In vitro experiments were performed on isolated brains of six adult grass frogs (Rana temporaria) and complied with the “Principles of animal care”, publication No. 86–23, revised 1985 by the National Institute of Health. As described in previous studies (Straka and Dieringer, 1993), animals were deeply anesthetized with 0.1% 3-aminobenzoic acid ethyl ester (MS-222), and perfused transcardially with iced Ringer solution (75 mM NaCl; 25 mM NaHCO₃; 2 mM CaCl₂; 2 mM KCl; 0.5 mM MgCl₂; 11 mM glucose; pH 7.4). Thereafter, the skull and bony labyrinth were opened ventrally. After dissecting the three semicircular canals on each side, the brain was removed with all labyrinthine end organs attached to the VIIIth nerve. Subsequently, the brain was submerged in iced Ringer and the dura, the labyrinthine end organs and the choroid plexus covering the IVth ventricle were removed. In all experiments the forebrain was disconnected. Brains were used as long as 4 days after their isolation and were stored overnight at 6°C in continuously oxygenated Ringer solution (Carbogen: 95% O₂, 5% CO₂) with a pH of 7.4 ± 0.1. The brains were directly fixed with insect pins to the sylgard floor of a chamber (volume 2.4 ml) or glued with cyanoacrylate to a plastic mesh that was fixed to the floor of the recording chamber. The chamber was continuously perfused with oxygenated Ringer solution at a rate of 1.3–2.1 ml/min. The temperature was electronically controlled and maintained at 14 ± 0.1°C.

External landmarks of the isolated frog whole brain served to identify the target sites for the intracellular recordings in the vestibular (Pfanzelt et al., 2008) and the abducens nuclei (Straka and Dieringer, 1993). Intracellularly recorded second-order vestibular neurons (2°VN) were identified by the activation of monosynaptic EPSPs following electrical stimulation of individual ipsilateral semicircular canal nerves (Straka et al., 1997) with single constant current pulses (duration: 0.2 ms; threshold: ∼1–3 μA). These pulses were produced by a stimulus isolation unit (WPI A 360) and applied across suction electrodes whose opening diameters (120–150 μm) were individually adjusted. All 2°VN were classified as phasic or tonic neurons based on their responses to the injection of long, positive current steps (Straka et al., 2004; Beraneck et al., 2007; Pfanzelt et al., 2008). Abducens motoneurons (AbMot) were identified by short-latency antidromic action potentials following electrical stimulation of the ipsilateral VIth cranial nerve with suction electrodes (diameter 100–150 μm). These neurons typically received a disynaptic crossed EPSP and an uncrossed IPSP following electrical stimulation of the bilateral horizontal canal nerve, compatible with previous results obtained after electrical stimulation of the entire VIIIth nerve (Straka and Dieringer, 1993). Only 2°VN and AbMot with resting membrane potentials that were more negative than −55 mV were included in this study.

Sharp high-resistant glass microelectrodes for intracellular recordings were made with a horizontal puller (P-87 Brown/Flaming) using filament-containing borosilicate glass with pre-fire-polished ends (GB150F-8P, Science Products GmbH, Hofheim, Germany). Electrodes were filled with a 3-M solution of KCl or a mixture of 3 M KCl and 2 M KAc (1:10), which gave final resistances of 70–90 MΩ and 80–120 MΩ, respectively. Bath application of the potassium channel blocker 4-aminopyridine (4-AP; 20 μM) confirmed in one experiment the putative contribution of voltage-dependent potassium conductances (Wu et al., 2001; Beraneck et al., 2007) to the non-linear response properties of phasic 2°VN.

Voltage recordings and current injections were performed with a single-electrode clamp amplifier (SEC-05L, NPI Electronic GmbH, Tamm, Germany) in the “bridge balance” operation mode with no resistance or capacitance compensation. For A/D and D/A conversion a National Instruments acquisition card (PCI-6052E) was used with a PC (Intel Pentium 4, 1.9 GHz, Windows XP); the signals were sampled at 5 kHz. Recordings were done with a custom Matlab (Mathworks Corp.) program.

