INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

Most electrophysiological studies designed to measure ionic currents from single cells use whole-cell patch-clamp techniques. While these techniques are suitable for a broad range of ionic currents, the limited bandwidth of the whole-cell configuration makes it difficult to measure rapid ionic currents, such as voltage-gated Na⁺ currents from nerve or muscle cells. The major factor that limits the voltage clamping bandwidth in the whole-cell configuration is the series resistance (R_s) introduced by the recording pipette (Sigworth, 1983). Therefore, to measure rapidly activating currents reliably, it is necessary to increase the voltage clamping bandwidth by compensating for R_s. The main difficulty with commonly used R_s compensation techniques is the instability that arises at high voltage clamping bandwidth. This instability makes measurements of rapid ionic currents with the whole-cell patch-clamp configuration extremely difficult.

To overcome the problem of instability, we have developed a novel approach to R_s compensation that is based on a membrane state estimator. State estimator R_s compensation overcomes the stability limitations of standard designs and is simple and straightforward to use. In this paper, we outline the theory of state estimator R_s compensation. Next, we demonstrate the high bandwidth and stable performance of state estimator R_s compensation and compare it with various methods currently in use to compensate for R_s. Finally, we show the utility of state estimator R_s compensation by measuring voltage-gated Na⁺ currents from dissociated neonatal rat sympathetic neurons. Our results clearly illustrate the advantages of state estimator R_s compensation for measuring rapidly activating ionic currents in single cells.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

Voltage clamping bandwidth and standard R_s compensation

The time constant that determines the whole-cell voltage-clamp bandwidth is given by _vclamp  R_sC_m when R_m R_s, where R_m is the cell membrane resistance and C_m is the cell membrane capacitance (Sigworth, 1983). With typical values for R_s and C_m (5-20 M; 15-100 pF), _vclamp is several hundred microseconds; this is too slow to voltage clamp rapid ionic currents.

The standard method for compensating for R_s and increasing bandwidth is to compute a scaled value of the pipette current (I_p) and add it as a correction signal to the command potential (V_c) (see Fig. 1). Ideally, when the scaling factor  approaches unity, corresponding to 100% R_s compensation, the membrane potential (V_m) follows V_c exactly. In practice, when  is greater than ~0.8, standard R_s compensation becomes unstable. For wide bandwidth R_s compensation, the instability results from two main factors: 1) limited bandwidth in the current-measuring circuitry and 2) the effects of stray pipette capacitance (C_p) (Sigworth, 1983; also see the Appendix). For stability at 90% R_s compensation, the bandwidth of the current measurement circuitry that generates the R_s correction signal must be greater than 300 kHz, which is difficult to achieve in practice. More importantly, C_p introduces an erroneous correction signal not modeled in Fig. 1 that destabilizes the R_s compensation feedback loop. For stable operation, C_p must be neutralized electronically to <0.05 pF (Sigworth, 1983); slight shifts in pipette capacitance, as happen when the pipette immersion depth changes, easily drive the R_s compensation circuitry from a marginally stable state into oscillation. Consequently, it is exceedingly difficult to achieve the wide bandwidth and stability necessary to measure rapidly activating currents with amplifiers that incorporate standard R_s compensation.

View larger version (10K): [in this window] [in a new window]   FIGURE 1   Standard Rs compensation for a single electrode voltage clamp: Vp = Vc + IpmeasRs, Vm = Vp  IpRs  Vm = Vc if Ipmeas = Ip and   1. Vc = command voltage, Vp = pipette voltage, Vm = membrane voltage, Ip = pipette current, Rs = pipette series resistance, Rm = cell resistance, Cm = cell capacitance.

The two-electrode configuration and the membrane voltage estimator

A successful approach to overcoming the effects of R_s is to use two electrodes instead of one. A two-electrode voltage clamp uses one electrode to measure V_m and the other electrode to pass current; V_m is clamped at V_c with a negative feedback loop (see Fig. 2 A). In this arrangement, R_s is contained within the feedback loop and produces little limitation on the voltage clamp bandwidth because its effects are attenuated by the large open-loop gain of the feedback circuit.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   (A) Two-electrode voltage clamp. (B) Single-electrode voltage clamp using a membrane state estimator. Vc = command voltage, Vp = pipette voltage, Vm = membrane voltage, Ip = pipette current, Ipmeas = measured pipette current, Vpmeas = measured pipette voltage, Vmmeas = measured membrane voltage, Vmest = estimated membrane voltage.

