Biological Cybernetics

, Volume 97, Issue 2, pp 137–149 | Cite as

Branching dendrites with resonant membrane: a “sum-over-trips” approach

  • S. Coombes
  • Y. Timofeeva
  • C. -M. Svensson
  • G. J. Lord
  • K. Josić
  • S. J. Cox
  • C. M. Colbert
Original Paper

Abstract

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dendrites. Another important contribution to the response characteristics of a single neuron comes from the intrinsic resonant properties of dendritic membrane. In this paper we combine the effects of dendritic branching and resonant membrane dynamics by generalising the “sum-over-trips” approach (Abbott et al. in Biol Cybernetics 66, 49–60 1991). To illustrate how this formalism can shed light on the role of architecture and resonances in determining neuronal output we consider dual recording and reconstruction data from a rat CA1 hippocampal pyramidal cell. Specifically we explore the way in which an Ih current contributes to a voltage overshoot at the soma.

Keywords

Dendrites Quasi-active membrane “Sum-over-trips” Cable theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abbott LF (1992) Simple diagrammatic rules for solving dendritic cable problems. Physica A 185:343–356CrossRefGoogle Scholar
  2. Abbott LF, Fahri E, Gutmann S (1991) The path integral for dendritic trees. Biol Cybern 66:49–60PubMedCrossRefGoogle Scholar
  3. Butz EG, Cowan JD (1974) Transient potentials in dendritic systems of arbitrary geometry. Biophys J 14:661–689PubMedGoogle Scholar
  4. Cao BJ, Abbott LF (1993) New computational method for cable theory problems. Biophys J 64:303–313PubMedCrossRefGoogle Scholar
  5. Carnevale NT, Hines ML (2006) The NEURON Book. Cambridge University Press, LondonGoogle Scholar
  6. Cox SJ, Griffith BE (2001) Recovering quasi-active properties of dendritic neurons from dual potential recordings. J Comput Neurosci 11:95–110PubMedCrossRefGoogle Scholar
  7. Cox SJ, Raol JH (2004) Recovering the passive properties of tapered dendrites from single and dual potential recordings. Math Biosci 190:9–37PubMedCrossRefGoogle Scholar
  8. Evans JD, Kember GC, Major G (1992) Techniques for obtaining analytical solutions to the multi-cylinder somatic shunt cable model for passive neurons. Biophys J 63:350–365PubMedGoogle Scholar
  9. Evans JD, Kember GC, Major G (1995) Techniques for the application of the analytical solutions to the multi-cylinder somatic shunt cable model for passive neurons. Math Biosci 125:1–50PubMedCrossRefGoogle Scholar
  10. Häusser M (2001) Dendritic democracy. Curr Biol 11:R10–R12PubMedCrossRefGoogle Scholar
  11. Hudspeth AJ, Lewis RS (1988) A model for electrical resonance and frequency tuning in saccular hair cells of the bull-frog, Rana Catesbeiana. J Physiol 400:275–297PubMedGoogle Scholar
  12. Hutcheon B, Miura RM, Puil E (1996) Models of subthreshold membrane resonance in neocortical neurons. J Neurophysiol 76: 698–714PubMedGoogle Scholar
  13. Hutcheon B, Yarom Y (2000) Resonance, oscillation and the intrinsic frequency preferences of neurons. Trends Neurosci 23:216–222PubMedCrossRefGoogle Scholar
  14. Johnson D, Magee JC, Colbert CM, Christie BR (1996) Active properties of neuronl dendrites. Ann Rev Neurosci 19:165–186CrossRefGoogle Scholar
  15. Koch C (1984) Cable theory in neurons with active, linearized membranes. Biol Cybern 50:15–33PubMedCrossRefGoogle Scholar
  16. Koch C, Poggio T (1985) A simple algorithm for solving the cable equation in dendritic geometries of arbitrary geometry. J Neurosci Methods 12:303–315PubMedCrossRefGoogle Scholar
  17. Kole MHP, Hallermann S, Stuart GJ (2006) Single I h channels in pyramidal neuron dendrites: Properties, distribution, and impact on action potential output. J Neurosci 26(6):1677–1687PubMedCrossRefGoogle Scholar
  18. Li X, Ascoli GA (2006) Computational simulation of the input–output relationship in hippocampal pyramidal cells. J Comput Neurosci 21:191–209PubMedCrossRefGoogle Scholar
  19. London M, Häusser M (2005) Dendritic computation. Annu Rev Neurosci 28:503–532PubMedCrossRefGoogle Scholar
  20. London M, Meunier C, Segev I (1999) Signal transfer in passive dendrites with nonuniform membrane conductance. J Neurosci 19:8219–8233PubMedGoogle Scholar
  21. Magee JC (1998) Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal CA1 pyramidal neurons. J Neurosci 18:7613–7624PubMedGoogle Scholar
  22. Mainen ZF, Sejnowski TJ (1996) Influence of dendritic structure on firing pattern in model neocortical neurons. Nature 382:363–366PubMedCrossRefGoogle Scholar
  23. Mauro A, Conti F, Dodge F, Schor R (1970) Subthreshold behavior and phenomenological impedance of the squid giant axon. J Gen Physiol 55:497–523PubMedCrossRefGoogle Scholar
  24. Migliore M, Ferrante M, Ascoli GA (2005) Signal propagation in oblique dendrites of CA1 pyramidal cells. J Neurophysiol 94:4145–4155PubMedCrossRefGoogle Scholar
  25. van Ooyen A, Duijnhouwer J, Remme MWH, van Pelt J (2002) The effect of dendritic topology on firing patterns in model neurons. Network 13:311–325PubMedCrossRefGoogle Scholar
  26. Pape HC (1996) Queer current and pacemaker: the hyperpolarization activated cation current in neurons. Annu Rev Physiol 58:299–327PubMedCrossRefGoogle Scholar
  27. Scott A (2002) Neuroscience: a mathematical primer. Springer, HeidelbergGoogle Scholar
  28. Segev I, London M (2000) Untangling dendrites with quantitative models. Science 290:744–750PubMedCrossRefGoogle Scholar
  29. Segev I, Rinzel J, Shepherd GM (eds) (1995) The theoretical foundations of dendritic function: selected papers of Wilfrid Rall with commentaries. MIT Press, CambridgeGoogle Scholar
  30. Stuart G, Spruston N, Häusser M. (eds.) (1999) Dendrites. Oxford University Press, New YorkGoogle Scholar
  31. Timofeeva Y, Lord GJ, Coombes S (2006) Dendritic cable with active spines: a modeling study in the spike-diffuse spike framework. Neurocomputing 69:1058–1061CrossRefGoogle Scholar
  32. Timofeeva Y, Lord GJ, Coombes S (2006) Spatio-temporal filtering properties of a dendritic cable with active spines. J Comput Neurosci 21:293–306PubMedCrossRefGoogle Scholar
  33. Tuckwell HC (1988) Introduction to theoretical neurobiology, vol 1. Cambridge University Press, LondonGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • S. Coombes
    • 1
  • Y. Timofeeva
    • 1
  • C. -M. Svensson
    • 1
  • G. J. Lord
    • 2
  • K. Josić
    • 3
  • S. J. Cox
    • 4
  • C. M. Colbert
    • 5
  1. 1.Department of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Department of MathematicsHeriot-Watt UniversityEdinburghUK
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA
  4. 4.Computational and Applied MathematicsRice UniversityHoustonUSA
  5. 5.Biology and BiochemistryUniversity of HoustonHoustonUSA

Personalised recommendations