set of pointsDefinition: The Mandelbrot set is defined as the set of points thatthat never divergesnever diverges!!
  • 69. 69 Example: The Mandelbrot Set Special cases:Special cases: 1.1. c = 0c = 0 (origin of complex plane):(origin of complex plane): xtxt = xt^2 = 0= xt^2 = 0 (stays bounded)(stays bounded) 2.2. c = ic = i: x[0] = 0; x[1] = i; x[2] =: x[0] = 0; x[1] = i; x[2] = --1 + i;1 + i; …… 3.3. c = 1+ic = 1+i:: x[tx[t]] unboudedunbouded (sequence diverges before(sequence diverges before tmaxtmax iterations)iterations) 4.4. c = 0.01+ic = 0.01+i (compare with 2): sequence diverges(compare with 2): sequence diverges beforebefore tmaxtmax iterationsiterations Butterfly effect!Butterfly effect!
  • 70. 70 Example: The Mandelbrot Set For each pixel on the screen do:For each pixel on the screen do: {{ x =x = aa = x co= x co--ordinate of pixelordinate of pixel y =y = bb = y co= y co--ordinate of pixelordinate of pixel x2 = x*xx2 = x*x y2 = y*yy2 = y*y iteration = 0iteration = 0 maxitermaxiter = 1000= 1000 while ( x2 + y2 < (2*2) AND iteration <while ( x2 + y2 < (2*2) AND iteration < maxitermaxiter )) {{ x = x2x = x2 -- y2 +y2 + aa y = 2*x*y +y = 2*x*y + bb x2 = x*xx2 = x*x y2 = y*yy2 = y*y iteration =iteration = iterationiteration + 1+ 1 }} if ( iteration ==if ( iteration == maxitermaxiter )) color = blackcolor = black elseelse color = iterationcolor = iteration }} xn+1 = xn 2 - yn 2 + a yn+1 = 2xnyn + b
  • 71. 71 On Compressibility The Mandelbrot set despite itThe Mandelbrot set despite it’’s complex nature iss complex nature is actually extremely compressible in the sense thatactually extremely compressible in the sense that an algorithm for the image is far simpler to storean algorithm for the image is far simpler to store than the image itself!than the image itself!
  • 75. 75 Diffusion-Limited Aggregation Is this a living organism?Is this a living organism?
  • 76. 76 Diffusion-Limited Aggregation What about this one?What about this one?
  • 77. 77 Diffusion-Limited Aggregation How was this pattern generated?How was this pattern generated?
  • 78. 78 Diffusion-Limited Aggregation Two processes:Two processes: -- DiffusionDiffusion -- AggregationAggregation
  • 79. 79 Diffusion (Random Walk) •• Diffusion is a random motionDiffusion is a random motion •• Although the motion of individual particles is totallyAlthough the motion of individual particles is totally random with respect to the direction, it may happen thatrandom with respect to the direction, it may happen that particles walk somewhat far relative to a starting pointparticles walk somewhat far relative to a starting point •• But, in contrary to a normal flow, where all particlesBut, in contrary to a normal flow, where all particles under investigation move more or less into the sameunder investigation move more or less into the same direction,direction, the average of walked distance of all particlesthe average of walked distance of all particles within a random walk is zerowithin a random walk is zero
  • 80. 80 •• When particles have the possibility to attract each otherWhen particles have the possibility to attract each other and stick together, they may formand stick together, they may form aggregatesaggregates •• The forces between the particles may be weak or strongThe forces between the particles may be weak or strong •• For particles which carry some electrical charge (ions) theFor particles which carry some electrical charge (ions) the forces are typically very strong and thus there is a gain inforces are typically very strong and thus there is a gain in energy when they build aggregatesenergy when they build aggregates Aggregation
  • 81. 81 Diffusion-Limited Aggregation In diffusionIn diffusion--limited aggregateslimited aggregates the shape of thethe shape of the structurestructure is controlled by the possibility of particles to reach theis controlled by the possibility of particles to reach the aggregateaggregate Starting with aStarting with a uniform distribution,uniform distribution, and someand some ‘‘seedseed’’ particles which start the growth processparticles which start the growth process The rate of growth is governed/limited by the diffusionThe rate of growth is governed/limited by the diffusion SSome particles might meetome particles might meet, and t, and the aggregate may growhe aggregate may grow as long there are particles moving aroundas long there are particles moving around Preferential attachment at tip of branchesPreferential attachment at tip of branches The volume is not filled in its entirety, but there areThe volume is not filled in its entirety, but there are many gapsmany gaps FractalityFractality!! Found in crystal growth, coral reefs, bacterial colonies,Found in crystal growth, coral reefs, bacterial colonies, and other natural systemsand other natural systems
  • 83. 83 How Do We Measure Self-Similarity and Scaling?
