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Unformatted text preview: artlmmmmmumnmlwflka Chapter 4 Electronic Design of the Patch Clamp F. J. SIGWORTH 1. Introduction The patch-clamp amplifier is fundamentally a sensitive current-to-voltage converter, con- verting small (picoampere to nanoampere) pipette currents into voltage signals that can be observed with an oscilloscope or sampled by a computer. This chapter describes the basic principles of the current-to-voltage (I—V) converter and describes some of the technology of.V converters having wide bandwidth and low noise. Also considered in this chapter are additions to the basic I—V converter design for capacitive-transient cancellation and series- resistance compensation, features that are particularly useful for whole-cell current recording. The most important property of an I—V converter for single-channel recording is its noise level. Under good patch-recording conditions, the background noise introduced by the patch membrane, together with the seal conductance and noise sources in the pipette, together corresponds to the movement of a few tens of elementary charges. It is important to maintain this low noise level in the circuitry of the I—V converter because there are many types of channels whose currents are near the limit of resolution of the recording system. In whole—cell recordings an important problem is the series access resistance between the pipette electrode and the cell interior. A high series resistance limits the speed with which voltage changes can be imposed on the cell membrane and limits the time resolution of current recordings. The access resistance is determined by the size of the pipette tip and the nature of the access to the cell interior; however, the effects of series resistance can be reduced by circuitry that is discussed later in this chapter. 2. Current-Measurement Circuitry 2.1. Current—Voltage Converter The standard way to measure small currents is to monitor the voltage drop across a large resistor. Figure 1 shows three circuits that accomplish this. In part A, a battery with voltage Vref is used to set the pipette potential, and the pipette current ID is measured from the voltage drop IpR across the resistor. The problem with this configuration is that the pipette F. J. SIGWORTH 0 Department of Cellular and Molecular Physiology, Yale University School of Medicine, New Haven, Connecticut 06510. fingle-Channel Recording, Second Edition, edited by Bert Sakmann and Erwin Neher. Plenum Press, New Wk, 1995. 95 F. J. Sigwonh re! e mi W "V I Figure 1. Current measurement circuits. A: A simple circuit in which the pipette potential is determined approximately by Vm. In B, the deviation Verr of the pipette potential from me is monitored, and VB is adjusted to keep it zero. In C, the op—amp makes the adjustment automatically. potential is not exactly equal to me but has an error that depends on the current; when the resistor R is made large to give high sensitivity, the voltage error also becomes large. The solution to this problem (Fig. IB) is to measure VP directly and to continuously adjust a voltage source VB to bring Vp to the correct value. Provided that the adjustment is made quickly and accurately, R can be made very large for high sensitivity. An operational amplifier (op—amp) can be used to automate the adjustment of V3. The op-amp can be thought of as a voltage-controlled voltage source; the output voltage changes in response to differences in the voltages V, and V. at the input terminals according to (1) The factor (1),. can be very large: for typical commercial op-amps, it is about 107 sec", which means that a l-mV difference on the input terminals causes the output voltage to slew at 104 V/sec. The value (9A is the gain—bandwidth product of the amplifier. In this chapter, a) will be used to represent an “angular frequency” in units of radians/sec. The relationship to the more familiar frequencies f (given in Hertz) is w = 2 17]", so that (1),, is 211 times the gain—bandwidth product fix that is normally given in amplifier specification sheets. (Sometimes mum:rwrinivmwMalawi:wrnxwwmmammw.. w H * ‘ Mmrwwmw-QWIWL99v‘ Design of the Patch Clamp 97 fA is given as the “unity-gain bandwidth" of an op-amp, which for our purposes is essentially the same thing.) At the same time, the op-amp draws essentially no current through its input terminals. This is an important feature, since such currents would disturb the measurement. (For a more complete discussion of op—amps and feedback, see, for example, Horowitz and Hill, 1989.) Part C of Fig. 1 shows the final current-to-voltage converter circuit. The op—amp varies its output to keep the pipette potential at Vref. This action can be made very rapid and precise, so that for practical purposes Vp can be assumed to be precisely 1%. This in turn allows us to measure VB — m; as showu, rather than VB — VP, to obtain the voltage drop across the resistor. The voltage differences should be the same, but the former measurement is preferable because it avoids an additional direct connection to the pipette electrode. The voltage differ- ence is usually measured using a standard differential amplifier circuit (not shown). 2.2. Dynamics of the I—V Converter For single-channel recording, suitable values for the current-measuring resistance Rf are on the order of 10—100 G0. Commercial resistors in this range typically have a shunt capacitance Cf of 0.1 pF (Figure 2A); the resulting time constant, 'rf = Rfo, is the order of l msec and limits the time resolution of the I—V converter. Assuming that the op-amp acts instantaneously (mA —9 00), the response characteristics of the I—V converter are given by a transfer function Zc(s), which can be used to give the response at Vom for any input current 1p, (2) This function can be used in two ways (see, for example, Aseltine, 1958). First, if the imaginary frequency jw is substituted for s, the resulting magnitude and phase of the (complex- valued) Zc give the amplitude ratio and phase shift of Vom relative to IP. Thus, Zc gives the “frequency response” of the circuit. For convenience, Zc can be rewritten as Zc(S) = Rf T(S) (3) where T is dimensionless and has the form _ 1 T(S) — TfS + 1 (4) which is the transfer function of a simple low-pass filter. T is unity at low frequencies (s —> 0) but rolls off at high frequencies. The “"corner frequency,” at which the response is down by 3 dB, occurs when no = 1. The second use of the transfer function is to calculate the time course of Vow for an arbitrary 1P. This is done using the inverse Laplace transform. For example, the response to a step of input current is found in this way to be exponential with a time constant Tf. The reason for going to all the trouble of introducing the transfer function is that things beCome more complicated when (”A is assumed to be finite. We define TA = l/wA, the Characteristic time constant of the op-amp and CI = C, + C,,,, the total capacitance on the 98 F. J. Sigwonh ‘ Relative Amplitude B 10 100 1k 10k 100k 1M Frequency , Hz Figure 2. A: Diagram of the I—V converter showing the stray feedback capacitance C, and the total input ‘ capacitance C,,.. A unity-gain differential amplifier on the output produces the current monitor signal. B: Magnitude of the transfer function T as the damping factor g is varied. The curves were calculated for the , circuit of part A with C," = 10 pF, Rr = 10 GO, and fA = 10 MHz (~rA = 16 nsec). The C values of 12.5, 2.5, l, and 0.2 correspond to C, values of 100, 20, 8, and 1.6 fF, respectively, The dashed curve is the response from a single time constant T. = 1 msec (corresponding to f. = 160 Hz as indicated, with C, = 100 fF). The corresponding 12 value is 1.6 usec (f2 = 100 kHz). C: Variation of the step response time course J with C. input terminal. Using equation 1 to describe the op-amp’s behavior, we can write the differen- tial equation for VOUT as 2 V TARfCt T?” + (TA + Tr) dVour dt + VOUT = erpm (5) The equivalent transfer-function representation of the response is Rf Zc = ———————— 6 (s) 'rARfC.s2 + (TA + 'rf)s + 1 ( ) Once again, let us write Zc = RJ} where T can be written in the form 1 T(s) = —— (7) ("ms + 1)(1'zs + l) i ,3 i 3 E mum—“4....“ Design of the Patch Clamp 99 This is the transfer function of two simple filters in cascade: 1'. and 72 are found as roots of a quadratic equation, but provided TA is sufficiently short, they can be approximated by Ti 2 Tt (8) Ct T2 2 a TA The frequency response now has two comer frequencies, a). = 7,", caused by the stray capacitance across the feedback resistor, and the higher cutoff (.02 = T2" which arises from the finite speed of the op-amp. To improve the frequency response of the I—V converter, one usually tries to reduce Cf. This reduces T, but lengthens 72 at the same time, usually by about the same factor, since Cf makes only a minor contribution to C,. If this is carried to an extreme such that r, and 1'2 become comparable in size, the approximations (equation 8) are no longer valid. Instead, it is more useful to write T in the equivalent form 1 : ____._.._ 9 T(s) 'rés2 + 2§Tos + l ( ) which is the equation for a damped harmonic oscillator with natural frequency 000 = To" and damping factor Q. These parameters are given by To = (”r/thCt)”2 (10) 17A + Tf 2 T0 The surprise here is that 10 does not depend directly on C, but only on the total capacitance Cf. However, since TA is typically very small compared to Tf, the damping factor is proportional to Cf. When C; is reduced beyond a certain point, then the bandwidth of the I—V converter does not increase further; instead, the frequency response begins to show a peak, and the step response shows “ringing” (Fig. 2B,C). For the best transient response, C should be kept in the vicinity of unity: g = 1 corresponds to a “critically damped” step response with no overshoot; Q = 0.71 gives the “maximally flat” frequency response but about 10% overshoot in the step response. It