Abstract

We used a variety of statistical measures to identify the point process that describes the maintained discharge of retinal ganglion cells (RGC’s) and neurons in the lateral geniculate nucleus (LGN) of the cat. These measures are based on both interevent intervals and event counts and include the interevent-interval histogram, rescaled range analysis, the event-number histogram, the Fano factor, the Allan factor, and the periodogram. In addition, we applied these measures to surrogate versions of the data, generated by random shuffling of the order of interevent intervals. The counting statistics reveal 1/f-type fluctuations in the data (long-duration power-law correlation), which are not present in the shuffled data. Estimates of the fractal exponents measured for RGC- and their target LGN-spike trains are similar in value, indicating that the fractal behavior either is transmitted from one cell to the other or has a common origin. The gamma-r renewal process model, often used in the analysis of visual-neuron interevent intervals, describes certain short-term features of the RGC and LGN data reasonably well but fails to account for the long-duration correlation. We present a new model for visual-system nerve-spike firings: a gamma-r renewal process whose mean is modulated by fractal binomial noise. This fractal, doubly stochastic point process characterizes the statistical behavior of both RGC and LGN data sets remarkably well.

© 1997 Optical Society of America

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1996 (4)

M. C. Teich, R. G. Turcott, R. M. Siegel, “Temporal correlation in cat striate-cortex neural spike trains,” IEEE Eng. Med. Biol. Mag. 15 (No. 5), 79–87 (1996).

O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
[Crossref] [PubMed]

S. B. Lowen, M. C. Teich, “The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains,” J. Acoust. Soc. Am. 99, 3585–3591 (1996).
[Crossref] [PubMed]

R. G. Turcott, M. C. Teich, “Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients,” Ann. Biomed. Eng. 24, 269–293 (1996).

1995 (3)

W. J. McGill, M. C. Teich, “Alerting signals and detection in a sensory network,” J. Math. Psychol. 39, 146–163 (1995).
[Crossref]

R. G. Turcott, P. D. R. Barker, 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).

S. B. Lowen, M. C. Teich, “Estimation and simulation of fractal stochastic point processes,” Fractals 3, 183–210 (1995).
[Crossref]

1994 (2)

B. J. West, W. Deering, “Fractal physiology for physicists: Lévy statistics,” Phys. Rep. 246, 1–100 (1994).
[Crossref]

J. B. Bassingthwaighte, G. M. Raymond, “Evaluating rescaled range analysis for time series,” Ann. Biomed. Eng. 22, 432–444 (1994).

1993 (3)

A. R. Kumar, D. H. Johnson, “Analyzing and modeling fractal intensity point processes,” J. Acoust. Soc. Am. 93, 3365–3373 (1993).
[Crossref] [PubMed]

S. B. Lowen, M. C. Teich, “Fractal renewal processes,” IEEE Trans. Inf. Theory 39, 1669–1671 (1993).
[Crossref]

F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
[Crossref]

1992 (2)

J. B. Troy, J. G. Robson, “Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance,” Visual Neurosci. 9, 535–553 (1992).
[PubMed]

H. E. Schepers, J. H. G. M. van Beek, J. B. Bassingthwaighte, “Four methods to estimate the fractal dimension from self-affine signals,” IEEE Eng. Med. Biol. Mag. 11, 57–71 (1992).

1991 (1)

S. B. Lowen, M. C. Teich, “Doubly stochastic Poisson point process driven by fractal shot noise,” Phys. Rev. A 43, 4192–4215 (1991).
[Crossref] [PubMed]

1990 (3)

M. C. Teich, T. Li, “The retinal rod as a chemical photomultiplier,” J. Visual Commun. Image Represent. 1, 104–111 (1990).

L. S. Liebovitch, T. I. Tóth, “Using fractals to understand the opening and closing of ion channels,” Ann. Biomed. Eng. 18, 177–194 (1990).
[PubMed]

M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
[PubMed]

1989 (1)

M. C. Teich, “Fractal character of the auditory neural spike train,” IEEE Trans. Biomed. Eng. 36, 150–160 (1989).
[Crossref] [PubMed]

1987 (2)

J. G. Robson, J. B. Troy, “Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat,” J. Opt. Soc. Am. A 4, 2301–2307 (1987).
[Crossref] [PubMed]

E. Kaplan, K. Purpura, R. M. Shapley, “Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus,” J. Physiol. (London) 391, 267–288 (1987).

1986 (1)

M. W. Levine, J. B. Troy, “The variability of the maintained discharge of cat dorsal lateral geniculate cells,” J. Physiol. (London) 375, 339–359 (1986).

1985 (3)

M. C. Teich, S. M. Khanna, “Pulse-number distribution for the neural spike train in the cat’s auditory nerve,” J. Acoust. Soc. Am. 77, 1110–1128 (1985).
[Crossref] [PubMed]

B. E. A. Saleh, M. C. Teich, “Multiplication and refractoriness in the cat’s retinal-ganglion-cell discharge at low light levels,” Biol. Cybern. 52, 101–107 (1985).
[Crossref]

B. G. Cleland, B. B. Lee, “A comparison of visual responses of cat lateral geniculate nucleus neurones with those of ganglion cells afferent to them,” J. Physiol. (London) 369, 249–268 (1985).

1984 (3)

E. Kaplan, R. Shapley, “The origin of the S (slow) potential in the mammalian lateral geniculate nucleus,” Exp. Brain Res. 55, 111–116 (1984).
[Crossref] [PubMed]

M. C. Teich, B. E. A. Saleh, J. Peřina, “Role of primary excitation statistics in the generation of antibunched and sub-Poisson light,” J. Opt. Soc. Am. B 1, 366–388 (1984).
[Crossref]

J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
[Crossref]

1983 (6)

J. B. Troy, “Spatial contrast sensitivities of X and Y type neurones in the cat’s dorsal lateral geniculate nucleus,” J. Physiol. (London) 344, 399–417 (1983).

R. W. McCarley, O. Benoit, G. Barrionuevo, “Lateral geniculate nucleus unitary discharge in sleep and waking: state- and rate-specific effects,” J. Neurophysiol. 50, 798–818 (1983).
[PubMed]

D. N. Mastronarde, “Correlated firing of cat retinal ganglion cells. I. Spontaneously active inputs to X- and Y-cells,” J. Neurophysiol. 49, 303–324 (1983).
[PubMed]

L. J. Frishman, M. W. Levine, “Statistics of the maintained discharge of cat retinal ganglion cells,” J. Physiol. (London) 339, 475–494 (1983).

P. R. Prucnal, M. C. Teich, “Refractory effects in neural counting processes with exponentially decaying rates,” IEEE Trans. Syst. Man Cybern. SMC-13, 1028–1033 (1983).

B. B. Lee, V. Virsu, O. D. Creutzfeldt, “Linear signal transmission from prepotentials to cells in the macaque lateral geniculate nucleus,” Exp. Brain Res. 52, 50–56 (1983).

1982 (4)

A. M. Derrington, P. Lennie, “The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cat,” J. Physiol. (London) 333, 343–366 (1982).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, W. J. McGill, “Multiplication noise in the human visual system at threshold: 1. Quantum fluctuations and minimum detectable energy,” J. Opt. Soc. Am. 72, 419–431 (1982).
[Crossref] [PubMed]

E. Kaplan, R. M. Shapley, “X and Y cells in the lateral geniculate nucleus of macaque monkeys,” J. Physiol. (London) 330, 125–143 (1982).

B. E. A. Saleh, M. C. Teich, “Multiplied-Poisson noise in pulse, particle, and photon detection,” Proc. IEEE 70, 229–245 (1982).
[Crossref]

1981 (1)

1980 (1)

M. C. Teich, P. Diament, “Relative refractoriness in visual information processing,” Biol. Cybern. 38, 187–191 (1980).
[Crossref]

1978 (2)

F. Frontera, F. Fuligni, “The effect of dead time on the power spectral density estimates of discrete time series,” Nucl. Instrum. Methods 157, 557–561 (1978).
[Crossref]

M. C. Teich, L. Matin, B. I. Cantor, “Refractoriness in the maintained discharge of the cat’s retinal ganglion cell,” J. Opt. Soc. Am. 68, 386–402 (1978).
[Crossref] [PubMed]

1976 (2)

T. Sato, M. Yamamoto, H. Nakahama, “Variability of interspike intervals of cat’s on-center optic track fibres activated by steady light spot: a comparative study on X- and Y-fibers,” Exp. Brain Res. 24, 285–293 (1976).
[Crossref] [PubMed]

S. Hochstein, R. M. Shapley, “Quantitative analysis of retinal ganglion cell classifications,” J. Physiol. (London) 262, 237–264 (1976).

1975 (1)

R. M. Shapley, S. Hochstein, “Visual spatial summation in two classes of geniculate cells,” Nature (London) 256, 411–413 (1975).
[Crossref]

1973 (1)

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
[PubMed]

1972 (1)

E. G. Merrill, A. Ainsworth, “Glass-coated platinum-plated tungsten microelectrodes,” Med. Biol. Eng. 10, 662–672 (1972).
[PubMed]

1971 (2)

B. G. Cleland, M. W. Dubin, W. R. Levick, “Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus,” J. Physiol. (London) 217, 473–496 (1971).

H. B. Barlow, W. R. Levick, M. Yoon, “Responses to single quanta of light in retinal ganglion cells of the cat,” Vis. Res. 11 (Suppl. 3), 87–101 (1971).
[Crossref]

1969 (3)

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,” J. Physiol. (London) 200, 1–24 (1969).

H. B. Barlow, W. R. Levick, “Changes in the maintained discharge with adaptation level in the cat retina,” J. Physiol. (London) 202, 699–718 (1969).

H. B. Barlow, “Scaling and refractoriness in pulse trains,” J. Opt. Soc. Am. 59, 1500 (1969).
[PubMed]

1967 (1)

R. W. Rodieck, “Maintained activity of cat retinal ganglion cells,” J. Neurophysiol. 30, 1043–1071 (1967).
[PubMed]

1966 (3)

G. Gestri, L. Maffei, D. Petracchi, “Spatial and temporal organization in retinal units,” Kybernetik (Biol. Cybern.) 3, 196–202 (1966).
[PubMed]

L. M. Ricciardi, F. Esposito, “On some distribution functions for non-linear switching elements with finite dead time,” Kybernetik (Biol. Cybern.) 3, 148–152 (1966).

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[Crossref]

1965 (1)

J. M. Fuster, A. Herz, O. D. Creutzfeldt, “Interval analysis of cell discharge in spontaneous and optically modulated activity in the visual system,” Arch. Ital. Biol. 103, 159–177 (1965).
[PubMed]

1964 (1)

P. O. Bishop, W. R. Levick, W. O. Williams, “Statistical analysis of the dark discharge of lateral geniculate neurones,” J. Physiol. (London) 170, 598–612 (1964).

1961 (2)

T. Lukes, “The statistical properties of sequences of stochastic pulses,” Proc. Phys. Soc. London 78, 153–168 (1961).
[Crossref]

D. H. Hubel, T. N. Wiesel, “Integrative action in the cat’s lateral geniculate body,” J. Physiol. (London) 155, 385–398 (1961).

1958 (1)

P. O. Bishop, W. Burke, R. Davis, “Synapse discharge by single fibre in mammalian visual system,” Nature (London) 182, 728–730 (1958).
[Crossref]

1957 (1)

S. W. Kuffler, R. FitzHugh, H. B. Barlow, “Maintained activity in the cat’s retina in light and darkness,” J. Gen. Physiol. 40, 683–702 (1957).
[Crossref] [PubMed]

1951 (2)

H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civ. Eng. 116, 770–808 (1951).

W. Feller, “The asymptotic distribution of the range of sums of independent random variables,” Ann. Math. Stat. 22, 427–432 (1951).

1947 (1)

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26–29 (1947).
[Crossref]

Agnati, L. F.

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
[PubMed]

Ainsworth, A.

E. G. Merrill, A. Ainsworth, “Glass-coated platinum-plated tungsten microelectrodes,” Med. Biol. Eng. 10, 662–672 (1972).
[PubMed]

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[Crossref]

Barker, P. D. R.

R. G. Turcott, P. D. R. Barker, 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).

Barlow, H. B.

H. B. Barlow, W. R. Levick, M. Yoon, “Responses to single quanta of light in retinal ganglion cells of the cat,” Vis. Res. 11 (Suppl. 3), 87–101 (1971).
[Crossref]

H. B. Barlow, W. R. Levick, “Changes in the maintained discharge with adaptation level in the cat retina,” J. Physiol. (London) 202, 699–718 (1969).

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,” J. Physiol. (London) 200, 1–24 (1969).

H. B. Barlow, “Scaling and refractoriness in pulse trains,” J. Opt. Soc. Am. 59, 1500 (1969).
[PubMed]

S. W. Kuffler, R. FitzHugh, H. B. Barlow, “Maintained activity in the cat’s retina in light and darkness,” J. Gen. Physiol. 40, 683–702 (1957).
[Crossref] [PubMed]

Barrionuevo, G.

R. W. McCarley, O. Benoit, G. Barrionuevo, “Lateral geniculate nucleus unitary discharge in sleep and waking: state- and rate-specific effects,” J. Neurophysiol. 50, 798–818 (1983).
[PubMed]

Bassingthwaighte, J. B.

J. B. Bassingthwaighte, G. M. Raymond, “Evaluating rescaled range analysis for time series,” Ann. Biomed. Eng. 22, 432–444 (1994).

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J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994).

Benoit, O.

R. W. McCarley, O. Benoit, G. Barrionuevo, “Lateral geniculate nucleus unitary discharge in sleep and waking: state- and rate-specific effects,” J. Neurophysiol. 50, 798–818 (1983).
[PubMed]

Bishop, P. O.

P. O. Bishop, W. R. Levick, W. O. Williams, “Statistical analysis of the dark discharge of lateral geniculate neurones,” J. Physiol. (London) 170, 598–612 (1964).

P. O. Bishop, W. Burke, R. Davis, “Synapse discharge by single fibre in mammalian visual system,” Nature (London) 182, 728–730 (1958).
[Crossref]

Breton, M. E.

Burke, W.

P. O. Bishop, W. Burke, R. Davis, “Synapse discharge by single fibre in mammalian visual system,” Nature (London) 182, 728–730 (1958).
[Crossref]

Cantor, B. I.

Cariani, P.

O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
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Cash, S. S.

S. B. Lowen, S. S. Cash, M.-m. Poo, M. C. Teich, “Neuronal exocytosis exhibits fractal behavior,” in Computational Neuroscience: Trends in Research 1966, J. M. Bower, ed. (Plenum, New York) (to be published).

