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The exact probabilities of branching patterns under terminal and segmental growth hypotheses

  • J. van Pelt
  • R. W. H. Verwer
Article

Abstract

Information about the way of branching of dendritic arborizations may be obtained by comparing the frequency distributions of observed branching patterns with theoretical distributions based on well-defined growth models. Two models usually get much attention in geomorphological and (neuro)biological studies, viz. terminal growth and segmental growth. Formulae to construct the exact probability distributions for both growth models are presented. It is shown that ranking and lumping of the individual branching patterns enable the analysis of very large arborizations with relatively few data. The application of the Kolmogorov goodness-of-fit test for discrete distributions to the analysis is discussed.

Keywords

Bifurcation Point Exact Probability Topological Type Terminal Segment Ranking Scheme 
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Literature

  1. Berry, M. and P. M. Bradley. 1976. “The Application of Network Analysis to the Study of Branching Patterns of Large Dendritic Fields.”Brain Res.,109, 111–132.CrossRefGoogle Scholar
  2. —, T. Hollingworth, E. M. Anderson and R. M. Flinn. 1975. “Application of Newwork Analysis to the Study of the Branching Patterns of Dendritic Fieldss” InAdvances in Neurology, Ed. G. W. Kreutzberg, Vol. 12, pp. 217–245, New York: Raven Press.Google Scholar
  3. — and D. Pymm. 1981. “Analysis of Neural Networks” InAdvances in Physiological Science Vol. 30.Neural Communications and Control, Eds. G. Szekely, E. Labos and S. Damjanovitch. 28th International Congress of Physiological Science. Budapest 1980. Budapest: Akadémiai Kiadó/Oxford: Pergamon Press, 1981.Google Scholar
  4. Conover, W. J. 1972. “A Kolmogorov Goodness-of-Fit Test for Discontinuous Distributions”J. Am. statist. Ass. 67, 591–596.MATHMathSciNetCrossRefGoogle Scholar
  5. Dacey, M. F. and W. C. Krumbein. 1976. “Three Growth Models For Stream Channel Networks”J. Geol.84, 153–163.CrossRefGoogle Scholar
  6. Harding, E. F. 1971. “The Probabilities of Rooted Tree-shapes Generated by Random Bifurcation”J. appl. Prob. 3, 44–77.MATHMathSciNetCrossRefGoogle Scholar
  7. Horn, S. D. 1977. “Goodness-of-Fit Tests for Discrete Data: A Review and an Application to a Health Impairment Scale”.Biometrics,33, 237–248.MATHMathSciNetCrossRefGoogle Scholar
  8. Scheidegger, A. E. 1967. “On the Topology of River Nets”.Water Resources Res.,3, 103–106.Google Scholar
  9. Shreve, R. L. 1966. “Statistical Law of Stream Numbers”J. Geol. 74, 17–37.Google Scholar
  10. Smart, J. S. 1969. “Topological Properties of Channel Networks”,Geol. Soc. Am. Bull. 80, 1757–1774.Google Scholar
  11. Smit, G. J., H. B. M. Uylings and L. Veldmaat-Wansink. 1972. “The Branching Pattern in Dendrites of Cortical Neurons”Acta Morphol. Neerl. Scand. 9, 253–274.Google Scholar
  12. Uylings, H. B. M., G. J. Smit and W. A. M. Veltman. 1975. “Ordering Methods in Quantitative Analysis of Branching Structures of Dendritic Trees”. InAdvances in Neurology, Ed. G. W. Kreutzberg, Vol. 12, pp. 247–254. New York: Raven Press.Google Scholar
  13. Verwer, R. W. H. and J. Van Pelt. 1983. “A New Method for the Topological Analysis of Neuronal Tree Structures.”,J. Neurosci. Meth. (submitted).Google Scholar

Copyright information

© Society for Mathematical Biology 1983

Authors and Affiliations

  • J. van Pelt
    • 1
  • R. W. H. Verwer
    • 1
  1. 1.Netherlands Institute for Brain ResearchAmsterdamThe Netherlands

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