Abstract

Probability densities for interevent time are obtained for a doubly stochastic Poisson point process (DSPP) in the presence of self-excitation. The DSPP is assumed to have a stochastic rate that is a filtered Poisson point process (shot noise). The model of a Poisson process driving another Poisson process produces a pulse-bunching effect. Self-excitation (relative refractoriness) results in a deficit of short time intervals. Both effects are observed in many applications of optical detection. The model is applicable to the detection of fluorescence or scintillation generated by ionizing radiation in a photomultiplier tube. It is also used successfully to fit the maintained discharge interspike-interval histograms recorded by Barlow, Levick, and Yoon [Vision Res. <b>11</b>, Suppl. 3, 87–101 (1971)] for a cat’s on-center retinal ganglion cell in darkness.

© 1981 Optical Society of America

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  1. D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).
  2. D. R. Cox, "Some statistical methods connected with series of events," J. R. Statist. Soc. B 17, 129–164 (1955).
  3. A relatively recent book in the area is B. E. A. Saleh, Photoelectron Statistics (Sprinsger-Verlag, New York, 1978).
  4. M. S. Bartlett, "The spectral analysis of two-dimensional point processes," Biometrika 51, 299–311 (1964).
  5. A. J. Lawrance, "Some models for stationary series of univariate events," in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, ed. (Wiley-Interscience, New York, 1972).
  6. B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," submitted to Proc. IEEE.
  7. Synder has shown that a doubly stochastic Poisson point process may be expressed in-terms of an equivalent self-exciting point process (see Ref. 1, pp. 292–293).
  8. W. L. Smith, "Renewal theory and its ramifications," J. R. Statist. Soc. B 20, 243–302 (1958).
  9. L. Takács, "On a probability problem in the theory of counters," Ann. Math. Stat. 29, 1257–1263 (1958).
  10. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).
  11. J. W. Müller, ed., "Bibliography on dead time effects," Rep. no. BIPM-75/6 (Bureau International des Poids at Mesures, Sèvres, France, 1975).
  12. M. C. Teich, L. Matin, and B. I. Cantor, "Refractoriness in the maintained discharge of the cat's retinal ganglion cell," J. Opt. Soc. Am. 68, 386–402 (1978).
  13. M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybernet. 38, 187–191 (1980).
  14. H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).
  15. D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1968).
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  17. J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, New York, 1964).
  18. The rate function is sometimes more precisely described as a sum of two time-decaying exponential functions, a strong fast component plus a much weaker slow component. See G. Walter and A. Coche, "Mesure de la composante lente de l'émission de lumière dans les scintillateurs organiques: application à la discrimination des particules selon leur pouvoir ionisant," Nucl. Instrum. Methods 23, 147–151 (1963).
  19. W. Viehmann and A. G. Eubanks, "Noise limitations of multiplier phototubes in the radiation environment of space," NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Maryland, 1976).
  20. W. Viehmann, A. G. Eubanks, G. F. Piper, and J. H. Bredekamp, "Photomultiplier window materials under electron irradiation: fluorescence and phosphorescence," Appl. Opt. 14, 2104–2115 (1975).
  21. M. C. Teich, "Role of the doubly stochastic Neyman Type-A and Thomas counting distributions in photon detection," Appl. Opt. (to be published).
  22. W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. London 206, 677–700 (1970).
  23. W. R. Levick, John Curtin School of Medical Research, Australian National University, Canberra City, Australia, personal communication.
  24. H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

1980 (1)

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybernet. 38, 187–191 (1980).

1978 (1)

1977 (1)

H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

1975 (1)

1971 (1)

H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

1970 (1)

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. London 206, 677–700 (1970).

1964 (1)

M. S. Bartlett, "The spectral analysis of two-dimensional point processes," Biometrika 51, 299–311 (1964).

1963 (1)

The rate function is sometimes more precisely described as a sum of two time-decaying exponential functions, a strong fast component plus a much weaker slow component. See G. Walter and A. Coche, "Mesure de la composante lente de l'émission de lumière dans les scintillateurs organiques: application à la discrimination des particules selon leur pouvoir ionisant," Nucl. Instrum. Methods 23, 147–151 (1963).

1958 (2)

W. L. Smith, "Renewal theory and its ramifications," J. R. Statist. Soc. B 20, 243–302 (1958).

L. Takács, "On a probability problem in the theory of counters," Ann. Math. Stat. 29, 1257–1263 (1958).

1955 (1)

D. R. Cox, "Some statistical methods connected with series of events," J. R. Statist. Soc. B 17, 129–164 (1955).

Barlow, H. B.

H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

Bartlett, M. S.

M. S. Bartlett, "The spectral analysis of two-dimensional point processes," Biometrika 51, 299–311 (1964).

Birks, J. B.

J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, New York, 1964).

Bredekamp, J. H.

Cantor, B. I.

Coche, A.

The rate function is sometimes more precisely described as a sum of two time-decaying exponential functions, a strong fast component plus a much weaker slow component. See G. Walter and A. Coche, "Mesure de la composante lente de l'émission de lumière dans les scintillateurs organiques: application à la discrimination des particules selon leur pouvoir ionisant," Nucl. Instrum. Methods 23, 147–151 (1963).

Cox, D. R.

D. R. Cox, "Some statistical methods connected with series of events," J. R. Statist. Soc. B 17, 129–164 (1955).

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1968).

Diament, P.

