Skip to main content
SpringerLink
Log in

Analysis of binary trees when occasional multifurcations can be considered as aggregates of bifurcations

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The geometrical properties of neurons are important for the way they function within neural circuits. The arborescent processes of neurons that are necessary for the transmission of the information are formed by branching and elongation of segments. In studies that model the outgrowth the tree structures have generally been considered as binary. However, multifurcations do occur. It will be shown that if the multifurcations can be considered as aggregates of bifurcations they may be included in the topological analysis of neuronal branching patterns.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Berry, M. and R. Flinn. 1984. Vertex analysis of Purkinje cell dendritic trees in the cerebellum of the rat.Proc. R. Soc. Lond. B221, 321–348.

    Article  Google Scholar 

  • Berry, M., T. Hollingworth, E. M. Anderson and R. M. Flinn. 1975. Application of network analysis to the study of the branching patterns of dendritic fields. In:Advances in Neurology, Vol. 12, G. W. Kreutzberg (Ed.), pp. 217–245. New York: Raven Press.

    Google Scholar 

  • Berry, M., P. McConnell and J. Sievers. 1981. Dendritic growth and the control of neuronal form. In:Current Topics in Developmental Biology, Vol. 15, R. Hunt (Ed.), p. 67–101. New York: Academic Press.

    Google Scholar 

  • Berry, M., M. Sadler and R. Flinn. 1986. Vertex analysis of neural tree structures containing trichotomous nodes.J. Neurosci. Meth. 18, 167–178.

    Article  Google Scholar 

  • Biggs, N. L. 1985.Discrete Mathematics, Oxford University Press.

  • Bray, D. 1973. Branching patterns of individual sympathetic neurons in culture.J. Cell Biol. 56, 702–712.

    Article  Google Scholar 

  • Bray, D. and M. B. Bunge. 1973. The growth cone in neurite extension. In:Locomotion of Tissue Cells. Ciba Foundation Symposium 14 (new series), pp. 195–209, Amsterdam: Elsevier.

    Google Scholar 

  • Calvet, M. J. and J. Calvet. 1984. Computer assisted analysis of HRP labelled and Golgi stained Purkinje neurons.Progr. Neurobiol. 23, 251–272.

    Article  Google Scholar 

  • Coates, P. W. 1986. Quantitation and morphological characterization of rapid axon and dendritic growth from single cerebral hemispheric neurons in hydrated collagen lattice culture.Dev. Brain Res. 25, 11–20.

    Article  Google Scholar 

  • Cohan, C. S. and S. B. Kater. 1986. Suppression of neurite elongation and growth cone motility by electrical activity.Science 232, 1638–1640.

    Google Scholar 

  • Comtet, L. 1974.Advanced Combinatorics. Dordrecht: Reidel.

    MATH  Google Scholar 

  • Conover, W. L. 1980.Practical Nonparametric Statistics (Second edn.). New York: Wiley.

    Google Scholar 

  • Eysel, U. T., L. Peichl and H. Wässle. 1985. Dendritic plasticity in the early postnatal feline retina: quantitative characteristics and sensitive period.J. Comp. Neurol. 242, 134–145.

    Article  Google Scholar 

  • Glaser, E. M., H. Van der Loos and M. Gisler. 1979. Tangential orientation and spatial order in dendrites of cat auditory cortex: a computer microscope study of Golgi-impregnated material.Exp. Brain Res. 36, 411–431.

    Article  Google Scholar 

  • Harding, E. F. 1971. The probabilities of rooted tree-shapes generated by random bifurcation.Adv. Appl. Prob. 3, 44–77.

    Article  MATH  MathSciNet  Google Scholar 

  • Hollingworth, T. and M. Berry. 1975. Network analysis of dendritic fields of pyramidal cells in neocortex and Purkinje cells in the cerebellum of the rat.Phil. Trans. R. Soc. Lond. B270, 227–264.

    Google Scholar 

  • Horsfield, K., M. J. Woldenberg and C. L. Bowes. 1987. Sequential and synchronous growth models related to vertex analysis and branching ratios.Bull. math. Biol. 49, 413–429.

    MATH  Google Scholar 

  • Kliemann, W. 1987. A stochastic dynamical model for the characterization of the geometrical structure of dendritic processes.Bull. math. Biol. 49, 135–152.