Our approach to R_s compensation for a single-electrode voltage clamp is based on the two-electrode topology in conjunction with a membrane state estimator, as shown in Fig. 2 B. The state estimator computes V_m and functions as a "virtual electrode" in place of the physical measuring electrode and voltage follower. We call this configuration state estimator R_s compensation.

Using only a single electrode, state estimator R_s compensation achieves a bandwidth and stability similar to those of a two-electrode voltage-clamp amplifier. The closed-loop bandwidth and steady-state error become independent of R_s, effectively achieving 100% R_s compensation. Stability is greatly improved over standard R_s compensation by eliminating the need to neutralize C_p electronically. Equally important, state estimator R_s compensation is independent of cell conductance changes, ensuring wide voltage-clamp bandwidth, even during the measurements of large ionic currents.

State estimator theory

The recording patch pipette electrode can be modeled as in Fig. 3. Given that I_p, V_p, R_s, and C_p are known values as defined in Fig. 3, V_m can be estimated as follows:

View larger version (10K): [in this window] [in a new window]   FIGURE 3   Lumped parameter RC model of patch pipette: Vp = pipette voltage, Vm = cell voltage, Ip = total pipette current, Ipc = pipette capacitive current, Ipr = pipette resistive current, Rs = pipette series resistance, Cp = pipette capacitance, Zcell = cell impedance.

Applying Kirchoff's current law gives

(1)

Applying Ohm's law gives

(2)

Substituting Eq. 1 into Eq. 2 gives

(3)

From generalized Ohm's law,

(4)

Substituting Eq. 4 into Eq. 3 gives

(5)

where _p = pipette time constant = R_sC_p and s is the Laplace transform frequency variable. Once R_s and C_p are determined for the patch pipette electrode, Eq. 5 can be solved in real time to compute V_m independently of cell conductance changes.

Bandwidth and stability of state estimator R_s compensation

Fig. 4 A shows the block diagram for a single-electrode voltage clamp incorporating state estimator R_s compensation. In Fig. 4 A, block 1 represents a controlled current source, with gain G₀ in units of conductance and bandwidth set by _cs. Block 2 gives the transfer function of pipette voltage to pipette current when the pipette is modeled as in Fig. 3 and the cell is modeled as in Fig. 1. Blocks 3, 4, and 5 implement Eq. 5, with _Vpmeas setting the pipette voltage measurement bandwidth, _Ipmeas setting the pipette current measurement bandwidth, and _vest setting the state estimator output bandwidth. Full R_s compensation occurs when R_sest in block 3 is set equal to R_s and _pest in block 4 is set equal to the pipette time constant _p.

View larger version (17K): [in this window] [in a new window]   FIGURE 4   Stability of state estimator Rs compensation. (A) s-domain block diagram of single-electrode voltage clamp using state estimator Rs compensation. (B) Open-loop Bode plot of Vmest (s)/Vc (s), where s = j2f, with the feedback path broken at X. (C) Closed-loop step response plot of Vm when Vc undergoes a stepwise transition from 0 to 100 mV at t = 0. Rs = 5 M, Cp = 1 pF, Rm = 500 M, Cm = 50 pF, Vpmeas, Ipmeas, vmest = 1.6 µs, cs = 0.32 µs, G0 = 3 µA/V. Vc = command voltage, Verr = error voltage, Ip = pipette current, Vp = pipette voltage, Vmest = computed membrane voltage, p = RsCp = pipette time constant, m = RmCm = membrane time constant, cs sets current source bandwidth, Ipmeas sets Ip measurement bandwidth, Vpmeas sets Vp measurement bandwidth, vest sets membrane state estimator bandwidth.

Fig. 4, B and C, shows the typical performance of state estimator R_s compensation in the frequency domain (Fig. 4 B) and the time domain (Fig. 4 C), using representative parameter values. The Bode plot in Fig. 4 B shows a gain margin of 17 dB and a phase margin of 62°, ensuring stable voltage clamping with full R_s compensation. The closed-loop voltage-clamping bandwidth is roughly equal to the open-loop 0-dB cross-over frequency (10 kHz). This 10-kHz bandwidth translates to a step response time of less than 50 µs, as shown in Fig. 4 C. These wide stability margins ensure stable performance even if parameters shift during an experiment. (A comparative stability analysis is given in the Appendix.)

