Capacitance is a fundamental neuronal property. In an isopotential neuron, the capacitance times the input resistance yields the cell’s time constant, which determines how quickly the neuron’s membrane potential responds to inputs (Rall ¹⁹⁵⁷). In nonisopotential cells, the specific capacitance and the specific membrane resistance play similar roles in determining the time constant at which the cell as a whole (i.e. averaged over space) responds to inputs (Rall ¹⁹⁶⁹). Additionally, a measurement of capacitance is often useful as a stand-in for a measurement of cell surface area, because the capacitance scales with the surface area (Koch ¹⁹⁹⁹, p. 8). Thus measurements of cellular capacitance have been used to normalize for variability in cell size, both in isopotential (Turrigiano et al. ¹⁹⁹⁵; Swensen and Bean ²⁰⁰⁵) and nonisopotential neurons (Schulz et al. ²⁰⁰⁶; Khorkova and Golowasch ²⁰⁰⁷).

Several methods are in widespread use for measuring cell capacitance. One common method is to voltage clamp the cell near its resting membrane potential, and to deliver a small (generally ≤10 mV) voltage-clamp step, long enough for the resulting current to come to steady state. One then subtracts off the steady-state current to determine the transient current, and this is then integrated to calculate the transient charge delivered during the voltage-clamp step. The transient charge is then divided by the size of the voltage-clamp step to yield an estimate of the cell capacitance (Hodgkin et al. ¹⁹⁵²; Golowasch et al. ²⁰⁰⁹, see also Gillis ²⁰⁰⁹, p. 160). I will call this the voltage-clamp step estimate of capacitance, or [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] (Fig. 1).

For an isopotential cell, the voltage-clamp step method yields an accurate estimate of the total cell capacitance. But for nonisopotential cells, it has not been clear exactly how [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] relates to the total capacitance. It has generally been considered to be a measurement of the capacitance of the part of the cell that is “well-clamped”, but this is vague. In previous work, I and my coauthors found (although it is a rather trivial result) that in a two-compartment model, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is equal to a weighted sum of the capacitance of the two compartments, where the weights are determined by the size of the steady-state voltage deflection in each compartment, measured as a fraction of the voltage-clamp command step (Golowasch et al. ²⁰⁰⁹). Here I generalize this result, showing that for a neuron with arbitrary geometry, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is equal to a weighted sum of the total cell capacitance, where the weights are determined by the size of the steady-state voltage deflection at each part of the neuron’s surface area, measured as a fraction of the voltage-clamp command step. This makes precise the idea that this method measures the “well-clamped” part of the capacitance.

I assume that the neuron being measured is effectively passive over the range of voltages used (see Section 3), and that it is a tree of cylinders, with each cylinder described by its capacitance per unit length, c_m; its membrane resistance for a unit length, r_m; its axial resistance per unit length, r_a; and its length, l. These parameters are assumed to be uniform in each cylinder, but may vary across cylinders. I also assume that the cylinder tree has a uniform resting membrane potential, which can be assumed to be zero without loss of generality.

The voltage-clamp step capacitance is measured by voltage clamping the cell at some convenient location (normally the soma) and delivering a step:

[Formula ID: Equ1]
1  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}(t) = {\Delta{\it v}} \, u(t) , $$\end{document}]

[Formula ID: Equ2]
2  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc} = \frac{1}{{\Delta{\it v}}} \int^{\infty}_{0} [i(t)-i(\infty)] \, dt , \label{C_vc_def} $$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}]

I now define a quantity which I call the “clamp-weighted capacitance”. The main result of this paper will be to show that the voltage-clamp step capacitance is equal to the clamp-weighted capacitance for any neuronal geometry. When the voltage-clamp step has been on long enough for the voltage at all points in the tree to come to steady-state, each point will have a voltage, denoted [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\it v}_{ss}^{k}(x)$\end{document}] , where k indexes the cylinders, and x represents the distance along a cylinder. The clamp-weighted capacitance is defined as

[Formula ID: Equ3]
3  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it w}=\sum\limits_{k} \int_{x} c_m^k \left[ \frac{{\it v}_{ss}^{k}(x)}{{\Delta{\it v}}} \right]^2 dx , \label{C_w_def} $$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_m^k$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Delta{\it v}}$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}]

The main result of this paper is that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for any cable tree.

Proof that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}]

The proof proceeds by induction on cable trees. The smallest possible cable tree is a single cable segment (Fig. 3(a)), and any tree can be constructed recursively by the two operations of (1) joining two subtrees at their roots (Fig. 3(b)), and (2) cascading a cable segment with a subtree (Fig. 3(c)). We therefore first prove that the theorem holds for a single cable segment, and then prove that (1) if the theorem holds for two subtrees, it holds for the tree formed by joining them at a common point, and (2) if the theorem holds for a subtree, it holds for the tree formed by cascading a single cable segment with the subtree.

Before presenting the main part of the proof, it will be useful to establish another identity involving [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] , namely

[Formula ID: Equ4]
4  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc} = Y'(0) , \label{C_vc_and_Y} $$\end{document}]

where Y(s) is the complex admittance of the cable tree, with s the complex frequency, and Y′(s) is its first derivative. To show this, we define a function that approaches [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] in the limit as t → ∞, given by

[Formula ID: Equ5]
5  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc}(t) = \frac{1}{{\Delta{\it v}}} \int^{t}_{0} \left [i(t')-i(\infty) \right] \, dt' . $$\end{document}]

The Laplace transform of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}(t)$\end{document}] is then given by

[Formula ID: Equ6]
6  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{C}_{\it vc}(s) = \frac{1}{{\Delta{\it v}}} \frac{1}{s} \left[ I(s) - \frac{i(\infty)}{s} \right] , $$\end{document}]

where I have used the fact that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L}\left\{ \int^{t}_{0} x(t') dt' \right\} = s^{-1} X(s)$\end{document}] (Siebert ¹⁹⁸⁶, p. 62). We can then show that

[Formula ID: Equ7]
7  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{C}_{\it vc}(s) = s^{-2} \left[ Y(s) - Y(0) \right] , $$\end{document}]

by substituting Y(s) V(s) for I(s), using the fact that

[Formula ID: Equ8]
8  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i(\infty)=\lim\limits_{s\rightarrow 0} s I(s) , $$\end{document}]

and doing some algebra. Expressing [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] in terms of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}(t)$\end{document}] , we then obtain

[Formula ID: Equ9]
9  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\it vc} = C_{\it vc}(\infty) $$\end{document}]

[Formula ID: Equ10]
10  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \lim\limits_{s \rightarrow 0} s \tilde{C}_{\it vc}(s) $$\end{document}]

[Formula ID: Equ11]
11  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \lim\limits_{s \rightarrow 0} s^{-1} \left[ Y(s) - Y(0) \right] $$\end{document}]

[Formula ID: Equ12]
12  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = Y'(0) . $$\end{document}]

This completes the proof that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=Y'(0)$\end{document}] .

I now show that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for a single finite cable segment. The input admittance of a finite cable segment with sealed end is given by

[Formula ID: Equ13]
13  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y(s) = G_{\infty} \sqrt{1+\tau_0 s} \, \tanh\left[L \sqrt{1+\tau_0 s} \right] , $$\end{document}]

where τ₀ = r_mc_m is the fundamental time constant of the cable, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{\infty}=1/\sqrt{r_a r_m}$\end{document}] , the input conductance of the cable if it were infinitely long, and L = ℓ/λ, the length of the cable in units of the length constant, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda=\sqrt{r_m/r_a}$\end{document}] (Koch and Poggio ¹⁹⁸⁵, Rule I). We can then determine [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] for the finite cable by taking the derivative of Y(s) and evaluating at zero to find

[Formula ID: Equ14]
14  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc}=Y'(0)=\frac{1}{2} \tau_0 G_{\infty} \left( \tanh L + L {\mathrm{sech}}^2 L \right) . \label{C_vc_cable} $$\end{document}]

The steady-state voltage distribution for a finite cable with a sealed end subjected to a voltage clamp at the near end (where x = 0) is given by

