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

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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.

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Abbreviations

R N :

input resistance at soma (MΩ)

A SD(D) = V D /V S :

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

A DS(D) = V S /V D :

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

η SD :

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

η DS :

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

P(D) = SA soma /SA total :

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

V S = V m,S− E leak :

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

V D  = V m,D − E leak :

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

I S :

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

I D :

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

G C, S :

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

G C, D :

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

G m :

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

G C :

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

G m,S :

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

G m,D :

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

C m :

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

τ 0=τ m :

passive membrane time constant for all models (ms)

τ 1 :

equalizing time constant for all models (ms)

C 0, C 1 :

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

r N, implicit , r N, 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

R eff :

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

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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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Correspondence to Kelvin E. Jones.

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Action Editor: J. Rinzel

Appendix

Appendix

1.1 Equations for implicit and explicit models in the case II

The system equations for the case II with inject current normalized by the entire surface area of cell can be derived by replacing current density injected at soma and dendrite in Eq. (6) and (12) with

$${\left[ {\frac{{I_{S} }}{{I_{D} }}} \right]} \to {\left[ {\frac{{{I_{S} } \mathord{\left/ {\vphantom {{I_{S} } P}} \right. \kern-\nulldelimiterspace} P}}{{{I_{D} } \mathord{\left/ {\vphantom {{I_{D} } {1 - P}}} \right. \kern-\nulldelimiterspace} {1 - P}}}} \right]}$$
(19)

To obtain an expression for the electrotonic properties for case II, we need to apply I D = 0 to Eq. (8) derived for the case II under steady-state conditions: the specific input resistance is,

$$r_{{\text{N,implicit}}} = \frac{{\text{1}}}{{G_C }} \cdot \left( {\frac{{G_{{\text{m,S}}} \cdot p}}{{G_C }} + \frac{{G_{{\text{m}},D} \cdot \left( {{\text{1}} - P} \right)}}{{G_C + G_{{\text{m}},D} \cdot \left( {{\text{1}} - P} \right)}}} \right)^{ - 1} $$
(20a)
$$r_{{\text{N,explicit}}} = \frac{{\text{1}}}{{G_{C,{\text{S}}} }} \cdot \left( {\frac{{G_{\text{m}} \cdot P}}{{G_{C,{\text{S}}} }} + \frac{{G_{\text{m}} \cdot \left( {{\text{1}} - P} \right)}}{{G_{C,D} + G_{\text{m}} \cdot \left( {{\text{1}} - P} \right)}}} \right)^{ - 1} $$
(20b)

the voltage attenuation for both directions with DC input is,

$$\begin{aligned} & A^{V}_{{{\text{SD,implicit}}}} \, = \,\,\frac{{G_{C} }}{{G_{C} + G_{{{\text{m}},D}} \cdot {\left( {{\text{1}} - P} \right)}}} \\ & A^{V}_{{{\text{DS,implicit}}}} \, = \,\,\frac{{G_{C} }}{{G_{C} + G_{{{\text{m,S}}}} \cdot P}} \\ \end{aligned} $$
(21a)
$$\begin{aligned} & A^{V}_{{{\text{SD,explicit}}}} \, = \frac{{G_{{C,D}} }}{{G_{{C,D}} + G_{{\text{m}}} \cdot {\left( {{\text{1}} - P} \right)}}} \\ & A^{V}_{{{\text{DS,explicit}}}} \, = \,\,\frac{{G_{{C,{\text{S}}}} }}{{G_{{C,{\text{S}}}} + G_{m} \cdot P}} \\ \end{aligned} $$
(21b)

The passive membrane time constant for implicit and explicit models is consistent with Eq. (13) and (14) in the case I because the system matrix A in Eq. (12) is not affected by the replacement with Eq. (19), thereby Reff of case II is identical to those of case I.

$$\tau _{{\text{m,implicit}}} = R_{{\text{eff,implicit}}} \cdot C_{\text{m}} $$
(22a)
$$\tau _{{\text{m}},{\text{explicit}}} = R_{{\text{eff}},{\text{explicit}}} \cdot C_{\text{m}} $$
(22b)

All passive membrane parameters can be derived analytically as in case I: G C , G m,S, G m,D in implicit models, from Eq. (20a) and (21a), and G C ,S, G C ,D, G m in explicit models, from Eq. (20b) and (21b), and are:

$$G_C = \frac{{A_{{\text{DS,implicit}}}^V }}{{r_{{\text{N,implicit}}} \cdot \left( {{\text{1}} - A_{{\text{SD,implicit}}}^V \cdot A_{{\text{DS,implicit}}}^V } \right)}}$$
(23a)
$$G_{{\text{m,S}}} = \frac{{{\text{1}} - A_{{\text{DS}}}^V }}{{P \cdot r_{{\text{N,implicit}}} \cdot \left( {{\text{1}} - A_{{\text{SD,implicit}}}^V \cdot A_{{\text{DS,implicit}}}^V } \right)}}$$
(23b)
$$G_{m,D} = \frac{{A_{{\text{DS,implicit}}}^V \cdot \left( {{\text{1}} - A_{{\text{SD,implicit}}}^V } \right)}}{{\left( {{\text{1}} - P} \right) \cdot r_{{\text{N,implicit}}} \cdot A_{{\text{SD,implicit}}}^V \cdot \left( {{\text{1}} - A_{{\text{SD,implicit}}}^V \cdot A_{{\text{DS,implicit}}}^V } \right)}}$$
(23c)
$$G_{C,{\text{S}}} = \frac{{A_{{\text{DS}},{\text{explicit}}}^V }}{{r_{{\text{N}},{\text{explicit}}} \cdot \left( {{\text{1}} - A_{{\text{SD}}}^V \cdot A_{{\text{DS}}}^V } \right)}}$$
(24a)
$$G_{C,D} = \frac{{\left( {{\text{1}} - P} \right) \cdot A_{{\text{SD,explicit}}}^V \cdot \left( {{\text{1}} - A_{{\text{DS,explicit}}}^V } \right)}}{{P \cdot r_{{\text{N,explicit}}} \cdot \left( {{\text{1}} - A_{{\text{SD,explicit}}}^V \cdot A_{{\text{DS,explicit}}}^V } \right) \cdot \left( {{\text{1}} - A_{{\text{SD,explicit}}}^V } \right)}}$$
(24b)
$$G_{\text{m}} = \frac{{{\text{1}} - A_{{\text{DS,explicit}}}^V }}{{P \cdot r_{{\text{N,explicit}}} \cdot \left( {{\text{1}} - A_{{\text{SD,explicit}}}^V \cdot A_{{\text{DS,explicit}}}^V } \right)}}$$
(24c)

Then C m in implicit and explicit models from Eq. (22a) and (22b) based on previously determined parameter values are

$${{C_{\text{m}} = \tau _{\text{m}} } \mathord{\left/{\vphantom {{C_{\text{m}} = \tau _{\text{m}} } {R_{{\text{eff,implicit}}} }}} \right.\kern-\nulldelimiterspace} {R_{{\text{eff,implicit}}} }}$$
(25a)
$${{C_{\text{m}} = \tau _{\text{m}} } \mathord{\left/{\vphantom {{C_{\text{m}} = \tau _{\text{m}} } {R_{{\text{eff,explicit}}} }}} \right.\kern-\nulldelimiterspace} {R_{{\text{eff,explicit}}} }}$$
(25b)

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Kim, H., Major, L.A. & Jones, K.E. Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites. J Comput Neurosci 27, 321–336 (2009). https://doi.org/10.1007/s10827-009-0145-7

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