To achieve the stability margins in Fig. 4, state estimator R_s compensation requires close phase matching of the I_p and V_p signals up to ~50 kHz. In contrast, obtaining similar stability margins with standard R_s compensation requires bandwidths greater than 300 kHz (Sigworth, 1983); such bandwidths are difficult to achieve in practice. In addition, standard R_s compensation requires near-perfect C_p nulling, whereas C_p nulling is not needed with state estimator R_s compensation. However, as with standard R_s compensation, the value of C_p, once determined, must not fluctuate for stability to be maintained.

Other approaches for R_s compensation

Moore et al. (1984) and Strickholm (1995b) each describe modifications to standard R_s compensation that feed back a scaled value of the steady-state pipette current as opposed to the total pipette current. The steady-state current is computed using an electronic bridge that subtracts the transient membrane capacitive current from the measured total pipette current. While this approach can achieve 100% R_s compensation in the steady state, it is not suitable for measurements during large, rapid (<1 ms) changes in membrane conductance.

Because the bridge is balanced only for a fixed value of R_m, large changes in membrane conductance will unbalance the bridge and voltage control returns slowly. As shown in Fig. 5 A, with typical pipette and cell parameters and a step conductance change from 2 to 50 nS, the voltage recovers in ~1 ms; this is too slow to voltage clamp fast ionic currents. Adding a "supercharging" potential to V_c speeds voltage-clamp control for a step change in V_c (Strickholm, 1995a) but does not improve the voltage recovery time when the membrane conductance changes (compare dotted and solid lines in Fig. 5 A). In contrast, state estimator R_s compensation is independent of cell conductance changes, ensuring rapid voltage-clamp recovery from changes in ionic conductance as well as changes in V_c, as shown in Fig. 5 B.

View larger version (10K): [in this window] [in a new window]   FIGURE 5   (A) Steady-state Rs compensation with () and without (·····) supercharging showing Vm (top) and Ip (bottom). Bridge balance set for Rm = 500 M, Cm = 50 pF. (B) State estimator Rs compensation showing Vm (top) and Ip (bottom). In both A and B, Vc steps form 0 to 100 mV at t = 0; Rm steps from 500 to 20 M at t = 3 ms (marked by arrow). All other values are as in Fig. 4.

Brennecke and Lindemann (1972) described another approach to overcoming R_s (see also Wilson and Goldner, 1975). Variously called switch-clamp, pulsed current clamp, or discontinuous feedback voltage-clamp amplifiers, these designs operate by repetitively cycling a single electrode between current-passing and voltage-measuring modes. During voltage-measuring mode the amplifier passes no current, ensuring that the measured voltage reflects V_m independently of R_s. The attainable bandwidth is limited by the maximum switching rate, yet this switching rate is itself limited by C_p (Finkel and Redman, 1984). Consequently, to increase bandwidth it is necessary to neutralize C_p electronically, as with standard R_s compensation. This C_p neutralization compromises the stability of the voltage clamp. The attainable voltage-clamp bandwidth using discontinuous feedback is generally insufficient to measure rapidly activating currents, such as voltage-gated Na⁺ currents from nerve or muscle.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

Amplifier benchmarks

We built a voltage-clamp amplifier implementing the state estimator R_s compensation described in the Theory section and tested its performance using the model membrane and the low-noise dynamic switch shown in Fig. 6. This single-pole single-throw (SPST) switch makes a clean, single-step transition with no detectable switching artifact. We constructed this switch by attaching two silver chlorided wires separated by ~1 cm to the inside of a microcentrifuge tube with a small hole drilled in the bottom. A dilute salt solution established the electrical contact; when the fluid drained from the tube the contact was broken (see Fig. 6). With the switch initially open, we adjusted the amplifier for full R_s compensation by giving repetitive voltage test pulses and adjusting R_sest and _pest (see Fig. 4) to minimize the capacity current transient decay time. The current source gain (G_o, Fig. 4) was then increased to 10 µA/V. The microcentrifuge tube was then filled with salt solution, and the current monitor output of the amplifier was captured using a Nicolet model 3091 digital oscilloscope (Nicolet Instrument Corporation, Madison, WI) when the switch made the closed to open state transition. As a comparison we repeated the test using an Axopatch 1D with standard R_s compensation (Axon Instruments, Foster City, CA).

