Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods

Wavelets in Medicine and Biology

Volumn , Issue , 2017, Pages 383-412

  Teich, Malvin C ^a,b   Heneghan, Conor ^a,b   Lowen, Steven B ^b   Turcott, Robert G ^b,c  

^a Boston University   (United States)

^b Columbia University ^*  (United States)

^c STANFORD UNIVERSITY SCHOOL OF MEDICINE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85130227123     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (41)

1. P. Abry, Transformées en ondelettes: Analyses multiresolution et signaux de presion en turbulence, Doctoral thesis, Université Claude Bernard-Lyon I, France (1994).
2. P. Abry and P. Flandrin, Wavelet-based Fano factor for long-range dependent point processes, Proc. 16th Annu. Int. Conf. IEEE Eng. Med. Biol. Soc. (Baltimore, MD, 1994), pp. 1330-1331.
3. D. W. Allan, Statistics of atomic frequency standards, Proc. IEEE 54, 221-230 (1966).
4. J. A. Barnes and D. W. Allan, A statistical model of flicker noise, Proc. IEEE 54, 176-178 (1966).
5. R. J. Barton and H. V. Poor, Signal detection in fractional Gaussian noise, IEEE Trans. Inform. Theory 34, 943-959 (1988).
6. J. Beran, Statistical methods for data with long-range dependence, Stat. Sei. 7, 404-427 (1992).
7. J. M. Berger and B. B. Mandelbrot, A new model for the clustering of errors on telephone circuits, IBM J. Res. Dev. 7, 224-236 (1963).
8. D. R. Cox and V. Isham, Point Processes (Chapman and Hall, London, 1980).
9. I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math. 41, 909-996 (1988).
10. L. J. DeFelice and A. Isaac, Chaotic states in a random world: relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels, J. Stat. Phys. 70, 339-354 (1993).
11. P. Flandrin, On the spectrum of fractional Brownian motions, IEEE Trans. Inf. Theory 35, 197-199 (1989).
12. P. Flandrin, Wavelet analysis and synthesis of fractional Brownian motion, IEEE Trans. Inf. Theory 38, 910-917 (1992).
13. P. Flandrin, Time-scale analyses and self-similar stochastic processes, Proc. NATO Adv. Study Inst. Wavelets and Their Applications (Il Ciocco, Italy, 1992).
14. P. Flandrin and P. Abry, Tracking long-range dependencies with wavelets, Proc. 1994 IEEE-IMS Workshop Inf. Stat. (Alexandria, VA, 1994), p. 54.
15. C. Heneghan, S. B. Lowen, and M. C. Teich, Wavelet analysis for estimating the fractal properties of neural firing patterns, Proc. Computational Neurosci., 1995.
16. A. H. Kumar and D. H. Johnson, Analyzing and modeling fractal intensity point processes, J. Acoust. Soc. Am. 93, 3365-3373 (1993).
17. L. S. Liebovitch and T. I. Tóth, Using fractals to understand the opening and closing of ion channels, Ann. Biomed. Eng. 18, 177-194 (1990).
18. S. B. Lowen and M. C. Teich, Fractal renewal processes as a model of charge transport in amorphous semiconductors, Phys. Rev. B 46, 1816-1819 (1992).
19. S. B. Lowen and M. C. Teich, Fractal renewal processes generate 1/f noise, Phys. Rev. E 47, 992-1001 (1993).
20. S. B. Lowen and M. C. Teich, Fractal renewal processes, IEEE Trans. Inf. Theory 39, 1669-1671 (1993).
21. S. B. Lowen and M. C. Teich, Fractal auditory-nerve firing patterns may derive from fractal switching in sensory hair-cell ion channels, in Noise in Physical Systems and 1/f Fluctuations (AIP Conference Proceedings 285), eds. P. H. Handel and A. L. Chung (American Institute of Physics, New York, 1993), pp. 781-784.
22. S. B. Lowen and M. C. Teich, Estimation and simulation of fractal stochastic point processes, Fractals 3, 183-210 (1995).
23. B. B. Mandelbrot, Self-similar error clusters in communication systems and the concept of conditional stationarity, IEEE Trans. Commun. Tech. 13, 71-90 (1965).
24. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983).
25. K. Matsuo, B. E. A. Saleh, and M. C. Teich, Cascaded Poisson processes, J. Math. Phys. 23, 2353-2364 (1982).
26. W. Rudin, Principles of Mathematical Analysis, 3rd ed. (McGraw Hill, New York, 1976), p. 197.
27. B. K. Ryu and S. B. Lowen, Modeling self-similar traffic with the fractal-shot-noise-driven Poisson process, Cent, for Telecomm. Res., Tech. Rep. 392-94-39 (Columbia University, New York, 1994).
28. B. Sakmann and E. Neher, Single-Channel Recording (Plenum, New York, 1983).
29. H. E. Schepers, J. H. G. M. van Beek, and J. B. Bassingthwaighte, Four methods to estimate the fractal dimension from self-affine signals, IEEE Eng. Med. Biol. Mag. 11, 57-64 (1992).
30. M. C. Teich, Fractal character of the auditory neural spike train, IEEE Trans. Biomed. Eng. 36, 150-160 (1989).
31. M. C. Teich, R. G. Turcott, and S. B. Lowen, The fractal doubly stochastic Poisson point process as a model for the cochlear neural spike train, in The Mechanics and Biophysics of Hearing (Lecture Notes in Biomathematics, Vol. 87), eds. P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, and C. R. Steele (Springer-Verlag, New York, 1990), pp. 354-361.
32. M. C. Teich, D. H. Johnson, A. R. Kumar, and R. G. Turcott, Rate fluctuations and fractional power-law noise recorded from cells in the lower auditory pathway of the cat, Hear. Res. 46, 41-52 (1990).
33. M. C. Teich, Fractal neuronal firing patterns, in Single Neuron Computation, eds. T. McKenna, J. Davis, and S. Zornetzer (Academic, Boston, 1992), pp. 589-625.
34. M. C. Teich and S. B. Lowen, Fractal patterns in auditory nerve-spike trains, IEEE Eng. Med. Biol. Mag. 13, 197-202 (1994).
35. M. C. Teich, R. G. Turcott, and R. M. Siegel, Variability and long-duration correlation in the sequence of action potentials in cat striate-cortex neurons, submitted to IEEE Eng. Med. Biol. Mag., 1996.
36. M. C. Teich, C. Heneghan, S. B. Lowen, T. Ozaki, and E. Kaplan, Fractal character of the neural spike train in the visual system of the cat, submitted to J. Opt. Soc. Am. A, 1996.
37. J. B. Troy and J. G. Robson, Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance, Visual Neurosci. 9, 535-553 (1992).
38. R. G. Turcott and M. C. Teich, Long-duration correlation and attractor topology of the heartbeat rate differ for healthy patients and those with heart failure, Proc. SPIE 2036 (Chaos in Biology and Medicine), 22-39 (1993).
39. R. G. Turcott, P. D. R. Barker, and M. C. Teich, Long-duration correlation in the sequence of action potentials in an insect visual interneuron, J. Stat. Comput. Simul. 52, 253-271 (1995).
40. R. G. Turcott and M. C. Teich, Fractal character of the electrocardiogram: Distinguishing heart-failure and normal patients, Ann. Biomed. Eng. 24, No. 2, 269-293 (1996).
41. G. W. Wornell and A. V. Oppenheim, Estimation of fractal signals from noisy measurements using wavelets, IEEE Trans. Sig. Process. 40, 611-623 (1992).