is quite difficult to obtain a high natural frequency in a sensitive I—V converter. For example, if we use a 10'0—9 feedback resistor, and the total input capacitance C, = 10 pF to obtain a bandwidth of 10 kHz—corresponding to To = 16 usec (equation 7)—-requires that TA = 2.6 X 10‘9 sec or a gain—bandwidth product of about 60 MHz for the amplifier. At the same time, the stray feedback capacitance would have to be kept to 3.2 X 10'15 F. Both of these requirements are not readily achieved. A better strategy for wide—band recording is to design the I—V converter to have a nonideal but well-defined frequency response characteristic and then to correct the frequency reSponse in a later amplifier stage. The best way to do this is to allow C, to be large enough, and choose the op-amp to be fast enough, so that the rolloff characterized by 1'; (equation 8) is well beyond the frequency range of interest. Within the range of interest, only the single rolloff of T] is then present, and this can be compensated as is described in the next section. 100 F. J. Sigwonh Making 72 small has several other advantages. First, 1'2 depends on C,, which includes contributions from the pipette and stray capacitances. Since these can vary, 72 depends 0n the experimental conditions; if T; is very small, these variations can be ignored. Second, 72 describes the response time of the I—V converter as a voltage clamp. The transfer function relating Vp to the command voltage chd is _ Vp(s) _ 1 + Tfs T _ __—.— VC chdm macs2 + (1,, + ms + 1 (11) The denominator of equation 11 is the same as that of T. The numerator approximately cancels the factor (1 + 7.3) in equation 7, so the clamp transfer function is approximately Tvc = 1/(T2s + 1) when 72 < T1. In a practical situation, the op-amp might have a gain—bandwidth product of 10 MHz. With Cf = 0.1 pF and CI = 10 pF, then T2 = 1.6 usec, giving a clamp bandwidth of 100 kHz. 7, in this case would be 1 msec, so the first rolloff in the frequency would occur at 160 Hz. The lowest curve in Fig. 23 shows this response, which corresponds to a damping factor of 12.5. 2.3. Correcting the Frequency Response When Cf is chosen as just described, the I—V converter will have a very nearly exponential step response with a fairly long time constant around 1 msec (Fig. 3, top trace). The role of the correction circuit is to perform an “inverse filtering” operation to recover a faster-rising response. One way of looking at the correction operation is to notice that the derivative of the exponential step response function is itself an exponential and to exploit this fact by summing the original response with a scaled copy of its derivative to recover the original form (Fig. 3, bottom trace). Fortunately, this particular strategy works for all possible input waveforms, not just steps. The operation just described has the transfer function T... = M + 1 (12) where Tc is the factor scaling the derivative. When the I—V converter transfer function T (equation 7) is multiplied by Tm, the rolloff caused by 'r, can be canceled exactly when Tc = T}. Derivative Figure 3. A strategy for correcting the response of the l—V converter. The input to the correction circuit shows a slow exponential step response. When it is summed with a scaled copy of its time derivative a nearly square step response results. :1 mmmmmm mamm- um WHW ‘ :I Design of the Patch Clamp 101 A practical compensation circuit is shown in Fig. 4A. Many implementations are possible, but this particular circuit is useful because its low-frequency gain is fixed at unity. At low frequencies, the capacitor C acts as an open circuit, and the op-amp acts just as a voltage follower. The increasing gain with frequency arises from the decreasing impedance of C; the “comer” time constant TC is equal to (RI + R2)C. The magnitude of the ideal transfer function (equation 12) increases without limit as the frequency variable s increases. This sort of behavior cannot be achieved in practice. The actual transfer function of the circuit in Fig. 4A is TCS+1 Too 2 —-—————__ TATCSZ + (TA + R2C)S + l (13) which is the desired response multiplied by a second-order transfer function that can be written in the form of equation 9 with To = JR (14) in + RZC 2 To The maximum useful frequency of this circuit is given immediately by m0 = To". If a 10-MHz op-amp were used in the circuit (TA = 0.016 nsec) and Tc = l msec, T0 would be 4 nsec, giving a useful bandwidth of about 40 kHz. The importance of R2 can be seen from the expression for g. Since TA is negligibly small, R2 should be chosen to give an RZC time constant on the order of T0. If R2 is zero, there will be no damping, and the circuit will oscillate. The correction circuit in effect “removes” the corner in the l—V converter’s frequency response (Fig. 4B) and replaces it with a new, second—order rolloff at a much higher frequency. Relative Magnitude to" 10" 10" i0 100 1% 10k mk 1M I“'glll‘e 4. A: Response-correction circuit. The variable resistor