View larger version (19K): [in this window] [in a new window]   FIGURE 6   (A) Model circuit used for dynamic conductance change. (B) Low-noise, bounceless SPST switch implementation. When the tube is filled with solution, the switch is closed; as the fluid level drops below wire A, the switch opens. (C, D) Measured current response of Axopatch 1D, using standard Rs compensation. (C) Custom-built amplifier using state estimator Rs compensation (D) to step conductance change while holding at 100 mV. The switch Sw1 opening marked by arrow. Rs = 4.7 M, Cp  1.5 pF, Cm = 47 pF, Rm1 = 180 M, Rm2 = 10 M. Traces in C: Low-pass filtered with a 4-pole Bessel filter at 10 kHz. Traces in D: Low-pass filtered with a 1-pole RC filter at 15 kHz.

Electrophysiological recordings

We made whole-cell patch-clamp recordings from neonatal rat superior cervical ganglia (SCG) neurons. Neonatal rat SCG neurons were cultured for 12-72 h as described by McFarlane and Cooper (1992). All experiments were done at 23°C, using the following solutions (in mM): Pipette solution: 50 KAc, 65 KF, 5 NaCl, 0.2 CaCl₂, 1 MgCl₂, 10 HEPES, 10 EGTA. (In some experiments K⁺ currents were blocked by replacing K⁺ with Cs⁺.) Extracellular solution: 140 NaCl, 5.4 KCl, 0.3 NaH₂PO₄, 0.44 KH₂P0₄, 2.8 CaCl₂, 0.18 MgCl₂, 10 HEPES, 5.6 glucose. Electrodes were made with a two-stage micropipette puller (model PP-83; Narishige Instrument Co., Tokyo, Japan) from Kimax-51 glass capillary tubes (Kimble Science Products, Chicago, IL). Resistance measured in the bath was 2-5 M; whole-cell access resistance was 3-13 M. In some experiments, the electrodes were coated with Sylgard (Dow Corning, Auburn, MI). Once in whole-cell mode, we obtained full R_s compensation by giving hyperpolarizing test pulses and adjusting R_sest and _pest (see Fig. 4) to minimize the capacity current transient decay time. Then we increased the current source gain (G_o, Fig. 4) to achieve a closed-loop bandwidth greater than 10 kHz. In all cases, G_o was greater than 5 µA/V, and the capacity current transient decay time was less than 70 µs. This tuning procedure took ~20 s. PatchKit software (Alembic Software, Montréal, Québec, Canada) was used for simultaneous on-line stimulation and acquisition. Voltage-clamp recordings were low-pass filtered at 15 kHz with a single-pole RC filter and digitized at 100 kHz with an IBM-compatible 33-MHz 486 computer equipped with a DAS-20 acquisition board (Omega Engineering, Stamford, CT).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

Amplifier benchmarks

To demonstrate the value of state estimator R_s compensation, we built a prototype single-electrode voltage-clamp amplifier that incorporated state estimator R_s compensation as described in the Theory section and compared its performance to a commercial amplifier with standard R_s compensation (Axopatch 1D). To test the performance of both amplifiers, we measured their responses to a step change in conductance. While it would have been simpler to test the performance using a fixed-conductance model membrane, such tests can be misleading when used to measure the performance of R_s compensation. With a fixed-conductance model membrane, the amplifier bandwidth is normally determined by measuring the capacity transient decay time in response to a voltage step. However, with commercial amplifiers that incorporate standard R_s compensation, the whole-cell charging current is delivered by an injection capacitor to avoid saturation before standard R_s compensation is applied (Sigworth, 1983); therefore, the current transient in response to a voltage step no longer reflects of the amplifier's bandwidth. Similarly, with amplifiers that use prediction or supercharging R_s correction circuits to speed up the response to a voltage step, the capacity transient cannot be used as a measure of the system's bandwidth (see Theory; Fig. 5). For these reasons, we measured R_s compensation performance in response to a step conductance change.

