Fractal Dynamics in PhysiologyAlterations with Disease and Aging

Presentation by Furkan KIRAÇ

OUTLINE

• What is fractal?– Its roots and motivation– Fractal Dimension– Mandelbrot Set– Julia Set

• What is chaos? How is it related to real life?– Logistic Equation– Feigenbaum’s Constant

• How are fractals related to chaos and life?– The paper ‘Fractal Dynamics in Physiology’

How long is the coast of Britain?

• Suppose the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat again:

What is fractal?

• Mandelbrot invented the word ‘fractal’ in 1967:– Latin adjective fractus.– The corresponding Latin verb frangere means

“to create irregular fragments”

• Fractal means a “composition of irregular fragments”.

Fractal Dimension (FD)• FD can be calculated by taking the limit of the quotient of the log change in object

size and the log change in measurement scale, as the measurement scale approaches zero.

Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal.

blow up the line by a factor of two.

The line is now twice as long as before.

FD = Log 2 / Log 2 = 1

Consider a straight line. Consider a square

FD = Log 4 / Log 2 = 2

Consider a snowflake curve formed by repeatedly replacing ___ with _/\_

FD = Log 4 / Log 3 = 1.261

Mandelbrot Set• How is it actually computed? The basic algorithm is:• For each pixel c, start with z=0.• Repeat z = z2 + c up to N times, exiting if the magnitude of z gets large.• If loop reaches to N, the point is probably inside the Mandelbrot set.• If point exits the view, it can be colored according to how many iterations

were completed.

Why do you start with z=0

• Zero is the critical point of z2+c, that is, a point where d/dz (z2+c) = 0. If you replace z2+c with a different function, the starting value will have to be modified.

• E.g. for z = z2+z+c, the critical point is given by 2z+1=0, so start with z=-1/2.

• In some cases, there may be multiple critical values, so they all should be tested.

What is the area of Mandelbrot set?

• Ewing and Schober computed an area estimate using 240,000 terms of the Laurent series. The result is 1.7274... However, the Laurent series converges very slowly, so this is a poor estimate.

• A project to measure the area via counting pixels on a very dense grid shows an area around 1.5066.

• Hill and Fisher used distance estimation techniques to rigorously bound the area and found the area is between 1.503 and 1.5701.

Julia Set

• The Mandelbrot set iterates z2+c with z starting at 0 and varying c.

• The Julia set iterates z2+c for fixed c and varying starting z values.

• Mandelbrot set is in parameter space (c-plane) • while the Julia set is in variable space (z-plane).

Julia Set Example

What is Chaos Theory?

• A non-linear dynamical system can exhibit one or more of the following types of behaviour: – forever at rest – forever expanding– periodic motion– quasi-periodic motion– chaotic motion

• The type of behavior may depend on the initial state of the system and the values of its parameters, if any.

Demonstration of chaos!• Chaos is unpredictable behavior arising in a deterministic

system because of great sensitivity to initial conditions. • Chaos arises if two arbitrarily close starting points diverge

exponentially, so that future behavior is unpredictable.

• Let’s consider the following sequence: xn+1 = 4.xn.(1-xn)Iter # Sequence #1 Sequence #2 difference#1 0.7000000000 0.7000000001 0.0000000001#2 0.8400000000 0.8399999998 0.0000000002#3 0.5376000000 0.5376000004 0.0000000004…#20 0.3793606672 0.3794161825 0.0000555153#21 0.9417846056 0.9418381718 0.0000535662#22 0.2193054491 0.2191161197 0.0001893294…

#80 0.3149359471 0.8782328851 0.5632969380#81 0.8630051853 0.4277595387 0.4352456466#82 0.4729089416 0.9791252630 0.5062163214#83 0.9970642982 0.0817559295 0.9153083687

A Well Known Chaotic System

• Weather is considered chaotic since arbitrarily small variations in initial conditions can result in radically different weather later. This may limit the possibilities of long-term weather forecasting.

• The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later.

• Lorenz Model was the model that proved this idea.

Visualizing Chaotic Motion & Attractors

• One way of visualizing chaotic motion, or indeed any type of motion,

is to make a phase diagram.

• In such a diagram time is implicit and each axis represents one

dimension of the state.

• A phase diagram for a given system may depend on the initial state

of the system (as well as on a set of parameters),

• Often phase diagrams reveal that

– the system ends up doing the same motion for all initial states in a region

around the motion

– almost as though the system is attracted to that motion

– Such attractive motion is fittingly called an attractor for the system

An Attractor Example

• A simple harmonic oscillation system is shown.