  • 84. 84 Dimension •• A quantitative measure of selfA quantitative measure of self--similarity andsimilarity and scalingscaling •• Intuitively: The dimension tells us how manyIntuitively: The dimension tells us how many pieces we see when we look at a finer resolution.pieces we see when we look at a finer resolution. •• Definitions:Definitions: - Topological dimension (always integer) - Self-similarity dimension (fractal) - Capacity dimension (fractal) - Hausdorff dimension - Embedding dimension
  • 85. 85 Topological Dimension -- Always an integer (not fractal!)Always an integer (not fractal!) - in a minimal covering, each point of the object is covered by no more than G sets - Topological dimension d = G-1 - For a plane: G = 3 d = 3-1 = 2
  • 86. 86 Self-Similarity Dimension Simplest fractals are selfSimplest fractals are self--similar!similar! How many times does scaledHow many times does scaled--downdown copy fit into the whole object?copy fit into the whole object? N = number of copiesN = number of copies rr = scale factor= scale factor SelfSelf--ssimilarity dimension =imilarity dimension = ln(N)/lnln(N)/ln(r(r)) E.g. square: N = 9, r = 3E.g. square: N = 9, r = 3 SsdSsd == ln(9)/ln(3) = 2ln(9)/ln(3) = 2 What is the selfWhat is the self--similarity dimension ofsimilarity dimension of the Koch curve?the Koch curve?
  • 87. 87 Fractal Dimension: Koch Curve Log (number of new pieces) Log (factor of finer resolution) D = Log 4 = 1.2619= Log 3Log 3
  • 88. 88 Box Dimension •• Used for fractals that are not selfUsed for fractals that are not self--similarsimilar •• NN (r)(r) balls or squaresballs or squares of radiusof radius 1/1/r needed to cover ther needed to cover the objectobject •• Box dimensionBox dimension •• D = 1.65 (?)D = 1.65 (?) r rN d r ln )(ln lim 0→ =
  • 89. 89 Definition of a Fractal Mathematical:Mathematical: d(fractald(fractal) >) > d(topologicald(topological)) Origin:Origin: Fractal = fragmented, many pieces, one alsoFractal = fragmented, many pieces, one also speaks of fractal dimensionspeaks of fractal dimension
  • 90. 90 Example: Koch Curve Perimeter:Perimeter: d(fractald(fractal) = 1.2619) = 1.2619 d(topologicald(topological) = 1.) = 1. In other words: Perimeter covers more space than aIn other words: Perimeter covers more space than a 11--D line; covers less space than a 2D line; covers less space than a 2--D area.D area.
  • 91. 91 Fractal Dimensions in Biology: Measured •• surfaces of proteinssurfaces of proteins •• surface of cell membranessurface of cell membranes •• growth of bacterial coloniesgrowth of bacterial colonies •• islands of types of lipids inislands of types of lipids in cell membranescell membranes •• dendrites of neuronsdendrites of neurons •• blood flow in the heartblood flow in the heart •• blood vessels in the eye,blood vessels in the eye, heart, and lungheart, and lung •• shape of herpes simplexshape of herpes simplex •• ulcers in the corneaulcers in the cornea •• textures of Xtextures of X--rays of bonerays of bone and teethand teeth •• texture of radioisotopetexture of radioisotope tracer in the livertracer in the liver •• MANY MORE!MANY MORE!
  • 92. 92 Pulmonary Hypertension HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100 D = 1.65 normal 20% 02
  • 93. 93 D = 1.53 hypoxic 10% 02 HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100 Pulmonary Hypertension
  • 94. 94 Pulmonary Hypertension D = 1.43 hyperoxic 90% 02 HIGH BLOOD PRESSURE IN THE LUNGS Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser 1990 Circ. Suppl, lll, 82: 100
  • 95. 95 Statistical Indicators of Fractal StructureMean More Data Mean More Data 0 More Data Mean ∞ areaarea 00 lengthlength infinf
  • 96. 96 Statistical Indicators of Fractal Structure Log avg density within radius r Log radius r r rN d r ln )(ln lim 0→ = Remember!
