Capacitance compensation and bridge balance adjustment in intracellular recording from dendritic neurons

Journal ~4' Neuroscience Metho&, 27 (1989) 51 75 51 Elsevier

NSM 00897

Capacitance compensation and bridge balance adjustment in intracellular recording from dendritic neurons

C.J. Wilson and M.R. Park

Department of AnatomY and NeurobiMogv. Unil~ersi O, of Tennessee. Memphis, TN 38163 (U.S.A.)

(Received 10 May 1988) (Revised 23 August 1988l

(Accepted 27 August 1988)

Key word~: Intracellular recording; Capacitance compensation: Dendrite: Electronic property neuron: Microelectrode

Under most experimental conditions in intracellular recording, the proper adjustment of the amplifier is essential for the interpretation of the signals recorded from neurons. It is considered possible to accurately align the capacitance compensation and bridge balance adjustments of the amplifier simultaneously with the recording of membrane potential of an impaled cell if a number of conditions are met. In the strictest sense, these conditions are met only if: (1) the neuron is isopotential and if its electrical behavior can be adequately described using a single exponential decay constant, and if (2) that decay time constant is much longer than that of the microelectrode. These conditions cannot usually be satisfied, Because intracellular adjustment of capacitance compensation and bridge balance is necessary in many circumstances, it is desirable, to know whether any of the methods for performing these adjustments are accurate when used under less strict constraints, and to assess the nature and degree of the error that can be expected when the constraints are ignored. The results of computer simulations of a simple intracellular recording amplifier, microelectrode and a dendritic neuron model consisting of an isopotential cell and terminated finite equivalent cylinder representation of the dendrites are presented here. These studies show that the introduction of fast components of the response to intracellular current transients by the redistribution of applied charge in dendrite neurons may sometimes make it impossible to correctly apply the conventional methods of capacitance compensat ion and bridge balance. If the high-frequency response of the intracellular recording amplifier has sufficient fidelity, however, these adjustments can be made to a sufficient degree of accuracy using the response to sine wave calibration signals of varying frequency.

Introduction

Measurement of the electrical potential across cell membranes presents a serious technical problem. The quantity that is available for measurement is the potential difference between the reference electrode and the input node of the amplifier, but this is not the quality we wish to measure. The interesting quantity is instead the potential between the reference electrode and the microelec-

Correspondence: C.J. Wilson, Department of Ana tomy and Neurobiology, University of Tennessee, Memphis, 875 Monroe Avenue, Memphis, TN 38163. U.S.A.

trode tip. We may either assume that the microelectrode makes a negligible contribution to the potential measured at the amplifier input node, or we must simulate the effect of the microelectrode and computationally remove the components of the response that the simulation indicates must be artefacts of the recording apparatus, Whenever possible, of course, the former approach should be applied. However, the condition required to make it valid (vanishingly small electrode time constant) is often not available in intracellular recording experiments. The experimental circumstances that necessitate the simulation approach, and the most convenient means for doing so, have been exten-

0165-0270/89/$03.50 ' 1989 Elsevier Science Publishers B.V. (Biomedical Division)

52

sively tested and discussed for the past 35 years (e.g. Araki and Otani, 1955; Frank and Fuortes. 1956; Ito, 1957: Fein, 1966). That work has demonstrated that the properties of microelectrodes are often sufficiently simple to allow an accurate description in terms of only two variables, the electrode's resistance and its capacitance. The electrode resistance is considered to be con- centrated at the tip, while capacitance is taken to be distributed along the wall of the electrode. These parameters of the electrode can be extracted from the waveform recorded at the amplifier input node during passage of simple current waveforms through the electrode. The simulation of an electrode having these properties can then be imple- mented in the analog circuitry of the recording amplifier and the effects of the electrode can be subtracted from the recorded signal in real time. The resulting processed signal should be a faithful reconstruction of the potential occurring at the tip of the microelectrode, but there is no direct way to confirm its accuracy.

There are reasons to suspect that the simulation, and the processed waveform, may not always be accurate. The most important of these is the observation, made by nearly everyone who has at tempted the procedure, that the properties of the microelectrode are not stationary. It is expected that as the microelectrode is advanced into the tissue, its capacitance will increase due to the increased immersed surface area of the microelectrode wall. The resistance of the tip is also observed to change substantially, presumably because it encounters tissue components that partly occlude the aperture at the tip. Some authors have reported large shifts of the microelectrode resistance that have accompanied the penetration of cell membranes (e.g. Schanne, 1969). These changes require constant attention to the accuracy of adjustment of the simulation parameters of the amplifier (usually capacitance compensation and bridge balance), and frequent compensation for changes in microelectrode properties. Most authors have advised that these parameters be measured and adjusted immediately after impaling a neuron, to insure that recent changes in microelectrode properties are taken into account (Purves, 1981: Park. 1983).

The requirement that the electrode parameters be measured intracellularly introduces an additional complication. The response to any signal injected into the nficroelectrode will now reflect the electrical properties of the impaled neuron, in addition to those of the microelectrode, and the electrical properties of the neuron arc unknown. Often, their measurement is the ultimate goal of the experiment. There is no general solution tc, this problem. Reconstruction of a time-varying cell membrane potential requires an accurate simulation of the microelectrode, yet measurement of the electrode properties requires previous knowledge of the electrical properties of the neuron. Note that this complication is independent of the need to record neuronal responses to injected current waveforms. It is therefore not eliminated by application of recording technique,~ that sep- arate current injection and membrane potential measurements in time (e.g. Brennecke and Linde- mann, 1974: Wilson and Goldner, 1975: Finkel and Redman, 1984).

Several practical strategies for intracellular measurement of microelectrode capacitance and resistance have been proposed. These all rely on some special electrical property which is believed to be common to all neurons. The history of these methods has recently been reviewed by Park et al. (1983). Currently, virtually all of the methods in use are variants of the one introduced by lto (1957). In this approach, it is assumed that the electrical contribution of the neuron, like that of the electrode, can be adequately represented by a parallel capacitance and resistance. It is further required that the product of these parameters (the time constant) for the microelectrode be much smaller than that for the cell. If these conditions are met, the response to sufficiently high frequency signals will be determined almost entirely by the electrode. The properties of the electrode can be measured using such signals and used to correct subsequent signals recorded at the amplifier head stage.

The assumption that the neuron will behave as a simple RC circuit is very restrictive. It is not likely to ever be strictly valid, and the degree to which the assumption is violated probably varies greatly among neurons. There are two known

sources for this variation. One is the non-linear behavior of neuronal membranes. Some neurons have very nonlinear membrane electrical properties due to voltage-sensitive ionic conductances that are engaged at or near the resting membrane potential. These conductances alter the membrane resistance dynamically in response to changes in membrane potential. If they respond relatively rapidly to changes in membrane potential (at rates comparable to the membrane time constant), they can introduce high-frequency components into the neuronal responses to injected current transients used to measure the microelectrode time constant. Depending upon their amplitudes and frequencies, these signals would be either attributed to the microelectrode (resulting in a systematic electrode compensation error) or make determination of the electrode's contribution to the response ambiguous (making accurate compensation of the electrode impossible). The second source of variation in the electrical responses of neurons is geometrical. Rail (e.g. 1977) has shown that the cylindrical processes of neurons introduce high-frequency components into the responses of neurons to t ransmembrane currents. These may likewise con- fuse the discrimination of microelectrode properties from those of the cell and make compensation for microelectrode properties ambiguous or even introduce systematic errors. The degree to which this may occur depends very much on cell shape.

For the present report we have employed a computer simulation of a common amplifier design, a simple microelectrode model, and the neuron model introduced by Rall (1969), to assess the degree to which neuronal geometry may confound measurement of linear microelectrode and neuronal electrical parameters. Some of these findings can also generalize to the case of electrical non- linearity of the cell membrane.

Methods

Simulations were run using an interactive pro- gram written for the Apple Macintosh computer. The amplifier simulation was based on the circuit described by Park et al. (1983). It is a simplified version of a common amplifier design including

53

the most common features of these instruments. It is shown in Fig. 1. The input node of the amplifier is attached to: (1) the head stage amplifier (A). with fixed unity gain and an ideally flat frequency response. (2) the input shunt capacitance of the amplifier, ~,ht, nt (a stray capacitance). (3) a capacitance compensation feedback circuit (through the capacitance neutralization capacitor Q,,,,~). and (4) the current injection circuit (through current injection resistor R ...... and its stray parallel capacitance Qut)- For the simulations reported here. Qhum was fixed at 1 pF, C~.,,mp at 1 pF. R ..... was 100 M~2. and Qur was 5 pF except where other- wise noted. Current injection was controlled with the command voltage V~om,~,n a. The potentiometer Transient Balance placed a portion of the command voltage into the inverting input of the capacitance neutralization amplifier (which has a gain of 10), to provide an adjustable neutralization of the stray capacitance Q.~,r. This implementation of transient balance is not used by all manufac- turers of intracellular recording amplifiers, but it is the simplest and the most effective. The Capaci- tance Compensation potentiometer applies an adjustable amount of positive feedback to the amplifier input node through the capacitance compensat ion capacitor. The Bridge Bahmce potentiometer subtracts a portion of the command voltage from the amplifier 's output. Note that this is the only adjustment that does not have implica- tions for the voltage at the input node of the amplifier, and so does not affect the amplitude or time course of the current injected by the amplifier.

The electrode is represented by a simple parallel RC circuit. The electrode capacitance is parallel with the amplifier shunt capacitance, and so these two values can be combined. The neuron (Z~.~n) is represented as an isopotential somatic compar tment and a finite uniform equivalent cylinder for the dendritic field, as originally proposed by Rail (see review in Rail, 1977).

The 4 currents summing at the input node of the amplifier can be appreciated from Fig. 1. They are: (1) current flowing from the electrode (/pa~p), (2) current from the capacitance compensation circuit ( I~ ) , (3) current from the current injection circuit (l~.u~), and (4) current flowing through the

54

Vcommand

- °

i vou pu Vinput i shunt~, l lC c

- t _ Relectrode Capacitance Compensation ~

Celectrod _ £ .

_ ~ Zcell

"-d"

Fig. 1. A schematic diagram showing the amplifier circuit used for the simulations. The input node of the headstage, V, npu t, is connected to: (1) the simulated electrode and cell (not shown), (2) the shunt capacitance Csh,n t associated with the headstage, (3) the output of the capacitance compensation amplifier (C) through the compensation capacitor C~.omp, (4) the current injection amplifier (B) through the current injection resistor Rcu r and through the stray shunt capacitance associated with the current injection resistor C~u ,, and (5) the input of the headstage amplifier (A). The gains of the headstage amplifier and the current injection amplifier were always one. The gain of the capacitance compensation amplifier, CCg~m, was adjustable. It was 10 for most of the simulations

presented here. The bridge amplifier (D) produces the final output signal, V,,mp~t.

amplifier's shunt capacitance (/shunt)" Current through the input resistance of the head stage amplifier is assumed to be zero. The amplifier is therefore represented by the expressions below:

/prep = /cur + I c c -- /shunt ( l )

Ice. = Ccomp • d[ ( CCgai n " ( Vjn- Pcc - V~ . . . . . d" Ptb ))

-- Vin ] f a l l (2)

I~.~ = V~ . . . . . d/Rcur + C~or" d[ V~ . . . . . d ]/dt (3)

/shunt = Cshunt' d[ Vin]/dt (4)

/prep = Ce~e~,rod." d [ V ~ . ] / d t

+ Vin/(Relectrod e + Zcell ) (5)

In these expressions, V m is the voltage at the input node of the amplifier, Ptb is the proportion of Vc . . . . . . d placed by the Transient Balance poten-

tiometer on the input of the capacitance compensation amplifier, and Pc~ is the proportion of the signal at the input of the headstage amplifier that is placed on the input of the capacitance compensation amplifier by the Capacitance Com- pensation potentiometer. Zce~t is the complex input impedance of the neuron as measured at the soma.

