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Introduction to Fractals

Introduction to Fractals. Larry S. Liebovitch. Lina A. Shehadeh. Florida Atlantic University Center for Complex Systems and Brain Sciences Center for Molecular Biology and Biotechnology Department of Psychology Department of Biomedical Sciences. Copyright 2003 by Larry S. Liebovitch.

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Introduction to Fractals

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  1. Introduction to Fractals Larry S. Liebovitch Lina A. Shehadeh Florida Atlantic University Center for Complex Systems and Brain Sciences Center for Molecular Biology and Biotechnology Department of Psychology Department of Biomedical Sciences Copyright 2003 by Larry S. Liebovitch

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

  3. Non-Fractal

  4. Fractal

  5. Non - Fractal Size of Features 1 cm 1 characteristic scale

  6. Fractal Size of Features 2 cm 1 cm 1/2 cm 1/4 cm many different scales

  7. Fractals Self-Similarity

  8. Self-Similarity Pieces resemble the whole. Water Water Water Land Land Land

  9. Sierpinski Triangle

  10. Branching Patterns air ways blood vessels in the retina in the lungs Family, Masters, and Platt 1989 Physica D38:98-103 Mainster 1990 Eye 4:235-241 West and Goldberger 1987 Am. Sci. 75:354-365

  11. Blood Vessels in the Retina

  12. PDF - Probability Density Function HOW OFTEN there is THIS SIZE Straight line on log-log plot = Power Law

  13. Statistical Self-Similarity The statistics of the big pieces is the same as the statistics of the small pieces.

  14. Currents Through Ion Channels

  15. Currents Through Ion Channels

  16. 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) FC = 10 Hz 5 sec 5 pA FC = 1k Hz 5 msec

  17. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  18. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  19. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  20. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  21. Fractals Scaling

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

  23. 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 AUSTRIALIAN COAST 4.0 CIRCLE SOUTH AFRICAN COAST Log10 (Total Length in Km) 3.5 GERMAN LAND-FRONTIER, 1900 WEST COAST OF BRITIAN 3.0 LAND-FRONTIER OF PORTUGAL 3.5 1.0 2.0 3.0 1.5 2.5 LOG10 (Length of Line Segments in Km)

  24. 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.

  25. k in Hz eff effective kinetic rate constant 70 pS K+ ChannelCorneal Endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68. 1000 1-D k = A t eff eff 100 10 1 1 10 100 1000 effective time scale t in msec eff

  26. 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.

  27. Scaling scaling relationship: much more interesting one measurement: not so interesting one value Logarithm of the measuremnt Logarithm of the measuremnt slope Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement

  28. Fractals Statistics

  29. Not Fractal

  30. Not Fractal

  31. Gaussian Bell Curve “Normal Distribution”

  32. Fractal

  33. Fractal

  34. Non - Fractal Mean pop More Data

  35. The Average Depends on the Amount of Data Analyzed

  36. The Average Depends on the Amount of Data Analyzed each piece

  37. Ordinary Coin Toss 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

  38. Ordinary Coin Toss

  39. Ordinary Coin Toss

  40. St. Petersburg Game (Niklaus Bernoulli) 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 $2N. With probability 2-N we win $2N. H $2 TH $4 TTH $8 TTTH $16 The average winnings are: 2-121 + 2-222 + 2-323 + . . . = 1 + 1 + 1 + . . . = Fractal

  41. St. Petersburg Game (Niklaus Bernoulli)

  42. St. Petersburg Game (Niklaus Bernoulli)

  43. Non-Fractal Log avg density within radius r Log radius r

  44. Fractal Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135 Log avg density within radius r 0 .5 -1.0 -1.5 -2.0 -2.5 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0 Log radius r

  45. Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 action potentials voltage time

  46. Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 Divide the record into time windows: Count the number of action potentials in each window: 2 6 3 1 5 1 Firing Rate = 2, 6, 3, 1, 5,1

  47. 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

  48. Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 150 The variation in the firing rate does not decrease at longer time windows. 140 T = 50.0 sec T = 5.0 sec 130 120 FIRING RATE 110 100 90 80 T = 0.5 sec 70 60 0 4 8 12 16 20 24 28 SAMPLE NUMBER (each of duration T sec)

  49. Fractals Power Law PDFs

  50. Heart Rhythms

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