# Presentation: 866297

## Introduction to Fractals

- Larry S. Liebovitch

- Florida Atlantic University
- Center for Complex Systems and Brain Sciences
- Center for Molecular Biology and Biotechnology
- Department of Psychology
- Department of Biomedical Sciences

- Lina A. Shehadeh

*Copyright 2003 by Larry S. Liebovitch*

**Notes:**
This presentation is about one nonlinear method called “fractals”.

## How fractals CHANGE the most basic ways we analyze and understand experimental data.

## Non-Fractal

**Non-Fractal**

**Notes:**
As we enlarge a non-fractal object, no new details appear.

## Fractal

**Notes:**
But, as we enlarge a fractal object we keep seeing ever smaller pieces.
For example, this series of pictures could show first the inside of the intestine, then the crypts between the cells, then the microvilli on each cell.
The smaller pieces are copies of the larger pieces.
They are not exact smaller copies, but they are smaller replicas that are kind-of-like the larger pieces.

## Non - Fractal

**Non - Fractal**

**Size of Features**

**1 cm**

***1 characteristic scale***

**Notes:**
A non-fractal object has pieces that are all about the same size.

## Fractal

**Size of Features**

**2 cm**

**1 cm**

**1/2 cm**

**1/4 cm**

***many different scales***

**Notes:**
A fractal object has pieces of all different sizes.
Typically, there are a few big pieces, many medium sized pieces, and a huge number of tiny pieces.

## Fractals

- Self-Similarity

## Self-Similarity

**Water**

**Land**

**Water**

**Land**

**Water**

**Land**

**Self-Similarity**
**Pieces resemble the whole.**

**Notes:**
Another property of fractals is called Self-Similarity.
Here the coastline, the border between the land and the sea, has bays and peninsulas.
As you magnify the coastline you see ever smaller bays and peninsulas.
The structure at a large scale is similar to the structure at a small scale.
It is similar to itself at different scales.
This is called Self-Similarity.

## Sierpinski Triangle

**Notes:**
This is the Sierpinski Triangle.
In this mathematical object each little piece is an exact smaller copy of the whole object.

## 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*

**Notes:**
The blood vessels in the retina are self-similar.
The branching of the larger vessels is like the branching of the smaller vessels.
The airways in the lung are self-similar.
The branching of the larger airways is like the branching of the smaller airways.
In real biologically objects like these, each little piece is not an exact copy of the whole object.
It is kind-of-like the whole object.
This is called Statistical Self-Similarity.

## Blood Vessels in the Retina

**Notes:**
Let’s try to understand statistical self-similarity.
Here is a simplified (actually unrealistically simplified) picture of the blood vessels in the retina.
We can ask, how many vessels there are of each different size.
Here, there are one that is 40mm long, two that are 20mm long, four that are 10mm long, and eight that are 5 mm long.

## PDF - Probability Density Function

**HOW OFTEN there is THIS SIZE**

**Straight line on log-log plot**
**= Power Law**

**Notes:**
We can plot HOW MANY vessels there are of each SIZE.
This is called the Probability Density Function, PDF.
This is a straight line on a plot of log(Number) versus Log(size).

## Statistical Self-Similarity

- The statistics of the big pieces is the same
- as the statistics of the small pieces.

**Notes:**
The PDF of the large vessels is a straight line on a plot of log(Number) versus Log(size).
There are a few big-big vessels, many medium-big vessels, and a huge number of small-big vessels.
The PDF of the small vessels is also a straight line on a plot of log(Number) versus Log(size).
There are a few big-small vessels, many medium-small vessels, and a huge number of small-small vessels.
The PDF of the big vessels has the same shape, is Similar to, the PDF of the small vessels.
The PDF is a measure of the Statistics of the vessels.
So, the PDF (the statistics) of the large vessels are similar to the PDF (the statistics) of the small vessels.
This is Statistical Self-Similarity.
The small pieces are not exact copies of the large pieces.
But, the statistics of the small pieces are similar to the statistics of the large pieces.

## Currents Through Ion Channels

**Notes:**
Fractals are not only objects in space, but can also be processes in time.
There are proteins, called “ion channels”, in the fatty membranes of living cells that let ions, like sodium and potassium, enter or exit the cell.