Responses of 2°VN and AbMot to a multi-sine current-clamp were measured at different membrane potentials. The multi-sine stimulus I_MS was composed of 52–55 discrete frequencies of interest with a uniform stimulus amplitude and randomized phase spectra over a range of either 0.2–988.8 Hz or 0.2–1935.7 Hz (Idoux et al., 2008). The constant magnitude was designed to drive the voltage responses uniformly at all frequencies of interest and the random phase spectrum was chosen to minimize the peak-to-peak dynamic amplitude of the stimulus waveform. The maximal half-amplitude of the multi-sine stimulus (h_Amp = half of peak-to-peak amplitude) ranged from 0.4 to 1.2 nA with a duration of 6 or 12 s. In order to measure neurons at membrane potentials of interest (just below spike threshold), the multi-sine stimulus I_MS was superimposed on a constant holding current I_hold, which preceded the multi-sine stimulus by 1.5 s. This holding current + multi-sine signal was the current clamp command or input current, I_ref(t), (Figure 1B) that led to an output voltage response V(t) (Figure 1C). Each measurement consisted of two stimuli, where the second multi-sine stimulus, I_MS, was multiplied by −1. A fast Fourier transform (FFT) was done on the difference of the two runs providing I_ref(f) and V(f), respectively. The FFT, without windowing, has been computed on all N data points of I_ref(f) and V(f), but only the frequencies present in the stimulus were used for the frequency-domain analysis.

The sum of the voltage responses V(t), V_AveM, revealed the mean membrane potential and non-linear membrane effects, such as subthreshold responses or action potentials, as well as recurrent synaptic potentials evoked by the current injection during the two runs. Since the polarity of the stimulus alternated, the sum removed all coherent linear responses and thus V_AveM is an excellent control for the quality of the measurement. The corresponding admittance transfer functions were computed as Y(f) = I_ref(f)/V(f). Admittance extends the concept of conductance, taking into account dynamic effects by describing not only the relative amplitudes of the voltage and current, but also the relative phases. Admittance is expressed as a complex number Y(f) = G + jB with j being the imaginary unit and G the real part. It is the inverse of the impedance Z(f) = 1/Y(f), which is also a complex number, Z = R + jC with resistance R being the real part.

All transfer functions are shown as complex admittance or impedance Bode plots. In the complex admittance plots, the real part is shown on the x-axis, whereas the imaginary part is plotted on the y-axis. Each data point in this plot thus expresses a different frequency, starting with 0 Hz at the point (x,0). The length of a line that can be drawn between each point and (0,0) corresponds to the magnitude of the admittance at a given frequency, whereas the angle between this line and the horizontal x-axis is the phase shift between the response and the stimulation. Therefore the units of both axes of complex admittance plots are nano-Siemens (nS). All transfer functions after electrode subtraction are shown as the more common impedance Bode plots.

Since no electrode resistance or capacitance compensation was used, the resulting voltage response (Figure 1C), and therefore the admittance transfer function, Y(f), refers to both the electrode and the neuron. In the following all admittance transfer functions are written as Y_i with i being e, e + n or e + m indicating that the electrode was measured alone, within a neuron or with a neuronal model, respectively.

