Biological Cybernetics

, Volume 107, Issue 6, pp 685–694 | Cite as

The Green’s function formalism as a bridge between single- and multi-compartmental modeling

  • Willem A. M. WyboEmail author
  • Klaus M. Stiefel
  • Benjamin Torben-Nielsen
Original Paper


Neurons are spatially extended structures that receive and process inputs on their dendrites. It is generally accepted that neuronal computations arise from the active integration of synaptic inputs along a dendrite between the input location and the location of spike generation in the axon initial segment. However, many application such as simulations of brain networks use point-neurons—neurons without a morphological component—as computational units to keep the conceptual complexity and computational costs low. Inevitably, these applications thus omit a fundamental property of neuronal computation. In this work, we present an approach to model an artificial synapse that mimics dendritic processing without the need to explicitly simulate dendritic dynamics. The model synapse employs an analytic solution for the cable equation to compute the neuron’s membrane potential following dendritic inputs. Green’s function formalism is used to derive the closed version of the cable equation. We show that by using this synapse model, point-neurons can achieve results that were previously limited to the realms of multi-compartmental models. Moreover, a computational advantage is achieved when only a small number of simulated synapses impinge on a morphologically elaborate neuron. Opportunities and limitations are discussed.


Morphological simplification Cable theory Interacting synapses Green’s function formalism  Transfer functions 



We thank Marc-Oliver Gewaltig for comments on the manuscript and Moritz Deger for helpful discussion. This work was supported by the BrainScaleS EU FET-proactive FP7 grant.


