# What we talk about when we talk about capacitance measured with the voltage-clamp step method

## Abstract

Capacitance is a fundamental neuronal property. One common way to measure capacitance is to deliver a small voltage-clamp step that is long enough for the clamp current to come to steady state, and then to divide the integrated transient charge by the voltage-clamp step size. In an isopotential neuron, this method is known to measure the total cell capacitance. However, in a cell that is not isopotential, this measures only a fraction of the total capacitance. This has generally been thought of as measuring the capacitance of the “well-clamped” part of the membrane, but the exact meaning of this has been unclear. Here, we show that the capacitance measured in this way is a weighted sum of the total capacitance, where the weight for a given small patch of membrane is determined by the voltage deflection at that patch, as a fraction of the voltage-clamp step size. This quantifies precisely what it means to measure the capacitance of the “well-clamped” part of the neuron. Furthermore, it reveals that the voltage-clamp step method measures a well-defined quantity, one that may be more useful than the total cell capacitance for normalizing conductances measured in voltage-clamp in nonisopotential cells.

## Keywords

Capacitance Neurons Voltage clamp## 1 Introduction

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).

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 \(C_{\it vc}\) 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, \(C_{\it vc}\) 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, \(C_{\it vc}\) 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.

## 2 Results

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.

*u*(

*t*) is the unit step function. The voltage-clamp step capacitance is then defined by

*i*(

*t*) is the clamp current delivered as a function of time (Fig. 1). The integral above yields the charge delivered by the transient part of the current, which is then divided by the size of the voltage step to yield \(C_{\it vc}\). It should be noted that computing this quantity does not require the “peeling” of exponentials (Rall 1969; Holmes et al. 1992).

*k*indexes the cylinders, and

*x*represents the distance along a cylinder. The clamp-weighted capacitance is defined as

*k*th cable segment. That is, the clamp-weighted capacitance is calculated by dividing the cell surface area up into many small patches, and then adding up the capacitance of all these small patches, but with each patch weighted by a factor reflecting how much the voltage is deflected at that point (Fig. 2). In particular, this factor is equal to the square of the voltage deflection at each point, measured as a fraction of the voltage-clamp step size, \({\Delta{\it v}}\). The square was introduced because in the two-compartment case, a square factor arises in the expression for \(C_{\it vc}\) in terms of the two compartmental capacitances (Golowasch et al. 2009). And it turns out that \(C_{\it w}\) defined in this way can be proven equal to \(C_{\it vc}\) for an arbitrary geometry.

The main result of this paper is that \(C_{\it vc}=C_{\it w}\) for any cable tree.

### 2.1 Proof that \(C_{\it vc}=C_{\it w}\)

*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 \(C_{\it vc}\) in the limit as

*t*→ ∞, given by

*Y*(

*s*)

*V*(

*s*) for

*I*(

*s*), using the fact that

This completes the proof that \(C_{\it vc}=Y'(0)\).

*τ*

_{0}=

*r*

_{ m }

*c*

_{ m }is the fundamental time constant of the cable, \(G_{\infty}=1/\sqrt{r_a r_m}\), 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, \(\lambda=\sqrt{r_m/r_a}\) (Koch and Poggio 1985, Rule I). We can then determine \(C_{\it vc}\) for the finite cable by taking the derivative of

*Y*(

*s*) and evaluating at zero to find

*x*= 0) is given by

Comparing with Eq. (14), we see that \(C_{\it vc}=C_{\it w}\) for a finite cable with a sealed end.

*Y*(

*s*) is the admittance of the joined tree, and the

*Y*

_{ i }s are the admittances of the two subtrees. By taking derivatives on both sides of this equation and evaluating at zero, we find that

*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

*Y*

^{sub}, is easily shown to be

*s*, evaluating at

*s*= 0, and simplifying yields

*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

*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

*x*= ℓ), the boundary condition involves the steady-state current leaving the cable and entering the subtree, which is given by

*G*

^{sub}is the input conductance of the subtree.

*A*and

*B*yields

*G*in Eq. (29) above as

*v*(ℓ) derived by substituting into Eq. (38), and then use the relation

*c*

_{ m }=

*G*

_{ ∞ }

*τ*

_{0}/

*λ*to arrive at

This expression is very similar to Eq. (30), and by invoking the inductive hypothesis that \(C_{\it vc}^{\rm sub}=C_{\it w}^{\rm sub}\), we can immediately conclude that \(C_{\it w}=C_{\it vc}\) for the cascade. This completes the proof that \(C_{\it w}=C_{\it vc}\) for any arbitrary cable tree.

### 2.2 \(C_{\it vc}\) is related to the centroid of the impulse response and *R* _{ in }

*f*= 0. In response to a current pulse,

*i*(

*t*) =

*q*

_{0}

*δ*(

*t*), that delivers an amount of charge

*q*

_{0}, the voltage response of the neuron is

*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

*R*

_{ in }is the input resistance of the cell. It is easy to show that

*Z*

_{ f }(

*f*) = 1/

*Y*(

*j*2

*πf*).

*τ*

_{ in }, is given by

(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 \(C_{\it vc}\) is the capacitance given by *τ* _{ in }/*R* _{ in }, and suggests that an alternative name for \(C_{\it vc}\) might be the “input capacitance”, because it serves as a sort of “overall” or “summary” capacitance.

### 2.3 Relation of \(C_{\it vc}\) to equalizing time constants

*I*

_{0},

*τ*

_{ k }s are the equalizing time constants, and the

*R*

_{ k }s are resistances, each associated with a particular time constant. This implies that the impulse response of the neuron can be written as

*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

*z*(

*t*) as

This expresses \(C_{\it vc}\) 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.

## 3 Discussion

I have shown that \(C_{\it vc}=C_{\it w}\) 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 \(C_{\it vc}\) 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 \(C_{\it vc}\).

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, \(C_{\it vc}\) is given by Eq. (19). For a long cable, the factor in parenthesis is close to one, and so \(C_{\it vc} \approx c_m \lambda /2\). Since *c* _{ m } is a per-unit-length quantity, this implies that \(C_{\it vc}\) is equal to the total membrane capacitance of half a length constant of cable. This illustrates the way in which \(C_{\it vc}\) 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 \(C_{\it vc}\) 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.

*Y*

_{measured}′(

*s*) =

*Y*′

_{real}(

*s*). Because \(C_{\it vc}=Y'(0)\), we then have that \(C_{\it vc}^{\rm measured}=C_{\it vc}^{\rm real}\).

*R*

_{ s }is the series resistance. Thus a typical

*R*

_{ s }/

*R*

_{ in }ratio of ~0.01 (Gillis 2009) will lead to an error of ~2% in the measurement of \(C_{\it vc}\).

Another noteworthy point is that the proof that \(C_{\it vc}=C_{\it w}\) 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 \(C_{\it vc}\) 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 \(C_{\it vc}\)) as the subtree. This underscores the usefulness and naturalness of \(C_{\it vc}\) as a measure of neuronal capacitance.

## Notes

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