[Formula ID: Equ15]
15  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}_{ss}(x)={\Delta{\it v}} \frac{\cosh\left[(x-\ell)/\lambda\right]}{\cosh L} . \label{v_ss_cable} $$\end{document}]

This result is given in Koch (¹⁹⁹⁹, p. 34). The clamp-weighted capacitance is then given by

[Formula ID: Equ16]
16  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\it w} = \int_{0}^{\ell} c_m \left[ \frac{{\it v}_{ss}(x)}{{\Delta{\it v}}} \right]^2 dx \\ $$\end{document}]

[Formula ID: Equ17]
17  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{c_m}{\cosh^2 L} \int_{0}^{\ell} \cosh^2\left[(x-\ell)/\lambda\right] \, dx \\ $$\end{document}]

[Formula ID: Equ18]
18  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{c_m}{4 \cosh^2 L} \left( 2 \ell + \lambda \sinh 2L \right) \\ $$\end{document}]

[Formula ID: Equ19]
19  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{1}{2} c_m \lambda \left( \tanh L + L {\mathrm{sech}}^2 L \right) \label{C_w_cable_lambda} \\ $$\end{document}]

[Formula ID: Equ20]
20  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{1}{2} \tau_0 G_{\infty} \left( \tanh L + L {\mathrm{sech}}^2 L \right) . $$\end{document}]

Comparing with Eq. (14), we see that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for a finite cable with a sealed end.

I now show that if [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for two subtrees, then it holds for the tree formed by joining the two trees at their roots. If [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^1$\end{document}] is the voltage-clamp capacitance of the first subtree, and [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^2$\end{document}] that of the second, it is easy to see that

[Formula ID: Equ21]
21  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc} = C_{\it vc}^1 + C_{\it vc}^2 , \label{C_vc_join} $$\end{document}]

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is the voltage-clamp capacitance of the joined tree. This is because admittances add in parallel, so

[Formula ID: Equ22]
22  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y(s) = Y_1(s) + Y_2(s) , $$\end{document}]

where Y(s) is the admittance of the joined tree, and the Y_is are the admittances of the two subtrees. By taking derivatives on both sides of this equation and evaluating at zero, we find that

[Formula ID: Equ23]
23  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y'(0) = Y_1'(0) + Y_2'(0) , $$\end{document}]

and so Eq. (21) follows from Eq. (4).

If we voltage-clamp the joined tree at the join point, then the steady-state distribution of voltage in each subtree will be the same as if the other subtree were absent. Thus

[Formula ID: Equ24]
24  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it w} = C_{\it w}^1 + C_{\it w}^2 , $$\end{document}]

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}] is the clamp-weighted capacitance of the joined tree. Thus if [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^1=C_{\it w}^1$\end{document}] and [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^2=C_{\it w}^2$\end{document}] , it follows that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for the joined tree.

I now prove that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for a cable segment cascaded with a subtree, given that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^{\rm sub}=C_{\it w}^{\rm sub}$\end{document}] , where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^{\rm sub}$\end{document}] and [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}^{\rm sub}$\end{document}] are the voltage-clamp capacitance and clamp-weighted capacitance of the subtree, respectively. The impedance of the cascade, Y(s), is determined by considering the cable segment as a two-port cascaded with the admittance of the subtree, Y^sub(s) (Siebert ¹⁹⁸⁶, p. 84). In particular, the ABCD representation of the cable considered as a two-port is

[Formula ID: Equ25]
25  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left[\begin{array}{cc} A & B \\ C & D \end{array}\right] = \left[\begin{array}{cc} \cosh(Lq) & -{G_{\infty}}^{-1} q^{-1} \sinh(Lq) \\ G_{\infty} q \sinh(Lq) & -\cosh(Lq) \end{array}\right] , $$\end{document}]

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q=\sqrt{1+\tau_0 s}$\end{document}] [this follows from the definition of the ABCD representation (Siebert ¹⁹⁸⁶, p. 97) and from a simple application of Rule I in Koch and Poggio (¹⁹⁸⁵)]. The input admittance of a two-port cascaded with a known admittance, Y^sub, is easily shown to be

[Formula ID: Equ26]
26  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y = -\frac{C+Y^{\rm sub} A}{D+Y^{\rm sub} B}. $$\end{document}]

Thus the admittance of a cable segment cascaded with a subtree is given by

[Formula ID: Equ27]
27  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y = \frac{{G_{\infty}} q \sinh(Lq) + Y^{\rm sub} \cosh(Lq)} {\cosh(Lq) + Y^{\rm sub} {G_{\infty}}^{-1} q^{-1} \sinh(Lq) }. \label{Y_cascade} $$\end{document}]

Taking the derivative of this expression with respect to s, evaluating at s = 0, and simplifying yields

[Formula ID: Equ28]
28  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} C_{\it vc} & = & {{\frac{1}{2}}} ({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^{-2} \\ & & \cdot\left[ 2 C_{\it vc}^{\rm sub} {G_{\infty}}^2 {G_{\infty}}^3 L \tau_0 + \right. \\ & & + \left. {G_{\infty}}^3 \tau_0 \sinh L \cosh L + {G_{\infty}}^2 {G_{\rm sub}} \tau_0 \sinh^2 L \right. \\ & & + \left. {G_{\infty}}^2 {G_{\rm sub}} \tau_0 \cosh^2 L - {G_{\infty}}^2 {G_{\rm sub}} \tau_0 - {G_{\infty}} {G_{\rm sub}}^2 L \tau_0 \right. \\ & & + \left. {G_{\infty}} {G_{\rm sub}}^2 \tau_0 \sinh L \cosh L \right] . \label{C_vc_cascade_proto} \end{array} $$\end{document}]

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^{\rm sub}$\end{document}] is the voltage-clamp capacitance of the subtree, and G^sub is its input conductance. This expression can be simplified somewhat by putting it in terms of the input conductance of the cascade, G. We can derive an expression for G by evaluating Eq. (27) for s = 0, yielding

[Formula ID: Equ29]
29  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G = G_{\infty} \frac{G^{\rm sub} \cosh L + G_{\infty} \sinh L} {G_{\infty} \cosh L + G^{\rm sub} \sinh L} . \label{G} $$\end{document}]

Using this expression, and after some algebra, we can rewrite Eq. (28) as

[Formula ID: Equ30]
30  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} C_{\it vc} & = & {{\frac{1}{2}}} G \tau_0 + {{\frac{1}{2}}} {G_{\infty}} \tau_0 \frac{({G_{\infty}}^2-{G_{\rm sub}}^2) L - {G_{\infty}} {G_{\rm sub}}} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} \\ & & +\, C_{\it vc}^{\rm sub} \frac{{G_{\infty}}^2} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} . \label{C_vc_cascade} \end{array} $$\end{document}]

We would now like to show that the right-hand side of Eq. (30) is equal to [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}] . To calculate [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}] for the cascade, we first solve for the steady-state distribution of voltage along the cable segment. The general expression for this is

[Formula ID: Equ31]
31  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}_{ss}(x) = A \exp(+x/\lambda) + B \exp(-x/\lambda) , \label{v_ss_general} $$\end{document}]

where A and B are determined by the boundary conditions (Johnston and Wu ¹⁹⁹⁵, p. 75). The boundary condition at the near end of the cable (x = 0) is determined in part by the steady-state current being injected into the cable. This is given by

[Formula ID: Equ32]
32  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i_{ss}=G {\Delta{\it v}} . $$\end{document}]

At the far end of the cable (x = ℓ), the boundary condition involves the steady-state current leaving the cable and entering the subtree, which is given by

[Formula ID: Equ33]
33  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i_{ss}^{\rm sub}=G^{\rm sub} v(\ell) , $$\end{document}]

where G^sub is the input conductance of the subtree.

These relations yield the following boundary conditions:

[Formula ID: Equ34]
34  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left. \frac{\partial {\it v}}{\partial x} \right|_{x=0 } = -r_{a} i_{ss} \\ $$\end{document}]

[Formula ID: Equ35]
35  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left. \frac{\partial {\it v}}{\partial x} \right|_{x=\ell} = -r_{a} i_{ss}^{\rm sub} . $$\end{document}]

Plugging Eq. (31) into these equations, and solving for A and B yields

[Formula ID: Equ36]
36  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = {{\frac{1}{2}}} {\Delta{\it v}} \, e^{-L} \frac{{G_{\infty}}-{G_{\rm sub}}}{{G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L} \label{A} \\ $$\end{document}]

[Formula ID: Equ37]
37  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B = {{\frac{1}{2}}} {\Delta{\it v}} \, e^{+L} \frac{{G_{\infty}}+{G_{\rm sub}}}{{G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L} \label{B} $$\end{document}]

Plugging these expressions into Eq. (31) and invoking the hyperbolic trigonometric identities, we can then write an expression for [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\it v}_{ss}(x)$\end{document}] involving only known quantities:

[Formula ID: Equ38]
38  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}_{ss}(x) = {\Delta{\it v}} \frac{{G_{\infty}} \cosh(L-x/\lambda) + {G_{\rm sub}} \sinh(L-x/\lambda)} {{G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L} \label{v_ss_final}\\ $$\end{document}]

We can now determine an expression for [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}] (for the cascade) based on this solution and upon Eq. (3):

[Formula ID: Equ39]
39  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\it w} = \sum\limits_{k} \int_{x} c_m^k \left[ \frac{{\it v}_{ss}^{k}(x)}{{\Delta{\it v}}} \right]^2 dx \\ $$\end{document}]

[Formula ID: Equ40]
40  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \int_{x} c_m \left[ \frac{{\it v}_{ss}(x)}{{\Delta{\it v}}} \right]^2 dx + \sum\limits_{k \in \mathrm{subtree}} \int_{x} c_m^k \left[ \frac{{\it v}_{ss}^{k}(x)}{{\Delta{\it v}}} \right]^2 dx \\ $$\end{document}]

[Formula ID: Equ41]
41  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{c_m}{{\Delta{\it v}}^2} \int_{x} {\it v}_{ss}^2(x) dx + \frac{1}{{\Delta{\it v}}^2} \sum\limits_{k \in \mathrm{subtree}} \int_{x} c_m^k \left[ {\it v}_{ss}^{k}(x) \right]^2 dx \\ $$\end{document}]

[Formula ID: Equ42]
42  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{c_m}{{\Delta{\it v}}^2} \int_{x} {\it v}_{ss}^2(x) dx + \frac{v^2(\ell)}{{\Delta{\it v}}^2} \sum\limits_{k \in \mathrm{subtree}} \int_{x} c_m^k \left[ \frac{{\it v}_{ss}^{k}(x)}{v(\ell)} \right]^2 dx \\ $$\end{document}]

[Formula ID: Equ43]
43  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{c_m}{{\Delta{\it v}}^2} \int_{x} {\it v}_{ss}^2(x) dx + \frac{v^2(\ell)}{{\Delta{\it v}}^2} C_{\it w}^{\rm sub} , \label{C_w_proto} $$\end{document}]

where the last step follows from the definition of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}$\end{document}] as applied to the subtree. The integral in Eq. (43) can be evaluated by substituting for [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\it v}_{ss}(x)$\end{document}] from Eq. (38) to find that

[Formula ID: Equ44]
44  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} \int_{x} {\it v}_{ss}^2(x) dx &=& {{\frac{1}{2}}} \lambda {\Delta{\it v}}^2 \left[ \frac{({G_{\infty}}^2-{G_{\rm sub}}^2) L - {G_{\infty}} {G_{\rm sub}}} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} \right. \\ &&{\kern25pt} \left. + \frac{{G_{\rm sub}} \cosh L + {G_{\infty}} \sinh L} {{G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L} \right] . \end{array} $$\end{document}]

This can be written more compactly using the expression for G in Eq. (29) above as

[Formula ID: Equ45]
45  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} \lefteqn{\int_{x} {\it v}_{ss}^2(x) dx } \\ & & ={{\frac{1}{2}}} \lambda {\Delta{\it v}}^2 \left[ \frac{({G_{\infty}}^2-{G_{\rm sub}}^2) L - {G_{\infty}} {G_{\rm sub}}} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} + \frac{G}{{G_{\infty}}} \right] . \label{int_v_ss} \end{array} $$\end{document}]

We now substitute this expression into Eq. (43) along with the expression for v(ℓ) derived by substituting into Eq. (38), and then use the relation c_m = G_∞τ₀ / λ to arrive at

[Formula ID: Equ46]
46  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} C_{\it w} & = & {{\frac{1}{2}}} G \tau_0 + {{\frac{1}{2}}} {G_{\infty}} \tau_0 \frac{({G_{\infty}}^2-{G_{\rm sub}}^2) L - {G_{\infty}} {G_{\rm sub}}} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} \\ & &+\, C_{\it w}^{\rm sub} \frac{{G_{\infty}}^2} {({G_{\infty}} \cosh L + {G_{\rm sub}} \sinh L)^2} . \end{array} $$\end{document}]

This expression is very similar to Eq. (30), and by invoking the inductive hypothesis that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^{\rm sub}=C_{\it w}^{\rm sub}$\end{document}] , we can immediately conclude that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}=C_{\it vc}$\end{document}] for the cascade. This completes the proof that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it w}=C_{\it vc}$\end{document}] for any arbitrary cable tree.

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is related to the centroid of the impulse response and R_in

There is an interesting relationship between [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] and the impulse response of the neuron in current clamp. This arises because of the close correspondence between the moments of a function and the derivatives of its Fourier transform at f = 0. In response to a current pulse, i(t) = q₀δ(t), that delivers an amount of charge q₀, the voltage response of the neuron is

[Formula ID: Equ47]
47  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}(t) = q_0 \, z(t) , $$\end{document}]

where z(t) is the impulse response of the system (also known as the Green’s function). The impulse response is the inverse Fourier transform of the impedance as a function of frequency, which I denote by Z_f(f). (The subscript f is a reminder that Z_f(·) is a function of the frequency f, not the complex frequency s.) An elementary property of the Fourier transform is that

[Formula ID: Equ48]
48  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int_{-\infty}^{+\infty} z(t) dt = Z_{f}(0) = R_{in} , $$\end{document}]

where R_in is the input resistance of the cell. It is easy to show that

[Formula ID: Equ49]
49  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{-\infty}^{+\infty} t z(t) dt = \left. \frac{1}{-j 2 \pi} \frac{\partial}{\partial f}Z_{f} \right|_{f=0}. \\ $$\end{document}]

[Formula ID: Equ50]
50  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = R_{in}^2 C_{\it vc} , $$\end{document}]

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$j=\sqrt{-1}$\end{document}] , and where the last follows from the fact that Z_f(f) = 1/Y(j 2 πf).

The centroid of the impulse response [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\it v}(t)$\end{document}] , which I will call τ_in, is given by

[Formula ID: Equ51]
51  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{in} = \frac{\int_{-\infty}^{+\infty} t {\it v}(t) dt} {\int_{-\infty}^{+\infty} {\it v}(t) dt} \\ $$\end{document}]

[Formula ID: Equ52]
52  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \frac{q_0 R_{in}^2 C_{\it vc}}{q_0 R_{in}} \\ $$\end{document}]

[Formula ID: Equ53]
53  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = R_{in} C_{\it vc} . \label{tau_in} $$\end{document}]

(It should be noted that τ_in is just the “input delay” as defined by Agmon Snir and Segev (¹⁹⁹³). See Section 3.)

The centroid of the impulse response is a natural way of describing the overall time scale of the neuron’s response to current input, just as the input resistance is a natural way of describing the overall magnitude of the neuron’s response to current input. It is therefore very interesting that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is the capacitance given by τ_in/R_in, and suggests that an alternative name for [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] might be the “input capacitance”, because it serves as a sort of “overall” or “summary” capacitance.

Relation of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] to equalizing time constants

Rall (¹⁹⁶⁹) and Major et al. (¹⁹⁹³) showed that the response of a passive nonisopotential neuron to a current step of size I₀,

[Formula ID: Equ54]
54  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i(t)=I_0 \, u(t) , $$\end{document}]

can be written as a sum of exponential charging curves, namely

[Formula ID: Equ55]
55  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it v}(t) = I_0 \, u(t) \sum\limits_{k=0}^{\infty} R_{k} [1-\exp(-t/\tau_{k})] , $$\end{document}]

where the τ_ks are the equalizing time constants, and the R_ks are resistances, each associated with a particular time constant. This implies that the impulse response of the neuron can be written as

[Formula ID: Equ56]
56  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ z(t) = u(t) \sum\limits_{k=0}^{\infty} C_{k}^{-1} \exp(-t/\tau_{k}) , $$\end{document}]

where C_k = τ_k/R_k is a capacitance associated each equalizing time constant. Using this form for z(t), we can write R_in as

[Formula ID: Equ57]
57  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{in} = \int_{-\infty}^{+\infty} z(t) dt \\ $$\end{document}]

[Formula ID: Equ58]
58  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \sum\limits_{k=0}^{\infty} R_{k} . \label{R_in_as_sum} $$\end{document}]

In a similar fashion, we can write the first moment of z(t) as

[Formula ID: Equ59]
59  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{in}^2 C_{\it vc} = \int_{-\infty}^{+\infty} t z(t) dt \\ $$\end{document}]

[Formula ID: Equ60]
60  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \sum\limits_{k=0}^{\infty} R_{k}^2 C_{k} . $$\end{document}]

Combining these, we have

[Formula ID: Equ61]
61  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\it vc} = R_{in}^{-2} \sum\limits_{k=0}^{\infty} R_{k}^2 C_{k} \\ $$\end{document}]

[Formula ID: Equ62]
62  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \sum\limits_{k=0}^{\infty} \frac{R_{k}^2}{R_{in}^2} C_{k} \\ $$\end{document}]

[Formula ID: Equ63]
63  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = \sum\limits_{k=0}^{\infty} \frac{R_{k}^2}{(\sum_{m} R_{m})^2} C_{k} . \label{C_vc_as_sum} $$\end{document}]

This expresses [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] as a weighted sum of the capacitances associated with each equalizing time constant, similar to the way Eq. (58) expresses R_in as a sum of the resistances associated with each equalizing time constant.

I have shown that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] for any arbitrary tree of passive cables. This explains exactly what is being measured by the voltage-clamp step method: a weighted sum of the total cell capacitance, where each small patch of capacitance is weighted by the square of the fraction of the voltage-clamp step “felt” by that patch. Thus [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] includes the capacitance of the well-clamped part of the cell, but excludes the poorly clamped part, with “partly-clamped” parts of the cell being counted at a rather severe discount (because of the square). For instance, a part of the cell that only feels half of the voltage-clamp step only has one-fourth of its capacitance included in [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] .

As a concrete example, consider a neuron that is well-approximated by a single cable segment that is many length constants long. In this case, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is given by Eq. (19). For a long cable, the factor in parenthesis is close to one, and so [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc} \approx c_m \lambda /2$\end{document}] . Since c_m is a per-unit-length quantity, this implies that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is equal to the total membrane capacitance of half a length constant of cable. This illustrates the way in which [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] only counts the capacitance of membrane that is electrotonically close to the point of voltage-clamp, i.e. the well-clamped part of the cell.

Whether this discounting of poorly-clamped parts of the cell is a good thing or a bad thing depends on the circumstances. Certainly, if one wants to measure the total cell capacitance, it is a bad thing, and these results are consistent with the fact that the voltage-clamp step method cannot be used to measure total cell capacitance in a nonisopotential cell. However, if one is using the capacitance to normalize the magnitudes of currents measured in voltage clamp, one may want to measure only the capacitance of the well-clamped part of the membrane. This is a very common reason to measure neuronal capacitance, and in this case [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] may be preferable to a measurement of the total cell capacitance.

All measurements of neuronal capacitance (not just the voltage-clamp step method) are based on the assumption that the neuronal response is passive. Of course, real neurons are not generally passive: they contain voltage-gated conductances, often many of them. In order to make accurate capacitance measurements, the active currents evoked by the voltage-clamp step must be small compared to the passive currents so evoked. Thus the voltage-clamp steps used to measure capacitance are best performed at potentials far from those at which voltage-gated channels are appreciably activated. If there is no window of membrane potential free from active conductances, such as might be the case if h or Kir currents (Hille ¹⁹⁹²) are present, pharmacological blockers should be used to block any currents that might disrupt the passive response of the neuron. These techniques have been used successfully in the past to allow for measurement of capacitance and other passive properties (Hodgkin et al. ¹⁹⁵²; Rall ¹⁹⁶⁴; Major et al. ¹⁹⁹⁴; Roth and Häusser ²⁰⁰¹; Gillis ²⁰⁰⁹). Of course, if there are appreciable unblocked active currents at either the holding or the test potential, these currents will corrupt the measurement, presumably to an extent that depends on the size of the evoked active currents relative to the evoked passive current.

When recording with sharp electrodes, as in two-electrode voltage clamp, impalement generally causes a non-negligible shunt conductance to be introduced. One interesting feature of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is that it is not affected by an electrode-induced shunt conductance. This is because the net somatic conductance only contributes to the steady-state part of the voltage-clamp current, not the transient part (Eq. (2)). Another way of seeing this is to note that

[Formula ID: Equ64]
64  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Y_{\rm measured}(s)=G_{\rm shunt}+Y_{\rm real}(s) , $$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=Y'(0)$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}^{\rm measured}=C_{\it vc}^{\rm real}$\end{document}]

Whole-cell patch clamp is generally thought to introduce negligible shunt conductance, but does introduce a series resistance between the measured somatic voltage and the true somatic voltage (Roth and Häusser ²⁰⁰¹; Jackson ¹⁹⁹²). However, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is not strongly affected by a series resistance, even if it is uncompensated. It can easily be shown that in this case

[Formula ID: Equ65]
65  [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{\it vc}^{\rm measured}= \frac{1}{\left(1+R_s/R_{in}^{r\rm eal}\right)^2} C_{\it vc}^{\rm real}, $$\end{document}]

[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}]

Another noteworthy point is that the proof that [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}=C_{\it w}$\end{document}] does not require that the neuron’s passive properties are the same in all cable segments, nor does it require that the cable segments have cylindrical cross-sections.

The voltage-clamp step method is not the only method of measuring capacitance, although it is probably the most commonly used one (Gillis ²⁰⁰⁹; Golowasch et al. ²⁰⁰⁹). At least one popular data acquisition program (Clampex 10, Molecular Devices) has a form of this method built-in. Other commonly-used methods differ in whether they use voltage- or current-clamp, what sort of input waveform they use, and how they use the resulting output to form a capacitance estimate (Gillis ²⁰⁰⁹; Golowasch et al. ²⁰⁰⁹). Of course, these choices have consequences with regard to what part of the cell capacitance is being measured (Golowasch et al. ²⁰⁰⁹). (And for some of these methods, it is not clear what is being measured in a nonisopotential cell.) Additionally, the methods have different technical advantages and disadvantages in different settings (Gillis ²⁰⁰⁹; Golowasch et al. ²⁰⁰⁹).

As mentioned above, τ_in is just the “input delay” of Agmon Snir and Segev (¹⁹⁹³). Among many elegant results describing signal propagation in passive dendrites in terms of temporal centroids, they showed that the centroid of the voltage response to any current input follows the centroid of the current input by the input delay (i.e. τ_in). Thus [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] is just the input delay over the input resistance. They also showed that when computing the delays between different points of a neuron, one could lump a subtree into a single compartment having the same input resistance and input delay (and thus the same [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] ) as the subtree. This underscores the usefulness and naturalness of [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\it vc}$\end{document}] as a measure of neuronal capacitance.

This work was supported by NIH grants NS17813 and MH46742, and by James S. McDonnell Foundation Grant 220020065. These grants were awarded to Eve Marder, who I thank for her generous support during this work.

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