During intracellular measurements, the properties of the electrode are best described as a frequency-, depth- (electrode tip with respect to the surface of the Ringer solution in the recording chamber), time- and current-dependent serial impedance, Z_ee(f, x, t, I_e), and a depth-dependent capacitance, C_e(x), parallel to the neuron. In addition to this parallel electrode capacitance, a constant parallel stray capacitance, C_s, (Figure 1A) is always present, i.e. the capacitance to ground at the input of the buffer operational amplifier (Molecular Devices, 2008). In order to obtain reliable data on frequency responses of single neurons recorded with sharp electrodes, a procedure called Piece-wise Non-linear Electrode Compensation (PNEC) was developed to remove the electrode properties from the total response in the frequency-domain. To test the PNEC method a passive neuronal circuit model composed of a thin film resistance with R_m = 10 MΩ (Tolerance: ±0.02%) (Vishay Sfernice, Pennsylvania, USA) parallel to a polystyrene capacitor with C_m = 1 nF (Tolerance: ±1%) was used. For neuronal or model measurements the root-mean-square (RMS) error was computed separately for impedance magnitude and phase. The degree of electrode nonlinearity (λ) was calculated as the slope of the linear regression between injected current and the steady state electrode resistance (Brette et al., 2008). Matlab (Mathworks Corp.) was used for computations. For time-domain simulations Simulink was used in combination with the SimPowerSystems Blockset to model the electrode and the passive neuron as electrical circuits. For AEC compensation the kernel computation and electrode kernel extraction was done using the latest Python implementation from http://audition.ens.fr/brette/HRCORTEX/AEC/AECcode.html. For linear regression calculations the Matlab function “regress” was used. A model for the electrode has been created by fitting two distributed capacitances and resistances (RC's) and a serial resistance and inductance (R–L) to the measured frequency response of an electrode using the Matlab function “lsqcurvefit”. Graphical presentations were made with Corel Draw (Corel Corporation Ltd.). Statistica 6.1 was used for the Wilcoxon matched pairs test.

Electrode properties were revealed by using a multi-sine analysis of the electrodes in the recording chamber and/or the tissue to determine the electrode transfer function, Y_e. In contrast to a simple electrode model (constant parallel capacitance and resistance), which approximates a theoretically expected straight line on an admittance plot (red line in Figure 2A1), the glass microelectrodes used for the present intracellular recordings show a complex frequency-dependent behavior of the real part (green and blue lines in Figure 2A1) but an essentially linear behavior of the imaginary part, similar to a simple RC electrode model (Figure 2A2). This behavior depended on the depth (x) of the electrode tip in the tissue and/or bath with respect to the level of the Ringer solution; in particular, the capacitance of the electrode increased with the depth (Figure 2A2). Even though the resistance of sharp electrodes, filled with KCl or KAc, could fluctuate randomly within the range of minutes (Figure 2B1), the shape of the admittance measurements and its imaginary component (Figure 2B2) did not change over time. This spontaneous impedance change was considerably reduced, yet not absent when the electrode tip was inside a neuron. Injections of holding currents I_hold through the electrode radically changed the real part of the admittance measurements, but had little effect on its imaginary part (Figures 2C1,C2). This behavior can be best seen in the corresponding impedance magnitude plot (Figure 2C1 inset). Also, note that some sharp electrodes showed a resonance (* in Figure 2C1 inset) at ∼50 Hz that occurred mostly for positive I_hold injections. Considering these results and taking into account the additional stray capacitance of the amplifier, the electrode transfer function can be described as Figure 2

Multi-sine analysis of electrode properties with discrete frequencies plotted for the range from 0. 2–300 Hz. (A1,A2) Admittance plots (A1) of the electrode model circuit with constant parallel capacitance and resistance (red line); admittance plots (A1) of KCl (3 M) filled electrode with the tip at 1.5 mm (green line) and 2 mm below the surface of the Ringer solution (blue line) show frequency-dependent changes of electrode properties influenced by the depth in the Ringer solution; plots of the imaginary part (A2) show that the electrode capacitance increases with increasing depth of the electrode tip in the bath (*); I_hold = 0 nA. (B1,B2) Admittance plots (B1) of three measurements of electrode properties in the recording chamber; measurements 1 and 2 were consecutive (green and blue lines), whereas the measurement 3 (red line) was taken 1 min later; note that the admittance fluctuates over time while the capacitance (B2) remains stable; I_hold = 0 nA. (C1,C2) Admittance plots (C1) of three electrode measurements with different holding currents I_hold = 0 nA (black line), −0.5 nA (green line), +0.5 nA (red line); injection of a constant current changes the real part of the electrode admittance response while the capacitance (C2) remains unchanged.

