10. VOLTAGE CLAMPING OFEXCITABLE MEMBRANES

By Francisco Bezani l la, Jul io Vergara,Robert E. Taylor

and

10.1 . In t roduc t ion

"Animal e lect r ic i ty" was d iscusscd by Galvani in h is Cornnrer tur t o f 1791(Eng l i sh t r ans la t i on by Green l )and . as po in ted ou t by Hodgk in2 i n h i sSherr ington Lecture. Vol ta, in 1800, compured h is bat tery to the stack ofp lates in Lhe e lect r ic f lsh. Thus the observat ion that animals produce orrespond to e lcct r ica l changes is very o ld. Most of the ear ly work on exci tablecel ls . sr . rch as nerve and musclc, r 5 u us done by appl1, ' ing known currents andrrreasur ins the e lect r ica l or mechanical rcsponses. Typical ly , no (or a verysmal l ) response occurs i f the st i r r ru lus is belou ' " threshold." and when thesystem " f i res." one loses contro l . The cxc i tabi l i ty res ides in a sur face mem-brane in the form of channels which are er lbedded in a l ip id b i layer wi th arather large capaci tancc (about 1.04F cmr) . They are e i ther open or c losed,and the proport ion of the t ime any one is open is potent ia l dependent .

To see why control of the membrane voltage. or voltage clamping. is soimportant in the study of exc i table membranes, consider the components ofthe current 1- through a membrane. There are two broad categor ies:current carried by ions and dielectric. or displacement. currents. For fre-quencies above a iew kilocycles per second (kHz), the capacitance C- oft l re membranc is u lmost independent of potent ia l / . . "

- so the capaci tat ive

current is C-ri l/-, 'r l t. There is a small component of capacitative current,uhich is potent ia l and t ime dependent , associated 'uv i th the movement of

' R . N I . ( i r een . Eng l i sh t r ans la t i on o f Lu ig i Ga l ran i ' s " de V i r i bus E l cc t r i c i t a t r s i n u to rurnuscu la r i co l nn ren l i l r i u s

' r r i t h i n t r oduc t i on and o the r t l an : l a t i ons . L i ch t . Cambr i dge . Massa -

chuse t t s . 195 .1 .: A . L . Ho t l gk i n . "The Conduc t i on o f t he Ne r rou : l r npu l se . " Tho rn : r s . Spnng f i c l d . I l l i r r o i s .

196.1.r B . K a t z . " E l e c t r i c E r c i t a t i o n o f N e r r e . " O r l b r d L l n i r . P r c s s . L ( ) n d o n a n t i N e w Y o r k . 1 9 1 9 ." A . L . H o d g k i n a n d W . A . H . R u s h t o n . P r o t . R . . S t x . L r t n d r u t . . S a r . B l 3 3 . 4 : 1 . 1 ( 1 9 . 1 6 ) .5 R . E . Tay )o r . Phy . s . Tcch . B io l . R t ' . s . 6 . l l 9 ( 1963 ) .n H . J . C ' u r t i s and K . S . Co l c . . / . Gc r . Ph . t . s i o l . 21 .757 ( l 9 . 1 t l ) .-

K . S . ( ' o l e . "Membranes . I ons . and I n rpu l scs . 'Un i r ' . o l 'Ca l i l i r r n i a P rcss . Bc rkc l cy , 1972 .

-+-+5

'Lrror)sor:r:\r,r,Rr\rr:\r: \r-r,rr\sr(s.\or. l0 ,, :"] l l l l ] i l i i l , , l : l : ; , i l l : i l l i l i i : , i ' l : : : ; ] i :l s l l \ o - l l ' l l 5 q 6 : 9

446 10. volracE cLAMpTNG oF EXCTTABLE MEMBRANES

charge within the membrane responsible for the init iation of the voltagc-dependent ionic conductance changes which wil l be considered in Section10.3.6.3. It is sornetimes necessary to include the lact that the capacitance islossy, probably because of t l-re loss in the penetrating protcins (sce Taylor( t u l .1 \ .

To a very large extent then, following a sudden change in Z- the membranecurrent wil l be purely ionic after the capacity transient. This transienf wil ldepend on the electronics employed in the feedback system used to clampthe voltage (Section 10.3.4).

The excitabil ity properties of nerve and muscle membranes result from thefact that the membrane ionic conductances are volta_qe dependent. Theseconductanccs result from the flow of ions (most often sodium, potassium,and calcium) through imperfectly selective membrane channels, and themagnitude of the conductance depends upon the fraction of channels thatare open (for revicws, see Ehrenstein and Lecar8 and Taylor.8" For thesquid axon membrane, the behavior is well described by the empiricalequations of Hodgkin and Huxley,' and an excellent introduction can befound in Katz.ro The important point for our purposes is that the current-voltage relations contain a region of negative resistance such that thesystem is stable under potential control.

One of the first attempts to determine the characteristics o[ a system ofth is k ind us ing potent ia l contro l was that of Bar t le t t r r for an i ron wire incontact wi th ac id. rz14 Bart le t t was led to do th is at the suggest ion ofK. S.Cole, 's who later in t roduced th is concept in the study of the squidgiant axon with the use of electronic feedback.T

In order to proceed experimentally, it is necessary to establish conditionswhere a patch of membrane can be isolated over which the current and voltageare unilcrrm (averaged over some smaller area)l and much of the article wil lbe concerned wi th th is quest ion. Colers and Marmontr6 in t roduced tech-niques for isolating small regions of the squid giant axon membrane with theuse of an internal current supplying electrode, and external guard system

t 'R. E. Taylor , J . M. Fernindez, and F. Bezani l la. l r r "The Biophysical Approach toExc i t ab le Svs ren rs " (W. J . Ade l r nan and L . Go ldman ,ec l s . ) . P l enu rn . Ne rv Yo rk . 1982 .

" G. Efrrenstc in arrd H. Lccar, Ann. Rer. Biopht ' .s . Bioena.1,347 (1972).' " R. E. Taykrr , Ann. Rer. Phrs. Chent.25,387 (1971).'A. L. Hodgkin and A. I : . l lux ley, .1. Ph. ts io! . ( l t tndut) l l7 . 500 (1951).

" 'B . Ka t z , "Ne rve , Musc l c and Synapse . " McGraw-H l l l , New Yo rk , 1966 .I ' J . H. Bart let t . Trun.s. EleLtrochem. Sot.87, 521 (1945).'2 R. S. L i l l ie . Bio l . Rt ' r . Cunhr i t lae Phi lo.s. Sot . l l . l8 l (1936).l r L l . L . F r i r nck unc l R . F i t zHugb . 1 . . E ! , ' k t r o t l n , l r . 65 , 156 (1961 ) .'4 R. Suzuki, IEEII Trun:;. Bio-mt'd. En(t. 11, I l,+ ( 1967)r s K . S . Co le , A r t h . S t i . Phys io ! . 3 .253 (1949 ) .r { 'G. Marmont, J. Cel l . C'onry. Ph_ts io l .34,35l (1949).

10.2. ceNEnnL PRINCIPLES oF voLTAGE cLAMP 441

and electronic feedback. This system was improved by Hodgkin et al-l ' ' t8

by the addition of an internal voltage measuring electrode. They introducedthe term "voltage clamp."

More recently, voltage clamp methods have been extended to other

aspects of the membrane current. Some of the most notable recent extensionsof the voltage clamp method are the measurements of channel noise, single-

channel cllrrent jumps, and gating currents. Channel noise is the excess

clectrical noise created by the random opening and closing of the molecular

ionic channels in the membrane. Under favorable circumstances the discrete

current jurrrps caused by the opening and closing of single channels can be

resolved.rs ' The gat ing currents are the d isplacement currents wi th in the

membrane which occur when charged groups of the channel macromoleculenrc rearranged dur ing channel opening or c los ing. t8b

In this paper we describe the principles of single-cell voltage clamping and

discuss the basic theory and dil l icult ies found in different preparations.

We do not consider results o[ voltage clamp experiments. For reviews of

vol tage c lamping in general , see Cole and Moore, t 'Moore and Cole,2oMoore,2r and Katz and Schwat't2.22 For reviews of results of voltage

clurr rp ing. scc Hodgkin, r El r renstc in and Lecar ,8 Taylor .8" and Bczar t i l la andVcrgara. : r "

1O.2, General Pr inciples of Vol tage Clamp

In an ideal system thc curretrt measured (withtlut distortion) wor"rld be

that which was flowing across a region of membrane, where the potential

would ins lantaneously i tnd accurate ly fo l low somc t ime sequence ( the

command potential) determined by the experimenter.The ideal system is never achieved in practice, but in many cases can be

approximated quite closely. Some of the l imitations of the real case can be

il lustrated by a simple example. Suppose we would l ike to voltage clamp a

region of a cylindrical cell. In order to control the potential, it is necessary

r r A . L . Hodgk in , A . F . Hux ley , and B . Ka t z , . 4 r ch . S t i . Ph . t " s i o l . 3 ' 129 (1919 ) .r 8 A . L . Hodgk in , A . F . Hux ley , and B . Ka t z , J . Ph t s i o l . ( London l116 ,424 (1952 \ .' t " E. Nehcr ar . rd B. Sakrrann. Nuture (London\260.779 (1976).' " ^ C . M . A rms t rong and F . Bezan i l l a . J . Gan . P l r . t s i o l . 63 . 513 (1974 ) .

" ' K. S. C'o le and J. W. Moore, J. Gen. Phvsio l . . l4 , 123 (1960)t" J. W. Moore and K. S. Cole, Ph. t 's . Teth. Bio l . Res.6,263 (1963).t t J . W. Moore. i r . ' B iophysics and Physio logy of Exci table Membranes" (W. J. Adelmart .

cd.) , p. l , l -1. Van Nostr i tnd-Rcinl ro ld, Pl i t rccton. Nr:w Jcrsey. 1971.

" C; . Katz and T. L. Schwartz, J. ML'nhr. Bio l .17.275 (1974).r r " F . Bcza r r i l l a and J . Vc rga ra . i r r "Mcn rb ranc S t ruc tu r c und Func t i on " (E . E . B i t t a r . cd . ) ,

Vol . 2, p. 51. Wi lcy ( l r ) tcrsci ! -nce ) , Nerv York. 1980.