  1. Abrahamsson T, Cathala L, Matsui K, Shigemoto R, Digregorio DA (2012) Thin dendrites of cerebellar interneurons confer sublinear synaptic integration and a gradient of short-term plasticity. Neuron 73(6):1159–1172PubMedCrossRefGoogle Scholar
  2. Agmon-Snir H, Carr CE, Rinzel J (1998) The role of dendrites in auditory coincidence detection. Nature 393:268–272Google Scholar
  3. Angelo K, London M, Christensen SR, Häusser M (2007) Local and global effects of I(h) distribution in dendrites of mammalian neurons. J Neurosci Off J Soc Neurosci 27(32):8643–8653CrossRefGoogle Scholar
  4. Ascoli GA, Donohue DE, Halavi M (2007) NeuroMorpho.Org: a central resource for neuronal morphologies. J Neurosci 27(35):9247–9251PubMedCrossRefGoogle Scholar
  5. Blackman R, Tukey J (1958) The measurement of power spectra. Dover publications, NYGoogle Scholar
  6. Branco T, Clark BA, Häusser M (2010) Dendritic discrimination of temporal input sequences in cortical neurons. Science (New York, N.Y.) 329(5999):1671–5CrossRefGoogle Scholar
  7. Brette R, Rudolph M, Carnevale T, Hines M, Beeman D, Bower JM, Diesmann M, Morrison A, Goodman PH, Harris FC, Zirpe M, Natschläger T, Pecevski D, Ermentrout B, Djurfeldt M, Lansner A, Rochel O, Vieville T, Muller E, Davison AP, El Boustani S, Destexhe A (2007) Simulation of networks of spiking neurons: a review of tools and strategies. J Comput Neurosci 23(3):349–398Google Scholar
  8. Bullock TH, Horridge GA (1965) Structure and function in the nervous systems of invertebrates/[by] Theodore Holmes Bullock and G. Adrian Horridge. With chapters by Howard A. Bern, Irvine R. Hagadorn [and] J. E. Smith. W. H. Freeman, San FranciscoGoogle Scholar
  9. Butz EG, Cowan JD (1974) Transient potentials in dendritic systems of arbitrary geometry. Biophys J 14:661–689PubMedCrossRefGoogle Scholar
  10. Carnevale NT, Hines ML (2006) The NEURON Book. Cambridge University Press, New York, NY, USACrossRefGoogle Scholar
  11. Gewaltig M-O, Diesmann M (2007) NEST (NEural Simulation Tool). Scholarpedia 2(4):1430CrossRefGoogle Scholar
  12. Gidon A, Segev I (2012) Principles governing the operation of synaptic inhibition in dendrites. Neuron 75(2):330–41PubMedCrossRefGoogle Scholar
  13. Giugliano M (2000) Synthesis of generalized algorithms for the fast computation of synaptic conductances with markov kinetic models in large network simulations. Neural Comput 931:903–931CrossRefGoogle Scholar
  14. Govindarajan A, Israely I, Huang S-Y, Tonegawa S (2011) The dendritic branch is the preferred integrative unit for protein synthesis-dependent LTP. Neuron 69(1):132–46PubMedCrossRefGoogle Scholar
  15. Gütig R, Sompolinsky H (2006) The tempotron: a neuron that learns spike timing-based decisions. Nat Neurosci 9(3):420–428PubMedCrossRefGoogle Scholar
  16. Hay E, Schürmann F, Markram H, Segev I (2013) Preserving axo-somatic spiking features despite diverse dendritic morphology. J Neurophys 108:2972–2981Google Scholar
  17. Jolivet R, Lewis TJ, Gerstner W (2004) Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. J Neurophysiol 92(2):959–976PubMedCrossRefGoogle Scholar
  18. Kellems AR, Chaturantabut S, Sorensen DC, Cox SJ (2010) Morphologically accurate reduced order modeling of spiking neurons. J Comput Neurosci 28(3):477–494PubMedCrossRefGoogle Scholar
  19. Koch C (1998) Biophysics of computation: information processing in single neurons (computational neuroscience), 1st edn. Oxford University Press, OxfordGoogle Scholar
  20. Koch C, Poggio T (1985) A simple algorithm for solving the cable equation in dendritic trees of arbitrary geometry. J Neurosci Methods 12:303–315Google Scholar
  21. Larkum ME, Zhu JJ, Sakmann B (1999) A new cellular mechanism for coupling inputs arriving at different cortical layers. Nature 398(6725):338–41PubMedCrossRefGoogle Scholar
  22. London M, Häusser M (2005) Dendritic computation. Annu Rev Neurosci 28:503–532PubMedCrossRefGoogle Scholar
  23. Magee JC (1999) Dendritic Ih normalizes temporal summation in hippocampal CA1 neurons. Nat Neurosci 2(9):848PubMedCrossRefGoogle Scholar
  24. Markram H (2006) The blue brain project. Nat Rev Neurosci 7(2):153–160PubMedCrossRefGoogle Scholar
  25. Mathews PJ, Jercog PE, Rinzel J, Scott LL, Golding NL (2010) Control of submillisecond synaptic timing in binaural coincidence detectors by K(v)1 channels. Nat Neurosci 13(5):601–609PubMedCrossRefGoogle Scholar
  26. Mauro A, Conti F, Dodge F, Schor R (1970) Subthreshold behavior and phenomenological impedance of the squid giant axon. J General Physiol 55(4):497–523CrossRefGoogle Scholar
  27. Migliore M, Shepherd GM (2002) Emerging rules for the distributions of active dendritic conductances. Nat Rev Neurosci 3(5):362–370PubMedCrossRefGoogle Scholar
  28. Norman RS (1972) Cable theory for finite length dendritic cylinders with initial and boundary conditions. Biophys J 12(1):25–45 PubMedCrossRefGoogle Scholar
  29. Ohme M, Schierwagen A (1998) An equivalent cable model for neuronal trees with active membrane. Biolog Cybern 78(3):227–243CrossRefGoogle Scholar
  30. Pissadaki EK, Sidiropoulou K, Reczko M, Poirazi P (2010) Encoding of spatio-temporal input characteristics by a CA1 pyramidal neuron model. PLoS Comput Biol 6(12):e1001038PubMedCrossRefGoogle Scholar
  31. Poirazi P, Brannon T, Mel BW (2003) Pyramidal neuron as two-layer neural network. Neuron 37:989–999PubMedCrossRefGoogle Scholar
  32. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes 3rd edition: the art of scientific computing, 3rd edn. Cambridge University Press, New York, NY, USAGoogle Scholar
  33. Richert M, Nageswaran JM, Dutt N, Krichmar JL (2011) An efficient simulation environment for modeling large-scale cortical processing. Front Neuroinform 5(September):19PubMedGoogle Scholar
  34. Rotter S, Diesmann M (1999) Exact digital simulation of time-invariant linear systems with applications to neuronal modeling. Biolog Cybern 81(5–6):381–402CrossRefGoogle Scholar
  35. Schoen A, Salehiomran A, Larkum ME, Cook EP (2012) A compartmental model of linear resonance and signal transfer in dendrites. Neural Comput 24(12):3126–3144PubMedCrossRefGoogle Scholar
  36. Spruston N (2008) Pyramidal neurons: dendritic structure and synaptic integration. Nat Rev Neurosci 9(3):206–221PubMedCrossRefGoogle Scholar
  37. Torben-Nielsen B, Stiefel KM (2010) An inverse approach for elucidating dendritic function. Front Comput Neurosci 4(September):128PubMedGoogle Scholar
  38. Traub RD, Contreras D, Cunningham MO, Murray H, LeBeau FEN, Roopun A, Bibbig A, Wilent WB, Higley MJ, Whittington MA (2005) Single-column thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. J Neurophysiol 93(4):2194–2232PubMedCrossRefGoogle Scholar
  39. Tuckwell HC (1988) Introduction to theoretical neurobiology. Cambridge studies in mathematical biology, 8. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  40. Ulrich D (2002) Dendritic resonance in rat neocortical pyramidal cells. J Neurophysiol 87(6):2753–2759PubMedGoogle Scholar
  41. Van Pelt J (1992) A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees. Biolog Cybern 21:15–21CrossRefGoogle Scholar
  42. Vervaeke K, Lorincz A, Nusser Z, Silver RA (2012) Gap junctions compensate for sublinear dendritic integration in an inhibitory network. Science (New York, N.Y.), 1624Google Scholar
  43. Wang Y, Gupta A, Toledo-Rodriguez M, Wu CZ, Markram H (2002) Anatomical, physiological, molecular and circuit properties of nest basket cells in the developing somatosensory cortex. Cerebral cortex (New York, N.Y. 1991) 12(4):395–410CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Willem A. M. Wybo
    • 1
    Email author
  • Klaus M. Stiefel
    • 2
  • Benjamin Torben-Nielsen
    • 1
  1. 1.Blue Brain Project, Brain Mind InstituteEPFLLausanneSwitzerland
  2. 2.MARCS InstituteUniversity of Western SydneySydneyAustralia

Personalised recommendations