Frequency-dependent responses of neuronal models to oscillatory inputs in current versus voltage clamp

Abstract

Action potential generation in neurons depends on a membrane potential threshold and therefore on how subthreshold inputs influence this voltage. In oscillatory networks, for example, many neuron types have been shown to produce membrane potential ($V_\mathrm{m}$) resonance: a maximum subthreshold response to oscillatory inputs at a nonzero frequency. Resonance is usually measured by recording $V_\mathrm{m}$ in response to a sinusoidal current ($I_\mathrm{app}$), applied at different frequencies (f), an experimental setting known as current clamp (I-clamp). Several recent studies, however, use the voltage clamp (V-clamp) method to control $V_\mathrm{m}$ with a sinusoidal input at different frequencies [$V_\mathrm{app}(f)$] and measure the total membrane current ($I_\mathrm{m}$). The two methods obey systems of differential equations of different dimensionality, and while I-clamp provides a measure of electrical impedance [$Z(f) = V_\mathrm{m}(f) / I_\mathrm{app}(f)$], V-clamp measures admittance [$Y(f) = I_\mathrm{m}(f) / V_\mathrm{app}(f)$]. We analyze the relationship between these two measurement techniques. We show that, despite different dimensionality, in linear systems the two measures are equivalent: $Z = Y^{-1}$. However, nonlinear model neurons produce different values for Z and $Y^{-1}$. In particular, nonlinearities in the voltage equation produce a much larger difference between these two quantities than those in equations of recovery variables that describe activation and inactivation kinetics. Neurons are inherently nonlinear, and notably, with ionic currents that amplify resonance, the voltage clamp technique severely underestimates the current clamp response. We demonstrate this difference experimentally using the PD neurons in the crab stomatogastric ganglion. These findings are instructive for researchers who explore cellular mechanisms of neuronal oscillations.

Appendices

Equivalence between the I-clamp impedance and the V-clamp admittance for linear systems

The I-clamp impedance and the V-clamp admittance are equivalent if the corresponding amplitudes are the reciprocal of one another and the corresponding phases have the same absolute value but different sign. Using the notation introduced in this paper, $Z(\omega ) = Y^{-1}(\omega )$ and $\Phi (\omega ) = -\Psi (\omega )$.

We illustrate this for the following linear system,

$$\begin{aligned} \left\{ \begin{array}{ll} V' = a V + b w_1 + \gamma w_2 + I,\\ w_1' = c V + d w_1, \\ w_2' = \alpha V + \beta w_2, \end{array} \right. \end{aligned}$$

(40)

where the “prime” sign represents the derivative with respect to time t and a, b, c, d, $\alpha$, $\beta$ and $\gamma$ are constants satisfying the condition that the eigenvalues of the characteristic polynomial for (40) with a constant value of I have non-positive real part. System (40) has the structure of the linearized conductance-based models (Richardson et al. 2003; Rotstein 2017c) for the voltage (V) and two gating variables ($w_1$ and $w_2$).

We assume

$$\begin{aligned} I = A_I(\omega ) \mathrm{e}^{i \omega t} \quad \hbox {and}\quad V = A_V(\omega ) \mathrm{e}^{i \omega t}, \end{aligned}$$

(41)

where $\omega$ is the frequency (a linear function of the input frequency f). In I-clamp $A_I = A_\mathrm{in}$ and $A_V = A_\mathrm{out}$, while in V-clamp $A_I = A_\mathrm{out}$ and $A_V = A_\mathrm{in}$. Typically, $A_\mathrm{in}$ is independent of $\omega$, but this need not be the case. Note that Eqs. (40) are forced 3D and 2D linear systems in I- and V-clamp, respectively.

The I-clamp impedance and the V-clamp admittance are defined as

$$\begin{aligned} \mathbf{Z}(\omega ) = \frac{A_V(\omega )}{A_I} \quad \hbox {and}\quad \mathbf{Y}(\omega ) = \frac{A_I(\omega )}{A_V}, \end{aligned}$$

(42)

respectively, where $\mathbf{Z}$ and $\mathbf{Y}$ are complex quantities with amplitude (Z and Y, respectively) and phase ($\Phi$ and $\Psi$, respectively).