Because of the particular complex electrode behavior, no electrode compensation (neither resistance nor capacitance compensation) was used, but a procedure called Piece-wise Non-linear Electrode Compensation (PNEC) was developed to subtract the electrode from the frequency-domain response offline after the measurements. The subtraction procedure involved measuring the electrode alone just after removing it from the neuron, but leaving it in the immediate vicinity (5–8 μm) of the cell and applying the same holding currents, I_hold, as used during the intracellular recording. The latter procedure was necessary because of the current-dependent non-linearities of the electrode. As indicated above, changes in the electrode impedance observed either spontaneously or after impalement (Brette et al., 2008) are most likely due to partial blocking or unblocking of the micropipette tip as suggested by the finding that this behavior can be described by translating the real part of Y_e along the real axis, which is equivalent to adding a conductance G_T to the admittance Y_e.

The impedance Z_n of a passive neuron (resistance R_n and capacitance C_n) can be expressed by: (2) Zn=(Rn−1+j2πfCn)−1=Rn−j2πfCnRn21+4π2f2Cn2Rn2 and lim⁡f→∞Zn=0 with τ_n = R_nC_n (3) real(Zn)=Rn1+4π2f2τn2

Thus at high frequencies, the neuron is shunted by its capacitance such that the electrode + neuron admittance measurement Y_e+n should overlap with the electrode measurement Y_e. Therefore, in order to compensate for the change in electrode impedance after leaving the neuron, Y_e was translated along the real axis until it superimposed on Y_e + n at high frequencies above f_fit. For f_fit, the frequency was chosen where the resistance of the neuron reaches 1% of its baseline resistance. Knowledge of the membrane time constant (τ_n) of a typical recorded neuron, f_fit can thus be calculated by: (4) ffit=10.01−14π2τn2

A τ_n = 2 ms, as measured in e.g. tonic and phasic 2°VN (Straka et al., 2004) gives a f_fit of 792 Hz, which was used in our experiments. With a typical neuronal resistance of 10 MΩ and typical electrode resistances of ∼100 MΩ the theoretical error calculated for the electrode resistance estimation is thus only ∼0.1% in our experiments.

After translating and determining the conductance G_T, the electrode admittance measurements can be used to remove the electrode. Thus, this electrode compensation procedure, referred to as PNEC, is as follows:

(1) To estimate the conductance G_T the error [real(G_T + Y_e) − real(Y_e + n)]² has to be minimized in the frequency range between f_fit and the maximal frequency (f_max) measured. Since G_T is real, the estimation of G_T simplifies to: (5) GT = mean [real (Ye+n) − real (Ye)]

(2) Fit 2πfC to imaginary(Y_e), using linear regression, in order to remove the parallel electrode capacitance C_e(x), and amplifier stray capacitance C_s: (6) Ye + n − c = Ye + n − j2πfC

(3) Remove the serial impedance Z_ee(f, x, t, I_e) = (G_T + Y_e − j2πfC)⁻¹ of the electrode: (7) Yn=[ Ye+n−c−1−(GT+Ye− j2πfC)−1 ]−1

The PNEC procedure has been performed with 19 neurons: the 19 electrodes showed a mean resistance of 85.5 ± 24.5 MΩ (mean ± SD) and a mean capacitance of 8.16 ± 2.01 pF; for G_T an average of 0.76 ± 1.2 nS (mean ± SD) was estimated, which resulted in an RMS of 0.0375 ± 0.018 nS (mean ± SD; from f_fit to f_max). The mean degree of nonlinearity (λ) could be evaluated in 12 electrodes and resulted in −2.56 ± 3.9 MΩ/nA (mean ± SD) which is comparable to nonlinearities found previously (Brette et al., 2008).