Cleland, B. G.

B. G. Cleland, B. B. Lee, “A comparison of visual responses of cat lateral geniculate nucleus neurones with those of ganglion cells afferent to them,” J. Physiol. (London) 369, 249–268 (1985).

B. G. Cleland, M. W. Dubin, W. R. Levick, “Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus,” J. Physiol. (London) 217, 473–496 (1971).

Cox, D. R.

D. R. Cox, Renewal Theory (Methuen, London, 1962), p. 8.

D. R. Cox, P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).

Creutzfeldt, O. D.

B. B. Lee, V. Virsu, O. D. Creutzfeldt, “Linear signal transmission from prepotentials to cells in the macaque lateral geniculate nucleus,” Exp. Brain Res. 52, 50–56 (1983).

J. M. Fuster, A. Herz, O. D. Creutzfeldt, “Interval analysis of cell discharge in spontaneous and optically modulated activity in the visual system,” Arch. Ital. Biol. 103, 159–177 (1965).
[PubMed]

Davis, R.

P. O. Bishop, W. Burke, R. Davis, “Synapse discharge by single fibre in mammalian visual system,” Nature (London) 182, 728–730 (1958).
[Crossref]

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B. J. West, W. Deering, “Fractal physiology for physicists: Lévy statistics,” Phys. Rep. 246, 1–100 (1994).
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O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
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A. M. Derrington, P. Lennie, “The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cat,” J. Physiol. (London) 333, 343–366 (1982).

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M. C. Teich, P. Diament, “Relative refractoriness in visual information processing,” Biol. Cybern. 38, 187–191 (1980).
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B. G. Cleland, M. W. Dubin, W. R. Levick, “Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus,” J. Physiol. (London) 217, 473–496 (1971).

Esposito, F.

L. M. Ricciardi, F. Esposito, “On some distribution functions for non-linear switching elements with finite dead time,” Kybernetik (Biol. Cybern.) 3, 148–152 (1966).

Fano, U.

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26–29 (1947).
[Crossref]

Feller, W.

W. Feller, “The asymptotic distribution of the range of sums of independent random variables,” Ann. Math. Stat. 22, 427–432 (1951).

FitzHugh, R.

S. W. Kuffler, R. FitzHugh, H. B. Barlow, “Maintained activity in the cat’s retina in light and darkness,” J. Gen. Physiol. 40, 683–702 (1957).
[Crossref] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988).

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L. J. Frishman, M. W. Levine, “Statistics of the maintained discharge of cat retinal ganglion cells,” J. Physiol. (London) 339, 475–494 (1983).

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F. Frontera, F. Fuligni, “The effect of dead time on the power spectral density estimates of discrete time series,” Nucl. Instrum. Methods 157, 557–561 (1978).
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Fuligni, F.

F. Frontera, F. Fuligni, “The effect of dead time on the power spectral density estimates of discrete time series,” Nucl. Instrum. Methods 157, 557–561 (1978).
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Fuster, J. M.

J. M. Fuster, A. Herz, O. D. Creutzfeldt, “Interval analysis of cell discharge in spontaneous and optically modulated activity in the visual system,” Arch. Ital. Biol. 103, 159–177 (1965).
[PubMed]

Galletti, C.

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
[PubMed]

Gestri, G.

G. Gestri, L. Maffei, D. Petracchi, “Spatial and temporal organization in retinal units,” Kybernetik (Biol. Cybern.) 3, 196–202 (1966).
[PubMed]

Grüneis, F.

F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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F. A. Haight, Handbook of the Poisson Distribution (Wiley, New York, 1967).

Hara, K.-i.

J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
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Heneghan, C.

M. C. Teich, C. Heneghan, S. B. Lowen, R. G. Turcott, “Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), Chap. 14, pp. 383–412.

Herz, A.

J. M. Fuster, A. Herz, O. D. Creutzfeldt, “Interval analysis of cell discharge in spontaneous and optically modulated activity in the visual system,” Arch. Ital. Biol. 103, 159–177 (1965).
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Hochstein, S.

S. Hochstein, R. M. Shapley, “Quantitative analysis of retinal ganglion cell classifications,” J. Physiol. (London) 262, 237–264 (1976).

R. M. Shapley, S. Hochstein, “Visual spatial summation in two classes of geniculate cells,” Nature (London) 256, 411–413 (1975).
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Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Integrative action in the cat’s lateral geniculate body,” J. Physiol. (London) 155, 385–398 (1961).

Hurst, H. E.

H. E. Hurst, “Long-term storage capacity of reservoirs,” Trans. Am. Soc. Civ. Eng. 116, 770–808 (1951).

Johnson, D. H.

O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
[Crossref] [PubMed]

A. R. Kumar, D. H. Johnson, “Analyzing and modeling fractal intensity point processes,” J. Acoust. Soc. Am. 93, 3365–3373 (1993).
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M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
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Kaplan, E.

E. Kaplan, K. Purpura, R. M. Shapley, “Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus,” J. Physiol. (London) 391, 267–288 (1987).

E. Kaplan, R. Shapley, “The origin of the S (slow) potential in the mammalian lateral geniculate nucleus,” Exp. Brain Res. 55, 111–116 (1984).
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E. Kaplan, R. M. Shapley, “X and Y cells in the lateral geniculate nucleus of macaque monkeys,” J. Physiol. (London) 330, 125–143 (1982).

E. Kaplan, P. Mukherjee, R. Shapley, “Information filtering in the lateral geniculate nucleus,” in Contrast Sensitivity, R. Shapley, D. Man-Kit Lam, eds. (MIT Press, Cambridge, Mass., 1993), Vol. 5, pp. 183–200.

Kelly, O. E.

O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
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Khanna, S. M.

M. C. Teich, S. M. Khanna, “Pulse-number distribution for the neural spike train in the cat’s auditory nerve,” J. Acoust. Soc. Am. 77, 1110–1128 (1985).
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Kimura, M.

J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
[Crossref]

Kuffler, S. W.

S. W. Kuffler, R. FitzHugh, H. B. Barlow, “Maintained activity in the cat’s retina in light and darkness,” J. Gen. Physiol. 40, 683–702 (1957).
[Crossref] [PubMed]

Kumar, A. R.

A. R. Kumar, D. H. Johnson, “Analyzing and modeling fractal intensity point processes,” J. Acoust. Soc. Am. 93, 3365–3373 (1993).
[Crossref] [PubMed]

M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
[PubMed]

Lee, B. B.

B. G. Cleland, B. B. Lee, “A comparison of visual responses of cat lateral geniculate nucleus neurones with those of ganglion cells afferent to them,” J. Physiol. (London) 369, 249–268 (1985).

B. B. Lee, V. Virsu, O. D. Creutzfeldt, “Linear signal transmission from prepotentials to cells in the macaque lateral geniculate nucleus,” Exp. Brain Res. 52, 50–56 (1983).

Lennie, P.

A. M. Derrington, P. Lennie, “The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cat,” J. Physiol. (London) 333, 343–366 (1982).

Levick, W. R.

B. G. Cleland, M. W. Dubin, W. R. Levick, “Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus,” J. Physiol. (London) 217, 473–496 (1971).