## Currents Through Ion Channels

**Notes:**
A small pipette can suck up a small piece of cell membrane with only one ion channel in it.
And even it tear it off away from the cell.
The movement of sodium or potassium through the ion channel produces an electrical current that can be measured.
It’s a pretty small current, a picoAmp, which is about one billionth (1/1,000,000,000) of the current from a “D” battery.
This is called the “Patch Clamp”.
What’s really interesting is that these ion channel proteins act like little electrical switches.
Either they are either fully open or fully closed to the movement of sodium or potassium.
And they switch, all the time, between these fully open and fully closed states.
It’s impressive to watch this technology measure the changes in a single molecule at at time.

## Currents Through Ion Channels

**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**

**F****C**** = 10 Hz**

**F****C**** = 1k Hz**

**Notes:**
These open and closed times are fractal!
If you record them and play them back slowly you see a sequences of open and closed times.
But if you take one time, and play in back at higher resolution, you see that it actually consists of many briefer open and closed times.
It is self-similar in time.

## Closed Time Histograms

**Closed Time Histograms**

**potassium channel in the corneal endothelium**

**Number of closed Times per Time Bin in the Record**

- Liebovitch et al. 1987 Math. Biosci. 84:37-68

**Closed Time in ms**

**Notes:**
Here is a histogram of the times that one channel was closed.
The recording was made at the fastest time resolution and so the briefest closed times were recorded.
The PDF is mostly a straight line on this log(number) versus t plot, but with an occasional longer closed time.

## Closed Time Histograms

**Closed Time Histograms**

**potassium channel in the corneal endothelium**

**Number of closed Times per Time Bin in the Record**

- Liebovitch et al. 1987 Math. Biosci. 84:37-68

**Closed Time in ms**

**Notes:**
Here is a another histogram of the closed times of that same ion channel.
This recording was made at a little slower time resolution and so longer closed times were recorded.
The PDF is mostly a straight line on this log(number) versus t plot, but with an occasional longer closed time.

## Closed Time Histograms

**Closed Time in ms**

**Number of closed Times per Time Bin in the Record**

**Closed Time Histograms**

**potassium channel in the corneal endothelium**

- Liebovitch et al. 1987 Math. Biosci. 84:37-68

**Notes:**
Here is a another histogram of the closed times of that same ion channel.
This recording was made at a even slower time resolution and so even longer closed times were recorded.
The PDF is mostly a straight line on this log(number) versus t plot, but with an occasional longer closed time.

## Closed Time Histograms

**Closed Time Histograms**

**potassium channel in the corneal endothelium**

**Number of closed Times per Time Bin in the Record**

- Liebovitch et al. 1987 Math. Biosci. 84:37-68

**Closed Time in ms**

**Notes:**
Here is a another histogram of the closed times of that same ion channel.
This recording was made at a much slower time resolution and so only the longest closed times were recorded.
The PDF is mostly a straight line on this log(number) versus t plot, but with an occasional longer closed time.
The PDF looks Similar at different time resolutions.
The PDF is a measure of the Statistics.
So, the Statistics is Similar to itself at different time resolutions.
This is Statistical Self-Similarity in time.

## Fractals

- Scaling

## Scaling
  The value measured depends on the resolution used to do the measurement.

**  ****The value measured depends on the resolution used to do the measurement.**

**Notes:**
If we measure the length of the west coast of Britain with a large ruler, we get a certain value for the length of the coastline.
If we now measure it again with a smaller ruler, we catch more of the smaller bays and peninsulas that we missed before, and so the coastline is longer.
The value we measure for the coastline depends on the size of the ruler that we use to measure it.

## How Long is the Coastline of Britain?

**How Long is the Coastline of Britain?**

- Richardson 1961 The problem of contiguity: An Appendix to *Statistics of Deadly Quarrels General Systems Yearbook  *6:139-187

**Log****10**** (Total Length in Km)**

- AUSTRIALIAN COAST

- CIRCLE

- SOUTH AFRICAN  COAST

- GERMAN LAND-FRONTIER, 1900

- WEST COAST OF BRITIAN

- LAND-FRONTIER OF PORTUGAL

**4.0**

**3.5**

**3.0**

**1.0**

**1.5**

**2.0**

**2.5**

**3.0**

**3.5**

**LOG****10**** (Length of Line Segments in Km)**

**Notes:**
Here is a plot of how the length of the west coast of Britain depends on the resolution that we use to measure it.
There is no one value that best describes the length of the west coast of Britain.
It depends on the scale (resolution) at which we measure it.
As we measure it an finer scale, we include the segments of the smaller bays and peninsulas, and the coastline is longer.

## Genetic Mosaics in the Liver

- P. M. Iannaccone. 1990. FASEB J. 4:1508-1512.
- Y.-K. Ng and P. M. Iannaccone.  1992.  Devel. Biol. 151:419-430.