Examples of electrode compensation for three different neuronal subtypes recorded with different holding currents (I_hold) in an isolated adult frog whole brain were used to illustrate the applicability of PNEC for typical neuronal recordings. Figure 4A illustrates the electrode adjustment procedure for an AbMot at I_hold = +1 nA, which required a shift in the electrode admittance response, Y_e, due to a difference in the electrode properties when it was removed from the neuron. As described above, the abscissa values were shifted prior to the subtraction in order for the curves to overlap at high frequencies above f_fit indicated as G_T + Y_e. Electrode corrected magnitude and phase plots for this AbMot at other holding currents illustrate the marked potential dependence of the neuronal impedance including a pronounced resonance (Figures 4B1,B2). Bode plot examples for previously described tonic (Figures 4C1,C2) and phasic (Figures 4D1,D2,E1,E2) 2°VN (Beraneck et al., 2007) indicate that the membrane resonance properties reported for phasic 2°VN (* in Figures 4D1,E1) are not an electrode artifact.

Multi-sine analysis allows direct visualization of the dynamic responses of functionally different neuronal phenotypes. For instance, the AbMot, which is plotted in Figures 4B1,B2 revealed a high degree of resonance during depolarizing currents, most likely caused by an interaction of delayed rectifier potassium and sodium channels (Hutcheon and Yarom, 2000). Frog tonic 2°VN have no membrane resonance behavior but exhibited an increasing impedance with membrane depolarization (Figures 4C1,C2). This latter effect is likely caused by non-inactivating calcium or sodium inward currents as suggested previously (Beraneck et al., 2007). In contrast, frog phasic 2°VN showed decreasing impedance and increasing resonance (* in Figures 4D1,E1) with membrane depolarization due to the activation of low-threshold voltage-dependent I_D-type potassium channels as reported earlier (Beraneck et al., 2007). Furthermore, since I_D-type potassium channels are specifically blocked by low concentrations of 4-AP (Wu et al., 2001), bath application of 20 μM 4-AP during multi-sine measurements of a phasic 2°VN (dotted lines in Figures 4D1,D2) leads to an increase in impedance magnitude for I_hold = 0 and I_hold = 0.5 nA.

To demonstrate the errors induced by insufficient bridge compensation (BC), another phasic 2°VN (Figures 4E1,E2) has been additionally measured during standard BC from the amplifier (dotted lines). While both compensation types match fairly well at the lowest frequencies for I_hold = 0 and −0.5 nA, the measurements at depolarized membrane potentials show an overcompensation due to electrode rectification. Furthermore incorrect compensation of the distributed electrode RCs in BC leads to bends in the magnitude and large jumps in the phase at higher frequencies (∼200 Hz).

To compare the PNEC method with standard bridge compensation (BC), both techniques have been applied in a total of 11 neurons at I_hold = 0 nA. Since the measurements were made at resting membrane potential, these neurons presumably behave like passive RC circuits with an impedance of Y_n = G_n + j2πfC_n. Thus C_n can be estimated by fitting j2πfC_n to imaginary(Y) using a linear regression and G_n = mean[real(Y)], where Y is the measured neuronal admittance after either PNEC or BC compensation. The root-mean-square error of this fit is calculated as RMS=mean(|Yn−Y|2). This test resulted in a better RMS for PNEC, median of RMS: 0.0458 μS, compared to BC, median of RMS: 0.1818 μS (difference highly significant with p = 0.0076, Wilcoxon matched pairs test).

Control experiments were performed to evaluate the validity of the assumption of piece-wise linearity for PNEC. Measurements of the electrode alone with different multi-sine half-amplitudes (h_Amp = half of peak-to-peak amplitude of multi-sine stimulus; h_Amp = 1 nA and h_Amp = 0.5 nA) showed that, apart from the time-dependent translational shift of the electrode, the frequency response itself (e.g. I_hold = +0.5 nA in Figure 5A) does not change with increasing amplitude. Nevertheless, as shown before, frequency responses are significantly affected by the holding I_hold current level (compare with Brette et al., 2008). A further indication of the linearity over the range of multi-sine amplitudes is the successful test subtraction procedure (Figures 3A–C2). Thus, this is a piece-wise linear analysis of two non-linear elements, the neuron and the electrode, both of which show marked non-linearities at different holding current levels. The non-linear effects of the neuron itself are due to its voltage-dependent ion channels. Measuring AbMot with an I_hold = 0 nA and multi-sine currents with three different half-amplitudes of h_Amp = 0.4 nA, 0.6 nA and 0.8 nA showed that, apart from some fluctuations in the phase using the largest multi-sine current (h_Amp = 0.8 nA), all measurements yield identical results (Figures 5B1,B2). The source of these phase fluctuations using h_Amp = 0.8 nA could be pinpointed by the observation of V_AveM, which indicated that the multi-sine current triggered a few spikes (not shown). This result illustrates an important restriction required for most linearization methods, namely that the multi-sine amplitude should be chosen such that it is small enough not to trigger action potentials, but large enough to have a reasonable signal-to-noise ratio.

