Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

  • Hojeong Kim
  • Lora A. Major
  • Kelvin E. JonesEmail author


Spinal motor neurons have voltage gated ion channels localized in their dendrites that generate plateau potentials. The physical separation of ion channels for spiking from plateau generating channels can result in nonlinear bistable firing patterns. The physical separation and geometry of the dendrites results in asymmetric coupling between dendrites and soma that has not been addressed in reduced models of nonlinear phenomena in motor neurons. We measured voltage attenuation properties of six anatomically reconstructed and type-identified cat spinal motor neurons to characterize asymmetric coupling between the dendrites and soma. We showed that the voltage attenuation at any distance from the soma was direction-dependent and could be described as a function of the input resistance at the soma. An analytical solution for the lumped cable parameters in a two-compartment model was derived based on this finding. This is the first two-compartment modeling approach that directly derived lumped cable parameters from the geometrical and passive electrical properties of anatomically reconstructed neurons.


Dendrites Membrane potential Spinal cord Cat Bistability Plateau potential 

Abbreviations and symbols


input resistance at soma (MΩ)


voltage attenuation factor from soma to dendrites at distance, D, from soma


voltage attenuation factor from dendrites to soma at distance, D, from soma


decay constant for voltage attenuation in the soma to dendrites direction (μm)


decay constant for voltage attenuation in the dendrites to soma direction (μm)

P(D) = SAsoma/SAtotal

morphological factor for two-compartment model; the ratio of somatic surface area to total surface area at distance, D, from soma

VS = Vm,S− Eleak

deviation of somatic membrane potential from reversal potential of leak ion channel in soma of two-compartment models (mV)

VD = Vm,D− Eleak

deviation of dendritic membrane potential from reversal potential of leak ion channels in dendrite of two-compartment models (mV)


injected current density at soma in two-compartment models, normalized by somatic surface area (μA/cm2)


injected current density at dendrite in two-compartment models, normalized by dendritic surface area (μA/cm2)


direction-dependent passive coupling conductance from soma to dendrite in explicit two-compartment model (μS/cm2)


direction-dependent passive coupling conductance from dendrite to soma in explicit two-compartment model (μS/cm2)


uniform passive membrane conductance in explicit two-compartment model (μS/cm2)


directionless passive coupling conductance in implicit two-compartment model (μS/cm2)


passive membrane conductance of soma in implicit two-compartment model (μS/cm2)


passive membrane conductance of dendrite in implicit two-compartment model (μS/cm2)


uniform passive membrane capacitance for two-compartment models (μF/cm2)


passive membrane time constant for all models (ms)


equalizing time constant for all models (ms)

C0, C1

coefficients used to form linearly independent combination of exponential decays (mV)

rN,implicit, rN,explicit

input resistance at somatic part in implicit and explicit models respectively, normalized by somatic surface area (MΩ-cm2)

\(A_{{\text{SD,implicit}}}^V \left( D \right),A_{{\text{SD,explicit}}}^V \left( D \right)\)

voltage attenuation factor for soma to dendrite direction at distance, D, from soma; used in implicit and explicit models

\(A_{{\text{DS,implicit}}}^V \left( D \right),A_{{\text{DS,explicit}}}^V \left( D \right)\)

voltage attenuation factor for dendrite to soma direction at distance, D, from soma; used in implicit and explicit models


effective membrane resistivity for calculating passive membrane time constant in two-compartment models (MΩ-cm2)

\(A_{{\text{SD,implicit}}}^I \left( D \right),A_{{\text{SD,explicit}}}^I \left( D \right)\)

current attenuation factor for soma to dendrite direction at distance, D, from soma; used in implicit and explicit models

\(A_{{\text{DS,implicit}}}^I \left( D \right),A_{{\text{DS,explicit}}}^I \left( D \right)\)

current attenuation factor for dendrite to soma direction at distance, D, from soma; used in implicit and explicit models



The study was supported by the Natural Sciences and Engineering Reseach Council of Canada [NSERC] with salary support for KEJ from the Alberta Heritage Foundation for Medical Research [AHFMR]. We thank Karl Jensen for preparing the models for submission to ModelDB.