    Article  MATH  MathSciNet  Google Scholar 

  • Percheron, G. 1979. Quantitative analysis of dendritic branching. I. Simple formulae for the quantitative analysis of dendritic branching.Neurosci. Lett. 14, 287–293.

    Article  Google Scholar 

  • Purves, D. and J. W. Lichtman. 1985.Principles of Neural Development. Sunderland: Sinauer.

    Google Scholar 

  • Sadler, M. and M. Berry. 1983. Morphometric study of the development of Purkinje cell dendritic fields of the mouse using vertex analysis.J. Microsc. 131, 341–354.

    Google Scholar 

  • 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 

  • Speidel, C. C. 1942. Studies of living nerves. VII. Growth adjustments of cutaneous terminal arborizations.J. Comp. Neurol. 76, 57–73.

    Article  Google Scholar 

  • Speidel, C. C. 1964.In vivo studies of myelinated nerve fibers.Int. Rev. Cytol. 16, 173–231.

    Article  Google Scholar 

  • Stuermer, C. A. O. 1984. Rules for retinotectal terminal arborizations in the goldfish optic tectum.J. Comp. Neurol. 229, 214–232.

    Article  Google Scholar 

  • Todd, J. A. 1944. A table of partitions.Proc. Lond. math. Soc. 48, 229–241.

    MathSciNet  Google Scholar 

  • Van Pelt, J. and R. W. H. Verwer. 1983. The exact probabilities of branching patterns under terminal and segmental growth hypotheses.Bull. math. Biol. 45, 269–285.

    Article  MATH  Google Scholar 

  • Van Pelt, J. and R. W. H. Verwer. 1985. Growth models (including terminal and segmental branching) for topological binary trees.Bull. math. biol. 47, 323–336.

    Article  MATH  MathSciNet  Google Scholar 

  • Van Pelt, J. and R. W. H. Verwer. 1986. Topological properties of binary trees grown with order-dependent branching probabilities.Bull. math. Biol. 48, 197–211.

    Article  MATH  MathSciNet  Google Scholar 

  • Van Pelt, J., R. W. H. Verwer and H. B. M. Uylings. 1986. Application of growth models to the topological structure of neuronal branching patterns.J. Neurosci. Meth. 18, 153–165.

    Article  Google Scholar 

  • Verwer, R. W. H. and J. Van Pelt. 1983. A new method for the topological analysis of neuronal tree structures.J. Neurosci. Meth. 8, 335–351.

    Article  Google Scholar 

  • Verwer, R. W. H. and J. Van Pelt. 1985. Topological analysis of binary tree structures when occasional multifurcations occur.Bull. math. biol. 47, 305–316.

    Article  MATH  MathSciNet  Google Scholar 

  • Verwer, R. W. H. and J. Van Pelt. 1986. Descriptive and comparative analysis of geometrical properties of neuronal tree structures.J. Neurosci. Meth. 18, 179–206.

    Article  Google Scholar 

  • Verwer, R. W. H. and J. Van Pelt. 1987. Multifurcations in topological trees: growth models and comparative analysis.Acta Stereol. 6/III, 399–404.

    Google Scholar 

  • Verwer, R. W. H., J. Van Pelt and A. J. Noest. 1987. Estimation of parameters in topological growth models for neuronal trees.Bull. math. Biol. 49, 363–378.

    Article  MATH  MathSciNet  Google Scholar 

  • Verwer, R. W. H., J. Van Pelt and H. B. M. Uylings. 1985. A simple statistical test for the vertex ratio using Monte Carlo simulation.J. Neurosci. Meth. 14, 137–142.

    Article  Google Scholar 

  • Woodward, D. J., A. Seiger, L. Olson and B. J. Hoffer. 1977. Intrinsic and extrinsic determinants of dendritic development as revealed by Golgi studies of cerebellar and hippocampal transplantsin oculo.Exp. Neurol. 57, 984–998.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Netherlands Institute for Brain Research, Meibergdreef 33, 1105 AZ, Amsterdam, The Netherlands

    Ronald W. H. Verwer & Jaap Van Pelt

Authors
  1. Ronald W. H. Verwer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaap Van Pelt
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Verwer, R.W.H., Van Pelt, J. Analysis of binary trees when occasional multifurcations can be considered as aggregates of bifurcations. Bltn Mathcal Biology 52, 629–641 (1990). https://doi.org/10.1007/BF02462102

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02462102

Keywords

Search

Navigation