• Assume no friction

• Then the red circle is the attractor of the system.

v

x

Strange Attractors

• While most of the motion types mentioned give rise to very simple attractors, chaotic motion gives rise to what are known as strange attractors that can have great detail and complexity.

• Strange attractors

have fractal structure.

Strange Attractors and Fractals

Strange Attractors in Action!

How are chaos and fractals are related to real life?

• Let’s answer this question with 2 distinct examples:

• Logistic Equation used for animal population modeling

• Iterated Function Systems for rendering plants

What is Logistic Equation?• It models animal populations. • The equation is xn+1 = A.xn.(1-xn)

where x is the population (between 0 and 1) and A is a growth constant

Iteration of this equation yields the period doubling route to chaos.

For A between 1 and 3, the population will settle to a fixed value.

At 3.00, the period doubles to 2; one year the population is very high, causing a low population the next year, causing a high population the following year.

At 3.45, the period doubles to 4, population has a four year cycle.The period keeps doubling, faster and faster, at 3.54, 3.564, 3.569, ...

Until At 3.57, chaos occurs; the population never settles to a fixed period. For most A values between 3.57 and 4, the population is chaotic, but there are also periodic regions.

Attractor of Logistic Equation

Periodic Behavior of Logistic Map with A=3.5 and X0=0.7

…89 : 0.382819683017324 90 : 0.82694070659143891 : 0.500884210307217 92 : 0.87499726360246493 : 0.38281968301732494 : 0.82694070659143895 : 0.50088421030721796 : 0.87499726360246497 : 0.38281968301732498 : 0.82694070659143899 : 0.500884210307217100 : 0.874997263602464

Visualization of Logistic Equation

Feigenbaum’s Constant

• In a period doubling chaotic sequence, such as the logistic equation, consider the parameter values where period-doubling events occur

• e.g. r1=3, r2=3.45, r3=3.54, r4=3.564… • Let’s look at the ratio of distances between consecutive

doubling parameter values;let deltan = (rn+1 - rn) / (rn+2 - rn+1)

• Then the limit as n goes to infinity is Feigenbaum's (delta) constant.

• It has the value 4.669201609102990671853...• The interpretation of the delta constant is that as you

approach chaos, each periodic region is smaller than the previous by a factor approaching 4.669...

Iterated Function Systems

• The Fern

• T1 x' = 0x + 0y +.16, y' = 0x + 0y +0 1%

• T2 x' = .85x + .04y + 0, y' = -.04x + .85y + 1.6 85%

• T3 x' = .2x - .26y + 0, y' = .23 x + .22y +1.6 7%

• T4 x' = -.15x + .28y + 0, y' = .26x + .24y + .44 7%

How does the Fractal Fern look?

Other IFS Examples #1

Other IFS Examples #2

Fractals are related to natural laws!

We find chaotic behavior in every part of life.

We find fractals in every part of chaotic behavior.

It seems reasonable that the modeling method the nature uses is fractal geometry.

Recursively applying same set of gravitational transformations to an initial condition can create a beautiful fern for example.

So a fern only stores those 24 numbers in its genetic code in a certain way the nature can make use of them.

Human’s are also chaotic!

• It is not so much suprising that the human body also acts chaotic, and uses fractal geometry.

• In fact human body seems to possess a lot of fractal geometry structures– Arterial & venous trees– Cradiac muscle bundles– His-Purkinje Conduction System– Blood veins– Nervous System

How to make use of this chaos!

• Scientists have shown that with aging and disease the fractal like (chaotic) behavior of the systems degrade.

• So we can use this degradation as a feature for our disease diagnostic systems.

Heart Rate Dynamics in Health and Disease:A Time Series Test

HEALTHY!

We can find patterns – it is very predictable, not chaotic.

We can find patterns – it is very predictable, not chaotic.

It acts in a random fashion, still not chaotic

Spatial vs Temporal Self Similarity

Wavelet Analysis of Human Heart Beat

Detrended Fluctuation Analysis (DFA)Human Heart Beat

Electrical Signals Captured

Integrated to convert Heart Beat Velocities to Heart Displacements

Slope Corresponds to different kinds of behavior: can be used as a feature

DFA in General

Fractal Dimension & Diagnosis

Human Gait (Walking) Dynamics

Conclusion

• Nature uses chaotic systems in every part of our lives.– Weather– Animal Population Growth

• Human body is also built up of chaotic elements– Nervous System– Blood veins

• We can make use of fractals and chaos theory for diagnosing certain diseases.– Heart Failures: Atrial fibrilation, …– Huntington Disease: Gait Dynamics

Thanks

Any Questions ?