Since we are considering piece-wise linear systems, the frequency-domain data can be translated into the time-domain, and therefore it is also possible to use the present PNEC framework to dissociate electrode from neuronal responses for arbitrary stimuli, as long as the current amplitude of the latter remain in the linear range. In the following, a method is presented that allows the determination of the electrode kernel for electrode compensation in the time-domain using the described PNEC framework. For this purpose, multi-sine measurements of the electrode alone, outside the cell, were used to calculate the electrode kernel K_{e_outside} via cross-correlation between input current and output voltage (similar to Brette et al., 2008). These electrode kernels however differ from the real electrode kernel inside the cell and were thus corrected using the conductance G_T, estimated via the second measurement inside the cell, as described before. For this correction, the kernel K_{e_outside} was transformed into the frequency domain via FFT, shifted by G_T, which was estimated using the first step of PNEC, and transformed back into the time-domain using an inverse FFT: (9) Ke_inside = IFFT{[FFT(Ke_outside)−1 + GT]−1}

This corrected electrode kernel K_{e_inside} can be further used to compute the electrode voltage by convolving it with the injected current I_e: (10) Ve = (Ke_inside * Ie)

However, since no electrode capacitance compensation was used in our experiments, the measured electrode + neuronal voltage V_e + n and the computed electrode voltage V_e were low-pass filtered by the electrode, but could be corrected using voltage deconvolution (Richardson and Silberberg, 2008) with: (11) V′=τdVdt+V

Here by τ = RC, with R being the impedance at the lowest frequency and C estimated from imaginary(Y_e) during the PNEC procedure.

After application of this voltage deconvolution to V_e + n and V_e, the neuronal voltage response could be computed as V_n = V_e + n − V_e. The results of this procedure were applied offline to the responses of a phasic (Figure 5C1, green traces) and a tonic 2°VN (Figure 5C2, green traces), measured during multi-sine stimulation (Figures 5C1,C2 black traces) with current amplitudes in the linear range, showed that the high-frequency electrode voltage had been removed leaving a low-frequency neuronal voltage, since the injected current was filtered by the neuronal time constant. Compensation of electrode voltage under noisy current stimulation is a demanding task for each electrode compensation mechanism: using a simple constant resistance R for the estimation of the electrode voltage and thus subtracting V_e = RI_e, equivalent to standard bridge compensation, leads to insufficient electrode compensation (Figures 5C1,C2 red traces) since small errors in the estimation of the electrode voltage result in high-frequency electrode artifacts (compare with Brette et al., 2008, supplemental data). In addition, using the uncorrected electrode kernels K_{e_outside} for offline compensation (Figures 5C1,C2 blue traces) causes an incorrect estimation of V_n since the overall electrode resistance is overestimated in this case.

In essence, this indicates that the PNEC procedure can be used for a direct estimation of the electrode kernel and offline subtraction of the electrode voltage in the time-domain. For some applications, such as voltage-, or dynamic-clamp experiments, an offline compensation, however, is not sufficient. Nevertheless it is clear that the PNEC procedure to determine the electrode kernel can also be used in combination with online AEC (Brette et al., 2008), as long as the electrode is measured twice: before entering a neuron and inside a neuron. It should also be noted, that for online AEC using the PNEC electrode kernel estimation it is more convenient to use capacitance compensation from the amplifier, as also suggested for normal AEC (Brette et al., 2008), which supersedes the voltage deconvolution step.