Skip to main content
SpringerLink
Log in

Poisson Processes in Biology

  • Conference paper
Stochastic Nonlinear Systems in Physics, Chemistry, and Biology

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 8))

Abstract

A simple Poisson process, [N(t), t ≥ 0], with parameter λ, is a random process with stationary independent increments such that if 0 ≤ t1 < t2, then

$$ {\rm{Pr[N(}}{{\rm{t}}_{\rm{2}}})\; - \;{\rm{N(}}{{\rm{t}}_{\rm{1}}}{\rm{)}}\;{\rm{ = k] = [\lambda (}}{{\rm{t}}_{\rm{2}}} - {{\rm{t}}_1}){]^{\rm{k}}}\;\exp [ - {\rm{\lambda (}}{{\rm{t}}_{\rm{2}}} - {{\rm{t}}_{\rm{1}}})]/{\rm{k! k = 0,1,2}}.. $$
(1)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brink, F., Bronk, D.W. &Larrabee, M.G., Ann.N.Y.Acad.Sci. 47, 457 (1946).

    Article  ADS  Google Scholar 

  2. Burns, B.D. &Webb, A.C., Proc.Roy.Soc. Lond.B. 194, 211 (1976)

    Article  ADS  Google Scholar 

  3. Cohen, I., Kita, H. &Van der Kloot, W., Brain Res. 54, 318 (1973).

    Article  Google Scholar 

  4. Cohen, I., Kita, H. &Van der Kloot, W., J. Physiol. 326, 327 (1974).

    Google Scholar 

  5. Cope, D.K. &Tuckwell, H.C., J.Theor.Biol . 80, 1 (1979).

    Article  MathSciNet  Google Scholar 

  6. Cox, D.R. &Lewis, P.A.W. The statistical analysis of series of events. Methuen, London (1966).

    MATH  Google Scholar 

  7. Eccles, J.C The physiology of synapses. Academic, New York (1964).

    Book  Google Scholar 

  8. Fatt, P. &Katz, B. J.Physiol. 117, 109 (1952).

    Google Scholar 

  9. Gihman, I.I. &Skorohod, A.V. Stochastic differential equations.. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 72. Springer, Berlin, Heidelberg, New York (1972).

    Google Scholar 

  10. Hagiwara, S., Jap.J.Physiol. 4, 234 (1954).

    Article  Google Scholar 

  11. Hanson, F.B. &Tuckwell, H.C., Theor.Pop.Biol. 14 , 46 (1978).

    Article  MathSciNet  Google Scholar 

  12. Hanson, F.B. &Tuckwell, H.C., Theor.Pop.Biol. in press (1980).

    Google Scholar 

  13. Hanson, F.B. &Tuckwell, H.C., in preparation.

    Google Scholar 

  14. Holden, A.V. Models of the stochastic activity of neurones. Lecture Notes in Biomathematics, Vol. 12. Springer, Berlin, Heidelberg, New York (1976).

    Google Scholar 

  15. Khintchine, A.Y. Mathematical methods in the theory of queueing. Griffin, London (1960).

    MATH  Google Scholar 

  16. Lapicque, L. J.Physiol. Pathol.Gen. 9, 620 (1907).

    Google Scholar 

  17. Levine, J.W. &Shefner, J.M., Biophys.J. 19, 241 (1977).

    Article  ADS  Google Scholar 

  18. MacGregor, R.J. &Lewis, E.R. Neural modeling. Plenum, New York (1977) .

    Google Scholar 

  19. O’Brien, J.H., Packham, S.C. &Brunnhoelzl, W.W.,J. Neurophysiol . 36, 1051 .

    Google Scholar 

  20. Quastel, D.M.J. Synaptic transmission and neuronal interaction, p23. Raven, New York (1974).

    Google Scholar 

  21. Rall, W., Exp.Neurol. 1, 491 (1959).

    Article  Google Scholar 

  22. Roy, B.K. &Smith, D.R., Bul1 . Math.Biophys. 31, 341 (1969).

    Article  MATH  Google Scholar 

  23. Scharstein, H., J.Math.Biol. 8, 403 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Stein, R.B., Biophys.J. 5, 173 (1965).

    Article  ADS  Google Scholar 

  25. Stein, R.B., Biophys,J. 7, 37 (1967).

    Article  ADS  Google Scholar 

  26. Tsurui, A. &Osaki, S., Stoch.Proc.Appl. 4, 79 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  27. Tuckwell, H.C., Biol .Cybernetics 18, 225 (1975) .

    Article  MATH  Google Scholar 

  28. Tuckwell, H.C., J.Appl.Prob. 13, 39 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  29. Tuckwell, H.C., Biophys.J. 21, 289 (1978).

    Article  ADS  Google Scholar 

  30. Tuckwell, H.C. J. Theor. Biol. 77, 65 (1979).

    Article  MathSciNet  Google Scholar 

  31. Tuckwell, H.C. &Cope, D.K., J.Theor.Biol . 83, 377 (1980).

    Article  Google Scholar 

  32. Tuckwell, H.C. &Richter, W., J.Theor.Biol . 71, 167 (1978).

    Article  Google Scholar 

  33. Tuckwell, H.C. &Wan, F.Y.M., J .Theor.Biol . in press (1980).

    Google Scholar 

  34. Tuckwell, H.C. &Wan, F.Y.M., in preparation.

    Google Scholar 

  35. Wan, F.Y.M. &Tuckwell, H.C., Biol .Cybernetics 33, 39 (1979).

    Article  MATH  Google Scholar 

  36. Wan, F.Y.M. &Tuckwell, H.C., in preparation.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Monash University, Clayton, Victoria, Australia, 3168

    Henry C. Tuckwell

Authors
  1. Henry C. Tuckwell
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, D-2800, Bremen 33, Fed. Rep. of Germany

    Ludwig Arnold

  2. Service de Chemie Physique II, Université Libre de Bruxelles, C.P. 231, Campus Plaine, B-1050, Bruxelles, Belgium

    René Lefever

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Tuckwell, H.C. (1981). Poisson Processes in Biology. In: Arnold, L., Lefever, R. (eds) Stochastic Nonlinear Systems in Physics, Chemistry, and Biology. Springer Series in Synergetics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68038-0_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-68038-0_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-68040-3

  • Online ISBN: 978-3-642-68038-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation