Generalized Cable Models of Neurons and Dendrites

Abstract

Cable theory was introduced for neurons by Wilfrid Rall more than half a century ago and is widely used today for modeling the voltage and current flow in neuronal and dendritic structures. This theory was derived assuming that the extracellular medium is either inexistent or modeled as a resistor. For modeling neurons in more realistic situations, where the extracellular medium has more complex electric properties, it is necessary to generalize Rall’s cable equations. We summarize here such generalized cable equations and show that the nature of the surrounding extracellular medium can exert nonnegligible influences on the cable properties of neurons.

Appendix

Method to Solve the Generalized Cable

The basis of the method to solve the generalized cable is that Eq. 9 is exact for a continuous cylindric compartment of constant diameter (which is equivalent to an infinite number of membrane RC circuits). In other words, we can use this property to simulate exactly the full cylindric compartment as a continuum with no need of spatial discretization into segments that is usually done in numerical simulators. This exact solution is only possible if the cylindric compartment has a constant diameter. This approach was called the “continuous compartment” method (Bédard and Destexhe 2013), and this leads to an efficient method to simulate the generalized cable formalism in complex cable structures.

To apply this formalism to more complex morphologies than a single cable, one must adjust the specific limit conditions of the different compartments (continuity of V _m and of the current ${i}_i^g=-\frac{1}{{\overline{z}}_i}\frac{\partial {V}_m}{\partial x}$; see details in Bédard and Destexhe 2013).

We must calculate the input impedances needed to compute the membrane voltage in complex morphologies. We consider the input impedance of the membrane as well as the impedance of the extracellular medium, both of which are needed to calculate the spatial profile of the V _m in a given cable segment.

In a first step, one separates the cable into a series of continuous compartments of constant diameter, where parameters 2a, z _i , r _m, and z ^(m) _e are constant and specific to each compartment.

In a second step, one calculates the (transmembrane) input impedance ${z}_{\mathrm{in}}^{n+1}=\frac{V_m(0)}{i_i(0)}$ at the beginning of each compartment by taking into account the auxiliary impedance at the end of this compartment, ${Z}_a={Z}_{\mathrm{out}}^{n+1}=\frac{V_m\left({l}_{n+1}\right)}{i_i\left({l}_{n+1}\right)}={Z}_{\mathrm{in}}^n$ (see Fig. 1a) if there is no branching point. At the branching points, the auxiliary impedances are simply equal to the equivalent input impedance of n dendritic branches in parallel (where n is the number of “daughter” branches; see Fig. 1b–c). Thus, because the input impedance at one end is equal to the input impedance of the other compartment connected to this end, one obtains a recursive relation (see details in Bédard and Destexhe 2013):

$$ {Z}_{\mathrm{in}}^{n+1}\left[{Z}_{\mathrm{in}}^n\right]=\frac{{\overline{z}}_{i_n}}{\kappa_{\lambda_n}}\frac{\left({\kappa}_{\lambda_n}{Z}_{\mathrm{in}}^n+{\overline{z}}_{i_n}\right){e}^{2{\kappa}_{\lambda_n}{l}_n}+\left({\kappa}_{\lambda_n}{Z}_{\mathrm{in}}^n-{\overline{z}}_{i_n}\right)}{\left({\kappa}_{\lambda_n}{Z}_{\mathrm{in}}^n+{\overline{z}}_{i_n}\right){e}^{2{\kappa}_{\lambda_n}{l}_n}-\left({\kappa}_{\lambda_n}{Z}_{\mathrm{in}}^n-{\overline{z}}_{i_n}\right)}, $$

(10)

where ${\kappa}_{\lambda_n}=\kappa /{\lambda}_n.{\overline{z}}_i$ takes its value according to the model considered.

Thus, we can write

$$ {Z}_{\mathrm{in}}^{n+1}=F\left[{Z}_{\mathrm{in}}^n;{\overline{z}}_{i_n}{\kappa}_{\lambda_n},{l}_n\right] $$

This leads to the following expression to relate the first to the nth segment:

$$ {Z}_{\mathrm{in}}^{n+1}=F\left[\dots F\left[F\left[{Z}_{\mathrm{in}}^1;{\overline{z}}_{i_1},{\kappa}_{\lambda_1},{l}_1\right];{\overline{z}}_{i_2},{\kappa}_{\lambda_2},{l}_2\right]\dots; {\overline{z}}_{i_n},{\kappa}_{\lambda_n},{l}_n\right] $$

(11)

In a third step, to calculate the profile of V _m along the cable, one must use the spatial transfer function $\frac{V_m\left({P}_{n+1},\omega \right)}{V_m\left({P}_n,\omega \right)}$ on a continuous cylindric compartment of arbitrary length and calculate the product of the transfer functions between each connected compartment. This leads to (see details in Bédard and Destexhe 2013):

$$ {F}_T\left(l,\omega; {Z}_{\mathrm{out}}^n\right)=\frac{\kappa_{\lambda_n}{Z}_{\mathrm{out}}^n}{\kappa_{\lambda_n}{Z}_{\mathrm{out}}^n \cos h\left({\kappa}_{\lambda_n}l\right)+{\overline{z}}_i \sin h\left({\kappa}_{\lambda_n}l\right)} $$

(12)

$$ \frac{V_m\left({P}_n,\omega \right)}{V_m\left({P}_1,\omega \right)}={\displaystyle \prod_{i=1}^{n-1}\frac{V_m\left({P}_{i+1},\omega \right)}{V_m\left({P}_1,\omega \right)}} $$

(13)

In a fourth step, one evaluates z _proximal. To do this, one must calculate the first impedance Z ¹_in which enters the recursive relation 12. This impedance corresponds to the impedance of the soma, which is given by:

$$ {Z}_{\mathrm{in}}^1={Z}_s+{Z}_{cs}, $$

(14)

where Z _s is the soma membrane impedance, and Z _cs is the cytoplasm impedance inside the soma. This relation is obtained under the hypothesis that the soma is isopotential, and the application of the generalized current conservation law implies ${i}^g=\frac{V_i-{V}_e}{Z_s+{Z}_{cs}}\approx \frac{V_m}{Z_s+{Z}_{cs}}$ where V _i and V _e are the electric potentials at both sides of the membrane, inside and outside, respectively relative to a reference located faraway.

The impedance of the bilipidic membrane is approximated by a parallel RC circuit where R = R _m is the resistance and ${\tau}_m={R}_m{C}_m$ is the membrane time constant. Thus, Z ¹_in can be written as:

$$ {Z}_{\mathrm{in}}^1={Z}_s+{Z}_{cs}=\frac{R_m}{1+i\omega {\tau}_m}+{Z}_{cs} $$

(15)

Finally, to evaluate z _distal, we use the “sealed end” boundary condition ${Z}_{\mathrm{in}}^1=\infty$. In this condition, we have ${Z}_{\mathrm{in}}^2=\frac{{\overline{z}}_{i_1}}{\kappa_{\lambda_1}} \coth \left({\kappa}_{\lambda_1}{l}_1\right)$ (see Eq. 10). In the case of a single dendritic branch, we can write:

$$ {Z}_{\mathrm{in}}^{\mathrm{distal}}=\frac{{\overline{z}}_i}{\kappa_{\lambda }} \coth \left({\kappa}_{\lambda }l\right), $$

(16)

where l is the total length of the cable.

Thus, this method allows one to calculate the value of the V _m at any point of the dendritic structure. The method is analytical besides approximating the structure by a finite set of continuous compartments, because each continuous compartment has an analytic solution. The connection between compartments is calculated by a finite iteration method. Thus, the V _m can be calculated at every coordinate with an excellent approximation, without the need of discretizing the dendritic tree into isopotential segments (like in simulators such as NEURON). It should also be faster and more accurate, because it does not rely on a specific integration method.