To create a step conductance change, we built a low-noise SPST switch (see Materials and Methods). We could not use ordinary field effect transistor or mechanical relay switches: the charge injection of field effect transistor switches obscures the current transients, and most mechanical switches are noisy and the contacts bounce.

First, we tested the performance of the Axopatch 1D, using the circuit and low-noise SPST switch described in Fig. 6, A and B. Set at 80% R_s compensation, the response time of the Axopatch 1D was >100 µs. At higher R_s compensation settings, the amplifier exhibited oscillations due to the inherent instabilities described above (see Theory), and surprisingly, the response time of the Axopatch 1D never decreased below 90 µs. From our experiments on Na⁺ currents (see below), if the amplifier response time is greater than 90 µs, one has difficulty measuring Na⁺ currents from mammalian neurons under voltage-clamp conditions.

To ensure that the measured performance of the Axopatch 1D was not obscured by a switch artifact, we verified the performance of the SPST switch by setting R_s = 10 k and using no R_s compensation. The recorded current trace showed a clean step transition in less than 20 µs with no measurable switch artifact (data not shown).

Next we repeated the test, using our amplifier. We adjusted for full R_s compensation by minimizing the capacity current transient decay time in response to voltage steps (see Materials and Methods). In contrast to what we observed for the Axopatch 1D, our amplifier responded in under 50 µs to a step conductance change, confirming the bandwidth and stability of state estimator R_s compensation predicted from the theory (Fig. 6 D). These results clearly demonstrate the advantages of using a state estimator to compensate for R_s.

Measurements of voltage-gated Na⁺ currents

To demonstrate further the utility of state estimator R_s compensation when recording whole-cell currents with a single-electrode voltage clamp, we used our amplifier to measure ionic currents from single cells. As a stringent test, we measured voltage-gated inward Na⁺ currents from nerve and muscle cells. Because these Na⁺ currents are large and activate extremely rapidly, they are difficult to measure accurately with conventional single-electrode patch-clamp amplifiers (Schofield and Ikeda, 1988; Nerbonne and Gurney, 1989; Hanck, 1995; Sakakibara et al., 1993).

Rat SCG neurons

View larger version (17K): [in this window] [in a new window]   FIGURE 7   Voltage-clamped Na+ current from a P1 SCG neuron in culture for 6 days (Cm = 25 pF; Rs = 6 M). (A, B) Na+ current activation and peak I-V curve using 0% and 100% Rs compensation, respectively. (C) Stimulation protocol for displayed traces in A and C.

View larger version (22K): [in this window] [in a new window]   FIGURE 8   Voltage-clamped Na+ current from a P1 SCG neuron in culture for 1 day (Cm = 12 pF; Rs = 5 M). (A, B) Na+ current activation and peak I-V curve using 80% and 100% Rs compensation, respectively. (C) Stimulation protocol for displayed traces in A and C.

Effective R_s using state estimator R_s compensation

View larger version (12K): [in this window] [in a new window]   FIGURE 9   Measuring the effective Rs using state estimator Rs compensation. Peak I-V curves of Na+ current activation from a SCG neuron in response to depolarizing voltage steps with Vh = 90 mV (curve 1, ) and Vh = 70 mV (curve 2, - - -). Rs = 6 M, effective Rs = 220 k.

	CONCLUSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX REFERENCES

In this paper, we demonstrate an improved method for compensating for R_s when recording in whole-cell patch-clamp configurations. This method is straightforward to implement, and amplifiers incorporating state estimator R_s compensation are extremely stable. The main aspect of this R_s compensation method is to compute V_m using a state estimator and to use this value in a feedback loop to implement a voltage clamp.

Using a model cell with parameter values representative of whole-cell recording, our benchtop tests show that state estimator R_s compensation responds to step conductance changes in well under 50 µs, whereas standard R_s compensation reaches a limit of ~90 µs before the onset of oscillations. Our experiments on neurons demonstrate that it is essential to have such a rapid response to voltage clamp Na⁺ currents reliably. Equally important, our experiments demonstrate that state estimator R_s compensation amplifiers perform well in actual physiological experiments. As such, these amplifiers should make it easier to measure large rapid ionic currents by whole-cell patch-clamp techniques.