Alternatively, in I-clamp

$$\begin{aligned} I = A_I \sin (\omega t) \quad \hbox {and} \quad V = A_V \sin (\omega t - \Phi ), \end{aligned}$$

(43)

and in V-clamp

$$\begin{aligned} V= A_V \sin (\omega t) \quad \hbox {and} \quad I = A_I \sin (\omega t - \Psi ), \end{aligned}$$

(44)

where $A_I$ and $A_{V}$ are real quantities. According to this formulation, $A_I = A_\mathrm{in}$ and $A_V = Z(\omega ) = | \mathbf{Z}(\omega )| \$ in I-clamp and $A_I = Y(\omega ) = | \mathbf{Y}(\omega )|$ and $A_V = A_\mathrm{in}$ in V-clamp.

The particular solutions (neglecting transients) of the second and third equations in (40) are given, respectively, by

$$\begin{aligned} w_1 = - \frac{c}{d-i \omega } V \quad \hbox {and} \quad w_2 = -\frac{\alpha }{\beta -i \omega } V. \end{aligned}$$

(45)

Substituting (45) into the first equation in (40) and rearranging terms yield

$$\begin{aligned} V' - a V - I - F(\omega ) V = 0, \end{aligned}$$

(46)

where

$$\begin{aligned} F(\omega ) = - \frac{(b c \beta + \alpha \gamma d) - i \omega (b c + \alpha \gamma )}{(d b - \omega ^2) - i \omega (d + \beta )}. \end{aligned}$$

(47)

Substituting (41) into (46) gives the condition

$$\begin{aligned} i \omega - a - F(\omega ) = \frac{A_I(\omega )}{A_V(\omega )}. \end{aligned}$$

(48)

Therefore,

$$\begin{aligned} \frac{1}{\mathbf{Z}(\omega )} = \mathbf{Y}(\omega ) \end{aligned}$$

(49)

and

$$\begin{aligned} Z(\omega ) = Y(\omega )^{-1} \quad \hbox {and} \quad \Phi (\omega ) = -\Psi (\omega ). \end{aligned}$$

(50)

Solutions to oscillatory forced linear ODEs

1.1 A system of two forced ODEs

Any system of ODEs of the form

$$\begin{aligned} \left\{ \begin{array}{ll} v' = a v + b w + F(t),\\ w' = c v + d w + G(t), \end{array} \right. \end{aligned}$$

(51)

can be written as

$$\begin{aligned} v'' - \eta v' + \Delta v = -d F(t) + b G(t) + F'(t), \end{aligned}$$

(52)

where

$$\begin{aligned} \eta = a + d \quad \hbox {and} \quad \Delta = a d - b c. \end{aligned}$$

(53)

If F(t) and G(t) are linear combinations of sinusoidal and cosinusoidal function of the same frequency ($k \omega$), there is the right-hand side of Eq. (52). Therefore, it suffices to solve

$$\begin{aligned} v'' - \eta v' + \Delta v = \alpha \sin (k \omega t) + \beta \cos (k \omega t). \end{aligned}$$

(54)

The solution of Eq. (54) is given by

$$\begin{aligned} v(t) = A \sin (k \omega t) + B \cos (k \omega t), \end{aligned}$$

(55)

where

$$\begin{aligned}&A = \frac{\alpha (\Delta - k^2 \omega ^2) - \beta \eta k \omega }{W(k \omega )} \quad \hbox {and} \quad \nonumber \\&B = \frac{\alpha \eta k \omega + \beta (\Delta - k^2 \omega ^2)}{W(k \omega )}, \end{aligned}$$

(56)

with

$$\begin{aligned} W(k \omega ) = (\Delta - k^2 \omega ^2)^2 + \eta ^2 k^2 \omega ^2. \end{aligned}$$

(57)

This solution satisfies

$$\begin{aligned} A^2 + B^2 = \frac{\alpha ^2 + \beta ^2}{W(k \omega )}. \end{aligned}$$

(58)

1.2 A single forced ODE

The solution to any ODE of the form

$$\begin{aligned} w' - d w = \alpha \sin (k \omega t) + \beta \cos (k \omega t) \end{aligned}$$