  • 97. 97 Statistical Indicators of Fractal Structure Log avg density within radius r Log radius r .5 -1.0 -2.0 -1.5 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00 -2.5 0 Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135
  • 100. 100 On Growth and Form On Growth and Form (1917) laid the foundations ofOn Growth and Form (1917) laid the foundations of biobio--mathematicsmathematics ““Organic form itself is found, mathematicallyOrganic form itself is found, mathematically speaking, to be a function of time [speaking, to be a function of time [……] We might call] We might call the form of an organism an event in spacethe form of an organism an event in space--time, andtime, and not merely a configuration in space.not merely a configuration in space.”” EqsEqs. to describe static patterns of living organisms. to describe static patterns of living organisms Transforming form changing a few parametersTransforming form changing a few parameters ““In many growth processes of living organism,In many growth processes of living organism, especially of plants, regularly repeated appearancesespecially of plants, regularly repeated appearances of certainof certain multicellularmulticellular structures are readilystructures are readily noticeable [noticeable [……] In the case of a compound lead, for] In the case of a compound lead, for instance, some of the leaflets which are parts of ainstance, some of the leaflets which are parts of a leaf at an advanced stage, have the same shape asleaf at an advanced stage, have the same shape as the whole leave has at an earlier stage.the whole leave has at an earlier stage.”” (Herman et(Herman et al.)al.)
  • 102. 102 Affine Transformations Theory of transformation:Theory of transformation: Comparison of relatedComparison of related morphological forms (Thompson,morphological forms (Thompson, 1917)1917)
  • 104. 104 On Growth Three mechanisms:Three mechanisms: -- IterationIteration -- RecursionRecursion -- DiffusionDiffusion
  • 105. 105 On Growth and Form: L-Systems •• The selfThe self--similarity of plants is the result ofsimilarity of plants is the result of developmental processesdevelopmental processes •• Mathematical formalism introduced in 1968Mathematical formalism introduced in 1968 byby AristidAristid LindenmeyerLindenmeyer (1925(1925--1989)1989) •• Essentially, production systemEssentially, production system •• Productions are rewriting rules which stateProductions are rewriting rules which state how new symbols (or cells) can be producedhow new symbols (or cells) can be produced from old symbols (or cells)from old symbols (or cells)
  • 106. 106 Example •• Axiom = seedAxiom = seed ‘‘cellcell’’ •• Rule = description of how to grow newRule = description of how to grow new ‘‘cellcell’’ F = draw forwardF = draw forward G = go forward (but do not draw)G = go forward (but do not draw) + = right turn+ = right turn -- = left turn= left turn [ = save turtle[ = save turtle’’s current poss current pos ] = remove last saved pos] = remove last saved pos
  • 111. 111 Book Available for free online!Available for free online!
  • 113. 113 L-Systems Needless to sayNeedless to say …… LL--Systems display fractalSystems display fractal structurestructure ☺☺
  • 114. 114 On Growth and Form: Morphogenesis Morphogenesis = morphology (shape, form) + genesisMorphogenesis = morphology (shape, form) + genesis (development)(development) Technically speaking: stage in embryonic developmentTechnically speaking: stage in embryonic development Cells begin to cluster and form patterns:Cells begin to cluster and form patterns: -- Cell differentiationCell differentiation -- Cell growthCell growth -- Cell divisionCell division -- Cell migrationCell migration -- Chemical secretion/diffusionChemical secretion/diffusion ……
  • 115. 115 Cell Differentiation: Vertebrates •• Three stagesThree stages •• Four cell typesFour cell types
  • 116. 116 On Growth and Form: Morphogenesis •• How does body structure arise?How does body structure arise? •• If arm cells had an extra cycle of cell division, our armsIf arm cells had an extra cycle of cell division, our arms would be 1.8m longwould be 1.8m long …… how is growth controlled?how is growth controlled? •• How is cell division turned on or off? The early embryoHow is cell division turned on or off? The early embryo is like a cancerous tumoris like a cancerous tumor –– growing very rapidlygrowing very rapidly –– butbut the embryothe embryo ““knowsknows”” when to stop growing and diving.when to stop growing and diving. •• What are theWhat are the mechanisms that determine themechanisms that determine the anatomical structure of an organismanatomical structure of an organism??
  • 117. 117 Striped Patterns and Spotted Bodies
  • 118. 118 Morphogenetic Model [Classic][Classic] ““The Chemical Basis of Morphogenesis,The Chemical Basis of Morphogenesis,”” 1952,1952, Phil. Trans.Phil. Trans. Roy. Soc. of LondonRoy. Soc. of London, Series B: Biological Sciences,, Series B: Biological Sciences, 237237:37:37——72.72.