.l4tr 10 . \ ' o t . I A ( i t : ( l _A t \ l t ' tN ( i o l , LX ( . tTA l l l _ t : l l t F .MBRANt -_s

( o )

10.2. cnNEp,q,L pRTNCTpLES oF voLrAcE cLAMp 449

from other portions of the ccll which may have nonuniforrn potentialdistributions. Under these conditions, the current is no longer being mea-sured from a region of known and controlled polential. The errors couldbe morc than just quant i ta t ive i f the uncontro l lcd regi t ' l ts exhib i t iusta-b i l i t ies.2s Spat ia l uni formi ty of current f low may be impossib le to obta infor a membrane wi th nonuni form propert ies, but spat ia l uni formi ty ofvoftage can be approximated; when it is. it is referred to as a " spuce cluntp."Depending on the typc and geometry of cells, the space clamp condition isobta incd wi th a var iety of methclds. Axia l wi re is considered in Chapter10.3; at tenrpts wi th two and t l r ree microelectrodes in Clrapter 10.4: patchisolat ion wi th p ipet tes in Chapter 10.5; and gap iso lat ion is d iscussed inChaptcr 10.6.

ln the following discussion we ilssume pcrfect space clamp, and thus wepicture the nrembrane subject to voltage clamp as a uniforrn patch of mem-brane. Later. we consider cases wherc the ideal is not met.

Thc basic c i rcu i t is p ic tured in F ig. 2. In th is d iagram we have representedone membranc patch as a box. The external e lcct rode has a res is tunce ofR"o and the in tcrnal e lcct rode a res is tance R" i . l t is very common for theelectrodes to be located at a certain distance from the surface of the membranebecause there is connect ive t issue or advent i t ious cc l ls surrounding themcrnbrane under s tudy. The e lect r ica l equivalenl of these st ructures is aresistance in series with the membrane represented as R, in Fig. 2. Theresis tance o l the current e lect rode is represerr ted by R. . . ,4 . is an operat ional 'ampl i f icr wi th open- loop gain ,4, and ,4o is a d i f l 'erent ia l ampl i t ier wi th gainof l . I t is inst ruct ive to s tar t wi th an ideal ized system where the input impe-dance of both amplif iers is infinite and the dil lerential amplif ier has a flatfrequency response. Under tlrese circunrstances it is possible to derive theequation that relates I/M (the actual membrane voltage) to - I/.o" (thecommand voltage). The final result is

Fr<; . l . Pr incip le o{ 'co ' t ro l l ing membranc vol tage. (a) Schernat ic d iagram of a systcnrwi thout teedback. ( b) Schcrnat ic t l iagram of a vol tage c lanlp systenl wi th nelar ive leeclback.

to measure it. we could introduce a microelectrode (Fig. la) and measure thcmembrane potential as the dilTerence between the internal potential z, anclthe extcrnal potential Li, and supply current by another impaled micro-electrode (1') and collect it with an external electrode (1"). I i the voltagegenerator (E*) had zero output impedance and thcre wcre no res is tance r r rthe current patlr through the electrodes, we might be able to record a mem-brane voltage z- identical to the command tr/.o". These conditions arcseldorn possib le to achieve.

we can improve th is . r rangcment by use of negat ive feedback (F ig. rb) .where the measured voltage z. is compared to the command 2.o", and thedifference is amplil ied by an amplif ier l" that supplies the curreni.necessaryto make v^- L ' ro* :0. The current 1. requi red to contro l the vol tagccan be measured as the voltage drop /, across a resistance R. clearly 1- :V , l R .

Fora cy l indr ica l por t ion o la cel l the arrangement of F ig. lb suf fers f roma serious defect. Althoueh the potential at the point of the inrpaled voltagemeasuring microelectrode may be well controlled, the measured currentcomes not only from the immediate neighborhood of this electrocJe. but also

( 10.2. r )

This equat ion shows that the accuracy o l ' the c lanrp not only depends on ,4but a lso inc ludes membrane propert ies (unless R"" and R. arc zero) . In theideirl case, A + t, for all frequencies, and

Vu : Vcou + 1- R-. (10.2.2)

Note that the actual membrane voltage is rrc,r 'er equul to the ttnununtl aoltuglettnless the ntenhtane (urrcnl or the series lcsislarit 'c is =aro. However, Ir '*,

r r R . E . Tay lo r . J . W . Moo rc , and K . S . ( ' o l e , B i oT r l r r ' , r . . / . l . l 6 l ( 1960 ) .

450 | 0. vol rece ctLAMptNG oF EXCITABLE MEMBRANES

l" tc; . 2. Basic d iagranr o l 'a vol tage c lamp systcr l . The box represents a patch of membrane.l ' " rs the aqtual mernbranc potcrr l ia l . I , , , , , is the rneasurcd mcmbranc potent ia l , / , " is the ment-b ranecu r ren ( ' and l ' . , r n , i s t hccon tn ranc l v ( ) l t i l g c . R , . , . R . , , an t l ( . . l r l c t hc r cs i s t a r r ccs ( ) l c x ten la lvol tage. internal vol tage. and current e lectrodcs. respect ively. R. is the resistancc in ser ies wrththe menlbrane.

the measured voltage, wil l be equal to the command voltage because I/- :V* - I ̂ R.. The efl 'ect of series resistance can be serious in the determinationof the current voltage characteristics of a membrane because the voltageacross the membrane is not controlled and becomes current dependent.r8For this reason the voltage electrodes should be as close as possible to themembrane sur face. However. in some cases the b io logical preparal ioncontains a permanent barrier that effectively prevents positioning theelectrodes right at the surface; in this case, some positive feedback can beintroduced to compensate for the effect of series resistance as we shall seein the chapter on axial-wire voltage clamping (Section 10.3.5).

In modern operational amplif iers ,4 is very large at dc and low frequencies,and Eq.(10.2.2) is a good approximation for that frequency range. How-ever, at high frequencies ;l decreases, and Eq. (10.2.1) should be used. Inpractice this means that all last changes in imposed 7..," wil l not be followedby the membrane as can be seen by inspect ion of Eq.(10.2.1) when.4 isdecreased, and the clamp will perform far from ideal. ln this situation thecurrent-electrode resistance R". becomes very important.

Another very important source of error in practical voltage clamp ori-ginates in the capacitances of the voltage electrodes, which. in combinationwith their resistances. act as low-pass fi l ters. This fi l tering efl 'ect is norrrrallyaggravated by the differential amplif ier, which never has a flat frequencyresponse as assumed in the example. Under these conditions, it is clear thatthe control amplif ier wil l not receive the correct measurement of membrane

10.3. nxrn l -wrRE voLrAGIr cLAMP

vol tage, but i t wi l l be d is tor ted at h igh f requencies; th is , in turn, wi l l have theef fect of producing a d is tor ted error s ignal at the input of the contro l ampl i -f ier, which wil l be unable to clamp the membrane voltage at the commandedvalue. In some extreme situations the phase lag introduced by the measuring-electrodes arnplif ier membrane combination may be enough to renderthe whole system unstable. Some stabil ity characteristics of axial-wirevoltage clamping are discussed in Section 10.3.5.3. Attempts to decrease theresponse time usually increase the tendency of the clamp to oscil late, asdoes series-resistance compensation. We cclnsider below some detailedstabil ity analyses designed to produce a clamp system with series-resistance

compensation and fast rise time without overshoots or oscil lations in themembrane potential.

The above example i l lustrates the main features of a basic voltage clampsyslen ' l which we can l low summarizc:

1. Measurement of membrane potential should be made with low-impedance electrodes and positioned as close as possible to the membrane.

2. The current should be measured from a region where the voltage iscontrolled and uniform.

3. Current electrodes should be of low impedance.4. Amplif iers should introduce minin.rum alterations and phase shift in

lhe f requency range of in tercst .

10.3 . Ax ia l -Wi re Vo l tage C lamp

In the case of long cylindrical cells, the space clamp condition can beapproximated with the use of a long wire inserted axially as described inSection 10.3.1. This technique is restricted to cells of diameters large enoughto allow penetration of the axial wire without damage to the membraneproperties.

The large diameter o[ the giant axon of the squid makes it almost an idealpreparation to study the electrical properties of excitable rnembranes. Infact, the first voltage clamp system was built to control the membranepotent ia l o f squid axons. t5 ' t t ' Hodgkin et a l . t l ' t8 and Hodgkin and Huxleyemade their classic description of the ionic currents using a voltage clampsystem in the squid giant axon. In the squid axon it is possible to introducerelatively large electrodes from one cut end of the fiber. A longitudinallow-resistance current electrode can also be introduced from the ends(axial wire) and serves the dual purpose of passing curreN and attainingspace clamp conditions.

451

t+

452 10. volrecr cLAMptNG oF ExcrrABlrr MEMBRANES

The axial-wire voltage clamp has been also used in other preparatirtnssuch as Mvxit 'ola axons,24 craylish axons,2s and barnacle muscle fibers.26.rl

10 .3 . ' 1 . Cab le Theo ry o f an Axon w i th Ax ia l W i re i nVo l t age C lamp

The axial wire is usually made of a solid platinum wire coated witlrp lat inum black. As a metal i ts res is tance is very low, but when i t is in con-tact wi t l r an e lect ro ly t ic so lut ion l ike the axoplasm, i ts sur face impedancecan be large, depending on frcquency and thc amount and direction ofcurrent passing through the mctal solution intcrface. The minimum repre-sentation of an axial wire inside an axon must include the surface resistanceof thc r . r ' i lc :an cquivalcnt c i rcu i t r l o la scgmr. :nt t r f lcnglh A.r is prcscntcd inFig. 3a, where we havc assumed zcro cxternal resistance to sinrplify treat-ment.t Note that points A and B are at t l.re sante potential V^: therefore therepresentation of Fig. 3b is sti l l equivalent which, in turn, can be representedin equivalent form by Fig. 3c. This is the familiar representation of a cable inwhich the membrane e lement ( the box in the f igure) is in para l le l wi th aser ies combinat ion o[ a vol tage generator l / , ( t l te ax ia l -wi re vol tage) and aresistance r, (the surfacc resistance per unit length of the axial wire).1. Nowwe can find the value of the space constant of the combination axon andaxial wire. For this purpose we represent the membrane by a Theveninequivalent of a bat tery r : - in scr ies wi th the membrane res is tance r_ aspictured in trig. 4a. The final step is to obtain the equivalent of Fig. 4a aspictured in Fig. 4b, where

r : r - r , / ( r - * ru) , 6 : (V^r^ * r ; - ro) / ( r , , * , 'n , ) ;

r is the paral le l conrbinat ion of the mernbrane res is tance and the wiresurface resistance. A good axial wire wil l have r'" much smaller than r-;therefore r wil l be practically equal to r"; furthermore, from the above