  1. Alvarez, F. J., Pearson, J. C., et al. (1998). Distribution of 5-hydroxytryptamine-immunoreactive boutons on alpha-motoneurons in the lumbar spinal cord of adult cats. The Journal of Comparative Neurology, 393(1), 69–83. doi: 10.1002/(SICI)1096-9861(19980330)393:1<69::AID-CNE7>3.0.CO;2-O.CrossRefPubMedGoogle Scholar
  2. Ascoli, G. A. (2006). Mobilizing the base of neuroscience data: the case of neuronal morphologies. Nature Reviews. Neuroscience, 7(4), 318–324. doi: 10.1038/nrn1885.CrossRefPubMedGoogle Scholar
  3. Ballou, E. W., Smith, W. B., et al. (2006). Measuring dendritic distribution of membrane proteins. Journal of Neuroscience Methods, 156(1–2), 257–266. doi: 10.1016/j.jneumeth.2006.03.014.CrossRefPubMedGoogle Scholar
  4. Booth, V., & Rinzel, J. (1995). A minimal, compartmental model for a dendritic origin of bistability of motoneuron firing patterns. Journal of Computational Neuroscience, 2(4), 299–312. doi: 10.1007/BF00961442.CrossRefPubMedGoogle Scholar
  5. Booth, V., Rinzel, J., et al. (1997). Compartmental model of vertebrate motoneurons for Ca2+-dependent spiking and plateau potentials under pharmacological treatment. Journal of Neurophysiology, 78(6), 3371–3385.PubMedGoogle Scholar
  6. Bras, H., Korogod, S., et al. (1993). Stochastic geometry and electronic architecture of dendritic arborization of brain stem motoneuron. The European Journal of Neuroscience, 5(11), 1485–1493. doi: 10.1111/j.1460-9568.1993.tb00216.x.CrossRefPubMedGoogle Scholar
  7. Bui, T. V., Ter-Mikaelian, M., et al. (2006). Computational estimation of the distribution of L-type Ca(2+) channels in motoneurons based on variable threshold of activation of persistent inward currents. Journal of Neurophysiology, 95(1), 225–241. doi: 10.1152/jn.00646.2005.CrossRefPubMedGoogle Scholar
  8. Burke, R. E., Levine, D. N., et al. (1973). Physiological types and histochemical profiles in motor units of the cat gastrocnemius. The Journal of Physiology, 234(3), 723–748.PubMedGoogle Scholar
  9. Cameron, W. E., He, F., et al. (1991). Morphometric analysis of phrenic motoneurons in the cat during postnatal development. The Journal of Comparative Neurology, 314(4), 763–776. doi: 10.1002/cne.903140409.CrossRefPubMedGoogle Scholar
  10. Carlin, K. P., Jiang, Z., et al. (2000). Characterization of calcium currents in functionally mature mouse spinal motoneurons. The European Journal of Neuroscience, 12(5), 1624–1634. doi: 10.1046/j.1460-9568.2000.00050.x.CrossRefPubMedGoogle Scholar
  11. Carlin, K. P., Jones, K. E., et al. (2000). Dendritic L-type calcium currents in mouse spinal motoneurons: implications for bistability. The European Journal of Neuroscience, 12(5), 1635–1646. doi: 10.1046/j.1460-9568.2000.00055.x.CrossRefPubMedGoogle Scholar
  12. Carnevale, N. T., & Hines, M. L. (2005). The NEURON book. Cambridge; New York: Cambridge University Press.Google Scholar
  13. Carnevale, N. T., & Johnston, D. (1982). Electrophysiological characterization of remote chemical synapses. Journal of Neurophysiology, 47(4), 606–621.PubMedGoogle Scholar
  14. Coombs, J. S., Eccles, J. C., et al. (1955). The electrical properties of the motoneurone membrane. The Journal of Physiology, 130(2), 291–325.PubMedGoogle Scholar
  15. Cullheim, S., Fleshman, J. W., et al. (1987a). Membrane area and dendritic structure in type-identified triceps surae alpha motoneurons. The Journal of Comparative Neurology, 255(1), 68–81. doi: 10.1002/cne.902550106.CrossRefPubMedGoogle Scholar
  16. Cullheim, S., Fleshman, J. W., et al. (1987b). Three-dimensional architecture of dendritic trees in type-identified alpha-motoneurons. The Journal of Comparative Neurology, 255(1), 82–96. doi: 10.1002/cne.902550107.CrossRefPubMedGoogle Scholar
  17. Elbasiouny, S. M., Bennett, D. J., et al. (2005). Simulation of dendritic CaV1.3 channels in cat lumbar motoneurons: spatial distribution. Journal of Neurophysiology, 94(6), 3961–3974. doi: 10.1152/jn.00391.2005.CrossRefPubMedGoogle Scholar