(59)

is given by

$$\begin{aligned} w(t) = C \sin (k \omega t) + D \cos (k \omega t), \end{aligned}$$

(60)

where

$$\begin{aligned} C = \frac{-\alpha d + \beta k \omega }{W_0(k \omega )} \quad \hbox {and} \quad D = -\frac{\alpha k \omega + \beta d}{W_0(k \omega )}, \end{aligned}$$

(61)

with

$$\begin{aligned} W_0(k \omega ) = d^2 + k^2 \omega ^2. \end{aligned}$$

(62)

This solution satisfies

$$\begin{aligned} C^2 + D^2 = \frac{\alpha ^2 + \beta ^2}{W_0(k \omega )}. \end{aligned}$$

(63)

Linear systems receiving oscillatory inputs in I-clamp and V-clamp

1.1 A linear system in I-clamp

System (51) with $F(t) = A_\mathrm{in} \sin (\omega t)$ and $G(t) = 0$ can be written as

$$\begin{aligned} v'' - \eta v' + \Delta v = -A_\mathrm{in} d\ \sin (\omega t) + A_\mathrm{in} \omega \cos (\omega t), \end{aligned}$$

(64)

whose solution is given by (“Appendix B.1”)

$$\begin{aligned} v(t) = A_0(\omega ) \sin (\omega t) + B_0(\omega ) \cos (\omega t), \end{aligned}$$

(65)

where

$$\begin{aligned}&A_0(\omega ) = -\frac{d (\Delta - \omega ^2) + \eta \omega ^2}{W(\omega )} A_\mathrm{in} \quad \hbox {and} \quad \nonumber \\&B_0(\omega ) = \frac{-d \eta \omega + \omega (\Delta - \omega ^2)}{W(\omega )} A_\mathrm{in} , \end{aligned}$$

(66)

with $W(\omega )$ given by (57) with $k = 1$. From (58)

$$\begin{aligned} A_0(\omega )^2 + B_0(\omega )^2 = \frac{d^2 + \omega ^2}{W(\omega )} A_\mathrm{in}^2 = Z(\omega )^2 A_\mathrm{in}^2. \end{aligned}$$

(67)

1.2 A linear system in V-clamp

System (51) with $F(t) = I$, $v(t) = A_\mathrm{in} \sin (\omega t)$ and $G(t) = 0$ can be written as

$$\begin{aligned} \left\{ \begin{array}{ll} w' - d w = A_\mathrm{in} c \sin (\omega t), \\ I = A_\mathrm{in} \omega \cos (\omega t) - A_\mathrm{in} a \sin (\omega t) - b w. \end{array} \right. \end{aligned}$$

(68)

The solution to the first equation in (68) is given by (“Appendix B.2”)

$$\begin{aligned} w(t) = - A_\mathrm{in} \frac{d c}{W_0(\omega )} \sin (\omega t) - A_\mathrm{in} \frac{\omega c}{W_0(\omega )} \cos (\omega t), \end{aligned}$$

(69)

with $W_0(\omega )$ given by (62) with $k = 1$. Substitution into the second equation in (68) yields

$$\begin{aligned} I = C_0 \sin (\omega t) + D_0 \cos (\omega t), \end{aligned}$$

(70)

where

$$\begin{aligned} C_0(\omega )= & {} \left( \frac{d b c}{W_0(\omega )} - a \right) A_\mathrm{in} \quad \hbox {and} \quad \nonumber \\&D_0(\omega ) = \left( \frac{\omega b c}{W_0(\omega )} + \omega \right) A_\mathrm{in}. \end{aligned}$$

(71)

It can be shown that these constants satisfy

$$\begin{aligned} C_0(\omega )^2 + D_0(\omega )^2= & {} \frac{( \Delta - \omega ^2)^2 + \eta ^2 \omega ^2}{W_0(\omega )} A_\mathrm{in}^2\nonumber \\= & {} \frac{W(\omega )}{d^2 + \omega ^2} A_\mathrm{in}^2 \nonumber \\= & {} \frac{A_\mathrm{in}^2}{ Z(\omega )^2} = Y(\omega )^2 A_\mathrm{in}^2. \end{aligned}$$

