Skip to main content
SpringerLink
Log in

A Statistical Theory of Dendritic Morphology

  • Chapter
  • First Online:
The Computing Dendrite

Part of the book series: Springer Series in Computational Neuroscience ((NEUROSCI,volume 11))

  • 2208 Accesses

Abstract

Since Santiago Ramon y Cajal, neuroscientists have been fascinated by the shapes of dendritic arbors for more than 100 years. However, the principle underlying these complex and diverse structures remains elusive. Here we propose that evolution has tinkered with brain design to maximize its functionality while minimizing the cost associated with building and maintaining it. We hypothesize that the functionality of a neuron benefits from a larger repertoire of connectivity patterns between dendrites and surrounding axons, and the cost of a dendritic arbor increases with its total length and path length from synapses to soma. We solved this optimization problem by drawing an analogy with maximization of the entropy for a given energy in statistical physics. The solution predicts several scaling relationships between arbor dimensions and closely fits with experimental data. Moreover, our theory may explain why basal dendrites of pyramidal cells and Purkinje cells, the two major cell types in the mammalian brains, exhibit distinct morphologies.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  • Anderson JC, Binzegger T, Douglas RJ, Martin KA (2002) Chance or design? Some specific considerations concerning synaptic boutons in cat visual cortex. J Neurocytol 31(3–5):211–229

    Article  PubMed  CAS  Google Scholar 

  • Ascoli GA (2006) Mobilizing the base of neuroscience data: the case of neuronal morphologies. Nat Rev Neurosci 7(4):318–324

    Article  PubMed  CAS  Google Scholar 

  • Attwell D, Laughlin SB (2001) An energy budget for signaling in the grey matter of the brain. J Cereb Blood Flow Metab 21(10):1133–1145

    Article  PubMed  CAS  Google Scholar 

  • Binzegger T, Douglas RJ, Martin KA (2005) Axons in cat visual cortex are topologically self-similar. Cereb Cortex 15(2):152–165

    Article  PubMed  Google Scholar 

  • Braitenberg V, Schuz A (1998) Cortex: statistics and geometry of neuronal connectivity. 2nd thoroughly rev. edn. Springer, New York

    Book  Google Scholar 

  • Caserta F, Stanley HE, Eldred WD, Daccord G, Hausman RE, Nittmann J (1990) Physical mechanisms underlying neurite outgrowth: a quantitative analysis of neuronal shape. Phys Rev Lett 64(1):95–98

    Article  PubMed  Google Scholar 

  • Cash S, Yuste R (1999) Linear summation of excitatory inputs by CA1 pyramidal neurons. Neuron 22(2):383–394

    Article  PubMed  CAS  Google Scholar 

  • Chklovskii DB, Mel BW, Svoboda K (2004) Cortical rewiring and information storage. Nature 431(7010):782–788

    Article  PubMed  CAS  Google Scholar 

  • Chklovskii DB, Vitaladevuni S, Scheffer LK (2010) Semi-automated reconstruction of neural circuits using electron microscopy. Curr Opin Neurobiol 20(5):667–675

    Article  PubMed  CAS  Google Scholar 

  • Cuntz H (2012) The dendritic density field of a cortical pyramidal cell. Front Neuroanat 6:2

    Article  PubMed  Google Scholar 

  • Cuntz H, Borst A, Segev I (2007) Optimization principles of dendritic structure. Theor Biol Med Model 4:21

    Article  PubMed  Google Scholar 

  • Cuntz H, Forstner F, Borst A, Häusser M (2010) One rule to grow them all: a general theory of neuronal branching and its practical application. PLoS Comput Biol 6(8)

    Google Scholar 

  • Cuntz H, Mathy A, Häusser M (2012) A scaling law derived from optimal dendritic wiring. Proc Natl Acad Sci USA 109(27):11014–11018

    Article  PubMed  CAS  Google Scholar 

  • Denk W, Horstmann H (2004) Serial block-face scanning electron microscopy to reconstruct three-dimensional tissue nanostructure. PLoS Biol 2(11):e329

    Article  PubMed  Google Scholar 

  • Fuerst PG, Koizumi A, Masland RH, Burgess RW (2008) Neurite arborization and mosaic spacing in the mouse retina require DSCAM. Nature 451(7177):470–474

    Article  PubMed  CAS  Google Scholar 

  • Fujita I, Fujita T (1996) Intrinsic connections in the macaque inferior temporal cortex. J Comp Neurol 368:467–486