H. B. Barlow, W. R. Levick, M. Yoon, “Responses to single quanta of light in retinal ganglion cells of the cat,” Vis. Res. 11 (Suppl. 3), 87–101 (1971).
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H. B. Barlow, W. R. Levick, “Changes in the maintained discharge with adaptation level in the cat retina,” J. Physiol. (London) 202, 699–718 (1969).

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,” J. Physiol. (London) 200, 1–24 (1969).

P. O. Bishop, W. R. Levick, W. O. Williams, “Statistical analysis of the dark discharge of lateral geniculate neurones,” J. Physiol. (London) 170, 598–612 (1964).

W. R. Levick, “Maintained discharge in the visual system and its role for information processing,” in Handbook of Sensory Physiology, Vol. VII/3, Central Processing of Visual Information, Part A, R. Jung, ed. (Springer-Verlag, New York, 1973), pp. 575–598.

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M. W. Levine, J. B. Troy, “The variability of the maintained discharge of cat dorsal lateral geniculate cells,” J. Physiol. (London) 375, 339–359 (1986).

L. J. Frishman, M. W. Levine, “Statistics of the maintained discharge of cat retinal ganglion cells,” J. Physiol. (London) 339, 475–494 (1983).

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S. B. Lowen, 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/fFluctuations, P. H. Handel, A. L. Chung, eds., AIP Conf. Proc.285 (American Institute of Physics, New York, 1993), pp. 781–784.

S. B. Lowen, S. S. Cash, M.-m. Poo, M. C. Teich, “Neuronal exocytosis exhibits fractal behavior,” in Computational Neuroscience: Trends in Research 1966, J. M. Bower, ed. (Plenum, New York) (to be published).

M. C. Teich, C. Heneghan, S. B. Lowen, R. G. Turcott, “Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), Chap. 14, pp. 383–412.

M. C. Teich, R. G. Turcott, 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, P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, C. R. Steele, eds., Vol. 87 of Lecture Notes in Biomathematics (Springer-Verlag, New York, 1990), pp. 354–361.
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T. Lukes, “The statistical properties of sequences of stochastic pulses,” Proc. Phys. Soc. London 78, 153–168 (1961).
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G. Gestri, L. Maffei, D. Petracchi, “Spatial and temporal organization in retinal units,” Kybernetik (Biol. Cybern.) 3, 196–202 (1966).
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Maioli, M. G.

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
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McCarley, R. W.

R. W. McCarley, O. Benoit, G. Barrionuevo, “Lateral geniculate nucleus unitary discharge in sleep and waking: state- and rate-specific effects,” J. Neurophysiol. 50, 798–818 (1983).
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McGill, W. J.

Meesmann, M.

F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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Mukherjee, P.

E. Kaplan, P. Mukherjee, R. Shapley, “Information filtering in the lateral geniculate nucleus,” in Contrast Sensitivity, R. Shapley, D. Man-Kit Lam, eds. (MIT Press, Cambridge, Mass., 1993), Vol. 5, pp. 183–200.

Munemori, J.

J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
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Musha, T.

F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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S. B. Lowen, S. S. Cash, M.-m. Poo, M. C. Teich, “Neuronal exocytosis exhibits fractal behavior,” in Computational Neuroscience: Trends in Research 1966, J. M. Bower, ed. (Plenum, New York) (to be published).

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E. Kaplan, K. Purpura, R. M. Shapley, “Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus,” J. Physiol. (London) 391, 267–288 (1987).

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J. B. Bassingthwaighte, G. M. Raymond, “Evaluating rescaled range analysis for time series,” Ann. Biomed. Eng. 22, 432–444 (1994).

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L. M. Ricciardi, F. Esposito, “On some distribution functions for non-linear switching elements with finite dead time,” Kybernetik (Biol. Cybern.) 3, 148–152 (1966).

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Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, “Multiplication and refractoriness in the cat’s retinal-ganglion-cell discharge at low light levels,” Biol. Cybern. 52, 101–107 (1985).
[Crossref]

M. C. Teich, B. E. A. Saleh, J. Peřina, “Role of primary excitation statistics in the generation of antibunched and sub-Poisson light,” J. Opt. Soc. Am. B 1, 366–388 (1984).
[Crossref]

B. E. A. Saleh, M. C. Teich, “Multiplied-Poisson noise in pulse, particle, and photon detection,” Proc. IEEE 70, 229–245 (1982).
[Crossref]

M. C. Teich, B. E. A. Saleh, “Interevent-time statistics for shot-noise-driven self-exciting point processes in photon detection,” J. Opt. Soc. Am. 71, 771–776 (1981).
[Crossref]

Salvi, R. J.

N. L. Powers, R. J. Salvi, “Comparison of discharge rate fluctuations in the auditory nerve of chickens and chinchillas,” in Abstracts of the XVth Midwinter Research Meeting, Association for Research in OtolaryngologyD. J. Lim., ed. (Association for Research in Otolaryngology, Des Moines, Ia., 1992), Abstract No. 292, p. 101.

N. L. Powers, R. J. Salvi, S. S. Saunders, “Discharge rate fluctuations in the auditory nerve of the chinchilla,” in Abstracts of the XIVth Midwinter Research Meeting, Association for Research in OtolaryngologyD. J. Lim, ed. (Association for Research in Otolaryngology, Des Moines, Ia., 1991); Abstract No. 411, p. 129.

Sanseverino, E. R.

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
[PubMed]

Sato, R.

J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
[Crossref]

Sato, T.

T. Sato, M. Yamamoto, H. Nakahama, “Variability of interspike intervals of cat’s on-center optic track fibres activated by steady light spot: a comparative study on X- and Y-fibers,” Exp. Brain Res. 24, 285–293 (1976).
[Crossref] [PubMed]

Saunders, S. S.

N. L. Powers, R. J. Salvi, S. S. Saunders, “Discharge rate fluctuations in the auditory nerve of the chinchilla,” in Abstracts of the XIVth Midwinter Research Meeting, Association for Research in OtolaryngologyD. J. Lim, ed. (Association for Research in Otolaryngology, Des Moines, Ia., 1991); Abstract No. 411, p. 129.

Schepers, H. E.

H. E. Schepers, J. H. G. M. van Beek, J. B. Bassingthwaighte, “Four methods to estimate the fractal dimension from self-affine signals,” IEEE Eng. Med. Biol. Mag. 11, 57–71 (1992).

Shapley, R.

E. Kaplan, R. Shapley, “The origin of the S (slow) potential in the mammalian lateral geniculate nucleus,” Exp. Brain Res. 55, 111–116 (1984).
[Crossref] [PubMed]

E. Kaplan, P. Mukherjee, R. Shapley, “Information filtering in the lateral geniculate nucleus,” in Contrast Sensitivity, R. Shapley, D. Man-Kit Lam, eds. (MIT Press, Cambridge, Mass., 1993), Vol. 5, pp. 183–200.

Shapley, R. M.

E. Kaplan, K. Purpura, R. M. Shapley, “Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus,” J. Physiol. (London) 391, 267–288 (1987).

E. Kaplan, R. M. Shapley, “X and Y cells in the lateral geniculate nucleus of macaque monkeys,” J. Physiol. (London) 330, 125–143 (1982).

S. Hochstein, R. M. Shapley, “Quantitative analysis of retinal ganglion cell classifications,” J. Physiol. (London) 262, 237–264 (1976).

R. M. Shapley, S. Hochstein, “Visual spatial summation in two classes of geniculate cells,” Nature (London) 256, 411–413 (1975).
[Crossref]

Siegel, R. M.