**Notes:**
Iannaccone and his colleagues study how organisms develop to understand and cure cancer in kids.
They mix cells from another animal into an embryo so that the fate of these marker cells can be traced out as the animal develops.
The cells that are added have a different enzyme which attaches to a radioactive marker that blackens a photographic film to make a picture.
Here are some of those pictures of the liver.
Look, the added cells are not in one clump.
They are in islands of all different sizes.
There is no one area that best describes the size of these islands.
The area measured depends on the resolution used.
This scaling relationship is a straight line on a plot of log(area) versus log(resolution).
There is no one perimeter that best describes the size of these islands.
The perimeter measured depends on the resolution used.
This scaling relationship is also a straight line on a plot of log(perimeter) versus log(resolution).
These are examples of Scaling Relationships. 

## Fractal Kinetics

- 		k = Prob. to  change states in the next ∆t.

**Effective Kinetic Rate Constant**:
- 		keff = Prob. to  change states in the next ∆t, 		given that we have already remained 		in the state for a time keff.

**k**

-       =  Pr ( T=t, t+∆t  |  T > t     ) / ∆t

- eff

- eff

*age-specific failure rate*

- =  –

- d

- dt

- ln P(t)

- P(t) = cumulative dwell time distribution

**Fractal Kinetics**

**Notes:**
So far, we’ve seen fractal scalings in space.
There are also fractal scalings in time.
The usual way to measure the switching of an ion channel is the “kinetic rate constant”.
That tells us the probability that the ion channel switches between open and closed states.
But, the ion channel must be closed (or open) long enough for us to see it as closed (or open).
A more appropriate measure is the probability that the ion channel switches between open and closed states, given that it has already remain in a state for a certain amount of time.
That certain amount of time defines the time resolution at which we measure the switching probability.
We called that probability the “effective kinetic rate constant”.

## 70 pS K+ ChannelCorneal Endothelium

- Liebovitch et al.  1987  Math. Biosci. 84:37-68.

*effective time scale*

- t

- eff

-    in  msec

- 100

- 1000

- 10

- 1

- 1

- 10

- 100

- 1000

- k

- eff

-  = A

- t

- eff

- 1-D

**Notes:**
We measured the open and closed times for an ion channel in the cells in the cornea, the clear part in the front of the eye that you look through to see these words.
The effective kinetic rate constant is a straight line on a plot of log(efffective kinetic rate constant) versus log(effective time used to measure it).
This is a fractal scaling relationship in time.
The faster we could look, the briefer open and closed times we would see.

## Fractal Approach

**New viewpoint:**
- 	Analyze how a property, the effective kinetic rate constant, keff, depends on the effective time scale, teff, at which it is measured.
**This Scaling Relationship:**
- 	We are using this to learn about the _structure_ and _motions_ in the ion channel protein.

**Notes:**
Fractals have given us a new way to analyze data from the patch clamp measurements of the open and closed times of ion channels.
Instead of measuring a property (the kinetic rate constant) at one time scale, we measure how a property (the efffective kinetic rate constant) changes when we measure it at different time scales.
We have been using the information in this fractal scaling relationship to give us clues about the structure and motions in ion channel protein.

## Scaling

- one measurement: ***not so interesting***

***slope***

**Scaling**

**Logarithm **of
- the measuremnt

**Logarithm **of
- the measuremnt

***one value***

**Logarithm** of the resolution used to make the measurement

**Logarithm** of the resolution used to make the measurement

- scaling relationship: ***much more interesting***

**Notes:**
Take Home Lesson:
We are used to thinking that there is ONE measurement that best describes a property of an object.
For a fractal object that extends over many scales, a property depends on the scale at which it is measured.
There is no ONE measurement that best describes the object.
The object is best described by HOW THE PROPERTY MEASURED DEPENDS ON THE RESOLUTION AT WHICH IT IS MEASURED.
This relationship is characterized by a parameter called the Fractal Dimension.

## Fractals

- Statistics

## Not Fractal

**Notes:**
Here is a set of numbers, maybe they are the values measured from an experiment.
I have drawn a circle to represent each number.
The diameter of the circle is proportional to the size of the number.
Here is a non-fractal set of numbers.
Most of them are about the size of an average number.
A few are a bit smaller than the average.
A few are bit larger than the average.

## Not Fractal

**Notes:**
Here is the PDF of these non-fractal numbers.
The PDF is how many numbers there are of each size.
The PDF here is called a “Bell Curve”, a “Gaussian Distribution”, or a “Normal Distribution”.
What an interestingly unloaded word to use, a “NORMAL” distribution.
We are about to see that much of the world is definitely not “normal”.