(72)

1.3 A linearized conductance-based model in I-clamp

The solution to systems (1)–(2) with $I(t) = A_\mathrm{in} \sin (\omega t)$ (I-clamp) is given by (“Appendix D”)

$$\begin{aligned} v(t) = A(\omega ) \sin (\omega t) + B(\omega ) \cos (\omega t), \end{aligned}$$

(73)

where

$$\begin{aligned}&A(\omega ) = \frac{\tau ^{-1} (\Delta - \omega ^2) - \eta \omega ^2}{W(\omega )} A_\mathrm{in} \quad \hbox {and} \quad \nonumber \\&B(\omega ) = \frac{\tau ^{-1} \eta \omega + \omega (\Delta -\omega ^2)}{W(\omega )} A_\mathrm{in}, \end{aligned}$$

(74)

with

$$\begin{aligned}&\Delta = \frac{g_\mathrm{L}+g}{\tau }, \quad \eta = - \frac{1+g_\mathrm{L} \tau }{\tau } \quad \hbox {and} \quad \nonumber \\&W(\omega ) = (\Delta - \omega ^2)^2 + \eta ^2 \omega ^2. \end{aligned}$$

(75)

From (58),

$$\begin{aligned} Z(\omega ) = \sqrt{\frac{A(\omega )^2 + B(\omega )^2}{A_\mathrm{in}}} = \sqrt{\frac{\tau ^{-2} + \omega ^2}{W(\omega )}} \frac{1}{A_\mathrm{in}}. \end{aligned}$$

(76)

1.4 A linearized conductance-based model in V-clamp

If, instead, we assume that $v(t) = A_\mathrm{in} \sin (\omega t)$ (V-clamp), then the solution to Eq. (2) is given by (“Appendix D”)

$$\begin{aligned} w(t) = A_\mathrm{in} \frac{\tau ^{-2}}{W_0(\omega )} \sin (\omega t) - A_\mathrm{in} \frac{\omega \tau ^{-1}}{W_0(\omega )} \cos (\omega t), \end{aligned}$$

(77)

with

$$\begin{aligned} W_0(\omega ) = \tau ^{-2} + \omega ^2. \end{aligned}$$

(78)

Substitution into the second equation in (68) yields

$$\begin{aligned} I = C(\omega ) \sin (\omega t) + D(\omega ) \cos (\omega t), \end{aligned}$$

(79)

where

$$\begin{aligned}&C(\omega ) = \left[ \frac{g \tau ^{-2}}{W_0(\omega )} + g_\mathrm{L} \right] A_\mathrm{in} \quad \hbox {and} \quad \nonumber \\&D(\omega ) = \left[ -\frac{\omega g \tau ^{-1}}{W_0(\omega )} + \omega \right] A_\mathrm{in}. \end{aligned}$$

(80)

It can be easily shown that

$$\begin{aligned} C(\omega )^2 + D(\omega )^2= & {} \frac{( \Delta - \omega ^2)^2 + \eta ^2 \omega ^2}{W_0(\omega )} A_\mathrm{in}^2 \nonumber \\= & {} \frac{W(\omega )}{\tau ^{-2} + \omega ^2} A_\mathrm{in}^2 \nonumber \\= & {} \frac{A_\mathrm{in}^2}{ Z(\omega )^2} = Y(\omega )^2 A_\mathrm{in}^2. \end{aligned}$$

(81)

Weakly nonlinear forced systems of ODEs in I- and V-clamp: asymptotic approach

1.1 Oscillatory input in I-clamp

We consider the following weakly perturbed system of ODEs

$$\begin{aligned} \left\{ \begin{array}{ll} v' = a v + b w + \epsilon \sigma _v v^2 + I_\mathrm{in}(t), \\ w' = c v + d w + \epsilon \sigma _v v^2, \end{array} \right. \end{aligned}$$

(82)

where

$$\begin{aligned} I_\mathrm{in}(t) = A_\mathrm{in} \sin (\omega t) \end{aligned}$$