Two different strategies for solving these equa- tions were employed, depending upon whether the solution was intended to be in the time or frequency domain. If the solution was to be in the time domain, a discrete time simulation was employed using a time increment of 100/~s. At each time point, the currents flowing in the previous time sample were used to calculate the voltage across the preparation (electrode and neuron) for the previous time point. The rate of change of voltage at the input node was calculated from the

difference in voltage from the previous sample. The command voltage and its rate of change were givem and together these were used to calculate a new value for the current in each of the 4 paths. These currents and the impulse response of the cell were used to continue the convolution, which was then completed for the next time point. The cell's impulse response was calculated using the method described by Rall (1977) for a lumped soma and finite equivalent cylinder representation of the dendritic tree. When a frequency domain solution was required, the real and imaginary components of the preparation (defined as the combination of cell and electrode and, for compu- tational convenience, the amplifier input shunt capacitance) input impedance were calculated and used to generate amplitude and phase values for steady state AC responses to sine wave inputs. The solutions for the cell input impedance are essentially the same as those obtained by previous authors (e.g. Rall and Rinzel, 1973; Rall and Segev, 1985). They are listed below:

1 q = (1 +j~0r):

1 + (~0~t2)~ + 1

2

1 + ( , o , f ) ~ - 1

+J 2

= RE. + j . IM~

R < = { Rd -JR<-sinh(2-La .d .RE t

- sin(2 r e . t ] }

I m d = - -

x {[ REq + IM2q] • [cosh(2. Ld~.d" REq)

-1 - C O S ( 2 " L d e n d ' I M q ) ] }

{ Rd~" [IMq" sinh(2, gdend" eEq)

R E q ' s i n ( 2 " L d ~ , d ' I M q ) ] }

X {[ R E 2 + IM2q] • [cosh(2. Ldend" REq)

55

1

- cos(2 - Ld~nU - 1Mq )] j

RE ........ = n e , - (1+I~ ) 2)

R ........ = R~.,, • (1 + p)

R NOllltl " ~ T I M ....... = 1M~ -

e ~OlI]~A Rdv c =

p" coth( Ld~,d )

R&d, = RE, = { ( RE,): . t REd + t RE~ ). t R & )'-

+(RE~) (IMd) ~- • + ( I M p ) 2 (REd) }

× { ( R E ~ + R E a ) 2 + ( I M + I M d ) 2}

).: IM~.~q, = IM,. = {( RE~)'-. ( I M d ) + ( RE,, . ( 1M

+ (IMp) 2"(1Md) + ( 1 U , ) ' ( I U d ) ' - }

): X {(RE, , - t - R E d -Jr-(Ira -t- lMd) 2} 1

In these, RE~ and IM~ are the real and imaginary parts of the somatic input impedance, RE d and I M d are the real and imaginary parts of the dendritic input impedance, and RE~, and lM~,~l I are the real and imaginary parts of input impedance of the entire cell. R~om, , is the input resistance of the soma, Rd~: is the input resistance of the dendrite if it were infinitely long, Lug,, d is the length of the dendrite in length constants, and 0 is the dendritic dominance for the finite equivalenl cylinder.

Having the real and imaginary parts of the cell input impedance, the amplitudes and phases for steady state AC signals at all of the amplifier nodes are easily obtained as a function of the frequency of the input signal. The expressions for the current through the current injection circuit 1 .. . . the voltage at the amplifier input node, and the voltage at the output node of the amplifier are given below:

[/c . . . . . . . . d" (1 + joa ( (7[,~,~- R ~.u~ ) ) /cur

R c u r

+j~0 Ccomp(( Vin "Pc, -- Vc ............ t" Ptb)

• c % , ~ - v , ~ )

56

Let

Aq

Ac 2

/cur

= ( Pcc" Cqain - - 1)" (,O~comp

= ~ " (~cur " Ccomp • Ptb" Cqa in )

{( 1 + / M p . AC~ = V~ . . . . and" R~or

- A c l ' A c 2 " R E p ) + j ( R E p . Ac, , R cu r

× {(1 + IMp. Ac,)2 + (RG- Ac,)2} ~

Phase/cur

=arc tan

+ AC 2 (1 + IMp. Acl) REp. Ac l

R~ur

1 + IMp. Ac~

+ ( A c 2 ) 2

R cur

Amplitude lcur

Ac 1 - Ac 2 . REp

(1 + I M p . Ac,)2+ ( REp" Acl) 2

Imp] Phase Vi, = a r c t a n [ ~ I + Phase 'cut

Amplitude Vin = [(REp)2 + (iMp)2]~

• Amplitude Icu r

Phase Vou t

= arctan[ {Amplitude Vin- sin(Phase Vi, )}

× {Amphtude Fin" cos(Phase Vi. )

- - Vcommand • P b b } - 1 ]

Amplitude Vout

= [(Amplitude Vin. sin(Phase V,n ))2

+ (Amplitude Vi,. cos(Phase Vin)

] - - g c o m m a n d • e b b ) 2 "~

Dendritic length and the dendritic dominance ratio were extracted from the time constants and intercepts of the first two exponential components of the derivative of the response to a step current using the method described by Kawato (1985).

Results

Capacitance Compensation The adjustment of capacitance compensation is

critical to the success of all subsequent steps in the electrical alignment of the amplifier and in the measurement of membrane potential both in the presence and absence of injected currents• The reason it is so critical is because it determines both the fidelity with which voltages at the electrode tip are represented at the input stage of the amplifier, and also the extent to which currents injected through the microelectrode represent the intended current waveform. The effect of capacitance compensation on the waveform of recorded signals is universally recognized. The importance of capacitance compensation for the current injection is less obvious, but its cause is the same. Some proportion of a transient current applied to the top of the microelectrode will be used to alter the charge on the glass wall of the microelectrode, rather than be passed through the tip and into the cell. To make the current injected into the neuron an accurate representation of the current command signal, the amplifier must inject a current waveform into the top of the electrode that is very different from that desired at the electrode tip. The difference between the two should accurately reflect the current that will be lost through the e l e c t r o d e c a p a c i t a n c e . The amplifier must accurately calculate this current waveform. To do so, some representation of the electrode time constant must be present in the circuitry of the amplifier. The source of this infor- mation is the setting of the capacitance compensation circuit. If it is incorrectly adjusted, the amplifier will not only fail to accurately measure the voltage at the electrode tip, but it will also inject an incorrect current waveform. The implementation of this in the amplifier circuit of Fig. 1, as in nearly all intracellular recording amplifiers,

follows directly from the design of the capacitance compensation circuit itself. The current injection circuit feeds directly into the input node of the amplifier, which is also the point at which the capacitances are nulled by positive feedback through the capacitance compensation circuit. If the compensation is complete at that point, any of the current applied through the current injection capacitor (Rcur) that is lost through the electrode or amplifier capacitances will be compensated precisely by the positive feedback.

Can all capacitance be compensated? The success or failure of any capacitance com-

pensation strategy depends critically upon the accuracy of the assumption that all the capacitative losses are lumped at the input node of the amplifier. There are two main sources of capacitance that cannot be compensated because they are not in parallel with the amplifier input capacitance. One of them is a stray capacitance that often occurs in parallel with the current injection resistor (Ccu r in Fig. 1). The second is electrode capacitance that behaves in a nonideal way (see e.g. Finkel and Redman, 1984). Electrodes whose capacitance cannot be compensated should be avoided whenever possible. Luckily, many electrodes do behave very closely to the ideal way required by intracellular recording amplifiers. There are a larger variety of solutions to the problem of nonideal resistors. One solution is included in the circuit shown in Fig. 1, and is represented by the Transient Balance control. It is intended to compensate specifically for the stray capacitance around the current injection resistor. The effect of that capacitance is to overemphasize high-frequency components of the current injection waveform. (It also acts as a capacitance to ground through the output impedance of the current injection amplifier, but this effect can be compensated by the capacitance compensation.) As a result, the onset and offset of injected current pulses will have sharp artifactual transients, and will produce an artifactually sharp voltage record. Fig. 2 shows examples of this effect on the response to a depolarizing current step in a simulation of the same amplifier, electrode and cell, with and without the addition of 0.5 pF uncorrected

57

A.

Recerded Vm

Aclual Vm

[20 mV

"; : I ~ .0 ms F

J l OnA

B.

Recorded Vm - -

Actual Vm

Fig. 2. The effect of an uncompensatible capacitance on the adjustment of capacitance compensat ion using the response to current steps. In both A and B, Recorded V m is the potential measured at the amplifier headstage, A c t u a l V m is the actual membrane potential, shown at the same scale, Actua l 1 m is the current injected into the cell, and C o m m a n d is the command voltage applied to the current injection amplifier. The electrode capacitance was 3 pF, the electrode resistance was 50 M~ . The neuronal input resistance was 40 M~2 and the time constant was 5 ms. In A, all capacitance can be compensated, and several traces at different levels of capacitance compensation are shown. At optimal compensat ion (the trace showing the discontinuity), the response component due to the electrode and amplifier shunt capacitance becomes effectively instantaneous, and the electrode's component of the response is purely resistive. Increasing the capacitance compensation further would cause the amplifier to oscillate. In B, a 0.5 pF shunt capacitance is present in parallel with the current injection resistor (Ccu r in Fig. 1). The high frequency positive feedback that results from this capacitor produces an apparently instantaneous component of the response at all levels of capacitance compensation. It is not possible to tell by simple inspection which is the optimal setting of capacitance compensation. It is in fact the one with the largest initial overshoot, both in injected current, and in recorded membrane potential. A response that appears correct, in the middle of the range shown, actually represents a substantial undercompensat ion of the

input capacitance.

58

shunt capacitance around the current injection resistor. The superimposed waveforms show the voltage and current waveforms observed with no bridge balance and with a series of values for capacitance compensation. The voltage waveforms are Recorded V m, the voltage recorded at the amplifier headstage, Actual V m, the actual membrane potential of the neuron, and Command, the command voltage driving the current injection circuit. Another trace, Actual Ira. shows the current flowing from the tip of the electrode (and therefore across the cell membrane) in response to the command voltage. The ideal case, in which there is no uncompensatible capacitance, is shown in A. At low levels of capacitance compensation, a significant proportion of the current injected at the amplifier input node is lost through the amplifier and electrode capacitances, causing the current flowing from the electrode tip to underestimate the command voltage during the beginning of the step. The size and duration of this lag in the injected current is decreased as the capacitance compensation is increased, until at the optimal setting of this control, the actual injected current matches the command waveform. Like- wise, the voltage recorded at the amplifier headstage shows a substantial capacitative loss of high frequency components that can be completely compensated. At the optimal setting of the capacitance compensation, the electrode appears as a pure resistance, causing an instantaneous step in the Recorded V m, followed by a slow response that reflects the time constant of the neuron (in this case a simple isopotential neuron with a single time constant of 10 ms). Increases of capacitance compensation beyond the optimal setting cannot be shown in the figure because they invariably caused the amplifier simulation to oscillate.

The effect of an uncompensatible capacitance ((~ur) is shown in Fig. 2B. Because this capacitance acts as a bypass of the current injection resistor R~u~ for high-frequency components of the injected waveform, it produces an initial boost to the initial transient of the injected current. When the capacitance was purposely undercompensated by 1 or 2 pF, there appeared to be an improvement of the response. As the capacitance compensation was increased, however, an over-

shoot in injected current and in the actual and recorded membrane potential resulted. At settings of the capacitance compensation that were still suboptimal for compensation of the known input capacitance of the system, these transients became very severe. The setting of capacitance compensation that appeared to produce the best looking response in the example in Fig. 1B was in fact one that produced an almost 1 pF undercompensation of the compensatible capacitance of the electrode and amplifier. The transient observed at the optimal setting of the capacitance compensation looked clearly in error, and would never be chosen as the correct setting of the control.

The appearance of good capacitance compensation in a particular situation, for example on the rising phase of an intracellular current pulse, does not guarantee proper alignment of the amplifier. In the case just described, the compensation is only correct for the particular case of the transient due to an injected square wave pulse of a certain size. Another signal, for example an action potential in the recorded neuron, would not be faithfully reproduced by the amplifier.

Any objective method that might be proposed for the adjustment of capacitance compensation requires prior knowledge that there is no uncompensatible capacitance in either the amplifier or the electrode. In addition to their adjustments for capacitance compensation, therefore, some amplifiers provide a means of adjusting their high- frequency response for current injection. There is no agreement about the frequency range that is required, or on the means to implement the adjustment. The simple control used in the simulation and called Transient Balance in Fig. 1 is designed to counteract the effect of the stray capacitance Ccur. It is not the same thing that is called Transient Balance on most commercial amplifiers. There is in fact no common agreement as to what a control called Transient Balance

should do, and it does something different on nearly every different make of amplifier. Regard- less of the implementation or the name given the controls, all amplifiers should either faithfully re- produce signals and accurately inject currents in a frequency range from 0 Hz to at least 10 kHz, or should provide some way of adjusting them so

that they do. The l inear behavior of an ampl i f i e r can be tested by app ly ing a series of sine wave signals to the input as descr ibed by Park et al. (1983). In that method , a s inusoidal vol tage of

10

GAIN

10

0.1

001 1.0 I'0 I00 1040 10060

+ 1 8 0

+ 9 0

k90

-180

FREQ (Hz)

~ F

ct

+ lpF

Fig. 3. Adjustment of the transient balance using injection of sinusoidal currents in the absence of capacitance compensation. In these simulations, capacitance compensation was disabled so that the electrode and amplifier input capacitances were uncompensated. The electrode resistance was 50 M£, and the total uncompensated input capacitance was 5 pF. There was no cell. The three curves show log of the gain (normalized to the electrode resistance) and the phase shift between the current command voltage and the voltage at the amplifier headstage as a function of the log of frequency. When all uncompensatible capacitances are nulled (Correct), the gain rolls off smoothly at a frequency determined by the time constant of the load at the amplifier headstage. The phase shifts smoothly beginning at a frequency likewise determined and approaches 90° Addition of 1 pF of uncompensatible capacitance (or the undercompensation of Ccu r by 1 pF using the transient balance) is shown in the curve labeled - 1 pF. Overcompensation of Qur by 1 pF (equivalent to adding a 1 pF negative capacitance around Rcur) is shown in the curve labeled +1 pF. Undercompensation of the current injection shunt capacitance causes a phase lead because the current injected into the electrode experiences a phase lead through Q,r- Overcompensation produces a phase lag relative to that in the perfectly compensated condition, because of the inversion (180 ° phase shift) at the inverting input of the capacitance compensation amplifier and then the phase lead through the current injection capacitor. Undercompensation and overcompensation of the current injection shunt capacitance produce

equal high-frequency boosts of the gain.

59

known phase is app l i ed di rec t ly to the input node of the ampl i f i e r in the absence of an electrode. and the ampl i f i e r inpu t capac i t ance is accura te ly measured and hulled. The sine wave genera to r is then connec ted to the cur ren t c o m m a n d input of the ampl i f i e r and the ampl i f ie r heads tage is connected to g round th rough a low resis tance load (whose t ime cons tan t will be vanishingly short). The cur ren t in jected th rough that resistor is then measu red and c o m p a r e d with the current com- m a n d waveform. Transient Balance is ad jus ted so that there is no phase di f ference between these two signals over the ent i re f requency range of interest . Since this is pure ly an ad jus tmen t of the amplif ier , it should not be necessary to readjust it dur ing an exper iment . A n a l ternat ive me thod that can be pe r fo rmed dur ing an exper imen t employs a series of sine wave current c o m m a n d s appl ied while the ampl i f i e r is a t t ached to a simple, l inear load such as a well behaved e lec t rode immersed in saline or ex t race l lu la r fluid. Both the capac i t ance neu t ra l i za t ion and b r idge ba lance circuits should be d i sab led dur ing this a l ignment procedure . The pe r fo rmance of the ampl i f i e r is assessed by examina t i on of the a m p l i t u d e and phase of the ou tpu t sine wave relat ive to that of the input sine wave as a func t ion of frequency. A s imula t ion showing the results ob t a ined in the la t ter test s i tua t ion is shown in Fig. 3. Because the e lec t rode and ampl i f i e r input capac i tances are uncom- pensa ted in this case, the amp l i t ude of the ou tpu t sine wave is a t t enua t ed at high frequencies, and it suffers a phase shift that should smooth ly ap- p roach 90 ° at high frequencies. The results of add i t i on of a 1 p F shunt capac i t ance a round the current in jec t ion res is tor ( - 1 pF) , or overcom- pensa t ion of the Trans i en t Balance control by an equivalent a m o u n t ( + 1 p F ) are c o m p a r e d with the expec ted response for a well ba lanced ampl i - fier (Correct). As seen in that figure, the phase is very sensit ive to u n c o m p e n s a t i b l e capac i t ance at high frequencies, and thus a very precise adjus t - men t of the t rans ien t ba lance cont ro l is possible. The shape of the gain response is also character is - tic of the p r o p e r behav io r of the amplif ier . No te tha t at the op t ima l sett ing, the gain of the ampl i - fier is min imized in the f requency range at which the phase shift is a p p r o a c h i n g 90 °.

60

Strategies for capacitance compensation There are a var ie ty of me thods for capac i t ance

c om pensa t i on in c o m m o n use. One of them is i l lus t ra ted in Fig. 2A. In this method , the br idge ba lance cont ro l is d i sab led so that the response of the e lect rode to a cur rent pulse is p rominen t . Because it boos t s the h igh-f requency componen t s of the waveform recorded at the ampl i f ie r input , the capac i t ance compensa t i on is increased unti l the response to cur ren t onset or offset appea r s to have an ins tan taneous componen t . If the e lec t rode is immersed in sal ine or is ext racel lu lar at the t ime of the test, it should be poss ib le to make the response to a cur rent pulse perfec t ly square. A n y further increase in capac i t ance c o m p e n s a t i o n should make the ampl i f ie r osci l late in response to a current pulse. A second me thod in c o m m o n use employs a pulsed cons tan t vol tage ca l ib ra t ion signal p laced in the record ing circuit in series with the p r epa ra t i on (on the reference side). In this method , the capac i t ance compensa t ion is in- c reased unti l the edges of the ca l ib ra t ion pulse are square. As prev ious ly descr ibed (Park et al., 1983), this m e t h o d sys temat ica l ly u n d e r c o m p e n s a t e s e lec t rode capac i t ance and should not be employed. It in fact will lead to exact compensa t i on of the inpu t capac i t ance of the amplif ier , wi th no compensa t ion for the e lec t rode capac i t ance at all. This is because the e lec t rode capac i t ance is in para l le l with the e lec t rode resis tance for the reference signal, ra ther than forming an a l ternat ive pa th for cur ren t as it does for in t race l lu la r po ten- tials. The reference vol tage pulse me thod is therefore an excel lent way to measure the inpu t capac i - tance of the amplif ier , bu t should not be used for compensa t ing the effect of the e lec t rode capaci - tance. A thi rd c o m m o n l y used me thod for capac i - tance compensa t i on is the phase me thod suggested by Park et al. (1983). The basis of this method , is i l lus t ra ted by the s imula t ions shown in Fig. 4. They s imulate on ly the e lec t rode and amplif ier , as in the ins tance of an e lec t rode in sal ine or ex- t racel lu lar fluid. The b r idge ba lance ad jus tmen t of the ampl i f ie r is d i sab led (ad jus ted to its m i n i m u m posi t ion) . The flat f requency response of the ampl i f ie r if it were a l igned perfec t ly is shown as a do t t ed line. The amp l i t ude and phase of the reco rded vol tage relat ive to the s inusoida l cur rent

lOT ̧

GAIN

10 1 I I 1 i No C o m p e n s a l J , ~ .

0.01 . . . . . ~ - 1 0 10

+180

s~

4-90 !

o ] , !

90 t

100 1000 10000 FREQ (Hz)

180 L

Fig. 4. Capacitance compensation adjustment in the absence of a cell. The arrangement is the same as in Fig, 2, except that transient balance is set to its optimal position, and the capacitance compensation is varied. The optimal capacitance compensation is shown as a dotted line in both the gain and phase plots. The responses observed with no capacitance compensation, and with 1 and 2 pF of capacitance undercompensation ( - 1 pF and - 2 pF, respectively), and with 1 and 2 pF of capacitance overcompensation ( + 1 pF and + 2 pF) are shown in both the gain and phase plots. The adjustment of capacitance compensation is particularly easy in the absence of a cell, because it is only necessary to adjust capacitance to give the smallest phase shift possible over the largest possible frequency range. The phase advance produced by capacitance compensation at high frequencies can be understood if it is appreciated that the voltage at the input node of the amplifier lags the current at the same point by some amount less than or equal to 90 o. The capacitance compensation amplifier applies an additional current signal that leads the voltage at the input node by 90 °, regardless of the amount of lag in the input node voltage. Thus it can produce a voltage phase lead of up to 90 o when overcompensated (as in the curves marked + l pF and + 2 pF), and there can be a voltage lag of up to 90 o when capacitance is undercompensated. The maximum gain will be achieved when the current injected through the capacitance compensation circuit is just sufficient to produce a zero voltage phase shift (dotted line), which is also the optimal setting for

the faithful reproduction of signals at the electrode tip.

c o m m a n d vol tage are shown for the case of no capac i t ance c o m p e n s a t i o n at all, and for var ious amoun t s of unde r and over compensa t ion . Several things should be n o t e d here. Firs t , when app ly ing sine wave c o m m a n d voltages, the ampl i f i e r does not osci l la te in an uncon t ro l l ed fash ion when the

c a p a c i t a n c e is o v e r c o m p e n s a t e d , bu t ins tead y ie lds

a pos i t ive phase shif t a n d a dec rease in gain. Smal l

e r rors in c a p a c i t a n c e c o m p e n s a t i o n are read i ly

d e t e c t e d in the phase shif t seen at h igh f r e q u e n -

cies, and ga in at h igh f r equenc i e s is m a x i m i z e d at

the pos i t i on of o p t i m a l c a p a c i t a n c e c o m p e n s a t i o n .

lntracellular capacitance compensation T h e phase sens i t ive m e t h o d m a y be m o r e

e l a b o r a t e t han w h a t is r equ i r ed for c o m p e n s a t i n g

c a p a c i t a n c e w h e n an e l ec t rode is i m m e r s e d in

sa l ine or ex t r ace l l u l a r fluid. In this s i tua t ion , s im-

p ly squa r ing up the r e sponse to a cu r r en t pu l se

c o m m a n d is p r o b a b l y e n o u g h ( a s s u m i n g tha t the

ampl i f i e r is k n o w n to be phase accu ra t e to fre-

quenc i e s b e y o n d the h ighes t ones of interes t ) . But

the d i f f i cu l ty o f c a p a c i t a n c e c o m p e n s a t i o n is

g rea t ly inc reased if it is a t t e m p t e d wi th an in-

t r ace l lu la r e lec t rode . Th is is because it a b s o l u t e l y

requi res the a s s u m p t i o n of a p a r t i c u l a r m o d e l for

the e lec t r ica l n a t u r e of the n e u r o n b e i n g r eco rded .

In a lmos t all t r e a t m e n t s of the p r o b l e m , it is

a s s u m e d tha t for p u r p o s e s of a m p l i f i e r a d j u s t m e n t

the n e u r o n can be r ep re sen t ed as a s ingle t ime

cons t an t , that is, as a pass ive i sopo ten t i a l m e m -

b r a n e pa tch . It is well k n o w n tha t in this case if

the e l ec t rode t ime c o n s t a n t is m u c h shor t e r t han

the t ime c o n s t a n t of the cell ( o n e - t e n t h o r less, it is

o f t en said) then a d j u s t m e n t of the c a p a c i t a n c e

c o m p e n s a t i o n , and the br idge , can be m a d e us ing

the r e sponse to the r is ing or fa l l ing edge o f a

cu r r en t c o m m a n d pulse. Th is m e t h o d is i l lus t r a t ed

by the s i m u l a t i o n s shown in Fig. 5. T h e u p p e r

t race ( labe led t9 = 0.05) shows the resul t for a 50

M ~ e l ec t rode wi th a t ime c o n s t a n t o f 0.015 ms

r e c o r d i n g f rom an i sopo ten t i a l n e u r o n wi th an

i n p u t res i s tance of 40 M g and a t ime c o n s t a n t o f

5 ms. T h e c a p a c i t a n c e was u n d e r c o m p e n s a t e d

s l ight ly (less t han 0.1 p F ) to r e m o v e d i s c o n t i n u i -

ties f rom the r e sponses and so c rea te a m o r e

real is t ic l ook ing trace. T h e e l ec t rode c a p a c i t a n c e

was e f fec t ive ly c o m p e n s a t e d , so that the r e sponse

of the n e u r o n was r e spons ib l e for all o f the g radu -

al ly inc reas ing vo l t age o b s e r v e d in the trace. T h e r e

is a c lear in f l ec t ion p o i n t tha t i nd ica t e s the beg in -

n ing of the c h a r g i n g cu rve of the cell. A d j u s t m e n t

of the c a p a c i t a n c e c o m p e n s a t i o n in this case

shou ld be p e r f o r m e d to sha rpen that i n f l ec t ion

61

Recorded Vm

Reco rded Vm - -

2 0 m s

C o m m a n d - -

Fig. 5, The problem of bridge balance in a dendritic neuron. Transient responses to intracellularly rejected current pulses in a simulation of an isopotential cell (O = 0.05l and in a cell dominated by its dendrites ( p - 50.0) are shown as they appear at the amplifier output. In both cases, the electrode resistance was 50 Mr2, the electrode capacitance was 3 pF, the cell input resistance was 40 M~-2, the membrane time constant was 5 ms, and the length of the dendrites was 1 length constant. ('apaci- tance compensation was adjusted to within 5¢~: of the optimal setting, and the bridge balance was completely disabled so that both the electrode's and the cell's components to the step response were apparent. This simulates the situation m which the bridge balance adjustment is made in intracellular recording experiments. In the case of the isopotential cell, the dramatic difference between the very fast transient due to the electrndc (after optimal capacitance compensationl and the very slov, charging transient of the neuronal membrane creates a sharp inflection point which can be used to determine the electrode resistance for bridge balance. The voltage drop attributable to the electrode is indicated by the arrow. Because of fast charge redistribution in the dendritic neuron, there is no ~,harp inflection point to distinguish the electrode's transient from that of the neuron. The fastest time constants of charge redistribution on the cell are comparable to those of the electrode. The electrode resistance is the same in the two simulations, so the contribution of the electrode is the same as m the upper case. The correct point to use in bridge balance is again indicated bv the arrow. No inflection point is apparent at that point. making accurate bridge balance "~ery difficuh to achieve in the usual way. In an actual experiment, this failure of the technique might be incorrectly attributed to undercompensation of capacitance or non-ideal behavior of the electrode or the

amplifier.

p o i n t as m u c h as poss ib le . T h e e x t r e m e l y r ap id

ini t ial de f l e c t i on is due to the res i s tance o f the

e lec t rode . T h e b r i d g e b a l a n c e con t ro l cou ld there-

fore be ad ju s t ed to b r i n g the in f l ec t ion p o i n t to

the base l ine vol tage . T h e s econd t race ( l abe led

O = 50.0) shows the s a m e s imu la t ion , bu t ins tead o f an i sopo t en t i a l sphere , the n e u r o n is a long

u n i f o r m cy l inder , o n e l eng th c o n s t a n t long. It still

has an i npu t r e s i s t ance of 40 MY2 and a m e m b r a n e

t ime c o n s t a n t of 5 ms. A cy l i nde r c a n n o t be

62

cannot be represented by a single time constant, but rather requires an infinite series of exponential processes for its description (Rall, 1969). The membrane time constant is still responsible for the later portions of the charging curve of the cell, but the initial portions of the curve are greatly stee- pened due to the effect of much more rapid charg-

A t0

GAIN

1.0

0 . 1

0.01 1.0

+180 "

+ 9 0 -

-90

-180

1 0 B

GAIN

1.o

0 . 1

001 10

+180 "

+90

90

180

L d e n d =

110 100 1000 10000 FREQ (Hz)

9=25

L d e n d = 0 . 5

t

10 100 1000 10000 FREQ (Hz)

p=0,1

ing curves of the equalizing time constants. The amplitudes and time constants that describe the high-frequency components of the responses of dendritic neurons are dependent upon the electronic length of the dendrites. In compact dendritic neurons, as are known to occur in the central nervous systems of man3.' species, these high- frequency components of the cell's charging curve are in the same range as the electrode time constant and are very likely to be confused with it. The result when adjusting capacitance compensation and bridge balance is seen in Fig 5. The charging curve is gradual at all settings of the capacitance compensation, and no inflection point appears to indicate the transition from the charging curve of the electrode to that of the cell. Thus capacitance compensation cannot be adjusted to optimize visualization of this inflection point. Even when the capacitance compensation is welt-adjusted, as in Fig. 5, there is no inflection point that can be used to set the bridge balance control, and the correct position to use for bridge balance alignment (indicated by the arrow) is not apparent.

Fig. 6. The effect of dendrites on the steady state frequency responses of passive simulated neurons. Gain in all cases is normalized to the input resistance of the neuron. In all plots, the neuron input resistance was 40 M~, and the time constant was 5 ms. A model neuron with infinite dendritic length is shown in the upper plots, and the lower plots show the responses of a cell with dendrites extending to 0.5 length constants. In both cases, the ratio of the dendritic to somatic conductance varies from 0.1 to 25 in 4 steps. A neuron model consisting of only a dendrite ( p - 25) tends toward a phase shift of 45 °, rather than 90" at high frequency. When there is any significant somatic component to the conductance, however, it will come to dominate the response of the neuron when the input frequency becomes high enough. At relatively high frequencies the distinction between short and long dendrites also becomes inconsequential. The cause of the phenomena demonstrated in Fig. 5 is apparent in these plots. As the dendrites become more and more influential, the response of the cell extends into higher and higher frequency ranges, and it becomes impractical to predict at what frequency the cell can be assumed to make no significant contribution to the response. These response components are present m the early phases of the responses to steps, such as the one shown in Fig. 5, and they are in the frequency range in which all signals are

normally attributed to the microelectrode.

The responses of dendritic neurons to sinusoidal intracellular currents

The amplitude and phase of the ideal response of dendritic neurons of a variety of forms to sinusoidal currents injected into the soma are shown in Figs. 6 and 7. These simulations all employed the standard finite terminated cylinder representation of the neuron introduced by Rall (1969) and by Jack and Redman (1971). This model assumes that the neuron can be treated as consisting of an isopotential compartment repre- senting the soma and a cylindrical compartment that is connected to the soma at one end and terminated in an open circuit at the other. The Bode plots shown in Fig. 6 were generated using two dendritic lengths: (1) 50 length constants (effectively infinite), shown in the upper plots, and (2) 0.5 length constants, shown in the lower plots. Within each plot the ratio of dendritic to somatic conductance (dendritic dominance, 0) is shown varying from 0.1 (an effectively isopotential neuron) to 25 (an almost totally dendritic neuron). In both plots, the effect of the dendritic membrane on the contribution of high-frequency response components is apparent in the gain plots. When the neuron is isopotential, the cell in these simulations contributes very little to any response seen beyond 2 kHz (the membrane time constant is 5 ms), and practically everything seen at that frequency or above could be accurately assumed to arise from the amplifier or electrode. The addition of dendritic membrane to the neuron, however, increases the frequency range of the neuron's response substantially, so that components in the range normally attributed to the electrode (com- pare with Fig. 4) are actually due to the neuron instead. Likewise, the range of frequencies over which the neuron creates changes in the phase of the output is also extended.

The high-frequency responses of dendritic neurons, and especially the phase of those responses, are also sensitive to the length of the dendrites. This is apparent in the comparison between Fig. 6A and B. A neuron consisting of only isopotential membrane (O = 0.1 in Fig. 6A and B) shows an especially simple response. At low frequencies it acts as though it were simply a resistance and has a gain determined by the input resistance and

63

a phase shift of zero. As the frequency is increased to the point at which the reactance of the membrane capacitance becomes comparable to the membrane resistance, the total impedance of the membrane drops and the membrane voltage begins to lag behind the current. When the reactance of the capacitance drops sufficiently so that it is the primary determinant of membrane impedance, the gain approaches zero and the phase shift approaches - 9 0 ° . When there is a dendrite, the response of the neuron is a composite of that due to the isopotential part, the soma, and the dendritic cable. The dendrite does not tend toward a phase shift of 90 ° at high frequency, but rather 45 ° (Fig. 6A, P = 25). (This is because at high frequency the resistive and reactive parts of the

10

GAIN

1 0

0.1

0 0 1 - - 1.0

+180

+90

0

9O

0,5

l i0 ~ ~ . 1 O0 1000 10000 FREQ (Hz)

i , i ,

5.0 1 ¸ • • ,

L=0.01

-180

Fig. 7. The effect of dendr i t i c length on the s teady state f requency responses of passive s imula ted neurons. In all plots,

the neuronal input res i s tance was 40 M~2, the dendr i t i c domi-

nance rat io was 5.0, and the t ime cons tan t was 5 ms. Dendri t ic

length de te rmines the lower l imi t to the frequency range in

which the dendr i t e causes the response of the neuron to devia te

from that expected for an i sopotent ia l neuron. This is true for the gain plot, where the dendr i t e causes an enhanced sens i t iv i t \

to high f requency signals, and also to the phase plot. where

there is a local m i n i m u m in the phase shift in a frequenc5 range de t e rmined by dendr i t i c length. This local m m i n m m , which is marked for each dendr i t i c length, is the range in which the dendr i t e causes the phase shift to approach 4 5 ° and

so is the region in which the phase of the response is most inf luenced by the presence of the dendri te .

64

dendritic response approach each other in magni- tude.) When the dendritic dominance is very large, the asymptotic phase value at high-frequency corresponds to that of the dendrite. If the somatic component is present, however, it will tend to become dominant at high frequency because the membrane capacitance of the soma becomes a short-circuit to very high-frequency signals, and less and less current is diverted out along the axial resistance of the dendrites. This progression is most plainly seen when the dendrite is infinitely long, as in Fig. 6A. When the dendrite is relatively short, the response is somewhat more complicated. This is because at low frequencies, redistribution of charge along the dendrite can stay in phase with the input waveform and the dendrite behaves in the fashion predicted by its DC steady state response. Although there are longitudinal dendritic currents flowing, they are in phase at all points along the dendrite, and the behavior of the dendrite is not qualitatively different from the soma. This is apparent in a comparison of Fig. 6A and B. Despite increasing dendritic dominance, the simulated neurons in Fig. 6B show responses that mimic that of a purely somatic neuron for a wide range of low frequencies. As the frequency of the input waveform increases, however, out of phase longitudinal currents begin to form in the dendrite and the responses of the neurons become more similar to those for the infinite dendrite. To sufficiently high-frequency input signals, all dendrites appear to be infinitely long, and the effect of dendritic length disappears when the frequency is high enough. The effect of dendritic length is shown more systematically in Fig. 7, where the dendritic dominance has been fixed at 5 and dendritic length has been varied. The frequency at which the dendrite takes over dominance of the input impedance of the neuron is seen to increase as the dendrite becomes shorter, and it can be appreciated that the effect of dendritic length on the response of a neuron to an input signal can only be appreciated using signals of a restricted frequency band. Below that frequency range dendrites all act approximately as though they were isopotential membrane patches, and above that range of frequencies they all act approximately as though they were infinitely long.

Capacitance compensation with a dendriti~ neuron

The principle underlying the use of the dec- trode's phase shift for compensating capacitance intracellularly is illustrated in Fig. 8. The upper plot was generated using an isopotential neuron model, with no uncompensatible capacitance, and with the bridge balance adjustment disabled. The response of the cell model alone is shown as a dotted line, and the responses obtained using the amplifier and 0-2.0 pF of uncompensated electrode capacitances are shown as solid lines. As always, the gain values are normalized to the input resistance of the cell. The response obtained with perfect capacitance compensation (labeled compensated) is the cell response summed with the resistive component of the microelectrode. At low frequencies the input impedance of the cell is substantial and plays a part in the response. At higher frequencies the response of the cell is seen to roll off, and the recorded response is almost completely due to the electrode resistance in isolation. Because resistances do not produce phase shifts, the phase of the recorded voltage signal approaches zero as the contribution of the neuron becomes negligible at high frequencies. Adding uncompensated capacitance, either in the form of undercompensation for the electrode capacitance (indicated as negative numbers) or overcompe~qsa- tion (shown as positive capacitance ~alues), decreases the amplitude of the responses seen at the amplifier headstage because the current injected through the positive feedback capacitance compensation pathway is not in phase with that in the current injection circuit. This phase shift is especially large at high frequencies, and so it is easy to detect even small errors in capacitance compensation. This is the basis of the phase method described previously by Park et al. (1983). Its application to a neuron consisting of only a dendrite is shown in the lower pair of graphs in Fig. 8. As expected from the behavior of dendritic neurons already described, the contribution of the neuron to the amplitude of the response observed at high frequency is much greater than that seen for an isopotential cell. Thus, even at frequencies above 2 kHz, a small but important phase shift is seen in the response with a perfectly compensated electrode. We cannot, in principle, adjust the phase

shift to zero at these frequencies and trust that this is the optimal setting of the capacitance compensation. In practice however, when the electrode resistance is approximately as large as the input resistance of the neuron or larger, this phase shift will be very small compared to that generated by even very small imperfections in the capacitance

10

GAIN

10

01

0 01 10

+180

+90

90

180

.d

10

GAIN

10

01

001

+180

+90

90

Soma Only

180

....................................................... ' " . . .

lqO

compensated

• - ~ - . ~ + o . s pF .... '............ +2 0 pF

.... "'-...

i

10~0 1000 100C)O FREQ (Hz)

+20 pF , . , - / ~ / .......... ~ ~ ~ 5 p~ ,

.......... ~ - " ~ - O 5 pF ............... -20 p ~

• ", ................. _ _

Dendrite Only

compensated

............. pF .......................... ~2.0 pF

",........ ................ '"'"'....

0 10 100 1000 100O0 FREQ (Hz)

~ _ . . . . . . . . + 0 ~ p~

-20 pF

65

compensation. Furthermore, the rate of change of phase with frequency is very low near the correct compensation, while small errors in capacitance compensation produce artifactual phase shifts that grow very rapidly with increasing frequency. In addition, the rapid decrease in the amplitude of recorded sine waves as capacitance becomes less compensated adds another dimension to the infor- mation used to detect these errors. Therefore, while some additional caution should be exercised when employing the phase method for capacitance compensation in dendritic neurons, the practice of this method is much less affected by deviations of the neuron from isopotentiality than are other tech- niques. In general, somewhat higher frequencies should be employed than those recommended by Park et al. (1983), however, to account for the wider frequency spectrum of the responses of dendritic neurons.

Bridge Bahmce Adjustment of the bridge balance control does

not affect the current passed through a microelectrode or the waveform appearing at the input node of an amplifier. Because it is nothing more than the subtraction of an attenuated version of the current command waveform from the final re-

Fig. 8. Intracellular capacitance compensation using sinusoidal currents. In all plots, the responses of the cell ahme. in the absence of any influence of the amplifier or electrode, arc shown as dotted lines. Gain is normalized to the cell input resistance, which was 40 M~'. The solid lines are responses observed at the output of the amplifier, ~ith bridge balance disabled and capacitance compensation adjusted to the optimal setting, and to 0.5 pF or 2.0 pF overcompensat ion ( +0.> pF and +2.0 pF) or undercompensation ( - 0 . 5 pf: and 2.() pF). The upper plots show the case of an isopotential neuron, while the lower plots show the responses of a neuron con~,isting of only dendritic conductance. Although the presence of the dendrite implies that the optimal capacitance compensation will not be exactly located at the position thal gives a zen~ phase shift at high frequencies, even very small errors in capacitance compensation produced very large and recogniz- able phase shifts that were clearly similar to those observed in the isopotential neuron. In practice, capacitance compensation bTy the phase method offers nearly as good accurac} in dendritic neurons as in isopolenlial neurons if the electrode resis- lance is large (comparable in size to the input resistance of the

cell or greater).

66

corded voltage, it could actually be done computationally after the experiment, and in some in- stances this may be a good strategy. It is very important for the measurement of the input resistance of neurons, however, and no matter when it is performed, it is critical that it be performed properly. If an inflection is prominent on the onset and offset transients of the response to a step current indicating the transition between the charging curve of the electrode and that of the cell, and if the time courses of these two charging curves are sufficiently different so that we can say that the neuron has not yet begun to charge at the time that the electrode is fully charged, then bridge balance is relatively easy. It is often noted in practice that this condition is not always met, and in fact the considerations described above show that it may not be expected to occur very fre- quently when recording from neurons that have dendrites, even when the electrode capacitance is properly compensated. An alternative method for adjusting bridge balance is therefore desirable. Park et al. (1983) suggested an alternative using the passage of sinusoidal intracellular currents that is complementary to the phase method for capacitance compensation. It relies on the assumption that the phase shift between an artifi- cially applied t ransmembrane current and the membrane potential waveform that it creates should approach 90 ° as the frequency of the sinusoidal current becomes large and the trans- membrane current comes to be dominated by that flowing through the membrane capacitance. The component of the recorded waveform that is due to the electrode resistance (the electrode's behavior is assumed to continue to be dominated by its resistance at this frequency) has no phase shift. It is therefore in phase with the current command signal and can be effectively nulled by the bridge balance, leaving only the out of phase cellular component. These relationships are clear in the upper plot in Fig. 9, which simulates the effects of bridge balance using an isopotential cell (Soma Only). Park et al. suggested the use of a phase lock-in amplifier, which could be used to precisely measure the in phase component so that it could be minimized, but this amplifier is probably not necessary in most cases. Instead, it can be seen

from Fig. 9 that the sine wave recorded at the output of the amplifier will be minimized at high frequency when the bridge is optimally balanced, and that the adjustment can be made by minimiz- ing the amplitude of the recorded output signal. It is only required that the minimization be made at a frequency that is high enough so that the overcompensated amplifier shows a higher gain than the compensated one. This happens at frequencies below 1000 Hz for most neurons.

The different phase shift observed for a dendritic neuron complicates this procedure however, as can be seen in the lower plot of Fig. 9. The reason for this is that the neuron continues to have a resistive, in phase, component to its response at quite high frequencies, and that this neuronal signal can be artifactually removed from the signal if the amplifier 's output is minimized over a wide range of very high frequencies (see the + 10% curve between 0 and about 2 kHz in the gain plot in the lower half of Fig. 9). If we were certain that the phase shift for this cell should be 45° at high frequencies, we could use the phase shift, but we cannot even be certain of that. The correct phase shift for a particular frequency (even a very high frequency) depends very much on the electrical and geometrical properties of the neuron that are not known at the time of the experiment (Figs. 6 and 7). It is still possible to find frequencies above which the optimal bridge balance is one that minimizes the amplitude of the output signal, but these frequencies may be much higher than seen for the isopotential cell. In the example shown in Fig. 9, using the bridge to minimize the apparent gain of the amplifier will lead to overcompensation of the bridge for sine wave signals up to about 2 kHz.

It is easy, however, to determine the frequency that is required to accurately balance the bridge. Beginning with the bridge undercompensated, increase the amount of bridge balance signal gradually. If the frequency of the test sine wave is high enough, the amplitude of the output signal should gradually approach a minimum, and then smoothly begin to increase again. At the point of minimum gain, the phase of the output signal should lag the input signal by an angle between 45 and 90 ° . If no clear minimum is reached, or if the phase lag

increases b e y o n d 90 ° as the a m p l i t u d e c o n t i n u e s to decrease, the f r equ e n cy shou ld be inc reased a n d the test repea ted . A l t h o u g h the m i n i m u m f requency requ i red for accura te b r idge b a l a n c e c a n n o t be abso lu te ly d e t e r m i n e d b e f o r e h a n d , in prac t ice it is a lmos t a lways be low 10 kHz, a n d is usua l ly sub- s tan t ia l ly be low this (Fig. 9 is an ex t r eme case

chosen to d e m o n s t r a t e the p h e n o m e n o n ) . At the

10

GAIN

1.0

01

001 10

+180

+90

90

180

Soma Only

10 100 1000 10000 FREQ (Hz)

compensated

10

GAIN

10

01

001 10

+ 180

+90

90

- 180

Dendrite Only

I'O to'o ~o+o 10060 FREQ (Hz)

- 10%

compensated

67

t ime of the a d j u s t m e n t , the test desc r ibed above can be used to d e t e r m i n e whe the r a n y pa r t i cu l a r f r equency is high enough . Th i s p r o c e d u r e al lows the phase m e t h o d to c o n t i n u e to be used with n e u r o n s of a r b i t r a r y geomet ry . N o c o r r e s p o n d i n g a p p r o a c h is a p p a r e n t for use wi th the responses to

t r ans ien t s .

The behavior of real amplifiers The ampl i f i e r r ep re sen t ed in the s i m u l a t i o n s

differs f rom real i n t r ace l lu l a r r eco rd ing ampl i f ie rs

in tha t its c o m p o n e n t s have perfec t ly l inear p rop - ert ies at all f requencies . At each stage in the

c i rcui t of a real ampl i f ie r , p e r f o r m a n c e is de- g r aded by n o n i d e a l b e h a v i o r of its c o m p o n e n t s ,

a n d a l t h o u g h these n o n i d e a l p roper t i e s m a y be

m i n i m i z e d by good des ign they c a n n o t be e l i m i n a t e d a l together . O n e i n d i c a t i o n of the dif- fe rence b e t w e e n real ampl i f i e r s a n d the ampl i f i e r

s i m u l a t i o n used here is the absence of any reso- n a n c e in the f r equency responses shown in Figs. 4 a n d 8. Whi l e the ampl i f i e r s i m u l a t i o n wou ld oscil-

late in r e sponse to a t r a n s i e n t inpu t , the ga in in the s teady s ta te r e sponse to s ine wave inpu t s never

Fig. 9. Intracellular bridge balance using sinusoidal currents. Simulations are as in Fig. 8, except that capacitance compensation is set at its optimal position, and the bridge balance is brought to with 10% of its optimal value. In each case the plots show the result with bridge balance set at its optimal position (compensated), slightly undercompensated ( 10%) and slightl\' overcompensated (+10%). In the upper plots the neuron is isopotential (Soma Only), while the lower plots show a neuron model in which all the conductance is dendritic (Dendritic Only). In the isopotential case, the correct setting for bridge balance is easily identified as the point at which gain is minimized over a wide range of higher frequencies starting around 100 Hz. Alternatively, the bridge could be balanced by adjusting for a constant, approximately 90 °, phase shift at high frequencies. In the dendritic neuron however, these strategies are much less reliable. At sufficiently high frequencies gain still reaches a minimum at the optimal bridge balance setting, but this may require the use of frequencies in excess of 5 kHz. The phase rule is even less reliable, and phase may not approach 90° in the practical experimental frequency range. In more realistic model neurons, which have some component of somatic conductance, the results are intermediate between those of the upper and lower plots, and bridge balance using the minimal gain criterion is possible with test frequencies

between 500 and 2000 Hz.

68

one, even with extreme capacitance overcompensation that caused phase leads up to 90 °. In addition, the simulated amplifier showed a perfect frequency response when properly aligned, and so could be made to contribute nothing to the response to inputs of any frequency. Real amplifiers have a limited frequency range. The effect of non-ideal behavior of amplification stages will also vary according to the role taken by the amplifier within the circuit. For example, the limited frequency response of the capacitance compensation amplifier will produce a very different kind of anomalous behavior than the same frequency response limitation when associated with the headstage. In real amplifiers, different amplifier implementations are often employed in the different parts of the circuit. It is therefore not feasible to simulate the nonideal behavior of an amplifier in a general way. Instead. we tested a real amplifier of a known design, which matches very closely the genetic amplifier used in the simulation. The amplifier was built by one of us (M.R.P.), and is used routinely for intracellular recording. We have also made less detailed ob- servations on two other (commercial) amplifiers. The characteristics of these amplifiers were qualitatively similar to the one we tested in detail. Their overall performance was no better, and usually was not as good, as the one we report on here.

Fig. 10 shows a Bode plot for the amplifier with its input node attached to a 10 MI2 resistance to ground. Input capacitance was undercompensated by 2.9 pF, producing an effective time constant at the input node of 29/~s. Precise capacitance compensation was unnecessary with the 10 MI2 load resistor. This strategy minimized the effect of the capacitance compensation amplification stage on the overall behavior of the circuit. The transient balance was adjusted so that the voltage record was flat and the voltage phase shift was minimized over the widest possible frequency range, which extended beyond 10 kHz. A slight non-tinearity produced a boost in current injection at the input node at frequencies above 5 kHz that could not be eliminated using the transient balance. Despite this deviation from ideal behavior, the results in Fig. 10 are very close to that obtained in the simulation. A more complete test of the amplifier.

10

GAIN

1

curren~ ~..

J voltage

0 1 ] 10 100 1000 10000 1SOl FREQ(Hz)

100 ~.s

Fig. 10. The practical frequency response of an intracellular recording amplifier. The gain and phase shift of the test amplifier were tested with a 10 M~2 resistor connected between the headstage and ground. The input capacitance of the amplifier was minimized, and was determined to be 2.9 pF. Because of the low input resistance it was not necessary to compensate this input capacitance, and so capacitance compensation was disabled. The transient balance adjustment (functionally equivalent to the one shown in Fig. 1) was adjusted to achieve the flattest possible frequency response. Both gain and phase of the output voltage signal were found to be almost perfectly linear to 10 kHz. There was a slight gain boost at frequencies above 5 kHz, however, as if there were a small residual uncompensatible capacitance in the current injection circuit. This boost is shown in the actual recorded sinusoidal signals in the inset. There the current command and actual current waveforms are both shown, slightly offset so that the command waveform is below the current waveform. The two frequencies

shown in the inset are 5 kHz (left) and 10 kHz (right).

including capacitance compensation is shown in Fig. 11. In this test the same amplifier was connected to a model electrode consisting of a 50 M~2 resistor and a 5 pF capacitor, and a model cell consisting of a 40 M~2 electrode and a 118 nF capacitor. Thus this model corresponds very closely to the simulation shown in Fig. 8 above.

10

GAIN

01

1sol

90

~, overcompensated

f(R electrode ) ~ p e n s a

10 100 1000 10000 FREQ (Hz)

overcompensated

\

Fig. 11. The frequency' responses recorded with the amplifier shown in Fig. 10, using a model neuron and electrode fabri- cated from resistors and capacitors. In these, the electrode resistance was set to 50 M~ and the electrode resistance was 5 pF. The cell model was isopotential (a simple parallel RC circuit) with a resistance of 40 M~2 and a capacitance of 0.118 p,F. The gain of the neuron circuit is shown as the plain solid line in the gain plot. Three plots are superimposed for different settings of the capacitance cornpensation near its optimal level, with a slight undercompensation of the current injection parallel capacitance (closed squares), and one plot (open squares) with optimal adjustment of the Transient Balance. These plots should be compared with those for the simulations shown in Fig. 8. For frequencies below 2 kHz they all follow the pattern set by the simulations very faithfully. With the small amount of uncompensatible capacitance, however, a resonance in the amplifier dominates the response starting at about 2 kHz, and produces a rapid phase lag associated with a peak in the gain and a subsequent loss of gain at high frequency. This resonance was eliminated by proper adjustment of the transient balance control, after which capacitance compensation was successful, Subsequently, adjustment of bridge balance was possible, and brought the recorded response to the line indicat-

ing the membrane response.

The curves p lo t t ed in Fig. 11 represent the input node voltage recorded at three set t ings of the capac i t ance compensa t ion close to the op t imal setting. Al though the response up to abou t 2 kHz

69

is a lmost ident ica l to the s imula t ion shown in Fig. 8, it devia tes subs tan t i a l ly f rom the s imula t ions for f requencies of 5 k H z and above. At these high frequencies, a boos t in gain is fol lowed by a rapid rol l -off and a r ap id phase shift that canno t be c o m p e n s a t e d with capac i t ance compensa t ion . In- creases in capac i t ance c o m p e n s a t i o n beyond the op t imal set t ing p r o d u c e d larger gain boos ts and f requency shifts b e y o n d 90° that were associa ted with resonance behav io r of the amplif ier . At the op t imal set t ing of capac i t ance compensa t ion , the s teady state AC behav ior of the ampl i f ie r could be made to be accep tab le for f requencies up to about 5 kHz. Af te r mak ing these observat ions , an at- t empt was made to improve the ampl i f ie r ' s performance by careful ad jus tmen t of the t ransient balance circuit to op t ima l ly compensa t e for stray capac i t ance in paral le l with the current inject ion resis tor (as shown in Fig. 1). These improvement s removed the h igh- f requency resonance in the amplif ier , and p r o d u c e d Bode plots that closely match those of the s imula ted ampl i f ie r (Fig. 11).

Other strategies

Al though capac i t ance c ompe nsa t i on and br idge ba lance are still poss ib le even in ext remely dendr i t ic neurons using the phase method , the proce- dures involved may seem compl i ca t ed and awk- ward. A poss ib le a l te rna t ive is the use of switching amplif iers . Because this me thod separa tes current inject ion and m e m b r a n e poten t ia l record ing in t ime, it avoids b r idge balance , and the ad jus tment of b r idge ba lance , a l together . Capac i t ance compensa t ion cont inues to be impor t an t in these amplif iers , and a new pa ramete r , the f requency at which the switching occurs, is in t roduced. The p r ima ry use for switching ampl i f iers is in single e lec t rode vol tage c lamp, and they offer many im- po r t a n t advan tages for that technique. The dy- namics of the ampl i f i e r in that method, and the hazards involved in its ad jus tment , have been descr ibed in deta i l s by others (e.g. Brennecke and L indemann , 1974: W i n d o w and Goldner , 1975: F inkel and Re dma n , 1984). It will be cons idered here only in the cur ren t inject ion (somet imes cal led current c l amp) mode.

The s imula t ions in Fig. 12 shows two cycles at the beg inn ing of a trial in which the ampl i f ie r was

70

A. optimal

capacitance compensation

Recorded Vm j . . . 1 Ac!uaJ,,Vm 0.5 ms I 100 mV

B.

3 pF uncompensated

capacitance

Recorded Vm J Aclual Vm

Fig. 12. The effect of undercompensated capacitance on signals recorded with a switching amplifier. In both A and B, the upper traces are voltages appearing at the headstage of the amplifier and the lower traces are the membrane potential. The electrode resistance in these simulations was 50 MY2. The neuron was an isopotential patch with an input resistance of 50 MY2 and a time constant of 5 ms. The traces show the first two switching cycles at low frequency, 1 kHz in an attempt to pass a step current. In A, capacitance compensation was set optimally, while in B there was 3 pF of uncompensated electrode

capacitance.

commanded to inject a positive step current. The cell in these simulations was a simple isopotential patch with a time constant of 5 ms and an input resistance of 50 M~2. The electrode had a resistance of 50 M~2. In A, the electrode and amplifier capacitances were nearly perfectly compensated, while in B there were 3 pF of uncompensated electrode capacitance. The switching frequency was 1000 Hz, and switching duty cycle was set at 36%. The waveforms shown are the voltage at the amplifier head stage, and below them, the actual membrane potential (slightly offset). They il- lustrate the similarity between the situation for the switching amplifier and that for the active bridge. The charging and discharging curves observed at the amplifier input and the actuaI injected current waveform depend critically upon the capacitance compensation. The rate at which switching can

safely occur is determined by the charging and discharging of the electrode and amplifier capacitance, and these are determined primarily b~ capacitance compensation. In the example shown in Fig. 12, it is a relatively easy task to determine the correct setting of capacitance compensation, because the cell has a single time constant which is very different from that of the electrode. Finkel and Redman (1984) prescribe that there be a 1 : 10 ratio of the time constants of the electrode and the cell, and that switching should occur at a rate that allows for the passage of several electrode time constants between offset of one current pulse and the onset of the next. As they point out, this prescription holds for the effectit~e electrode capacitance. If the electrode capacitance were compensated completely, the switching rate could be increased considerably. This is important because both low and high switching rates introduce serious artifacts (Finkel and Redman, 1984). At too low switching rates, the cell membrane discharges somewhat during the part of each cycle in which the current is off. This effect is shown in Fig. 13A. The desired membrane potential is the smooth charging curve, the actual membrane potential is the superimposed stairstep curve, and the voltage at the amplifier headstage is shown by the transients that decay to the stairsteps. l-he arrows indicate the samples of the membrane potential that are taken at the end of each cycle. Because the membrane potential is relaxing between current pulses, the sampled membrane potential shows a cumulative underestimate of the membrane potential compared with that which should ideally be obtained (the smooth curve). The effective current injection is, because of leakage across the membrane capacitance, effectively less than that intended and the membrane potential is too low for that reason. Increasing the switching rate can remedy this error (Fig. 13B) but because the electrode and amplifier capacitance are not sufficiently compensated, the sampled membrane potential (arrows) exhibit an error in the opposite direction from that in A. Because of these two different kinds of artifacts, there is only a narrow band of switching rates that will produce both an accurate current injection and an accurate measurement of the membrane potential, and it is

not readily apparent from the waveform at the headstage what this switching frequency should be. The procedure employed by most investigators using this equipment is to be certain that there is a period after the discharge of the electrode capacitance in which the headstage membrane potential is flat (see e.g. Fig. 12). The frequency can be set to sample the membrane potential during that flat part of the cycle at the headstage, and to repeat the cycle as often as possible consistent with that sampling. Capacitance neutralization is used to speed recovery from the transient so that a sufficiently flat portion of the response can be created (as in Fig. 12A) and to increase its proportion of the cycle so that the sampling rate can be made to be as fast as possible. The existence of the flat response during part of the cycle is due to the extreme difference between the time constant of the electrode (after capacitance neutralization) and that of the cell. It is easy to tell when the electrode's charging curve is over and the cell's (practically flat) charging curve begins in the trace for the capacitance compensated case in Fig. 12A for

A .

Switching Too Slowly

Actual Vm ', m •

B. Switching Too Fast

71

the same reason that it is easy to see the inflection point in the charging curve to a step current with the bridge amplifier (Fig. 2, upper trace). It is because the cell models in these simulations have no dendrites. When the high frequency components generated by dendrites are present, the charging curves in both cases become more gradual, even when the capacitance is perfectly compensated. Adjusting a switching amplifier so that it samples the membrane potential waveform after completion of these transients is equivalent to attributing these dendritic time constants to the electrode, and will produce an error in the measurement of membrane potential. In the worst

Fig. 13. The two kinds of errors of adjustment that can occur in switching amPlifiers when used in the current injection mode. The simulation is the same as in Fig. 12B, except that the sweep rate is decreased and the switching rate is varied. In each case three waveforms are superimposed. One is the voltage at the headstage, which is recognized because of the large deflections (which are clipped) produced during the current passing portion of the cycle. The second is a stairstep waveform that is the actual membrane potential. The smooth curve is the membrane potential that would occur if a perfect amplifier applied a perfect step current to the cell. The figure of merit of the amplifier is its ability to achieve a membrane potential that matches the smooth curve, and to sample that membrane potential accurately once in each cycle. The sampling point for each cycle is indicated by the arrows. Because the uncompensated electrode and amplifier capacitance produces a long offset transient, correct adjustment of the switching amplifier is not possible. If a slow switching rate is employed, such as in A, the cell membrane discharges somewhat on each cycle during the settling period prior to the sampling point and the onset of the next current injection pulse, the membrane potential decays below the desired value and the samples undershoot the actual membrane potential. This error accumulates during the pulse. In B, the switching rate has been increased, and the actual membrane potential is much closer to the desired value, but the failure of the electrode capacitance to discharge completely between cycles causes the membrane potential samples to be inaccurate. Accurate capacitance compensation is required to find the inflection point on the discharge curve of the electrode that distinguishes the response of the electrode from that of the cell, and then to set the switching frequency so that samples are taken just after the inflection point is reached. This is in some ways equivalent to the bridge balance method. If fast dendritic components obscure the inflection point, which they often may, the adjustment of the switching amplifier will suffer the same difficulties that beset

the active bridge amplifier.

72

case, the dendritic response components may be so prominent that there is no flat portion of the headstage waveform that can be used to judge the best switching cycle (see e.g. the transient in Fig. 2, lower trace). This dilemma is analogous to that confronting adjustment of bridge amplifiers based on the responses to current steps. The use of switching amplifiers in the current injection mode is therefore not a panacea for the problems of amplifier alignment in intracellular recording. This is especially true in situations where fidelity in the representation of fast response components due to charge redistribution in the dendrites is desired.

The impact of amplifier misadjustment on estimates of cellular characteristics

In any real experimental situation there will always be some amplifier misadjustment. The effects of such misadjustment is obvious in some cases. For example, the effect of capacitance compensation misadjustment on measurement of membrane resistance will be negligible, since it affects only fast transient components of the response to current steps, while a bridge imbalance will cause a predictable error in the measurement of input resistance. In other cases, for example in the measurement of membrane time constant, or especially in the estimation of dendritic length and dendritic dominance ratio from the onset and offset transients of current steps, the expected effects of amplifier alignment error are not immediately evident. To examine those, step current onset transients generated from simulations with various amounts of capacitance compensation error were analyzed to determine the membrane time constant, dendritic electrotonic length, and dendritic dominance ratio using the Kawato (1985) variant of the time constant method of Rall (1977). Examples showing the results of these tests are presented in Fig. 14. Because these measurements are made using the derivative of the transient, they do not require the absolute measurement of membrane potential and are therefore not sensitive to errors of bridge balance. This is an advantage over variants of the method that employ absolute values of voltage during the transient responses. The measurement of membrane time constant is also relatively insensitive to errors in capacitance com-

pensation, assuming that the measurement is made from the latest available portion of the transient. As can be seen in Fig. 14, the error introduced by even a relatively large error of capacitance compensation decreases dramatically with time, while the slope used for estimation of the time constant can (and actually must) be measured as late as 1.5 or 2.0 time constants into the response. At earlier times it may be confounded by dendritic equalizing time constants as well as by errors of amplifier adjustment. Measurement of the first equalizing time constant is required for the estimation of electrotonic length of the dendrites. This time constant is much shorter and much more likely to be affected by errors of capacitance compensation. The two cases in Fig. 14 show extreme cases of neuronal geometry using a constant capacitance compensation error. In the upper plot, the dendrite is long and the somatic conductance is very significant. These two factors conspire to create a very long first equalizing time constant. At relatively late times (after the arrow) il is easy to distinguish this time constant from that of the electrode, which only confounds the earliest time points. In the lower plot a more typical case is shown, in which the dendrites are considerably shorter, 0.75 length constants, and account for a much greater proportion of the conductance. In this case, although measurement of the membrane time constant is not compromised, the first equalizing time constant is indistinguishable from the electrode time constant. The estimate of dendritic length will be in error, and there is no objective indication that anything is amiss.

Several methods for estimation of the dendritic dominance also rely on the extraction of equalizing time constants (see review in Duran& 1984; Kawato~ 1985). These rely on measurement of the intercept, as well as the slope of the equalizing time constant, and they are even more sensitive than electrotonic length to errors in capacitance compensation as well as noise and other sources of error. Finally, it is possible in principle to estimate the degree to which somatic membrane resistivity differs from dendritic resistivity using a combination of neuron input resistance and the time constants and coefficients of exponential components of the transient response to current pulses or steps

( D u r a n d , 1984; K a w a t o , 1985). T h e s e tes t s a r e the

m o s t d i f f i c u l t of all f r o m t h e p o i n t o f v iew of

a m p l i f i e r a d j u s t m e n t , r e q u i r i n g a ve ry p r e c i s e

c o m p e n s a t i o n for b o t h t he r e a c t i v e a n d r e s i s t i ve

c o m p o n e n t s of t he e l e c t r o d e a n d a m p l i f i e r ' s re-

s p o n s e .

Discussion

A l t h o u g h ac t ive b r i d g e c a p a c i t a n c e c o m p e n s a -

t i on a m p l i f i e r s a re in c o m m o n use a n d h a v e b e e n

for s o m e t ime , t h e r e r e m a i n m i s c o n c e p t i o n s a b o u t

t h e i r o p e r a t i o n a n d s o m e i n c o r r e c t f o l k l o r e a b o u t

t h e i r a d j u s t m e n t . O n e c o m m o n b u t p e s s i m i s t i c

v iew is t h a t they s i m p l y d o n o t w o r k , e i t h e r be -

c a u s e e l e c t r o d e c a p a c i t a n c e is t oo c o m p l e x to b e

Ldend = 2.0 p = 0.5

dV/dt (volts/sec)

10

OA , , , , , , , , , , , , , , , , , , , , , , , ,

5 10 15 20 25 30 Ttme (ms)

1°°1~:-~ Ldend = 0.75 !

t :: p = 5.0

dV/dl 101 !"'\

v°UIO 0 . 1 ~

0 5 10 15 20 25 30 Time (ms)

73

c o m p e n s a t e d , o r b e c a u s e o f l i m i t a t i o n s of the im-

p l e m e n t a t i o n of t he a m p l i f i e r des ign . C u r i o u s l y ,

m a n y of t he p r o p o n e n t s of th i s v iew c o n t i n u e to

use t he a m p l i f i e r , o r the s w i t c h i n g v a r i a n t t h a t is

k n o w n to b e s u b j e c t to the s a m e d i f f i cu l t i es .