    Article  PubMed  CAS  Google Scholar 

  • Gilbert CD, Wiesel TN (1989) Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex. J Neurosci 9(7):2432–2442

    PubMed  CAS  Google Scholar 

  • Gutin AM, Grosberg AY, Shakhnovich EI (1993) Polymers with annealed and quenched branchings belong to different universality classes. Macromolecules 26(6):1293–1295

    Article  CAS  Google Scholar 

  • Hattori D, Demir E, Kim HW, Viragh E, Zipursky SL, Dickson BJ (2007) Dscam diversity is essential for neuronal wiring and self-recognition. Nature 449(7159):223–227

    Article  PubMed  CAS  Google Scholar 

  • Hillman D (1979) Neuronal shape parameters and substructures as a basis of neuronal form. In: The Neurosciences: fourth study program. MIT Press, Cambridge, MA, pp vii, 1185 p

    Google Scholar 

  • Hsu A, Tsukamoto Y, Smith RG, Sterling P (1998) Functional architecture of primate cone and rod axons. Vision Res 38(17):2539–2549

    Article  PubMed  CAS  Google Scholar 

  • Hughes ME, Bortnick R, Tsubouchi A, Baumer P, Kondo M, Uemura T, Schmucker D (2007) Homophilic Dscam interactions control complex dendrite morphogenesis. Neuron 54(3):417–427

    Article  PubMed  CAS  Google Scholar 

  • Jelinek H, Elston GN, Zietsch B (2005) Fractal analysis: pitfalls and revelations in neuroscience. In: Losa GA, Merlini D, Nonnenmacher TF, Weibel ER (eds) Fractals in biology and medicine, vol 4. Birkhauser Verlag AG, Switzerland, pp 85–94

    Chapter  Google Scholar 

  • Jelinek HF, Fernandez E (1998) Neurons and fractals: how reliable and useful are calculations of fractal dimensions? J Neurosci Methods 81(1–2):9–18

    Article  PubMed  CAS  Google Scholar 

  • Kawaguchi Y, Karube F, Kubota Y (2006) Dendritic branch typing and spine expression patterns in cortical nonpyramidal cells. Cereb Cortex 16(5):696–711

    Article  PubMed  Google Scholar 

  • Kisvarday ZF, Eysel UT (1992) Cellular organization of reciprocal patchy networks in layer III of cat visual cortex (area 17). Neuroscience 46(2):275–286

    Article  PubMed  CAS  Google Scholar 

  • Kleinfeld D, Bharioke A, Blinder P, Bock DD, Briggman KL, Chklovskii DB, Denk W, Helmstaedter M, Kaufhold JP, Lee WC, Meyer HS, Micheva KD, Oberlaender M, Prohaska S, Reid RC, Smith SJ, Takemura S, Tsai PS, Sakmann B (2011) Large-scale automated histology in the pursuit of connectomes. J Neurosci 31(45):16125–16138

    Article  PubMed  CAS  Google Scholar 

  • Larkman AU (1991) Dendritic morphology of pyramidal neurones of the visual cortex of the rat: III. Spine distributions. J Comp Neurol 306(2):332–343

    Article  PubMed  CAS  Google Scholar 

  • Lichtman JW, Livet J, Sanes JR (2008) A technicolour approach to the connectome. Nat Rev Neurosci 9(6):417–422

    Article  PubMed  CAS  Google Scholar 

  • Llinás RR, Walton KD, Lang EJ (2004) Cerebellum. In: Shepherd GM (ed) The synaptic organization of the brain, 5th edn. Oxford University Press, New York, p xiv, 719 p

    Google Scholar 

  • Losonczy A, Magee JC (2006) Integrative properties of radial oblique dendrites in hippocampal CA1 pyramidal neurons. Neuron 50(2):291–307

    Article  PubMed  CAS  Google Scholar 

  • Lund JS, Yoshioka T, Levitt JB (1993) Comparison of intrinsic connectivity in different areas of macaque monkey cerebral cortex. Cereb Cortex 3:148–162

    Article  PubMed  CAS  Google Scholar 

  • Magee JC, Cook EP (2000) Somatic EPSP amplitude is independent of synapse location in hippocampal pyramidal neurons. Nat Neurosci 3(9):895–903

    Article  PubMed  CAS  Google Scholar 

  • Markram H, Lubke J, Frotscher M, Roth A, Sakmann B (1997) Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. J Physiol 500(Pt 2):409–440