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybernet. 38, 187–191 (1980).

Eubanks, A. G.

W. Viehmann, A. G. Eubanks, G. F. Piper, and J. H. Bredekamp, "Photomultiplier window materials under electron irradiation: fluorescence and phosphorescence," Appl. Opt. 14, 2104–2115 (1975).

W. Viehmann and A. G. Eubanks, "Noise limitations of multiplier phototubes in the radiation environment of space," NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Maryland, 1976).

Kuiper, J. W.

H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

Lawrance, A. J.

A. J. Lawrance, "Some models for stationary series of univariate events," in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, ed. (Wiley-Interscience, New York, 1972).

Levick, W. R.

H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. London 206, 677–700 (1970).

W. R. Levick, John Curtin School of Medical Research, Australian National University, Canberra City, Australia, personal communication.

Lewis, P. A. W.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1968).

Mastebroek, H. A. K.

H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

Matin, L.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).

Piper, G. F.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," submitted to Proc. IEEE.

A relatively recent book in the area is B. E. A. Saleh, Photoelectron Statistics (Sprinsger-Verlag, New York, 1978).

Smith, W. L.

W. L. Smith, "Renewal theory and its ramifications," J. R. Statist. Soc. B 20, 243–302 (1958).

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

Takács, L.

L. Takács, "On a probability problem in the theory of counters," Ann. Math. Stat. 29, 1257–1263 (1958).

Teich, M. C.

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybernet. 38, 187–191 (1980).

M. C. Teich, L. Matin, and B. I. Cantor, "Refractoriness in the maintained discharge of the cat's retinal ganglion cell," J. Opt. Soc. Am. 68, 386–402 (1978).

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," submitted to Proc. IEEE.

M. C. Teich, "Role of the doubly stochastic Neyman Type-A and Thomas counting distributions in photon detection," Appl. Opt. (to be published).

Viehmann, W.

W. Viehmann, A. G. Eubanks, G. F. Piper, and J. H. Bredekamp, "Photomultiplier window materials under electron irradiation: fluorescence and phosphorescence," Appl. Opt. 14, 2104–2115 (1975).

W. Viehmann and A. G. Eubanks, "Noise limitations of multiplier phototubes in the radiation environment of space," NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Maryland, 1976).

Walter, G.

The rate function is sometimes more precisely described as a sum of two time-decaying exponential functions, a strong fast component plus a much weaker slow component. See G. Walter and A. Coche, "Mesure de la composante lente de l'émission de lumière dans les scintillateurs organiques: application à la discrimination des particules selon leur pouvoir ionisant," Nucl. Instrum. Methods 23, 147–151 (1963).

Yoon, M.

H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

Zaagman, W. H.

H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

Zacks, J. L.

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. London 206, 677–700 (1970).

Ann. Math. Stat. (1)

L. Takács, "On a probability problem in the theory of counters," Ann. Math. Stat. 29, 1257–1263 (1958).

Appl. Opt. (1)

Biol. Cybernet. (1)

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybernet. 38, 187–191 (1980).

Biometrika (1)

M. S. Bartlett, "The spectral analysis of two-dimensional point processes," Biometrika 51, 299–311 (1964).

J. Opt. Soc. Am. (1)

J. Physiol. London (1)

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. London 206, 677–700 (1970).

J. R. Statist. Soc. B (2)

D. R. Cox, "Some statistical methods connected with series of events," J. R. Statist. Soc. B 17, 129–164 (1955).

W. L. Smith, "Renewal theory and its ramifications," J. R. Statist. Soc. B 20, 243–302 (1958).

Nucl. Instrum. Methods (1)

The rate function is sometimes more precisely described as a sum of two time-decaying exponential functions, a strong fast component plus a much weaker slow component. See G. Walter and A. Coche, "Mesure de la composante lente de l'émission de lumière dans les scintillateurs organiques: application à la discrimination des particules selon leur pouvoir ionisant," Nucl. Instrum. Methods 23, 147–151 (1963).

Suppl. (1)

H. B. Barlow, W. R. Levick, and M. Yoon, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

Vision Res. (1)

H. A. K. Mastebroek, W. H. Zaagman, and J. W. Kuiper, "Intensity and structure of visually evoked neural activity: rivals in modeling a visual system," Vision Res. 17, 29–35 (1977).

Other (13)

W. R. Levick, John Curtin School of Medical Research, Australian National University, Canberra City, Australia, personal communication.

M. C. Teich, "Role of the doubly stochastic Neyman Type-A and Thomas counting distributions in photon detection," Appl. Opt. (to be published).

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Methuen, London, 1968).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, New York, 1964).

W. Viehmann and A. G. Eubanks, "Noise limitations of multiplier phototubes in the radiation environment of space," NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Maryland, 1976).

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).

J. W. Müller, ed., "Bibliography on dead time effects," Rep. no. BIPM-75/6 (Bureau International des Poids at Mesures, Sèvres, France, 1975).

D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

A relatively recent book in the area is B. E. A. Saleh, Photoelectron Statistics (Sprinsger-Verlag, New York, 1978).

A. J. Lawrance, "Some models for stationary series of univariate events," in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, ed. (Wiley-Interscience, New York, 1972).

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," submitted to Proc. IEEE.

Synder has shown that a doubly stochastic Poisson point process may be expressed in-terms of an equivalent self-exciting point process (see Ref. 1, pp. 292–293).

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