  • 119. 119 Reaction-Diffusion Systems Key idea: Morphology is the result of two substancesKey idea: Morphology is the result of two substances ‘‘aa’’ andand ‘‘bb’’ ((morphogensmorphogens) diffusing at different rates) diffusing at different rates MorphogenMorphogen:: ““is simply the kind of substance concerned in thisis simply the kind of substance concerned in this theorytheory…”…” in fact, anything thatin fact, anything that diffusesdiffuses into the tissue andinto the tissue and ““somehow persuades it to develop along different lines fromsomehow persuades it to develop along different lines from those which would have been followed in its absencethose which would have been followed in its absence”” qualifies.qualifies. (Generator of form)(Generator of form) Investigation of pattern due to breakdown of symmetry andInvestigation of pattern due to breakdown of symmetry and homogeneity in initially homogeneous continuous mediahomogeneity in initially homogeneous continuous media Breakdown captured by set of differential equationsBreakdown captured by set of differential equations Equations capture dynamics of autocatalytic andEquations capture dynamics of autocatalytic and antagonistic chemical reactions with diffusionantagonistic chemical reactions with diffusion Result = Turing patternsResult = Turing patterns
  • 120. 120 Pattern Formation from Homogenous I.C. ActivatorActivator--Inhibitor ModelInhibitor Model Requirements:Requirements: -- Activator:Activator: local selflocal self--enhancement (positive feedback)enhancement (positive feedback) -- Inhibitor:Inhibitor: LongLong--range antagonistic effect (negative feedback)range antagonistic effect (negative feedback) -- Diffusion:Diffusion: Diffusion of inhibitor >> diffusion of activator (>7 timesDiffusion of inhibitor >> diffusion of activator (>7 times faster)faster) inhibitor must act rapidly (rapid adaptation of inhibitorinhibitor must act rapidly (rapid adaptation of inhibitor concentration)concentration) –– otherwise oscillations will occurotherwise oscillations will occur -- Decay (removal):Decay (removal): Decay inhibitor > decay activatorDecay inhibitor > decay activator Pattern formation:Pattern formation: -- Small deviations from a homogeneous distribution create a strongSmall deviations from a homogeneous distribution create a strong positive feedbackpositive feedback which causes deviations to grow even more.which causes deviations to grow even more. -- AA longlong--range antagonistic effectrange antagonistic effect restricts the selfrestricts the self--enhancing reactionenhancing reaction and causes a localization.and causes a localization.
  • 122. 122 Diffusion: 1-D Case x a D t a txa a 2 2 ),( ∂ ∂ = ∂ ∂ •• Constant concentration difference betweenConstant concentration difference between neighboring cellsneighboring cells a(x,ta(x,t) and a(x+1,t): net) and a(x+1,t): net exchange of molecules by diffusion is 0exchange of molecules by diffusion is 0 •• Linear concentration gradient: net exchangeLinear concentration gradient: net exchange of molecules by diffusion is also 0 (whatof molecules by diffusion is also 0 (what goes out come in)goes out come in) •• Hence, second derivate!Hence, second derivate! first temporal derivative: rate second spatial derivative: flux
  • 124. 124 Activator-Inhibitor System: Sea Shells 2 2 2 2 22 x b Dbbrsa t b x a Darb b a s t a bba aaa ∂ ∂ ++−= ∂ ∂ ∂ ∂ +−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ += ∂ ∂ a(x,ta(x,t),), b(x,tb(x,t): concentration of activator and inhibitor): concentration of activator and inhibitor s: ability of cells to perform autos: ability of cells to perform auto--catalysiscatalysis DaDa, Db: diffusion constants, Db: diffusion constants rara,, rbrb: decay rates: decay rates sasa22 /b : non/b : non--linear autocatalytic influencelinear autocatalytic influence --rraaaa : rate of removal (proportional to concentration): rate of removal (proportional to concentration) bbaa, b, bbb: basic activator/inhibitor production: basic activator/inhibitor production
  • 125. 125 Turing Patterns in a Chemical Medium Patterns arise spontaneously through competition between localizPatterns arise spontaneously through competition between localizeded autocatalytic chemical reaction and longautocatalytic chemical reaction and long--ranged diffusion of aranged diffusion of a substance inhibiting the reactionsubstance inhibiting the reaction
  • 127. 127 Activator-Inhibitor System •• ShortShort--range autocatalysis / longrange autocatalysis / long--range inhibitionrange inhibition •• Local instability but global stabilizationLocal instability but global stabilization •• Behavior of system is often difficult to predictBehavior of system is often difficult to predict (fluctuations in initial conditions render accurate(fluctuations in initial conditions render accurate predictions impossible)predictions impossible)