  18. Fleshman, J. W., Segev, I., et al. (1988). Electrotonic architecture of type-identified alpha-motoneurons in the cat spinal cord. Journal of Neurophysiology, 60(1), 60–85.PubMedGoogle Scholar
  19. Frank, K., & Fuortes, M. G. (1955). Potentials recorded from the spinal cord with microelectrodes. The Journal of Physiology, 130(3), 625–654.PubMedGoogle Scholar
  20. Grande, G., Bui, T. V., et al. (2007). Estimates of the location of L-type Ca2 + channels in motoneurons of different size: a computational study. Journal of Neurophysiology, 97, 4023–4035. doi: 10.1152/jn.00044.2007.CrossRefPubMedGoogle Scholar
  21. Heckman, C. J., & Lee, R. H. (1999a). Synaptic integration in bistable motoneurons. Progress in Brain Research, 123, 49–56. doi: 10.1016/S0079-6123(08)62843-5.CrossRefPubMedGoogle Scholar
  22. Heckman, C. J., & Lee, R. H. (1999b). The role of voltage-sensitive dendritic conductances in generating bistable firing patterns in motoneurons. J Physiol Paris, 93(1–2), 97–100. doi: 10.1016/S0928-4257(99)80140-5.CrossRefPubMedGoogle Scholar
  23. Heckman, C. J., Lee, R. H., et al. (2003). Hyperexcitable dendrites in motoneurons and their neuromodulatory control during motor behavior. Trends in Neurosciences, 26(12), 688–695. doi: 10.1016/j.tins.2003.10.002.CrossRefPubMedGoogle Scholar
  24. Heckmann, C. J., Gorassini, M. A., et al. (2005). Persistent inward currents in motoneuron dendrites: implications for motor output. Muscle & Nerve, 31(2), 135–156. doi: 10.1002/mus.20261.CrossRefGoogle Scholar
  25. Holmes, W. R., & Rall, W. (1992). Electrotonic length estimates in neurons with dendritic tapering or somatic shunt. Journal of Neurophysiology, 68(4), 1421–1437.PubMedGoogle Scholar
  26. Hounsgaard, J., Hultborn, H., et al. (1984). Intrinsic membrane properties causing a bistable behaviour of alpha-motoneurones. Experimental Brain Research, 55(2), 391–394. doi: 10.1007/BF00237290.CrossRefGoogle Scholar
  27. Jack, J. J. B., Noble, D., et al. (1975). Electric current flow in excitable cells. Oxford: Clarendon.Google Scholar
  28. Jones, K. E., Carlin, , K. P., et al. (2000). Simulation techniques for localising and identifying the kinetics of calcium channels in dendritic neurons. Neurocomputing, 32, 173–180. doi: 10.1016/S0925-2312(00)00160-0.CrossRefGoogle Scholar
  29. Kim, H., Major, L. A., et al. (2008). Voltage attenuation in reconstructed type-identified motor neurons as a constraint for reduced models. BMC Neuroscience, 9(Suppl 1), 55. doi: 10.1186/1471-2202-9-S1-P55.CrossRefGoogle Scholar
  30. Korogod, S., Bras, H., et al. (1994). Electrotonic clusters in the dendritic arborization of abducens motoneurons of the rat. The European Journal of Neuroscience, 6(10), 1517–1527. doi: 10.1111/j.1460-9568.1994.tb00542.x.CrossRefPubMedGoogle Scholar
  31. Lee, R. H., & Heckman, C. J. (1996). Influence of voltage-sensitive dendritic conductances on bistable firing and effective synaptic current in cat spinal motoneurons in vivo. Journal of Neurophysiology, 76(3), 2107–2110.PubMedGoogle Scholar
  32. Lee, R. H., & Heckman, C. J. (1998a). Bistability in spinal motoneurons in vivo: systematic variations in persistent inward currents. Journal of Neurophysiology, 80(2), 583–593.PubMedGoogle Scholar
  33. Lee, R. H., & Heckman, C. J. (1998b). Bistability in spinal motoneurons in vivo: systematic variations in rhythmic firing patterns. Journal of Neurophysiology, 80(2), 572–582.PubMedGoogle Scholar
  34. Lee, R. H., & Heckman, C. J. (1999). Enhancement of bistability in spinal motoneurons in vivo by the noradrenergic alpha1 agonist methoxamine. Journal of Neurophysiology, 81(5), 2164–2174.PubMedGoogle Scholar
  35. MacGregor, R. J. (1987). Neural and brain modeling. San Diego, CA: Academic.Google Scholar
  36. Mainen, Z. F., & Sejnowski, T. J. (1996). Influence of dendritic structure on firing pattern in model neocortical neurons. Nature, 382(6589), 363–366. doi: 10.1038/382363a0.CrossRefPubMedGoogle Scholar