    PubMed  CAS  Google Scholar 

  • Marks WB, Burke RE (2007a) Simulation of motoneuron morphology in three dimensions. I. Building individual dendritic trees. J Comp Neurol 503(5):685–700

    Article  PubMed  Google Scholar 

  • Marks WB, Burke RE (2007b) Simulation of motoneuron morphology in three dimensions. II. Building complete neurons. J Comp Neurol 503(5):701–716

    Article  PubMed  Google Scholar 

  • Martone ME, Zhang S, Gupta A, Qian X, He H, Price DL, Wong M, Santini S, Ellisman MH (2003) The cell-centered database: a database for multiscale structural and protein localization data from light and electron microscopy. Neuroinformatics 1(4):379–395

    Article  PubMed  Google Scholar 

  • Mauseth JD (2003) Botany : an introduction to plant biology, 3rd edn. Jones and Bartlett Publishers, Boston

    Google Scholar 

  • Milosevic NT, Ristanovic D, Stankovic JB (2005) Fractal analysis of the laminar organization of spinal cord neurons. J Neurosci Methods 146(2):198–204

    Article  PubMed  Google Scholar 

  • Mooser F, Bosking WH, Fitzpatrick D (2004) A morphological basis for orientation tuning in primary visual cortex. Nat Neurosci 7(8):872–879

    Article  PubMed  CAS  Google Scholar 

  • Napper RM, Harvey RJ (1988a) Number of parallel fiber synapses on an individual Purkinje cell in the cerebellum of the rat. J Comp Neurol 274(2):168–177

    Article  PubMed  CAS  Google Scholar 

  • Napper RM, Harvey RJ (1988b) Quantitative study of the Purkinje cell dendritic spines in the rat cerebellum. J Comp Neurol 274(2):158–167

    Article  PubMed  CAS  Google Scholar 

  • Nevian T, Larkum ME, Polsky A, Schiller J (2007) Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study. Nat Neurosci 10(2):206–214

    Article  PubMed  CAS  Google Scholar 

  • Nicol D, Meinertzhagen IA (1982) An analysis of the number and composition of the synaptic populations formed by photoreceptors of the fly. J Comp Neurol 207(1):29–44

    Article  PubMed  CAS  Google Scholar 

  • Poirazi P, Brannon T, Mel BW (2003a) Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell. Neuron 37(6):977–987

    Article  PubMed  CAS  Google Scholar 

  • Poirazi P, Brannon T, Mel BW (2003b) Pyramidal neuron as two-layer neural network. Neuron 37(6):989–999

    Article  PubMed  CAS  Google Scholar 

  • Ramón y Cajal S (1899) Textura del Sistema Nervioso del Hombre y de los Vertebrados (Texture of the nervous system of man and the vertebrates), vol 1. Springer, New York

    Google Scholar 

  • Rapp M, Segev I, Yarom Y (1994) Physiology, morphology and detailed passive models of guinea-pig cerebellar Purkinje cells. J Physiol 474(1):101–118

    PubMed  CAS  Google Scholar 

  • Rothnie P, Kabaso D, Hof PR, Henry BI, Wearne SL (2006) Functionally relevant measures of spatial complexity in neuronal dendritic arbors. J Theor Biol 238(3):505–526

    Article  PubMed  CAS  Google Scholar 

  • Rubinstein M, Colby RH (2003) Polymer physics. Oxford University Press, Oxford, NY

    Google Scholar 

  • Samsonovich AV, Ascoli GA (2003) Statistical morphological analysis of hippocampal principal neurons indicates cell-specific repulsion of dendrites from their own cell. J Neurosci Res 71(2):173–187

    Article  PubMed  CAS  Google Scholar 

  • Samsonovich AV, Ascoli GA (2005) Statistical determinants of dendritic morphology in hippocampal pyramidal neurons: a hidden Markov model. Hippocampus 15(2):166–183

    Article  PubMed  Google Scholar 

  • Schmucker D, Clemens JC, Shu H, Worby CA, Xiao J, Muda M, Dixon JE, Zipursky SL (2000) Drosophila Dscam is an axon guidance receptor exhibiting extraordinary molecular diversity. Cell 101(6):671–684

    Article  PubMed  CAS  Google Scholar 

  • Shepherd GM, Svoboda K (2005) Laminar and columnar organization of ascending excitatory projections to layer 2/3 pyramidal neurons in rat barrel cortex. J Neurosci 25(24):5670–5679