## Gaussian

- Bell Curve
- “Normal Distribution”

**Notes:**
Here is a picture of Gauss on the old 10 Deustche Mark German bill.
He has now been replaced by the 5 Euro.
You can see his curve and even the equation for it on this bill!
There are no equations on American money.
There is a scientist on American money.
Do you know who it is?

## Fractal

**Notes:**
Here is a set of numbers from a fractal distribution.
The diameter of each circle is proportional to the size of the number.
These numbers could be from the room around you.
Look around your room.
There are few big things: people and chairs.
Many medium sized things: pens and coins.
And a huge number of tiny things: dust and bacteria.
It is NOT at all like that “Normal” distribution.
There are a few big things, many medium sized things, and a huge number of tiny things.
Sets of data from many things in the real world are just like this.
We call this a FRACTAL distribution of numbers because it has the same statistical properties as the sizes of the pieces in fractal objects.

## Fractal

**Notes:**
Here is the PDF of these fractal numbers.
The PDF is how many numbers there are of each size.
There are a few big numbers, many medium sized numbers, and a huge amount of tiny numbers.
The PDF is a straight line on a plot of log(How Many Numbers, the PDF) versus log(value of the numbers).

## Non - Fractal

**Non - Fractal**

**More Data**

- pop

**Notes:**
The statistics of a fractal set of numbers is VERY DIFFERENT from the statistics of “normal” numbers that they taught you about in Statistics 101.
The statistics you learned in Statistics 101 is only about non-fractal numbers.
Take the average of a sample of non-fractal numbers.
This is called the Sample Mean.
As you include ever more data, the sample means get ever closer to one value.
We call that value the Population Mean.
We think that the population mean is the “real” value of the mean.

## The Average Depends on the Amount of Data Analyzed

**Notes:**
But the statistics of fractal numbers is very different.
Take the average of a sample of fractal numbers.
This is called the Sample Mean.
As you include ever more data, the sample means do NOT get ever closer to one value.
Either the the sample means keep increasing OR the sample means keep decreasing as you include more data.
There is NO Population Mean.
There is NO one value that best describes the data.
The data extends over a range of many different values.

## The Average Depends on the Amount of Data Analyzed

- each piece

**Notes:**
Here is why that happens.
Again, here is a set of fractal numbers.
The diameter of the circles are proportional to the size of the numbers.
As you include more data, either:
	-you include many smaller values and so the sample means get smaller and smaller, or
	-you include a few bigger values and so the sample means get bigger and bigger.
You can kind-of-see that which happens depends on the ratio of the amount of small numbers to the amount of big numbers.
That ratio is characterized by a parameter called the Fractal Dimension.

## The average winnings are:  2-1.1  =  0.5 

**Toss a coin. If it is tails win $0, If it is heads win $1.**
**The average winnings are:  2****-1****.1  =  0.5 **

**1/2**

***Non-Fractal***

**Notes:**
Let’s play a non-fractal game of chance.
Toss a coin, if it comes up tails we win nothing, if it comes up heads we win $1.
The average winnings are the probability of each outcome times how much we win on that outcome.
The average winnings are (1/2) x ($0) + (1/2) x ($1) = 50 cents.
Let’s go to a fair casino to play this game.
Fair casinos exist only in math textbooks, it means the bank is willing only to break even and not make a profit.
We and the casino think it’s fair for us to be charged 50 cents to play one game.
Seems reasonable, half the time we win nothing, half the time we win a $1, so if it costs 50 cents to play each time, on average, we and the casino will break even.

## Ordinary Coin Toss

**Notes:**
Here is the PDF of that non-fractal game of chance.
It shows HOW OFTEN you will win HOW MUCH $ if you play 100 times.
It’s a Bell Curve, Gaussian, Normal distribution.
Just the kind of distribution they taught you about in Statistics 101.

## Ordinary Coin Toss

**Notes:**
Here’s what happens when I played that non-fractal game, over and over again.
A computer (actually a Macintosh Plus running Microsoft BASIC!) picked a random number to simulate flipping the coin.
Here, the average winnings per game is shown after n games.
For a while I (the Mac) was lucky.
I was winning more than an average 50 cents in each game.
But, as you might suspect (this is called the Law of Large Numbers) after a while, my luck ran out.
In the long run, I was winning exactly an average of 50 cents in each game.