A n o t h e r c o m m o n , b u t o p p o s i t e , v iew is t h a t the

d i f f i c u l t i e s in a d j u s t m e n t of the a m p l i f i e r de-

s c r i b e d in m o s t m e t h o d o l o g i c a l p a p e r s a re exag -

g e r a t e d , a n d t h a t in fac t the a m p l i f i e r s are very

r e l i a b l e a n d c a n b e u s e d w i t h o u t r e g a r d for p r o b -

l ems s u c h as t h o s e d i s c u s s e d here . N e i t h e r of these

is s u p p o r t e d b y o u r resu l t s . T h e c a p a c i t a n c e of

s o m e e l e c t r o d e s m a y b e u n c o m p e n s a t i b l e , a n d

s o m e a m p l i f i e r s m a y n o t h a v e the f r e q u e n c y re-

s p o n s e o r p h a s e l i n e a r i t y r e q u i r e d to c o m p e n s a t e

e l e c t r o d e c h a r a c t e r i s t i c s proper ly , , b u t c o r r e c t ad -

j u s t m e n t of w e l l - b u i l t a m p l i f i e r s is p o s s i b l e for

m o s t k i n d s of e l e c t r o d e s . A c a v a l i e r a t t i t u d e to-

w a r d a m p l i f i e r i m p l e m e n t a t i o n or a d j u s t m e n t

Fig. 14. Peeling exponentials in the presence of uncompensated electrode capacitance. The transient to a step current was differentiated and plotted on a log scale against time in the way described by Rail (e.g. 1977). The fine line is a best fit to the time constant governing the last portion of the decay of tile transient. In both simulations the membrane time constant was 10 ms and was correctly fit despite the presence of 2 pF of uncompensated electrode capacitance. The difference between the membrane potential transient and the response predicted from the time constant is also plotted on the same coordinates in the conventional way for extraction of a second time constant governing the deviation from a simple exponential decay In the upper case, with a long dendrite and low dendritic dominance, the equalizing time constant is much slower than the electrode time constant, and a clear reflection ix seen (at arrow). If a line is fit to the later portion of that curve, it will be an accurate estimate of the first equalizing time constant and the dendritic length can be accurately determined. "l'he remaining fast component (at times earlier than the inflection point indicated by the arrow) is a combination of higher order time constants of the neuron and the electrode time constant, and the higher order equalizing time constants of the cell cannot be extracted in isolation. In the lower case, howcver, with a shorter dendrite and a greater dendritic domimmce, the first equalizing time constant is comparable to the time constant of the electrode. No inflection point is present, and so there is no portion of the second exponential process that i, uncontamimtted by the electrode's portion of the responses. In this case, the transient response to a step current cannot bL' used to make an accurate estimate of the electrotonic length of

the cell.

74

strategy is certain to produce serious errors with measurements of many kinds, however, and these are often not evident from a superficial examina- tion of the recorded data. Implicit in the procedure for alignment of the amplifier is an assumption about the nature of the neuron being studied. If this assumption is incorrect, the alignment of the amplifier could be systematically in error, and the results could be largely determined by that systematic error. Although some assumptions are apparently always required, it is important to make the least restrictive ones possible, to minimize the chances of error of this kind.

The signals of interest in intracellular recording are mostly contained in the frequency range from 0 to 2000 Hz, but the correct alignment of the amplifier during intracellular recording relies on the use of signals well beyond this frequency range. If this fact is not adequately appreciated by amplifier designers, and they sacrifice fidelity at high frequencies to optimize some other design parameter (e.g. noise immunity or the cost of components), the usefulness of the amplifier is badly compromised. The importance of phase linearity at high frequencies has not been sufficiently recognized, and some commercial amplifiers we have examined have substantial phase shifts at high frequency that severely limit their usefulness in situations requiring precise amplifier adjustment. Any useful amplifier should accurately represent the phase, as well as the amplitude, of a sine wave appearing at its input at frequencies up to at least 5 kHz. Amplifiers that can be made to be accurate to 10 kHz or beyond would be v e ~ desirable. It is not sufficient to correct phase shifts arising in the amplifier headstage by phase correc- tion in later stages of the amplifier or in the bridge balance circuit, because these will not correct the phase of intracellularly injected currents. Neuro- physiologists considering the purchase of a new amplifier should insist upon testing the frequency response of any new amplifier and should com- pare it with the results presented here.

Although the use of sine wave signals for the adjustment of capacitance compensation and bridge balance is not common, our results suggest that these signals offer a much more sensitive way of determining the correct alignment of the

amplifier. More importantly, they provide much less subtle indications of misalignments of the amplifier and deviations from ideal behavior of the amplifier and the electrode. The difficulty of implementation of these methods is less than it appears. The measurement of relative phases of sine waves to the precision required here is made simple by the use of an x v oscilloscope display, in which the sinusoidal current injection signal and the recorded signal are used together to form a Lissajous figure, as described by Park et al. (1983). A very accurate impression of the variation of amplitude and phase shifts across a large range of frequencies, that can be readily compared with the Bode plots shown here can be obtained by observing these Lissajous figures during relatively rapid frequency sweeps. Points of minimum phase shift or gain are surprisingly apparent with this method, as are relatively more complex features of the performance of the system, such as regions of relatively shallow or steep slopes in the Bode plots. At the same time, important features of the electrotonic structure of the recorded neuron are also revealed.

Our results show that there is no assumption- free strategy for amplifier adjustment. There is no frequency guaranteed to be high enough so that the component of the response that is due to the neuron can be ignored. There is not even a rule of thumb that can be depended upon entirely. The ratio of the membrane time constant to the time constant of the electrode, the figure of merit used so widely in the past, cannot be used because the amplitude and time constant of fast cellular components due to dendritic charge redistribution is dependent upon factors other than the membrane time constant. Despite the absence of an absolute standard, it is still possible to align the amplifier in practice. This is because the behavior of the electrode and amplifier is (or can be) simple and predictable over the entire frequency range. Above a sufficiently high frequency, the behavior of the entire system, including the neuron, will match that of the electrode and amplifier alone. No guarantee can be made concerning the frequency at which that will occur, so frequencies in a wide range should be tested to determine it empirically. It is possible that in some neurons the cellular

f requency ' r e s p o n s e will e x c e e d t h a t of t he a m p l i -

fier, or will e x t e n d so h i g h t h a t t he r e s p o n s e o f t h e

e l e c t r o d e is u n m e a s u r a b l y sma l l in t he f r e q u e n c y

r a n g e in w h i c h it d o m i n a t e s the ove r a l l r e c o r d e d

r e s p o n s e of t he sys t em. N e u r o n s of th i s k i n d , if

t h e y exist , m u s t b e a d d e d to p o o r a m p l i f i e r c o n -

s t r u c t i o n a n d n o n l i n e a r e l e c t r o d e s as c i r -

c u m s t a n c e s k n o w n to f r u s t r a t e a c c u r a t e a m p l i f i e r

a d j u s t m e n t . N e u r o n s w i t h e l ec t r i ca l p a r a m e t e r s in

the r a n g e b e l i e v e d to b e c o m m o n in t he n e r v o u s

s y s t e m s of m o s t spec ies s h o u l d , a c c o r d i n g to the

s i m u l a t i o n s p r e s e n t e d here , l e n d t h e m s e l v e s r e a d -

ily to th is m e t h o d .

Acknowledgements

S u p p o r t e d by N I H G r a n t N S 2 0 4 7 3 a n d N I H

R C D A N S 0 1 0 7 8 ( to C . J .W. ) a n d N I H G r a n t

N S 2 0 8 4 1 ( to M . R . P . ) .

References

Araki, T. and Otani, T. (1955) Response of single motoneurons to direct stimulation in toad's spinal cord, J. Neurophysioh, 18:472 485.

Brennecke, R. and Lindemann, B. (1974) Theory of membrane-voltage clamp with discontinuous feedback through a pulsed current clamp, Rev. Sci. Instr., 45: 184-188.

Durand, D. (1984) The somatic shunt cable model for neurons. Biophys. J., 46:645 653.

Fcin, H. (1966) Passing current through recording glass micro- pipette electrodes, IEEE Trans. Biomed. Eng., 13:211-212.

Finkel, A.S. and Redman, S. (1984) Theow and operation of a single microelectrode voltage clamp, J. Neurosci. Methods. 11:101-127.

75

Frank, K. and Fuortes, M.G.F. (1956) Stimulation of spinal motoneurones with intracellular electrodes, J. Physiol. (Lond.), 134:451 470.

lto, M. (1957) The electrical activity of spinal ganglion cells investigated with intracellular microelectrodes, Jap. J. Phys- iol., 7:297 323.

Jack, J.J.B. and Redman. S.J. (1971) An electrical description of the motoneurone, and its application to the analysis of synaptic potentials, J. Physiol. (Lond.), 215:321 352,

Kawato, M. (1985) Cable properties of a neuron model with non-uniform membrane resistivity, ,I. Theor. Biol.. 111: 149-169.

Park, M.R., Kita, H.. Klee, M.R. and Oomura. Y. (1983) Bridge balance in intracellular recording: introduction of the phase-sensitive method, J. Neurosci. Methods, 8: 105 125.

Purves, R.D. (1981) Microelectrode Methods for lntracellular Recording and lontophoresis, Academic. London.

Rail, W. (1969) Time constants and electrotonic length of membrane cylinders and neurons, Biophys. J., 9: 1483-15(/8.

Rail, W, (1977) Core conductor theory and cable properties of neurons. In J.M. Brookhart, V.B. Mountcastle and E.R. Kandel (Eds.), Handbook of Physiology, Sect. l, Vol. 1. Cellular Biology of Neurons, American Physiological Society, Bethesda, MD, pp. 39-97.

Rail, W. and Rinzel, J. (1973) Branch input resistancc and steady attenuation for input to one branch of a dendritic neuron model, Biophys. J., 13: 648-688.

Rall. W. and Segev, I. (1985) Space-clamp problems when voltage clamping branched neurons with intracellular microelectrodes. In T.G. Smith, H. Lecar, S.J. Redman and P.W. Gage (Eds.), Voltage and Patch Clamping with Mi- croelectrodes, American Physiological Society, Bethesda, MD, pp. 191-215.

Schanne, O.F. (1969) Measurement of cytoplasmic resistivity by means of the glass microelectrode. In M. Lavallde, O.F. Schanne and N.C. Hdbert (Eds.), Glass Microelectrodes, Wiley, New York, pp. 299-321.

Wilson, W.A. and Goldner. M.M. (1975) Voltagc clamping with a single microelectrode, ,1. Neurobiol.. 6:411 422.

Documents

Capacitance compensation and bridge balance adjustment in intracellular recording from dendritic neurons