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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
Article

Abstract

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.

Keywords

Dendrites Membrane potential Spinal cord Cat Bistability Plateau potential 

Abbreviations and symbols

RN

input resistance at soma (MΩ)

ASD(D) = VD/VS

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

ADS(D) = VS/VD

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

ηSD

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

ηDS

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)

IS

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

ID

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

GC,S

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

GC,D

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

Gm

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

GC

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

Gm,S

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

Gm,D

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

Cm

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

τ0=τm

passive membrane time constant for all models (ms)

τ1

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

Reff

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

Notes

Acknowledgments

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.

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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

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