(83)

and $\epsilon$ is assumed to be small. We expand the solutions of (82) in series of $\epsilon$

$$\begin{aligned}&v(t) = v_0(t) + \epsilon v_1(t) + \mathcal{O}(\epsilon ^2) \quad \hbox {and} \quad \nonumber \\&w(t) = w_0(t) + \epsilon w_1(t) + \mathcal{O}(\epsilon ^2). \end{aligned}$$

(84)

Substituting into (82) and collecting the terms with the same powers of $\epsilon$, we obtain the following systems for the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon )$ orders, respectively,

$$\begin{aligned} \left\{ \begin{array}{ll} v_0' = a v_0 + b w_0 + A_\mathrm{in} \sin (\omega t), \\ w_0' = c v_0 + d w_0, \end{array} \right. \end{aligned}$$

(85)

and

$$\begin{aligned} \left\{ \begin{array}{ll} v_1' = a v_1 + b w_1 + \sigma _v v_0^2, \\ w_1' = c v_1 + d w_1 + \sigma _w v_0^2. \end{array} \right. \end{aligned}$$

(86)

Solution to the $\mathcal{O}(1)$system

The solution to system (85) is given in “Appendix C.1” with v substituted by $v_0$.

Solution to the $\mathcal{O}(\epsilon )$ system

System (86) can be rewritten as

$$\begin{aligned} v_1'' - \eta v_1' + \Delta v_1= & {} -\sigma _v d v_0^2(t) + \sigma _w b v_0^2(t) \nonumber \\&+ 2 \sigma _v v_0(t) v_0'(t), \end{aligned}$$

(87)

where

$$\begin{aligned} v_0^2(t)= & {} \frac{A_0(\omega )^2+B_0(\omega )^2}{2} + \frac{B_0(\omega )^2-A_0(\omega )^2}{2} \cos (2 \omega t)\nonumber \\&+ A_0(\omega ) B_0(\omega ) \sin (2 \omega t). \end{aligned}$$

(88)

The solution to (87) is given (“Appendix B.1”) by

$$\begin{aligned} v_1(t)= & {} \frac{A_0(\omega )^2+B_0(\omega )^2}{2 \Delta } ( -\sigma _v d + \sigma _w b) \nonumber \\&+ A_1(\omega ) \sin (2 \omega t) + B_1(\omega ) \cos (2 \omega t), \end{aligned}$$

(89)

where

$$\begin{aligned} A_1(\omega )= & {} \frac{\alpha _1(\omega ) (\Delta - 4 \omega ^2) - 2 \beta _1(\omega ) \eta \omega }{W(2 \omega )} \quad \hbox {and} \quad \nonumber \\&B_1(\omega ) = \frac{2 \alpha _1(\omega ) \eta \omega +\beta _1(\omega ) (\Delta - 4 \omega ^2) }{W(2 \omega )}, \end{aligned}$$

(90)

with $W(2 \omega )$ given by (57) with $k = 2$,

$$\begin{aligned} \alpha _1(\omega ) = \sigma _v [ -d A_0 B_0 - \omega (B_0^2-A_0^2) ] + \sigma _w b A_0 B_0, \end{aligned}$$

(91)

and

$$\begin{aligned} \beta _1(\omega )= & {} \sigma _v [ -d (B_0^2-A_0^2)/2 + 2 \omega A_0 B_0 ] \nonumber \\&+ \sigma _w b (B_0^2-A_0^2)/2. \end{aligned}$$

(92)

1.2 Oscillatory input in V-clamp

We consider the following weakly perturbed system of ODEs

$$\begin{aligned} \left\{ \begin{array}{ll} v' = a v + b w + \epsilon \sigma _x v^2 + I, \\ w' = c v + d w + \epsilon \sigma _y v^2. \end{array} \right. \end{aligned}$$

(93)

where

$$\begin{aligned} v = v_\mathrm{in}(t) = A_\mathrm{in} \sin (\omega t) \end{aligned}$$

(94)

and $\epsilon$ is assumed to be small. We expand the solutions of (93) in series of $\epsilon$

$$\begin{aligned}&w(t) = w_0(t) + \epsilon w_1(t) + \mathcal{O}(\epsilon ^2) \quad \hbox {and} \quad \nonumber \\&I(t) = I_0(t) + \epsilon I_1(t) + \mathcal{O}(\epsilon ^2). \end{aligned}$$