M. C. Teich, R. G. Turcott, R. M. Siegel, “Temporal correlation in cat striate-cortex neural spike trains,” IEEE Eng. Med. Biol. Mag. 15 (No. 5), 79–87 (1996).

Teich, M. C.

M. C. Teich, R. G. Turcott, R. M. Siegel, “Temporal correlation in cat striate-cortex neural spike trains,” IEEE Eng. Med. Biol. Mag. 15 (No. 5), 79–87 (1996).

R. G. Turcott, M. C. Teich, “Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients,” Ann. Biomed. Eng. 24, 269–293 (1996).

S. B. Lowen, M. C. Teich, “The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains,” J. Acoust. Soc. Am. 99, 3585–3591 (1996).
[Crossref] [PubMed]

R. G. Turcott, P. D. R. Barker, 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).

S. B. Lowen, M. C. Teich, “Estimation and simulation of fractal stochastic point processes,” Fractals 3, 183–210 (1995).
[Crossref]

W. J. McGill, M. C. Teich, “Alerting signals and detection in a sensory network,” J. Math. Psychol. 39, 146–163 (1995).
[Crossref]

S. B. Lowen, M. C. Teich, “Fractal renewal processes,” IEEE Trans. Inf. Theory 39, 1669–1671 (1993).
[Crossref]

S. B. Lowen, M. C. Teich, “Doubly stochastic Poisson point process driven by fractal shot noise,” Phys. Rev. A 43, 4192–4215 (1991).
[Crossref] [PubMed]

M. C. Teich, T. Li, “The retinal rod as a chemical photomultiplier,” J. Visual Commun. Image Represent. 1, 104–111 (1990).

M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
[PubMed]

M. C. Teich, “Fractal character of the auditory neural spike train,” IEEE Trans. Biomed. Eng. 36, 150–160 (1989).
[Crossref] [PubMed]

M. C. Teich, S. M. Khanna, “Pulse-number distribution for the neural spike train in the cat’s auditory nerve,” J. Acoust. Soc. Am. 77, 1110–1128 (1985).
[Crossref] [PubMed]

B. E. A. Saleh, M. C. Teich, “Multiplication and refractoriness in the cat’s retinal-ganglion-cell discharge at low light levels,” Biol. Cybern. 52, 101–107 (1985).
[Crossref]

M. C. Teich, B. E. A. Saleh, J. Peřina, “Role of primary excitation statistics in the generation of antibunched and sub-Poisson light,” J. Opt. Soc. Am. B 1, 366–388 (1984).
[Crossref]

P. R. Prucnal, M. C. Teich, “Refractory effects in neural counting processes with exponentially decaying rates,” IEEE Trans. Syst. Man Cybern. SMC-13, 1028–1033 (1983).

B. E. A. Saleh, M. C. Teich, “Multiplied-Poisson noise in pulse, particle, and photon detection,” Proc. IEEE 70, 229–245 (1982).
[Crossref]

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, W. J. McGill, “Multiplication noise in the human visual system at threshold: 1. Quantum fluctuations and minimum detectable energy,” J. Opt. Soc. Am. 72, 419–431 (1982).
[Crossref] [PubMed]

M. C. Teich, B. E. A. Saleh, “Interevent-time statistics for shot-noise-driven self-exciting point processes in photon detection,” J. Opt. Soc. Am. 71, 771–776 (1981).
[Crossref]

M. C. Teich, P. Diament, “Relative refractoriness in visual information processing,” Biol. Cybern. 38, 187–191 (1980).
[Crossref]

M. C. Teich, L. Matin, B. I. Cantor, “Refractoriness in the maintained discharge of the cat’s retinal ganglion cell,” J. Opt. Soc. Am. 68, 386–402 (1978).
[Crossref] [PubMed]

M. C. Teich, R. G. Turcott, 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, P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, C. R. Steele, eds., Vol. 87 of Lecture Notes in Biomathematics (Springer-Verlag, New York, 1990), pp. 354–361.
[Crossref]

M. C. Teich, C. Heneghan, S. B. Lowen, R. G. Turcott, “Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), Chap. 14, pp. 383–412.

M. C. Teich, “Fractal neuronal firing patterns,” in Single Neuron Computation, T. McKenna, J. Davis, S. Zornetzer, eds. (Academic, Boston, 1992), pp. 589–625.

S. B. Lowen, 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/fFluctuations, P. H. Handel, A. L. Chung, eds., AIP Conf. Proc.285 (American Institute of Physics, New York, 1993), pp. 781–784.

R. G. Turcott, M. C. Teich, “Long-duration correlation and attractor topology of the heartbeat rate differ for normal patients and those with heart failure,” in Chaos in Biology and Medicine, W. L. Ditto, ed., Proc. SPIE2036, 22–39 (1993).
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S. B. Lowen, S. S. Cash, M.-m. Poo, M. C. Teich, “Neuronal exocytosis exhibits fractal behavior,” in Computational Neuroscience: Trends in Research 1966, J. M. Bower, ed. (Plenum, New York) (to be published).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988).

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L. S. Liebovitch, T. I. Tóth, “Using fractals to understand the opening and closing of ion channels,” Ann. Biomed. Eng. 18, 177–194 (1990).
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Troy, J. B.

J. B. Troy, J. G. Robson, “Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance,” Visual Neurosci. 9, 535–553 (1992).
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J. G. Robson, J. B. Troy, “Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat,” J. Opt. Soc. Am. A 4, 2301–2307 (1987).
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M. W. Levine, J. B. Troy, “The variability of the maintained discharge of cat dorsal lateral geniculate cells,” J. Physiol. (London) 375, 339–359 (1986).

J. B. Troy, “Spatial contrast sensitivities of X and Y type neurones in the cat’s dorsal lateral geniculate nucleus,” J. Physiol. (London) 344, 399–417 (1983).

Turcott, R. G.

M. C. Teich, R. G. Turcott, R. M. Siegel, “Temporal correlation in cat striate-cortex neural spike trains,” IEEE Eng. Med. Biol. Mag. 15 (No. 5), 79–87 (1996).

R. G. Turcott, M. C. Teich, “Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients,” Ann. Biomed. Eng. 24, 269–293 (1996).

R. G. Turcott, P. D. R. Barker, 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).

M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
[PubMed]

R. G. Turcott, M. C. Teich, “Long-duration correlation and attractor topology of the heartbeat rate differ for normal patients and those with heart failure,” in Chaos in Biology and Medicine, W. L. Ditto, ed., Proc. SPIE2036, 22–39 (1993).
[Crossref]

M. C. Teich, R. G. Turcott, 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, P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, C. R. Steele, eds., Vol. 87 of Lecture Notes in Biomathematics (Springer-Verlag, New York, 1990), pp. 354–361.
[Crossref]

M. C. Teich, C. Heneghan, S. B. Lowen, R. G. Turcott, “Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), Chap. 14, pp. 383–412.

van Beek, J. H. G. M.

H. E. Schepers, J. H. G. M. van Beek, J. B. Bassingthwaighte, “Four methods to estimate the fractal dimension from self-affine signals,” IEEE Eng. Med. Biol. Mag. 11, 57–71 (1992).

Vannucci, G.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988).

Virsu, V.

B. B. Lee, V. Virsu, O. D. Creutzfeldt, “Linear signal transmission from prepotentials to cells in the macaque lateral geniculate nucleus,” Exp. Brain Res. 52, 50–56 (1983).