## 2-121  +  2-222  +  2-323  +  . . .  =

**Toss a coin. If it is heads win $2, if not, keep tossing it until it falls heads.**
**If this occurs on the N-th toss we win $2****N****.**
**With probability 2****-N**** we win $2****N****.**

- H        $2
- TH      $4
- TTH    $8
- TTTH  $16

**The average winnings are:**

**2****-1****2****1****  +  2****-2****2****2  ****+  2****-3****2****3  ****+  . . .  =**

**1      +       1    +       1     +  . . .  =**

***Fractal***

**Notes:**
Now, let’s play a fractal game of chance.
This game was invented by Niklaus Bernoulli who lived in St. Petersburg, Russia and published by his uncle Daniel Bernoulli who lived in Germany, about 350 years ago.
Here, we toss a coin UNTIL it comes up heads.
If it comes up heads on the first toss, we win $2.
If it comes up tails first, and then heads on the second toss, we win $4.
If it comes up tails twice, and then heads on the third toss, we win $8.
And so on.
The average winnings are the probability of each outcome times how much we win on that outcome.
The average winnings are (1/2) x ($2) + (1/4) x ($4) + (1/8) x ($8) + (1/16) X ($16) + .... = 1 + 1 + 1 + 1 ...  = INFINITY
We say to the casino, “Half the time we’ll win $2, the median winnings of this game is $2” because half the time the coin comes up heads on the first toss and we win $2.
“So, we think it is very fair to put up twice the median winnings, $4, to play each game”.
To our surprise, the now angry casino owner says, “No!”
He adds, “The average winnings of this game are INFINITE, you must put up more than all the money in the universe to play this game, even once!”
This game became known as the St. Petersburg Paradox, because we and the casino cannot agree on the fee to play this game.
This game is well known amongst mathematicians, which means its well known amongst the people who know it well.
But, because of its unusually mathematical character, it was not one of the threads of probability theory that was woven into the fabric of statistics that became popular in the natural sciences that is taught in Statistics 101.
Now, with the popularity of fractals, it is being re-discovered and its importance in analyzing and understanding real data increasingly appreciated.

## St. Petersburg Game (Niklaus Bernoulli)

**Notes:**
Here is the PDF of that fractal game of chance.
It shows HOW OFTEN you will win HOW MUCH.
It’s NOT a Bell Curve, Gaussian, or Normal distribution.
Most often you win only a small amount, more often you win a bigger amount, very rarely you win a huge amount.
It is just like the fractal PDF of the blood vessels in the retina, or any fractal object!
It is a straight line on a plot of log(How Often) versus log(How Much).

## St. Petersburg Game (Niklaus Bernoulli)

**Notes:**
Here’s what happens when I played that fractal game, over and over again.
Here, the average winnings per game is shown after n games.
The more I played, the more often there was sometimes a lot of tails before that first head.
When there are a lot of those tails, I won a huge jackpot.
As more and more of those jackpots happened, the average winnings per game kept increasing.
There is NO average (population mean) for this game.
The more I played, the more the average kept changing.
They told you in Statistics 101 that the more data you have, the closer the sample means are to the population mean.
Not here!
There is no population mean.
The more data we have (the more games I played) the more the sample means keep changing.
Very different than what you learned in Statistics 101.Welcome to fractals.

## Non-Fractal

**Log avg**
**density within **
**radius r**

**Log radius r**

**Notes:**
Here is a non-fractal object.
It is a checkerboard.
Actually, I’m only showing you a piece of it, it should really extend forever in each direction.
Place a circle on it.
Count all the black pixels in that circle, and divide by the total number of pixels.
That is the average density within that circle.
The graph shows how that density changes as the circles get bigger and bigger.
The average density fluctuations a bit, after all, we are putting a round circle over a square grid.
But, as the circles get bigger and bigger, the average density gets closer and closer to 1/2.
Seems reasonable, the checkerboard is 1/2 black and 1/2 white.

## Fractal

**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**

**6.0**

**0**

**-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**

**Notes:**
Here is a fractal object.
It is called a Diffusion Limited Aggregation, a DLA.
It is statistically self-similar.
It has little spaces between its little arms.
And medium spaces between its medium sized arms.
And large spaces between its large arms.
I’m only showing you a piece of it, it should also really extend forever in each direction.
Place a circle on it.
Count all the black pixels in that circle, and divide by the total number of pixels.
That is the average density within that circle.
The graph shows how that density changes as the circles get bigger and bigger.
As the circles get bigger we catch more of the ever larger spaces between the arms, and so the density gets smaller.
As the circles get ever bigger, the density gets ever smaller.
There is no one density that describes this object.
What’s more, the local density on a big arm is very high.
The local density inbetween a big arm is very low.
Yet, the same mechanism makes the arms and the spaces between them.
We’re used to thinking, that Statistics 101 again, that when the local average changes, when there is a difference in the mean value between an experiment and a control, or between now and then, that the system MUST have changed.
Here, fractals, with INFINITE variance, have moments, like means, that can be very different in space and time or between experiments and controls, even though the basic process has NOT change at all!