I L . B i ns tock and L . G t i l dma n . . l . G tn . P l t t . . y i o l . 54 . 710 (1969 ) .r t P. Shragcr, . l . ( i t ' t t . Ph. t s io l . 64.666 ( lg71l .r " S. Hagiwara. H. Hayashi , and K. Takahashi , . , f . P/rr , . r lo/ . ( t ,ont lut l205, I l5 (1969).2r R. D. Kcyncs. E. Ro. jas. R. E. Taylor . and J. Vergara, .1. Physio l . (Londun)229,409 (1973).

' i .The- casc of f in i tc cxtcr : ra l resistancc is g iven in Taylor t , t u1.23

i . ln a t t to le general t reatnrent the sLrr l i rce inrpedance per uni t length ol ' the axia l wire, : , ,rcplaccs r . , : i t is dcf i ncd as t hc L:rp lacc t rarnstbrrn of t he vol tage d iv ided by ' thc Laplacc t ransformof thc currcnt pcr uni t lcngth of wirc. Thc sarnc cquat i ( ) l ls can bc uscd by s imply rcplacingr. , by : , ar . rd thc t ime solut ion can be obtained by the use of tables or the complex inversronl i r rmula.

F t c .3 . Schema t i cd i ag ra rno fanaxonseg rnen tw i t hax ia l w i r e : ( a ) s imp l i f i ed rep resen ta t i on

before reduct ion: (b) and (c) equivalent c i rcui t at ier c i rcui t reduct ion. For detai ls see text .

equation it can be seen that c will be very close toof the axon axial wire combinatiotr will be

- -A : \ / r l r l .

The above equations show that as the axial wire is made of lower surfaceresistance, ,t decreases. This is in direct contradiction with the general ideathat the introduction of an axial wire in an axon makes its space constantlarger. lt is clear, however, lrom the derivation presented that t lre effect ofthe axial wire is exactly the opposite;it shortens the space constant, makingeach patch of membrane more independent of its neighbors but at the sametime imposing a voltage Voin each patch. This effect is desirable to isolate

10.3. a.xt,q. l-wrRE voLTAcE cLAMP 4s3

a x t a l w l r e

a x o p l a s m

m e m b r a n e

e x t e r n a Is o l u l r o n

V^. The space constant

( r 0 . l . l )

454 10. volra.cg cLAMpING oF EXCITABLE MEMBRANES

Frc . 4 . Theven in cqu i va l cn t o l ' a segmen t o f an axon o f l eng th A . r con ta i n i ng an ax i a l w i r e

corrncctcd to a vol tage source at potent ia l / , . (a) Equivalent c i rcui t o l 'a patch wi th axia l wirc.(b) Thevenin equivalcnt of i r ntcmbrarte patch wi th axia l u i re.

patches of membrane that do not have exactly the same properties as therest of the membrane; in particular, it isolates end effects. This can best beil lustrated solving the cable equation for a terminated cable using the circuitelements indicated in Fig. 4b.

The equation to be solved is

d 2 v l d x 2 : ( v - t ) 1 7 2 , ( 10.3.2)

whcrc steady statc and a constant va luc for r are assumed. f Normal ly .the axon is penetrated in both ends with electrodes and perfusion cannula.We can approximate the boundary conditions by the expression

V(x ) : - ! n ^Yo , a t r : 0 and r : / ' ( 10 ' 3 ' 3 )ri 0x

where R* is the short-circuit resistance at the ends and ri is the internalresistance per unit length.

' lNote that when the membrane resistance is negat ive, a wi l l be i rnaginary. However, theequat ions st i l l hold and the solut io l ts wi l l contain t r igonomctr ic s ines and cosincs. For a d is-cus : i t ' r t o l l h i 5 casc sce C r ) l c - ( p . 14 : ) .

The solution of the differential equation ( 10.3.2) with the above boundarycondit ion (10.3.3) is

I . sinh(x/,l)v : r l t -\ [1 -r Rrlrtlfsinh(l/2)

_ sinhr(/ - p/ij + (Rj4:!T::llfll r4l) (r0 3 4)[ l - (R , r ' , , 1 )2 - l s i nh ( / ' , 1 ) I

To simplify the situation, let us assume further that R, : 0, which is

equivalent to a short circuit af both ends. The membrane potential distribu-

tion is given bysinh(xl t) + sinhL(/ - r t l i l \-

s inni t i r )

10.3. ,qxtaL-wtRE voLrAGE cLAMP 455

( r0.3.s), : , ( , -

As the membrane is voltage clamped, the membrane potential in the center

of the fiber is equal to the command voltage /.o"

then

and the voltage distribution in the voltage clamp will be

v(x) : u.o"(''nn'ur,,;;illl'llilllj!l- ",,4) (,03 6)Figure 5 shows the distribution of VlVco* along the axon for different

values of the space constant. It is clear that the smaller the space constant

the more homogeneous is the potential along the axon, because end effects

are circumscribed to smaller regions. Therefore the best space clamp is

achieved when the surface resistance of the axial wire ru is lowest. Very low-

resistance axial wires can be prepared by platinizing the platinum wire,2o

but their resistance becomes very high when steady current is passed. This

latter situation appears frequently when an axon is held at a potential differ-

ent from its resting potential ;under these conditions the space clamp may

be far from ideal.

10 .3 .2 . G ian t Axon P repa ra t i on

As explained above, the axial-wire technique can only be used on large

cells, and it has been successful with squid giant axon, crayfish axon, M-v-xi-

cola axon, and barnacle muscle fibers. The experimental procedures are

Vcou: V(U2);

| / z s inh(/12,1)\r : : Izcov/( l -

, i "h(/ / i ) / .

l / I = 100

456 10. volra,ce cLAMptNG oF EXCITABLE MEMBRANES

1 . 0

v ( x ) 0 .5

Vcot,l

o

F r t ; . 5 . Po te -n t i l l d i s t r i bu t i on i ns i dc an a ron con ta i n i r r g an ax i a l r v i r c . ( a ) Thc no rma l i zedstcxdy-st l tc p() tent iu l d ist r ibut ion as calculatcd f rorn Eq. (10.3.6) lor thrcc di lTcrcnt r i l lucs of1 ' ' ,1. whcrc / is the length of thc f ibcr and , l is the spacc col ls(ant . The uron is vol tagc c larnpcd, as

indicatcd in (b) , and the potcnt ia l rs contro l led in the centcr .

similar for all these preparations. To familiarize the reader with this tech-nique, we shal l br ie f ly descr ibe the squid g iant axon prL-p i r rut ion.

10.3.2.1. lso lat ion of the Giant Axon. The g iant axon f rom the ste l -late ganglion has been obtained from several species of squid: Loli11o./brbesi,Loliglo pealei, Loligo tulqaris, Loliqo opulescens. Dosidicus 11ig1as, Dori-theutus plei, and others. The axon diameter varies among diflerent speciesand ranges between 200nm and l.3mm. Normally, axons are dissectedout o[ the squid mantle in running seawater, and they are later cleaned ofother fibers and connective tissue under microscope dissection. For a moredetailed description see, for example, Gilbert.2s Cleaned segments from2 to 8 cm long are tied at the ends with threads and can be kcpt for scveralhours at 4 8 'C in seawater .

10.3.2.2. Exper imenta l Chamber and Internal Per fus ion. The deta i ls ofthe experimental chambcr differ depending on whether the axon is held horiz-onta l ly or vcr t ica l ly . Several designs have bcen used successfu l ly . '8 '2o '2e '3o

28 D. L. Gi lbert , i r r "A Cuide to Laboratory Use o[ the Squrd. Lol iuo pcale i . " Mar. BIol .

Lab. , Woods Hole, Massachusetts, 1974.2o W. K. Chant l ler and H. Meves. J. Phr.s io l . (London\ 180, 788 (1965).ro C. M. Armstrong, F. Bezani l la, and E. Rojas, .1. Gcn. Ph.v 's io l .62,375 (1973).

10.3. .qxral-wrRE voLTAGE cLAMP

The methcld used to intcrnally perfuse the axon has also influenced thedesign of the chamber. Two general types of perfusion methods have beendescribcd: the roller technique originally described by Baker et al.,3t inwhich thc axoplasm is ext ruded by a ro l ler and the axon later re inf la tcd bythc per fus ion solut io t t ; the cannulat ion technique or ig inal ly dcscr ibed byOikawa et a1. ,32 in which thc axoplasm is suckcd into a cannula as thecannula is advanced a long the axon.

In the following, we briefly describe the Tasaki technique,32 as modil ledby Fishman,rr to i l lustrate one of the methods to perfuse and voltageclamp a squid axon. The axon segment is positioned in the chamber asil lustrated in Fig. 6, which is based on the chanrber designed by Armstrong.r0A cut is made wi th a microscissors at point A, and a g lass cannula (PC) isinserted into the axon and advanced with the aid of a micromanipulator.As the cannula is advanced, microscope observation is required to maintainthe cannula centered in the ax is of the axon. Normal ly , a smal l pr ism is usedtc l v isual izc thc posi t ion of thc cannula in thc ver t ica l p lane. Simul taneouslywith the advancing of the cannula, mouth suction is applied, and the axo-plasm is sucked into the cannula lumen. At point B another cut is made, andthe cannula is pushed outs ide the axon. The axoplasrn ins ide the cannulais blown away, and the flow t-rf the internal perfusion solution is startedthrough the cannula. The composite electrode described in Section 10.3.2.3is introduced partially inside the cannula, and then the cannula is pulledback as the electrode is pushed in. Normally. the air gap (region betweenB and C) is trcated with a protcasc, such as papain or pronase, for about oncand a half minutes by positioning the tip of the cannula at C and letting theinternal perfusion solution containing the enzyme flow freely. Finally, thecannula is retrievcd almost to point A as the electrode is pushed to thefinal position that requires the tip of the voltage pipette to be in the centerof the chamber.

10.3.2.3. Internal Electrodes. Several types of internal electrodes havebeen used for voltage clamping. The classical combination electrode des-cribed by Hodgkin et q1.18 consists of two silver-chlorided silver wirestwisted into a double spiral on the glass rod. One of the wires was used tomeasure potential (voltage electrode) and the other was used to pass current(axial wire). The main disadvantage of this electrode is that inside theunperfused axon it does not measure steady dc potential because the voltageelectrode is not stable. One of the most commonly used combination

3' P. F. Baker, A. L. Hodgkin, and T. I . Shaw, " / . Physbl . (London) 164, 330 (1962).32 T. Oikawa, C. S. Sypropoulos, I. Tasaki, and

'f. Teorell, At'ta Ph,,-siol. Scand. 52, 195

i l 961 ) .t t H. M. Fishman, Biophl 's- " / 10, 799(1970).

( o )

( b )

458 10. vor rac l ( 'LAMptNc oF Ex( . r rABLE MEMBRANES

Ftc. 6 Squid axon chatnber. The rnain body of the charnber (MB) is nrade of 'Lucrte. Thelwo s i lver p lates SP' ant l SP' are movable and cun be approached to enclose thc ax()p (AX). Sp,is therrnal ly ct l t t t tcc(ccl t ( ) a Pel t ier cooler that is connected to a f 'eet lback loop to contro l (hechanlber tcmpcrature nreasured by thcrnr is tor T. The external solut ion (ES) is precoolccl bypassing i t ins ide the s i lver b lock belbre i t enlers the chamber through in let I . Thc solut ion issucked lwuv by out lc l O.

' fhe pl : r tes are nraclc o i several sect ions (G. guard; AG, auxi l iary

guard; ( 'P. cen(ral or measur i t ]g p late)separatcd among them by th in shects of Mylar . The f i rcesof thc platc 's in col l tact wi th thc cxternal solut ion are plat in ized (darker regions). Thc pcrfusioncat t l l t t l l t (PC) cntcrs iu the axon at point A and t i ie cornbinat ion electrodc (CE) at point B.lnsidc thc conrbinat ion electrode thc s i lver s i lver chlor ide pel let (SS) can be observe4. The t ipof the external e lectrode is at the bol l r im in thc ccnter Of the chantber (EE). (The length of theper lusion car lnul i t PC has been drau'n shorter to f i t the diagrant . l ts real length should span atlcast f ront point A to B, which is about 20 mrn.) lnset : c lc ta i l of the internalcombinat ion electrodeinside the axol t . A W is thc axia l wirs. EI is the cannula of thc potent ia l -measur ing electrodc. a ldFP is a f ioat ing pl : t t inum u' i re. The glass cannula EI is g lued to r he plat inrzed axia l wire A W withstra l l cpoxy beat ls (EB).

electrodes is that described by chandler and Meves,2e which consists of aplat in ized p lat inum wire (ax ia l wi re) used to pass current and a cannulaat tached to i t to meusure the potent ia l . The cannula is normal ly f i l led wi th0.6 M KCI and acts as a sal t br idgc between the axon intcr ior and a revers ib lehemicell of Ag AgCl or calomel. To decrease rhe high-lrequency impedanceof the cannula. a floating platinum wire is positioned inside (trig. 6). Sorne-t imes the f loat ing Pt wi re is p lat in ized.3a

10.3.2.4. External Electrodes. The external vo l tage e lec l rode is s implya salt bridge with one end positioned as close as possible to the axon and theother end in contact wi th a Ag AgCl or ca lomel hemicel l .

14 H. M. Fishrnan. IEI l t i ' t

runs. lJ iotnL, t l . Arrr l . BN,IE-20, luO (197.1) .

10.3 . ,qx ra . l -wrRE voLrAGE cLAMP 459

The external current electrode has been made in several dif l 'erent ways.The most common approach is to position two platinized surfirces made ofsilver or platinum on either side of the axon. E,ach surface is divided intothree electrically isolated plates: a central electrode or measuring electrodeand two lateral electrodes also called guards. These large metal electrodesboth pass current and, at the same t ime, contr ibute to the space c lamping.Norrnally. the guard electrodes are grounded and the central electrode isheld at virtual ground and current is measured only with that electrodc.See Fig. 6 for an example of a cerrtral plale and several guards used toverify homogeneity along the axon.

10.3.3. Measurement of Membrane Current

As mentioned earlier, one would l ike to collect all of the current f lowingthrough a region of membrane over which the voltage is uniform; severaldifferent methods have evolved for doing this. We shall discuss the use oflateral guards, external differential electrodes, external pipettes, and gap

isolation techniques (Chapter 10.6).When an axial wire is used to supply current. the effects of voltage non-

uniformities near the ends can be reduced with the use of lateral guards.