West, B. J.

B. J. West, W. Deering, “Fractal physiology for physicists: Lévy statistics,” Phys. Rep. 246, 1–100 (1994).
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J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994).

Wiesel, T. N.

D. H. Hubel, T. N. Wiesel, “Integrative action in the cat’s lateral geniculate body,” J. Physiol. (London) 155, 385–398 (1961).

Williams, W. O.

P. O. Bishop, W. R. Levick, W. O. Williams, “Statistical analysis of the dark discharge of lateral geniculate neurones,” J. Physiol. (London) 170, 598–612 (1964).

Wise, M. E.

M. E. Wise, “Spike interval distributions for neurons and random walks with drift to a fluctuating threshold,” in Statistical Distributions in Scientific Work, C. E. A. Taillie, ed. (Reidel, Boston, 1981), Vol. 6, pp. 211–231.

Yamamoto, M.

F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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T. Sato, M. Yamamoto, H. Nakahama, “Variability of interspike intervals of cat’s on-center optic track fibres activated by steady light spot: a comparative study on X- and Y-fibers,” Exp. Brain Res. 24, 285–293 (1976).
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Yoon, M.

H. B. Barlow, W. R. Levick, M. Yoon, “Responses to single quanta of light in retinal ganglion cells of the cat,” Vis. Res. 11 (Suppl. 3), 87–101 (1971).
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Ann. Biomed. Eng. (3)

L. S. Liebovitch, T. I. Tóth, “Using fractals to understand the opening and closing of ion channels,” Ann. Biomed. Eng. 18, 177–194 (1990).
[PubMed]

R. G. Turcott, M. C. Teich, “Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients,” Ann. Biomed. Eng. 24, 269–293 (1996).

J. B. Bassingthwaighte, G. M. Raymond, “Evaluating rescaled range analysis for time series,” Ann. Biomed. Eng. 22, 432–444 (1994).

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B. E. A. Saleh, M. C. Teich, “Multiplication and refractoriness in the cat’s retinal-ganglion-cell discharge at low light levels,” Biol. Cybern. 52, 101–107 (1985).
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J. Munemori, K.-i. Hara, M. Kimura, R. Sato, “Statistical features of impulse trains in cat’s lateral geniculate neurons,” Biol. Cybern. 50, 167–172 (1984).
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F. Grüneis, M. Nakao, Y. Mizutani, M. Yamamoto, M. Meesmann, T. Musha, “Further study on 1/f fluctuations observed in central single neurons during REM sleep,” Biol. Cybern. 68, 193–198 (1993).
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M. C. Teich, P. Diament, “Relative refractoriness in visual information processing,” Biol. Cybern. 38, 187–191 (1980).
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Brain Res. (1)

E. R. Sanseverino, L. F. Agnati, M. G. Maioli, C. Galletti, “Maintained activity of single neurons in striate and non-striate areas of the cat visual cortex,” Brain Res. 54, 225–242 (1973).
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Exp. Brain Res. (3)

B. B. Lee, V. Virsu, O. D. Creutzfeldt, “Linear signal transmission from prepotentials to cells in the macaque lateral geniculate nucleus,” Exp. Brain Res. 52, 50–56 (1983).

E. Kaplan, R. Shapley, “The origin of the S (slow) potential in the mammalian lateral geniculate nucleus,” Exp. Brain Res. 55, 111–116 (1984).
[Crossref] [PubMed]

T. Sato, M. Yamamoto, H. Nakahama, “Variability of interspike intervals of cat’s on-center optic track fibres activated by steady light spot: a comparative study on X- and Y-fibers,” Exp. Brain Res. 24, 285–293 (1976).
[Crossref] [PubMed]

Fractals (1)

S. B. Lowen, M. C. Teich, “Estimation and simulation of fractal stochastic point processes,” Fractals 3, 183–210 (1995).
[Crossref]

Hear. Res. (1)

M. C. Teich, D. H. Johnson, A. R. Kumar, 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).
[PubMed]

IEEE Eng. Med. Biol. Mag. (2)

M. C. Teich, R. G. Turcott, R. M. Siegel, “Temporal correlation in cat striate-cortex neural spike trains,” IEEE Eng. Med. Biol. Mag. 15 (No. 5), 79–87 (1996).

H. E. Schepers, J. H. G. M. van Beek, J. B. Bassingthwaighte, “Four methods to estimate the fractal dimension from self-affine signals,” IEEE Eng. Med. Biol. Mag. 11, 57–71 (1992).

IEEE Trans. Biomed. Eng. (1)

M. C. Teich, “Fractal character of the auditory neural spike train,” IEEE Trans. Biomed. Eng. 36, 150–160 (1989).
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IEEE Trans. Inf. Theory (1)

S. B. Lowen, M. C. Teich, “Fractal renewal processes,” IEEE Trans. Inf. Theory 39, 1669–1671 (1993).
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IEEE Trans. Syst. Man Cybern. (1)

P. R. Prucnal, M. C. Teich, “Refractory effects in neural counting processes with exponentially decaying rates,” IEEE Trans. Syst. Man Cybern. SMC-13, 1028–1033 (1983).

J. Acoust. Soc. Am. (4)

M. C. Teich, S. M. Khanna, “Pulse-number distribution for the neural spike train in the cat’s auditory nerve,” J. Acoust. Soc. Am. 77, 1110–1128 (1985).
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A. R. Kumar, D. H. Johnson, “Analyzing and modeling fractal intensity point processes,” J. Acoust. Soc. Am. 93, 3365–3373 (1993).
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O. E. Kelly, D. H. Johnson, B. Delgutte, P. Cariani, “Fractal noise strength in auditory-nerve fiber recordings,” J. Acoust. Soc. Am. 99, 2210–2220 (1996).
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S. B. Lowen, M. C. Teich, “The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains,” J. Acoust. Soc. Am. 99, 3585–3591 (1996).
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S. Hochstein, R. M. Shapley, “Quantitative analysis of retinal ganglion cell classifications,” J. Physiol. (London) 262, 237–264 (1976).

E. Kaplan, R. M. Shapley, “X and Y cells in the lateral geniculate nucleus of macaque monkeys,” J. Physiol. (London) 330, 125–143 (1982).

L. J. Frishman, M. W. Levine, “Statistics of the maintained discharge of cat retinal ganglion cells,” J. Physiol. (London) 339, 475–494 (1983).

P. O. Bishop, W. R. Levick, W. O. Williams, “Statistical analysis of the dark discharge of lateral geniculate neurones,” J. Physiol. (London) 170, 598–612 (1964).

J. B. Troy, “Spatial contrast sensitivities of X and Y type neurones in the cat’s dorsal lateral geniculate nucleus,” J. Physiol. (London) 344, 399–417 (1983).

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B. G. Cleland, M. W. Dubin, W. R. Levick, “Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus,” J. Physiol. (London) 217, 473–496 (1971).

E. Kaplan, K. Purpura, R. M. Shapley, “Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus,” J. Physiol. (London) 391, 267–288 (1987).

J. Stat. Comput. Simul. (1)

R. G. Turcott, P. D. R. Barker, 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).

J. Visual Commun. Image Represent. (1)

M. C. Teich, T. Li, “The retinal rod as a chemical photomultiplier,” J. Visual Commun. Image Represent. 1, 104–111 (1990).