## Electrical Activity of Auditory Nerve Cells

*Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52*

**voltage**

**time**

*action potentials*

**Notes:**
Here is yet another example of fractal data.
There is a lot of fractal data in the real world.
Maybe even more things have fractal data than that “Normal” statistics that you learned in Statistics 101.
Maybe.
Here are the action potentials, the little electrical sparks, that encode information sent down the nerves in your body.
Teich et al. Measured them in the nerve that brings information about sounds from your ear to your brain.

## Electrical Activity of Auditory Nerve Cells

*Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52*

**2**

- Count the number of action potentials in each window:

**6**

**3**

**1**

**5**

**1**

**Firing Rate = 2, 6, 3, 1, 5,1**

***Divide the record into time windows:***

**Notes:**
They divided the time record into windows and counted the number of action potentials in each window.
Here, 2 in the first window, 6 in the second, and so on.
The “firing rate” is the number of action potentials in each window, divided by the time duration of that window.

## Electrical Activity of Auditory Nerve Cells

*Teich, Johnson, Kumar, and Turcott 1990 Hearing Res. 46:41-52*

**Repeat for different lengths of time windows:**

**8**

**4**

**6**

**Firing Rate = 8, 4, 6**

**Notes:**
Here they made the windows twice as long in time, and counted the number of action potentials in each window.
Again, the “firing rate” is the number of action potentials in each window, divided by the time duration of that window.

## Electrical Activity of Auditory Nerve Cells

**Electrical Activity of Auditory Nerve Cells**

*Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52*

**0**

**The variation in the firing rate does not decrease at longer time windows.**

**4**

**8**

**12**

**16**

**20**

**24**

**28**

**70**

**60**

**80**

**90**

**100**

**120**

**130**

**140**

**110**

**150**

**T = 50.0 sec**

**T = 5.0 sec**

**T = 0.5 sec**

**FIRING RATE**

**SAMPLE NUMBER (each of duration T sec)**

**Notes:**
In Statistics 101 they taught you that as you collect more data, the fluctuations average out.
You were taught to expect that the fluctuations in the firing rate should be LESS as the time windows get LONGER.
But look here, the variations don’t change much as the time windows go from 0.5 sec to 5.0 sec to 50.0 sec!
(Actually, the real deal here is that the variance of the fluctuations falls much slower than 1 / sqrt (n) ).
You include more data, but you don’t get any closer to the real firing rate.
There is no one single value, like a population mean, that best describes the firing rate.
The increase in variation at longer time windows is real.
It represents correlations in the action potentials, that may tell how information is encoded in the timing of the action potentials.

## Fractals

- Power Law PDFs

## Heart Rhythms

**Notes:**
These are electrocardiograms, EKGs, that record the voltage from the heart.
First, a normal heart.
Then, a heart that is beating dangerously fast.
Dangerous, because it can lead to ventricular fibrillation, where the heart no longer contracts in a regular way, does not pump blood, and you’re dead in 3 minutes.

## Inter-event Times

- 	Episodes of Ventricular Tachycardia (v-tach)

- t

- 1

- t

- 2

- t

- 3

- t

- 4

- t

- 5

*time ->*

- Cardioverter Defibrillator

**Notes:**
To prevent ventricular tachycardia from leading to fibrillation and death, a small device, called a “cardioverter defibrillator”, can be placed just under the skin in the chest.
Small wires are snaked around through the blood vessels into the heart.
It listens, electronically, carefully, to the heartbeat.
When the heart beats 3 times its normal rate for 5 beats, it sends an electrical shock strong enough to kick the heart back into a normal rhythm.
But, it’s also a computer, with a small memory.
It can remember, WHEN it triggered.
Back in the hospital, a small coil is placed on the patient’s chest.
Over this radio link, the cardioverter defibrillator, can play back when it triggered.
We have been analyzing the times between when the device was triggered.
Just to be clear, we are not analyzing the time between heartbeats.
We are analyzing the durations between consecutive triggerings of the cardioverter defibrillator.

## Patient #33

**Interval (in days)**

**Relative**
**Frequency**

**Patient #33**

**Notes:**
Here is the PDF from one patient, how often there was each time between these events.
The plot is a straight line on a log(How Often) versus log(time between events).
This is fractal scaling.
The events of rapid heart rate happen with a fractal timing.