  37. Major, G., Evans, J. D., et al. (1993). Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt. Biophysical Journal, 65(1), 423–449. doi: 10.1016/S0006-3495(93)81037-3.CrossRefPubMedGoogle Scholar
  38. Nitzan, R., Segev, I., et al. (1990). Voltage behavior along the irregular dendritic structure of morphologically and physiologically characterized vagal motoneurons in the guinea pig. Journal of Neurophysiology, 63(2), 333–346.PubMedGoogle Scholar
  39. Pinsky, P. F., & Rinzel, J. (1994). Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. Journal of Computational Neuroscience, 1(1–2), 39–60. doi: 10.1007/BF00962717.CrossRefPubMedGoogle Scholar
  40. Rall, W. (1957). Membrane time constant of motoneurons. Science, 126(3271), 454. doi: 10.1126/science.126.3271.454.CrossRefPubMedGoogle Scholar
  41. Rall, W. (1959). Branching dendritic trees and motoneuron membrane resistivity. Experimental Neurology, 1, 491–527. doi: 10.1016/0014-4886(59)90046-9.CrossRefPubMedGoogle Scholar
  42. Rall, W. (1969). Time constants and electrotonic length of membrane cylinders and neurons. Biophysical Journal, 9(12), 1483–1508. doi: 10.1016/S0006-3495(69)86467-2.CrossRefPubMedGoogle Scholar
  43. Rall, W., & Rinzel, J. (1973). Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophysical Journal, 13(7), 648–687. doi: 10.1016/S0006-3495(73)86014-X.CrossRefPubMedGoogle Scholar
  44. Rall, W., Segev, I., et al. (1995). The theoretical foundation of dendritic function: selected papers of Wilfrid Rall with commentaries. Cambridge, MA: MIT.Google Scholar
  45. Rinzel, J., & Rall, W. (1974). Transient response in a dendritic neuron model for current injected at one branch. Biophysical Journal, 14(10), 759–790. doi: 10.1016/S0006–3495(74)85948–5.CrossRefPubMedGoogle Scholar
  46. Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. Methods in neuronal modeling: from ions to networks. C. Koch and I. Segev. Cambridge, MA, MIT Press: 251–91.Google Scholar
  47. Rose, P. K., & Cushing, S. (2004). Relationship between morphoelectrotonic properties of motoneuron dendrites and their trajectory. The Journal of Comparative Neurology, 473(4), 562–581. doi: 10.1002/cne.20137.CrossRefPubMedGoogle Scholar
  48. Saltelli, A. (2004). Sensitivity analysis in practice: a guide to assessing scientific models. Hoboken, NJ: Wiley.Google Scholar
  49. Schoenen, J. (1982). Dendritic organization of the human spinal cord: the motoneurons. The Journal of Comparative Neurology, 211(3), 226–247. doi: 10.1002/cne.902110303.CrossRefPubMedGoogle Scholar
  50. Segev, I., Fleshman Jr., J. W., et al. (1990). Computer simulation of group Ia EPSPs using morphologically realistic models of cat alpha-motoneurons. Journal of Neurophysiology, 64(2), 648–660.PubMedGoogle Scholar
  51. Thurbon, D., Luscher, H. R., et al. (1998). Passive electrical properties of ventral horn neurons in rat spinal cord slices. Journal of Neurophysiology, 80(1), 2485–2502.PubMedGoogle Scholar
  52. Williams, S. R., & Mitchell, S. J. (2008). Direct measurement of somatic voltage clamp errors in central neurons. Nature Neuroscience, 11(7), 790–798. doi: 10.1038/nn.2137.CrossRefPubMedGoogle Scholar
  53. Yakovenko, S., Mushahwar, V., et al. (2002). Spatiotemporal activation of lumbosacral motoneurons in the locomotor step cycle. Journal of Neurophysiology, 87(3), 1542–1553.PubMedGoogle Scholar
  54. Zengel, J. E., Reid, S. A., et al. (1985). Membrane electrical properties and prediction of motor-unit type of medial gastrocnemius motoneurons in the cat. Journal of Neurophysiology, 53(5), 1323–1344.PubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Hojeong Kim
    • 1
  • Lora A. Major
    • 2
  • Kelvin E. Jones
    • 1
    • 2
    • 3
    Email author
  1. 1.Department of Biomedical EngineeringUniversity of AlbertaEdmontonCanada
  2. 2.Faculty of Physical Education and RecreationUniversity of AlbertaEdmontonCanada
  3. 3.Faculty of Physical Education and RecreationUniversity of AlbertaEdmontonCanada

Personalised recommendations