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F. Frontera, F. Fuligni, “The effect of dead time on the power spectral density estimates of discrete time series,” Nucl. Instrum. Methods 157, 557–561 (1978).
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B. J. West, W. Deering, “Fractal physiology for physicists: Lévy statistics,” Phys. Rep. 246, 1–100 (1994).
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S. B. Lowen, M. C. Teich, “Doubly stochastic Poisson point process driven by fractal shot noise,” Phys. Rev. A 43, 4192–4215 (1991).
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D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
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B. E. A. Saleh, M. C. Teich, “Multiplied-Poisson noise in pulse, particle, and photon detection,” Proc. IEEE 70, 229–245 (1982).
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H. B. Barlow, W. R. Levick, M. Yoon, “Responses to single quanta of light in retinal ganglion cells of the cat,” Vis. Res. 11 (Suppl. 3), 87–101 (1971).
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J. B. Troy, J. G. Robson, “Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance,” Visual Neurosci. 9, 535–553 (1992).
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Other (19)

W. R. Levick, “Maintained discharge in the visual system and its role for information processing,” in Handbook of Sensory Physiology, Vol. VII/3, Central Processing of Visual Information, Part A, R. Jung, ed. (Springer-Verlag, New York, 1973), pp. 575–598.

S. B. Lowen, 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/fFluctuations, P. H. Handel, A. L. Chung, eds., AIP Conf. Proc.285 (American Institute of Physics, New York, 1993), pp. 781–784.

N. L. Powers, R. J. Salvi, S. S. Saunders, “Discharge rate fluctuations in the auditory nerve of the chinchilla,” in Abstracts of the XIVth Midwinter Research Meeting, Association for Research in OtolaryngologyD. J. Lim, ed. (Association for Research in Otolaryngology, Des Moines, Ia., 1991); Abstract No. 411, p. 129.

N. L. Powers, R. J. Salvi, “Comparison of discharge rate fluctuations in the auditory nerve of chickens and chinchillas,” in Abstracts of the XVth Midwinter Research Meeting, Association for Research in OtolaryngologyD. J. Lim., ed. (Association for Research in Otolaryngology, Des Moines, Ia., 1992), Abstract No. 292, p. 101.

J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994).

M. C. Teich, “Fractal neuronal firing patterns,” in Single Neuron Computation, T. McKenna, J. Davis, S. Zornetzer, eds. (Academic, Boston, 1992), pp. 589–625.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983).

D. R. Cox, P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1966).

F. A. Haight, Handbook of the Poisson Distribution (Wiley, New York, 1967).

S. B. Lowen, S. S. Cash, M.-m. Poo, M. C. Teich, “Neuronal exocytosis exhibits fractal behavior,” in Computational Neuroscience: Trends in Research 1966, J. M. Bower, ed. (Plenum, New York) (to be published).

M. E. Wise, “Spike interval distributions for neurons and random walks with drift to a fluctuating threshold,” in Statistical Distributions in Scientific Work, C. E. A. Taillie, ed. (Reidel, Boston, 1981), Vol. 6, pp. 211–231.

R. G. Turcott, M. C. Teich, “Long-duration correlation and attractor topology of the heartbeat rate differ for normal patients and those with heart failure,” in Chaos in Biology and Medicine, W. L. Ditto, ed., Proc. SPIE2036, 22–39 (1993).
[Crossref]

D. R. Cox, Renewal Theory (Methuen, London, 1962), p. 8.

E. Kaplan, P. Mukherjee, R. Shapley, “Information filtering in the lateral geniculate nucleus,” in Contrast Sensitivity, R. Shapley, D. Man-Kit Lam, eds. (MIT Press, Cambridge, Mass., 1993), Vol. 5, pp. 183–200.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988).

D. E. Nelsen, “Calculation of power spectra for a class of randomly jittered waveforms,” in (MIT Research Laboratory of Electronics, Cambridge, Mass., 1964), pp. 168–179.

M. C. Teich, C. Heneghan, S. B. Lowen, R. G. Turcott, “Estimating the fractal exponent of point processes in biological systems using wavelet- and Fourier-transform methods,” in Wavelets in Medicine and Biology, A. Aldroubi, M. Unser, eds. (CRC Press, Boca Raton, Fla., 1996), Chap. 14, pp. 383–412.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1964).

M. C. Teich, R. G. Turcott, 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, P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, C. R. Steele, eds., Vol. 87 of Lecture Notes in Biomathematics (Springer-Verlag, New York, 1990), pp. 354–361.
[Crossref]

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Figures (12)

Fig. 1
Fig. 1

Rate estimates formed by dividing the number of events in successive counting windows by the counting time T. (a) Rate estimate for a cat lateral-geniculate-nucleus (LGN) cell (file name MD2-LGN) generated with the use of four different counting times (T=1, 5, 10, and 50 s). The fluctuations in the rate estimate converge relatively slowly as the counting time is increased. This is characteristic of fractal processes. The convergence properties are quantified by event-number measures such as the rate standard deviation (RSD), the Fano factor (FF), the Allan factor (AF), and the periodogram (PG). (b) Rate estimates from the same recording as that in (a) after the intervals are randomly reordered (shuffled). This maintains the same relative frequency of interval sizes but destroys long-term correlations (and therefore the fractal behavior) arising from other sources, such as rate fluctuations. For such nonfractal signals the rate estimate converges more quickly as the counting time T is increased. The stimulus was a uniformly illuminated screen (with no temporal or spatial modulation) of luminance 33 cd/m2. The data presented here are typical of the 26 data sets examined.

Fig. 2
Fig. 2

(a) The sequence of action-potential waveforms is reduced to a set of event occurrence times {ti} that form a point process. (b) A sequence of interevent intervals {τi} is formed from the time between successive events, resulting in a discrete-time, positive, real-valued stochastic process. All information contained in the original point process is preserved in this representation, but the discrete-time axis of the sequence of interevent intervals is randomly distorted relative to the real-time axis of the point process. (c) The sequence of counts {Ni}, a discrete-time, nonnegative, integer-valued stochastic process, is formed from the point process by recording the numbers of events in successive counting windows of duration T. Information is lost in mapping the point process to the sequence {Ni}, but the amount lost can be made arbitrarily small by reducing T. An advantage of this representation is that no distortion of the time axis occurs.

Fig. 3
Fig. 3

Semilogarithmic plots of the interevent-interval histograms (IIH’s) for retinal-ganglion-cell (RGC) and lateral-geniculate-nucleus (LGN) nerve-spike trains (file names MD2-RET and MD2-LGN, respectively). The spike trains were simultaneously obtained from the same LGN relay neuron. The histograms were normalized to have unit area. Also shown are IIH’s for two candidate models (dotted and dashed curves). (a) IIH for the RGC data (solid curve), along with the best-fitting theoretical curve for the gamma-renewal-process (GRP) model (dotted curve) and for the fixed-dead-time-modified Poisson process (DTMP) model (dashed curve). The GRP model provides a good fit to the IIH. (b) IIH for the LGN data (solid curve), along with the best-fitting theoretical curve for the GRP model (dotted curve) and for the DTMP model (dashed curve). Neither of these two models provides a particularly good fit for short times, but both do well at longer times.

Fig. 4
Fig. 4

Rescaled range analysis (R/S) for the same data sets as those analyzed in Fig. 3 and for shuffled surrogates of the data. (a) R/S plot for the RGC data (upper solid curve) along with the mean of ten R/S curves generated by independent shufflings of the original data set (lower solid curve). The standard deviation of the surrogate R/S curve is not shown, since it is less than the thickness of the curve. (b) R/S plot for the LGN data (upper solid curve) along with the mean of ten R/S curves generated by independent shufflings (lower solid curve). The standard deviation of the surrogate R/S curve is again not shown because of its small value. In both (a) and (b), for sufficiently large k, the R/S curves have a slope greater than 0.5 on this doubly logarithmic plot, indicating the presence of positive correlation. The shuffled-surrogate curves have a dependence very close to k (dotted curves), as expected for sequences of random variables that are essentially independent.

Fig. 5
Fig. 5

Event-number histograms (ENH’s) for the same data sets as those analyzed in Figs. 3 and 4, with the use of a counting time of T=1.0 s, shown together with the same measures for shuffled surrogates of the data and their best-fitting GRP ENH’s. The RGC [solid curve in (a)] and LGN [solid curve in (b)] histograms were normalized so that they summed to unity. The original ENH’s in (a) and (b) are wider than the shuffled and model ENH’s, revealing their greater count variances on this time scale and indicating the presence of long-duration correlation in the original sequence of interevent intervals. For these particular data sets the firing rate of the RGC is approximately twice that of its target LGN cell.

Fig. 6
Fig. 6

Doubly logarithmic plots of the FF’s (solid curves) for the same data sets as those analyzed in Figs. 35, along with the FF’s for shuffled surrogates of the data (gray areas indicate the region bounded by ±1 standard deviation about the mean of the set of ten shuffled FF’s, represented by dotted curves), and asymptotic values of the FF’s for the best-fitting GRP models (short horizontal bars at the right-hand ordinates). The FF’s for the shuffled surrogates always lie near unity, indicating that long-duration correlation associated with the ordering of the intervals has been eliminated by the shuffling process, leaving only the correlation intrinsic to the form of the IIH. Dashed lines of unity slope [indicating F(T)∝T] are included for comparison.

Fig. 7
Fig. 7

Doubly logarithmic plots of the AF’s (solid curves) for the same data sets as those analyzed in Figs. 36, along with the AF’s for shuffled surrogates of the data (gray areas indicate the region bounded by ±1 standard deviation about the mean of the set of ten shuffled AF’s, represented by dotted curves), and asymptotic values of the AF’s for the best-fitting GRP models (short horizontal bars at the right-hand ordinates). The AF’s for the shuffled surrogates always lie near unity, indicating that long-duration correlation associated with the ordering of the intervals has been eliminated by the shuffling process, leaving only the correlation intrinsic to the form of the IIH. Dashed lines of unity slope [indicating A(T)∝T] are included for comparison.

Fig. 8
Fig. 8

Averaged PG’s for the same spike trains as those analyzed in Figs. 37 (solid curves), presented on doubly logarithmic coordinates. The units on the ordinate are spikes squared per seconds squared per Hertz. The shuffled-surrogate PG’s are shown as dotted curves. The high-frequency asymptote on the ordinate is numerically equal to the firing rate μ, whereas the low-frequency asymptote for the shuffled data (representing the renewal process) is numerically equal to μC2. The PG’s for the GRP with parameters that best fit the IIH’s are shown as the white curves. The function 1/f is included for comparison (dashed curves).

Fig. 9
Fig. 9

Estimates of the fractal exponents obtained from (a) the FF, (b) the AF, and (c) the PG. In each plot the exponent estimated from the RGC spike train is plotted along the abscissa, while the exponent estimated from the LGN spike train is plotted along the ordinate. The exponent associated with a given RGC spike train is significantly correlated with that of its target LGN spike train.

Fig. 10
Fig. 10

Construction of the fractal binomial-noise-driven doubly stochastic gamma-r process (FBNDG). This model characterizes the statistical behavior of RGC and LGN spike trains remarkably well. Fractal binomial noise (FBN), generated from the sum of K alternating fractal renewal processes, serves as the time-varying rate of a GRP with fixed order r.

Fig. 11
Fig. 11

Comparison of data presented in Figs. 3(a)8(a) for RGC data set MD2-RET with the predictions of the FBNDG model illustrated in Fig. 10. The model uses four parameters drawn from the data: μ, C, α, and T0. The FBNDG does just about as well as the GRP for (a) the IIH, and incomparably better for (b) R/S, (c) the ENH, (d) the FF, (e) the AF, and (f) the PG. Thus the FBNDG model preserves the IIH statistics of the original GRP model while accommodating the long-term correlations manifested in R/S and in the ENH, the FF, the AF, and the PG.

Fig. 12
Fig. 12

Comparison of AF’s (normalized-time abscissa) for several biological systems exhibiting fractal behavior: cat striate-cortex spike train (solid curve, adapted from Teich et al.25), cat LGN spike train [long-dashed curve, data from Fig. 7(b)], cat primary auditory-nerve-fiber spike train (VIII-NERVE, medium-dashed curve, adapted from Lowen and Teich31), cat RGC spike train [short-dashed curve, data from Fig. 7(a)], and sequence of human heartbeats (HEART, dotted curve, adapted from Turcott and Teich43). For longer counting times all of these AF’s show the power-law increase characteristic of fractal behavior. The dip in some of these curves, in the vicinity of a normalized counting time of 10, is associated with refractoriness, which serves to regularize the events and reduce the variance in the vicinity of these counting times.

Tables (2)

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Table 1 Characteristics of the Retinal-Ganglion-Cell Recordings

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Table 2 Characteristics of the Lateral-Geniculate-Cell Recordings

Equations (24)

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s(t)=i δ(t-ti).
pτ(τ)=λ exp(-λτ),
pτ(τ)=0λ exp[-λ(τ-τd)]0τ<τdττd,
μ=λ1+λτd.
pτ(τ)=(μr)rτr-1 exp(-μrτ)Γ(r),τ0,
pN(N; T)=(λT)N exp(-λT)N!.
pN(N; T)r2πμT exp-r2µT(N-μT)2.
F(T)=var[Ni(T)]Ni(T).
var[Ni(T)]=Ni(T)=λT.
limT0 F(T)=limT0 p(1-p)p=1,
limT F(T)=(1-μτd)2=(1+λτd)-2,
limT F(T)=1r.
F(T)=var[Ni(T)]Ni=T var[Ri(T)]Ri=TσR2Ri,
σR(T)=[μF(T)T]1/21T(1-α)/2.
A(T)=[Ni+1(T)-Ni(T)]22Ni(T).
A(T)=2F(T)-F(2T),
limT A(T)=1r.
Ssegment(f)1M|W˜(f)|2,
S(f)=λ(1+λτd)3forf0λ1+λτdforf;
S(f)=μ Re(1+j2πf/μr)r+1(1+j2πf/μr)r-1,
g(2)(τ)Pr[E(t, t+dt)andE(t+τ, t+τ+dt)]Pr[E(t, t+dt)]Pr[E(t+τ, t+τ+dt)],
F(T)=1+2µT0T(T-τ)[g(2)(τ)-1]dτ
g(2)(τ)=1+12µ2T2[TF(T)]|T=τ,
P(τ)=0τ pτ(t)dt=(μr)rΓ(r)0τ tr-1 exp(-μrt)dt=1Γ(r)0μrτ νr-1 exp(-ν)dν=γ(r, μrτ)Γ(r).

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