    Article  PubMed  CAS  Google Scholar 

  • Silver RA, Lubke J, Sakmann B, Feldmeyer D (2003) High-probability uniquantal transmission at excitatory synapses in barrel cortex. Science 302(5652):1981–1984

    Article  PubMed  CAS  Google Scholar 

  • Sjöström PJ, Turrigiano GG, Nelson SB (2001) Rate, timing, and cooperativity jointly determine cortical synaptic plasticity. Neuron 32(6):1149–1164

    Article  PubMed  Google Scholar 

  • Smith TG Jr, Marks WB, Lange GD, Sheriff WH Jr, Neale EA (1989) A fractal analysis of cell images. J Neurosci Methods 27(2):173–180

    Article  PubMed  Google Scholar 

  • Stepanyants A, Hof PR, Chklovskii DB (2002) Geometry and structural plasticity of synaptic connectivity. Neuron 34(2):275–288

    Article  PubMed  CAS  Google Scholar 

  • Stepanyants A, Martinez LM, Ferecsko AS, Kisvarday ZF (2009) The fractions of short- and long-range connections in the visual cortex. Proc Natl Acad Sci USA 106(9):3555–3560

    Article  PubMed  CAS  Google Scholar 

  • Stepanyants A, Tamas G, Chklovskii DB (2004) Class-specific features of neuronal wiring. Neuron 43(2):251–259

    Article  PubMed  CAS  Google Scholar 

  • Stevens CF (2008) All arbors have the same shape. In: Neuronal circuits: from structure to function, CSHL and private communication, 2008

    Google Scholar 

  • Thomas PA (2001) Trees: their natural history, 1st edn. Cambridge University Press, UK

    Google Scholar 

  • Thomson AM, Bannister AP (2003) Interlaminar connections in the neocortex. Cereb Cortex 13(1):5–14

    Article  PubMed  Google Scholar 

  • Trachtenberg JT, Chen BE, Knott GW, Feng G, Sanes JR, Welker E, Svoboda K (2002) Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex. Nature 420(6917):788–794

    Article  PubMed  CAS  Google Scholar 

  • van Pelt J, Uylings HB (2002) Branching rates and growth functions in the outgrowth of dendritic branching patterns. Network 13(3):261–281

    PubMed  Google Scholar 

  • Vetter P, Roth A, Häusser M (2001) Propagation of action potentials in dendrites depends on dendritic morphology. J Neurophysiol 85(2):926–937

    PubMed  CAS  Google Scholar 

  • Wen Q, Chklovskii DB (2008) A cost-benefit analysis of neuronal morphology. J Neurophysiol 99(5):2320–2328

    Article  PubMed  Google Scholar 

  • Wen Q, Stepanyants A, Elston GN, Grosberg AY, Chklovskii DB (2009) Maximization of the connectivity repertoire as a statistical principle governing the shapes of dendritic arbors. Proc Natl Acad Sci USA 106(30):12536–12541

    Article  PubMed  CAS  Google Scholar 

  • Wittenberg GM, Wang SS-H (2007) Evolution and scaling of dendrites. In: Stuart G, Spruston N, Häusser M (eds) Dendrites, 2nd edn. Oxford University Press, USA, pp 43–67

    Google Scholar 

  • Yamagata M, Sanes JR (2008) Dscam and Sidekick proteins direct lamina-specific synaptic connections in vertebrate retina. Nature 451(7177):465–469

    Article  PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Center for Brain Science, Harvard University, 17 Oxford Street, Cambridge, MA, 02138, USA

    Quan Wen

Authors
  1. Quan Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quan Wen .

Editor information

Editors and Affiliations

  1. Institute of Clinical Neuroanatomy, Ernst Strüngmann Institute (ESI) for Neuroscience in Cooperation with Max Planck Society & Goethe-University, Frankfurt, Germany

    Hermann Cuntz

  2. Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany

    Michiel W.H. Remme

  3. Department of Neurobiology, Hebrew University of Jerusalem, Jerusalem, Israel

    Benjamin Torben-Nielsen

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wen, Q. (2014). A Statistical Theory of Dendritic Morphology. In: Cuntz, H., Remme, M., Torben-Nielsen, B. (eds) The Computing Dendrite. Springer Series in Computational Neuroscience, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8094-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation