Extended morphometric analysis of neuronal cells with Minkowski valuations


Minkowski valuations provide a systematic framework for quantifying different aspects of morphology. In this paper we apply vector- and tensor-valued Minkowski valuations to neuronal cells from the cat's retina in order to describe their morphological structure in a comprehensive way. We introduce the framework of Minkowski valuations, discuss their implementation for neuronal cells and show how they can be used to characterize cells of different morphological categories. We also provide a comparison to a Sholl analysis.

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  1. S. Douady, Y. Coulder, Phys. Rev. Lett. 68, 2098 (1992)

    Article  ADS  Google Scholar 

  2. L. da F. Costa, E.T.M. Manoel, Neuroinformatics 1, 65 (2003)

    Article  Google Scholar 

  3. H. Wässle, B.B. Boycott, R.B. Illing, Phil. Trans. R. Soc. 212, 177 (1981)

    Google Scholar 

  4. Y. Fukuda, C.F. Hsiao, M. Watanabe, H. Ito, J. Neurophysiol. 52, 999 (1984)

    Google Scholar 

  5. A. van Ooyen, J. Duijnhouwer, M.W.H. Remme, J. van Pelt, Network: Comput. Neural Syst. 13, 311 (2002)

    Article  Google Scholar 

  6. H.B.M. Uylings, J. van Pelt, Network: Comput. Neural Syst. 13, 397 (2002)

    Article  Google Scholar 

  7. L. da F. Costa, T.J. Velte, Journal of Comparative Neurology 404, 33 (1999)

    Article  Google Scholar 

  8. G.A. Ascoli, J.L. Krichmar, Neurocomputing 48, 1003 (2000)

    Article  Google Scholar 

  9. R.C. Coelho, L. da F. Costa, Neurocomputing 48, 555 (2001)

    Article  MathSciNet  Google Scholar 

  10. S. Peng, B. Urbanc, L. Cruz, B.T. Hyman, H.E. Stanley, Proc. Nat. Ac. Sci. 100, 3847 (2003)

    Article  ADS  Google Scholar 

  11. F. Rieke, D. Warland, R. de Ruyyter van Steveninck, W. Bialek, Spikes: Exploring the Neural Code (Bradford Books, 1999)

  12. K. Morigiwa, M. Tauchi, Y. Fukuda, Neurosci. Res. 10, S131 (1989)

  13. R.C. Coelho, L. da F. Costa, Applied Signal Processing 3, 163 (1996)

    Google Scholar 

  14. L. da F Costa, E.T.M. Manoel, F. Faucereau, J. Chelly, J. van Pelt, G. Ramakers, Network: Comput. Neural Syst. 13, 283 (2002)

    Article  Google Scholar 

  15. T.G. Smith, G.D. Lange, W.B. Marks, J. Neuroci. Methods 69, 133 (1996)

    MATH  Google Scholar 

  16. E.P. Rodrigues, M.S. Barbosa, L. da F. Costa, Phys. Rev. E (2004)

  17. R.M. Cesar, L. da F. Costa, Biological Cybernetics 79, 347 (1998)

    MATH  Article  Google Scholar 

  18. M.S. Barbosa, E.S. Bernardes, L. da F. Costa, Phys. Rev. E 67 (2003)

  19. M.S. Barbosa, L. da F. Costa et al., Eur. Phys. J. B 37, 109 (2003)

    Article  ADS  Google Scholar 

  20. P. Soille, Morphological Image Analysis. Principles and Applications (Springer, Berlin, 1999)

  21. K. Michelsen, H. de Raedt, Phys. Rep. 347, 461 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  22. S. Alesker, Geom. Dedicata 74, 241 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  23. R. Schneider, Rend. Circ. Mat. Ser II, Suppl. 65, 355 (2000)

    Google Scholar 

  24. R. Schneider, R. Schuster, Rend. Circ. Mat. Ser II, Suppl. 70, 295 (2002)

    Google Scholar 

  25. C. Beisbart, R. Dahlke, K. Mecke, H. Wagner, in Morphology of Condensed Matter. Physics and Geometry of Spatial Complex Systems, edited by K. Mecke, D. Stoyan (Springer, 2002), Vol. 600 of Lecture Notes in Physics, pp. 238–260, arXiv:physics/0203072

  26. C. Beisbart, T. Buchert, H. Wagner, Physica A 293, 592 (2001)

    MATH  Article  ADS  Google Scholar 

  27. R.H. Masland, Nature Neuroscience 4, 877 (2001)

    Article  Google Scholar 

  28. B.J. O'Brien, T. Isayama, R. Richardson, D.M. Berson, J. Phys. 538.3, 787 (2002)

    Google Scholar 

  29. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer Verlag, Berlin, 1957)

  30. W. Weil, in Convexity and its Applications, edited by P.M. Gruber, J.M. Wills (Birkhäuser, Basel, 1983), pp. 360–412

  31. S. Alesker, Ann. of Math. 149, 977 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  32. H. Hadwiger, C. Meier, Mathematische Nachrichten 56, 361 (1974)

    MathSciNet  Google Scholar 

  33. C. Beisbart, K. Mecke, unpublished Manuscript 2005

  34. C. Beisbart, Ph.D. thesis, Ludwig–Maximilians–Universität München (2001), http://edoc.ub.uni-muenchen.de/ archive/00000483/01/Beisbart_Claus.pdf

  35. K. Mecke, T. Buchert, H. Wagner, Astronomy and Astrophysics 288, 697 (1994)

    ADS  Google Scholar 

  36. M. Pu, D.M. Berson, T. Pan, The journal of Neuroscience 14 (1994)

  37. D.M. Berson, M. Pu, E.V. Famiglietti, Journal of Comparative Neurology 399, 269 (1998)

    Article  Google Scholar 

  38. R.C.C. et al., Real-Time Imaging 8, 213 (2002)

    Article  Google Scholar 

  39. D. Fenchel, C.R. Acad. Sci. Paris 203, 647 (1936), in French

    MATH  Google Scholar 

  40. A.D. Alexandrov, Matem. Sb. SSSR 2, 1205 (1937), in Russian, summary in German

    Google Scholar 

  41. J. Schmalzing, T. Buchert, A.L. Melott, V. Sahni, B.S. Sathyaprakash, S.F. Shandarin, ApJ 526, 568 (1999)

    Article  ADS  Google Scholar 

  42. L. da F. Costa, R.M.C. Jr., Shape Analysis and Classification: Theory and Practice (CRC, 2000)

  43. N. Schmitz, Iwarp: Gimp Plugin, www.gimp.org

  44. J.H. Ward, J. Am. Stat. Ass. 58, 236 (1963)

    Article  Google Scholar 

  45. K.L. Whitford, P. Dijkhuizen, F. Polleux, A. Ghosh, Annu. Rev. Neurosci. 25, 127 (2002)

    Article  Google Scholar 

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Beisbart, C., Barbosa, M., Wagner, H. et al. Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur. Phys. J. B 52, 531–546 (2006). https://doi.org/10.1140/epjb/e2006-00328-1

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  • 07.05.Kf Data analysis: algorithms and implementation; data management
  • 87.19.La Neuroscience
  • 02.40.Ft Convex sets and geometric inequalities