(95)

Substituting into (93) and collecting the terms with the same powers of $\epsilon$ we obtain the following systems for the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon )$ orders, respectively,

$$\begin{aligned} \left\{ \begin{array}{ll} w_0' - d w_0 = A_\mathrm{in} c \sin (\omega t), \\ I_0 = A_\mathrm{in} \omega \cos (\omega t) - A_\mathrm{in} a \sin (\omega t) - b w_0, \end{array} \right. \end{aligned}$$

(96)

and

$$\begin{aligned} \left\{ \begin{array}{ll} w_1' - d w_1 = \sigma _w A_\mathrm{in}^2 [1 - \cos (2 \omega t )] / 2, \\ I_1 = - b w_1 - \sigma _v A_\mathrm{in}^2 [1 - \cos (2 \omega t )] / 2. \end{array} \right. \end{aligned}$$

(97)

Solution to the$\mathcal{O}(1)$system

The solution to system (96) is given in “Appendix C.2” with w and I and substituted by $w_0$ and $I_0$, respectively.

Solution to the $\mathcal{O}(\epsilon )$ system

The solution to the first equation in (97) is given by (“Appendix B.2”)

$$\begin{aligned} w_1(t)= & {} -\frac{\sigma _w A_\mathrm{in}^2}{2 d} - \frac{\sigma _w \omega A_\mathrm{in}^2}{W_0(2 \omega )} \sin (2\, \omega t) \nonumber \\&+ \frac{\sigma _w d A_\mathrm{in}^2}{2 W_0(2 \omega )} \cos (2 \omega \, t), \end{aligned}$$

(98)

with $W_0(2\, \omega )$ given by (62) with $k = 2$. Substitution into the second equation in (97) yields

$$\begin{aligned} I_1= & {} -\left( \sigma _v - \frac{b}{d}\, \sigma _w \right) \, \frac{A_\mathrm{in}^2}{2} + C_1(\omega )\, \sin (2\, \omega \, t) \nonumber \\&+ D_1(\omega )\, \cos (2\, \omega \, t), \end{aligned}$$

(99)

where

$$\begin{aligned}&C_1(\omega ) = \sigma _w\, \frac{b\, \omega }{W_0(2\, \omega )}\, A_\mathrm{in}^2, \quad \hbox {and} \quad \nonumber \\&D_1(\omega ) = \left( \sigma _v - \sigma _w\, \frac{b\, d}{W_0(2\, \omega )} \right) \, \frac{A_\mathrm{in}^2}{2}. \end{aligned}$$

(100)

Asymptotic formulas for large values of $\tau$

1.1 Impedance zeroth-order approximation in I-clamp

For large enough values of $\tau$, the coefficients of the solutions to the linear system (16) satisfy $A_0(\omega ) = \mathcal{O}(1)$ and $B_0(\omega ) = \mathcal{O}(1)$, and

$$\begin{aligned} A_0(\omega ) = \frac{1}{1 + \omega ^2} A_\mathrm{in}, \quad \quad B_0(\omega ) = -\frac{\omega }{(1 + \omega ^2)}A_\mathrm{in}, \end{aligned}$$

(101)

and

$$\begin{aligned} Z_0(\omega ) = \frac{\sqrt{A_0(\omega )^2+B_0(\omega )^2}}{A_\mathrm{in}} = \sqrt{\frac{1}{1+\omega ^2}}. \end{aligned}$$

(102)

We begin with Eqs. (74) and (75) and assume all other parameter values are $\mathcal{O}(1)$. For large enough values of $\tau$, these quantities behave as follows

$$\begin{aligned}&\Delta = \frac{1}{\tau }, \quad \eta = - \left( \frac{1}{\tau }+1 \right) \quad \hbox {and} \quad \nonumber \\&W(\omega ) = \left( \frac{1}{\tau } - \omega ^2\right) ^2 + \left( \frac{1}{\tau }+1 \right) ^2\, \omega ^2, \end{aligned}$$