## Patient #53

**Interval (in days)**

**Relative**
**Frequency**

*Relative Frequency =*
*(3.2545) Interval**-1.3664*

- 10

- 3

- 10

- 2

- 10

- 1

- 10

- 0

- 10

- -1

- 10

- -2

- 10

- -3

- 10

- -4

- 10

- -5

- 10

- -4

- 10

- -3

- 10

- -2

- 10

- -1

- 10

- 0

- 10

- 1

- 10

- 2

- 10

- 3

- 10

- 4

- 10

- 5

- 10

- 6

**Patient #53**

**Notes:**
Here is the PDF from another patient, how often there was each time between these events.
The plot is also a straight line on a log(How Often) versus log(time between events).
This is also fractal scaling.
The events of rapid heart rate here too also happen with a fractal timing.

## 6 Patients

**6 Patients**

*Liebovitch et al. 1999 Phys. Rev. E59:3312-3319.*

**Notes:**
In fact, for many patients, their events of ventricular tachycardia happen with a fractal pattern.
Most of the times between events are short, sometimes they are longer, very rarely they are very long.
Typical of fractals.
There is NO AVERAGE time between events.
If you measured the “average rate” of events you would get a different answer if you measured them over a day or a week or a year.
There is no one number that best describes the time between these events.
The time between events happens over many different time scales.
We are working on other ways, fractal ways, of characterizing these times to assess the status of patients and the effectiveness of medical therapies.

## Inter-arrival Times of E-mail Viruses

- t

- 1

- t

- 2

- t

- 3

- t

- 4

- t

- 5

*time ->*

*Liebovitch and Schwartz 2003 Phys. Rev. E68:017101.*

- AnnaKournikova

- "Hi: Check This!” AnnaKournikova.jpg vbs.

- Magistr

- Subject, body, attachment from other files: erase disk, cmos/bios.

- Klez

- E-mail from its own phrases: infect by just viewing in Outlook Express.

- Sircam

- “I send you this file in order to have your advice.”

**Notes:**
We are also analyzing the times at which different e-mail viruses arrive at the gateway into an Internet service provider.
Here the events are not the onset of ventricular tachycardia, they are the arrival times of e-mail viruses.
We are looking at the duration of times between the arrival of each virus.
We have studied 4 viruses:
AnnaKournikova doesn’t have a jpg picture of her, it’s a file that you wouldn’t want to open.
 Magistr can erase sectprs on your hard disk or your cmos/bios.  If you don’t know what the cmos/bios is, you don’t want me to tell you what happens if it gets erased.
 Klez puts together messages by joining fragments of phrases that it contains.
 Sircam tempts you to open and execute its attached file.
Much is known about the structure of the Internet.
Less is known about the dynamics of the Internet.
The arrival times of these viruses depend on both the structure and dynamics of the Internet.
We are hoping that our study of these arrival times will tell us how the structure interacts with the dynamics if the Internet.

## E-mail Viruses

- 20,884 viruses

- 153,519 viruses

**Notes:**
We have data, unfortunately, from many, many virus events.
Here are the PDFs from AnnaKournikova and Magistr, how often there was each time between the arrivals.
These plots are straight lines on a log(How Often) versus log(time between arrivals).
These are fractal scalings.
The arrival of these viruses happens with a fractal timing.

## E-mail Viruses

- 413,183 viruses

- 781,626 viruses

**Notes:**
Here are the PDFs from Klez and Sircam, how often there was each time between the arrivals.
These plots are also straight lines on a log(How Often) versus log(time between arrivals).
These are also fractal scalings.
The arrival of these viruses too also happens with a fractal timing.

## Determining the PDFfrom a Histogram

- Bins ∆t Small

- Good at small t.
- BAD at large t.

- Bins ∆t Large

- BAD at small t.
- Good at large t.

**Notes:**
It’s not so easy to make a good PDF from the histogram of How Many events there are in each time bin.
If we choose the bins narrow, we get good resolution for the small events.
But, at long times, because the bins are so narrow, we may see only 1 or 2 events, or even 0 events, in some of those bins.
If we try to fix this by making the bins wider, we do get more events in the bins at longer times.
But, at short times, we now have poor resolution.