R. allows the time constant TC to be varied ‘0 match the time constant T. of the 17V converter. B: Frequency response of the l—V converter (T), the COrrection circuit (Tc), and the composite (TR). The correction circuit response was computed for T, = l msec, TA = 16 nsec (fA = 10 MHz), and t: = 0.7. The resulting overall bandwidth is 40 kHz. 102 F. J. sigw0th , p The effective removal of the low-frequency comer requires that the time constant To be . precisely matched to the time constant T1 of the I—V converter. For this reason, R. is best made a variable resistance that is trimmed for the proper step response. The error in the Step ' ’ response when a mismatch is present is a relaxation of relative amplitude ('rc - T|)/T| and time constant 7.. How far can the frequency response be extended in this way? One limitation is the second time constant T2 in the [—V converter, which can be shortened by using a sufficiently fast op-amp there. As we saw in Section 2.2, an intrinsic bandwidth of 100 kHz is not 5 difficult to achieve. The other limitation is To of the correction circuit; op-amp speed is the ' limiting factor here also. To increase the gain—bandwidth product, two op-amps can be operated in cascade, with one of them provided with local feedback to make it an amplifier with fixed gain A. Provided the bandwidth of this amplifier stage is large compared to To", the composite can be used in the circuit as a “super op-amp” with the gain—bandwidth product ‘ increased by the factor A. Extension of the I-V converter’s bandwidth by a factor of 1000 (e.g., to 100 kHz or more) can be performed, provided care is taken in the grounding and layout of the circuitry. Up to this point, we have assumed that the I—V converter’s response in the frequency V range of interest can be represented as a single-time-constant rolloff caused by the stray - capacitance of the feedback resistor. We have modeled that capacitance as a “lumped" . quantity, Cr, but it is actually distributed along the length of the resistor. If the distribution « is uniform (Fig. 5A), the impedance is the same as that in the parallel R—C model. If it is l not, the impedance of the combination, and therefore the transfer function of the I—V converter, will show an additional “step” in its frequency dependence. Figure 5C shows a circuit for performing a two-time-constant correction for a response function of this kind. The circuit is relatively complex (requiring three op-amps) but has the advantage of being relatively ' easy to adjust, with minimal interaction among the controls. Testing the frequency response requires a precise, high-impedance source of picoampere ‘ currents. Since commercial high-value resistors have substantial stray capacitance, injecting a current into the I—-V converter with a resistor is suitable only for testing the DC gain. For dynamic testing, a capacitor is the circuit element of choice, since nearly ideal capacitors in the appropriate range of 0.01 to l pF are easy to make by bringing two conductors near , each other. Such homemade capacitors are usually better than commercial ones, which often 1 do not have low enough leakage conductance. For estimating small capacitances, recall that L for two parallel conducting plates of area A having a small spacing d, C = €0A/d where, in appropriate units, £0 = 0.089 pF/cm. Connecting a wire to the output of a signal generator and holding it near the pipette or “ i the input terminal of the I—V converter often makes an acceptable signal source. An appropriate “ waveform to apply in this way is a triangle wave, because the coupling capacitance carries . a current that is the time derivative of the applied voltage, and the derivative of a triangle . wave is a square wave. For critical use, such as in adjusting the compensation circuitry, 21 1 function generator with especially good linearity should be used. A nonlinearity of a few percent (not uncommon) results in a “drooping” of the injected current by the same amount. 5' I. An arbitrary current waveform could be injected if the test signal were electronically ' integrated before being differentiated by the coupling capacitance. A standard integrator f circuit (Fig. 6A) will not work in practice because unavoidable DC offsets in the input signal are also integrated and quickly drive the output into saturation. A slowly acting feedback : Design of the Patch Clamp Figure 5. Distributed capacitance in the feedback resistor. A: An evenly distributed capacitance results in the same frequency dependence of Z (and thermal noise) as a lumped capacitance. B: An uneven distribution introduces an extra dispersion in Z. C: A circuit for correcting the frequency response as would result from the network in B. R; is chosen small compared to Rl to minimize interaction between the controls. Because a large-valued capacitor is then needed to set the slow time constant, a large variable capacitor is synthesized using ...
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