(103)

and

$$\begin{aligned}&A_0(\omega ) = \frac{ (\tau ^{-1} - \omega ^2) + (1+\tau )\, \omega ^2}{\tau \, W(\omega )}\, A_\mathrm{in} \quad \hbox {and} \quad \nonumber \\&B_0(\omega ) = \frac{-(\tau ^{-1}+1)\, \omega + \omega \, (1 -\tau \, \omega ^2)}{\tau \, W(\omega )}\, A_\mathrm{in}, \end{aligned}$$

(104)

where

$$\begin{aligned} \tau \, W(\omega ) = \left( \frac{\tau ^{1/2}}{\tau } - \tau ^{1/2} \omega ^2\right) ^2 + \left( \frac{\tau ^{1/2}}{\tau }+\tau ^{1/2} \right) ^2\, \omega ^2, \end{aligned}$$

which can be reduced to

$$\begin{aligned} \tau \, W(\omega ) = \tau \, \omega ^4 + \tau \, \omega ^2. \end{aligned}$$

Substituting into (104) and rearranging terms yields (101) and 102.

1.2 Admittance first-order approximation in V-clamp

For large enough values of $\tau$

$$\begin{aligned} C_1(\omega ) = 0 \quad \hbox {and} \quad D_1(\omega ) = \sigma _v\, \frac{A_\mathrm{in}^2}{2}. \end{aligned}$$

(105)

From (36), this implies that

$$\begin{aligned} Y_1(\omega ) = \sigma _v\, \frac{A_\mathrm{in}^2}{2}. \end{aligned}$$

(106)

We begin with Eqs. (37) for $C_1(\omega )$ and $D_1(\omega )$ and Eq. (62) with $k = 2$ and $d = -\tau ^{-1}$ for $W_0(2\, \omega )$. Multiplication of the latter by $\tau$ and $\tau ^2$ yields, respectively,

$$\begin{aligned}&\tau \, W_0(2\, \omega ) = \frac{1}{\tau } + 4 \tau \, \omega ^2 \quad \hbox {and} \quad \nonumber \\&\tau ^2\, W_0(2\, \omega ) = 1+ 4 \tau ^2\, \omega ^2. \end{aligned}$$

(107)

For large values of $\tau$

$$\begin{aligned}&\frac{1}{\tau \, W_0(2\, \omega )} = \frac{1}{ \frac{1}{\tau } + 4 \tau \, \omega ^2} = \mathcal{O}(\tau ^{-1}) \quad \hbox {and} \quad \nonumber \\&\frac{1}{\tau ^2\, W_0(2\, \omega )} = \frac{1}{1+ 4 \tau ^2\, \omega ^2} = \mathcal{O}(\tau ^{-2}). \end{aligned}$$

Therefore, for large enough values of $\tau$ in (37) we obtain (105).

1.3 Impedance first-order approximation in I-clamp

For large enough values of $\tau$,

$$\begin{aligned} Z_1(\omega ) = \sigma _v\, \frac{1}{2 \sqrt{1+4 \omega ^2}} \frac{1}{1 + \omega ^2}. \end{aligned}$$

(108)

From (24) and (25) and the fact that $A_0(\omega ) = \mathcal{O}(1)$ and $B_0(\omega ) = \mathcal{O}(1)$ (“Appendix E.1”), it follows that for large enough values of $\tau$

$$\begin{aligned}&\alpha _1(\omega ) = -\sigma _v\,\omega \, [ B_0(\omega )^2-A_0(\omega )^2] \quad \hbox {and} \quad \nonumber \\&\beta _1(\omega ) = \sigma _v\, [ 2\, \omega \, A_0(\omega )\, B_0(\omega )\, ]. \end{aligned}$$

(109)

Substituting into (22) and rearranging terms, we obtain

$$\begin{aligned} Z_1(\omega )^2 = \sigma _v^2\, \omega ^2 \frac{[ B_0(\omega )^2+A_0(\omega )^2]^2}{W(2\, \omega )}\, \frac{1}{A_\mathrm{in}^2}. \end{aligned}$$

(110)

From (103) (and large enough values of $\tau$),

$$\begin{aligned} W(2 \omega ) = 16\, \omega ^4 + 4\, \omega ^2. \end{aligned}$$

(111)

Substituting (111) and (101) into (110), we obtain (108).