## Determining the PDF

*Liebovitch et al. 1999 Phys. Rev. E59:3312-3319.*

- Solution:
- Make ONE PDF
- From SEVERAL Histograms of DIFFERENT Bin Size

- Choose ∆t = 1, 2, 4, 8, 16 ... seconds

- ∆t = bin size

**Notes:**
We figured out a nice algorithm to get a better PDF.
Narrow bins are good at short times.
Wide bins are good at long times.
So, we make histograms of different bin sizes.
But, we cannot combine histograms of different bin sizes.
However, we can compute the PDF from each histogram.
And, we can then combine the PDFs.
So, the histograms with narrow bins give us good resolution in the PDF at short times.
And the histograms with wide bins give us good values in the PDF at long times.
We’ve found that this methods yields accurate and reliable PDFs for tails of many different kinds of distributions.
See Liebovitch et al. 1999 for details.

## Determiningthe PDF

- New multi-histogram

- Standard fixed ∆t

**Notes:**
Here, PDFs were measured from a set of fractal data.
The red boxes is the PDF made in the usual way from one histogram.
You can see where there is only 1 or 2 events in the largest bins.
The black boxes is the PDF, generated from the same data, using the new multi-histogram method to make the PDF.
Pretty impressive difference.

## Fractals

- Summary

## Summary of Fractal Properties

**Self-Simialrity**
**Pieces resemble the whole.**

## Summary of Fractal Properties

**Scaling**
**  ****The value measured depends on the resolution.**

## Summary of Fractal Properties

**Statistical Properties**
**  ****Moments may be zero or infinite.**

## 400 years ago:
	Gambling Problems     Probability Theory

200 years ago:
	Statistics     How we do experiments. 

100 years ago:
	Student’s t-test, F-test, ANOVA

Now:
	Still changing

- 	Gambling Problems     Probability Theory
*200 years ago:*
- 	Statistics     How we do experiments.
*100 years ago:*
- 	Student’s t-test, F-test, ANOVA
- Now:
- 	Still changing

- Statistics is NOT a dead science.

**Notes:**
Probability theory started from solving gambling problems about 400 years ago.
About 200 years ago, those results were used to develop basic statistics.
Most of the statistical tests we use were developed less than 100 years ago.
We show you this to emphasize that statistics is NOT a dead science, although it’s often presented like that in Statistics 101.
It has changed a lot.
It is still changing.
It will change even more in the future.
The statistical properties of fractals are examples of new ideas that are now being incorporated into and are changing statistics.

## Fractals CHANGE the most basic ways we analyze and understand experimental data.

- Fractals

- Measurements over _many_ scales.
- What is real is not _one_ number, but how the measured values change with the scale at which they are measured (fractal dimension).

- No Bell Curves
- No Moments
- No mean ± s.e.m.

**Notes:**
The take-home lesson here is NOT that fractals some arcane super-sophisticated mathematical tool that only needs to be used in some strange circumstance.
Fractals change the MOST BASIC WAY WE LOOK AT EXPERIMENTAL DATA.
They allow us to analyze and make sense out of the huge amount of real data that just ain’t a bell curve.
The most common use of mathematics and statistics in all science is those means ± s.e.m.
Fractals tell us that if the data is fractal, those means are meaningless!
A pretty basic change in the simplest way we handle data.
That’s what revolutions in science are about.
Not about changing the complex stuff, but about changing the simplest stuff.The stuff that we were taught so firmly that we never thought it would change. 

## References:

***Fractals and Chaos and ***
***Simplified for the Life ***
***Sciences***
**Larry S. Liebovitch**
**Oxford Univ. Press, 1998**

***The Mathematics and***
***Science of Fractals***
**Larry S. Liebovitch and**
**Lina Shehadeh**
**www.ccs.fau.edu/~liebovitch/larry.html**

- CD ROM

- NSF
- DUE-9752226
- DUE-9980715

**Notes:**
If you want to learn more about fractals:
 A book called Fractals and Chaos Simplified for the Life Sciences.
This book consists of facing pages, the left TEXT, and the right a PICTURE.
It leads you, one concept at a time, through the material.
2) A CD-ROM of curricula materials for a mathematics course for college students who never liked and never did well in math.
These materials emphasize what mathematics is, how mathematicians do mathematics, and how mathematics is used in science.
This project was funded, in part, by the National Science Foundation, Division of Undergraduate Education.
We’re almost finished with it and would be happy to send you a free demo, if you get in touch at:	Dr. Larry Liebovitch
	Center for Complex Systems and Brain Sciences
	Florida Atlantic University
	777 Glades Road
	Boca Raton, FL 33431
	561.297.2239
	liebovitch@clifford.ccs.fau.edu
	 www.ccs.fau.edu/~liebovitch/larry.html