In this method three external electrodes are employed and the currentmeasured only from the central one, either by passing it through an externalresistance and measuring the potential across it, t 6' r 7 or by means of a current-to-voltage converter.re Various modifications of the precise form of theexternal electrodes have been used,le'3s arrd the type rn most common usetoday:o':r consists of three pairs of electrodes forming the sides of a rect-angular t rough. In some systcms t l . re three chambcrs arc se 'paratcd bypartit ior.rs. and sr.rt.t.rc Vlsclineicli l mixture is nscd for insula(ittt l . Exccptfor considerat ions of noisc (Scct ion 10.3.4) , i t is not c lear that th is is rnimprovcr .nent ovcr us ing no par t i t ions. l f the axot . t is uni [orm and the gr"rardsarc long cnough, thc par t i t ions are not r - reeded; unless the insulat ion is qui tepcrfect, any currcnt f lowing through the gap could producc voltage clropsacross the mernbrane which nr ight aggravate thc spat ia l r lonul l i [ormi t ies. Asystem wi th th in p last ic par t i t ions Lrsed to record ionic and gat ing currentshas bcen recently rcported.on

Regardless of the arrangement for measuring currcnt, there is the importantexperimental question of how one knows that the current distribution isindeed uniform. This has been investigatcd with the use of two small closelyspaced external dif lerential electrodes for measuring current density over

r s l . Tasak iand S . Hag iwa r \ . 1 . Gan . Ph . r . s i t t l . 40 . t { 59 ( I 957 ) .r" E. Rofas ancl ( i . Ehrenstein. . l . ( - t l l . ( 'ontp. Phv.s io l .66, Suppl . 2. 7 l (196-5).- 1 r E . Ro i i l s . R . E . Ta t l o r , I . A t r va t c r . a r r d F . Bcz ; rn i l l u . J . ( j e r . l ) h t s i o / . 54 .511 (1969 ) .

4KgrOt \?v , / l -

460 10. volrlcp cLAMpTNG oF EXCTTABLE MEMBRANES

small regions of the squid giant axon. 1e'2o'23'37 At f irst this would seem anideal way to measure current, but there are problems of sensitivity, noise,and calibration. It is sti l l true, however, that we know of no other way tocheck membrane current uniformity except with the use of multiple externalfongitudinal electrodes (Section 10.3.2.4 and Fig. 6). Narrow C-shapedexternal electrodes have been used for current measurement2e and for check-ing uniformity in the case of radioactive tracer measurements where the useof guards was not possib le.3T'

We do not know of a theoretical analysis for the precise geometry employedwith external differential electrodes. Chandler and Meves2e present anapproximate solution in connection with their method of calibration usingthe simplifying assumption that the two external electrodes see the currentarising from a point within the axon, as if they were on two equipotentialconcentric spheres. We have derived that an electrode in a medium ofresistivity R" (Q cm) at a radial distance r from the center of a cylinder ofradius a and longitudinal position z has a potential due to a thin ring ofc u r r e n t a t : : 0 o f

v(r. :) : ( l0 .3 .7 )

where Ko and K, are modified Bessel functions. Numerical computations ofEq. (10.3.7) give results that are different from those obtained with the simpli-f ied model of Chandler and Meves.2e

Davies3s and Jaf fe and Nucci te l l i3e have used a s inglc- v ibrat ing e lect ro i lein placc of the differential pair. It would seem from tl.re analyses that havebeen done that the potential of a single external electrode close to themembrane would be a good approximation to the underlying membranecurrent.

Another way to mezrsure membrane current is to apply an externalpipette to thc surfacc. If the internal resistance of the pipctte is small com-pared to the leak around the tip, the method is successful (Chapter 10.5).One must s i ther usc l r ce l l whosc membranc is not covered by "extraneouscoats" (i.e., single muscle fibers treated with collagenase or t issue-culturedcells) or bathe the region around the tip with sucrose solutions which canpenetrate the connective tissue or Schwann cell layer. The use of suctioncan be helpful (Chapter 10.5). If the tip is small enough, it is possible toobserve single channels.r8u This is extremely important in that it not onlyallows direct determination of the conductance of single channels but alsoprovides a powerful tool for studying channel mechanisms.

r r ' I . Atwatcr , F. Bczani l la. and E. Rojas. .1. Pht 's iot . (Lont lon) 201. 657 ( 1969).38 P. W. Davies, Fcr l . Pr<t t . , Fcd. Am. Soc. Exp. 8 io1.25,332 (1966).3e L. F. Jaf fe and R. Nucci te l l i , J . Cel t Bio l .63,614 ( t974).

10.3. nxrel-wrRE voLrAGE cLAMp 461

The last, and very l itt le tried, method is the use of a pipette (Westerfield,aoLlano and Bezanil laat) or metal electrode (Taylor and Bezanil la4r"; whichis connected to a current-to-voltage converter and draws off some of thecurrent on its way to the main external electrode. The so-called gap isolationtechniques are probably the most widely used methods of measuring currentat this time and are extensively discussed in Chapter 10.6.

10.3.4. Electronics for the Vol tage Clamp System

Many voltage clamp diagrams used for squid axons have been published.The system used by Hodgkin et al.t8 has a very short settl ing time, but itcannot control the membrane potential for long periods of t ime becausethe voltage electrode does not measure steady-state potentials (see Section10.3.2.3). However, the system is perfectly appropriate to impose changes ofpotential across the axonal membrane. The major changes introduced byMoore and Cole2o were to incorporate a microelectrode to measure thetrue membrane potential and the use of operational amplif iers. Since then,opcrat ional -ampl i f icr per formancc has improved not iceably, and veryfast clamps can now be built with commercially available units using thecombination electrode described in Section 10.3.2.3.

Figure 7 is a schematic diagram of one possible voltage clamp systemthat has been used successfully in our laboratory. The membrane potentialis measured by the differential combination A,, Ar, and 43. Special caremust be exercised to minimize capacitative loading at the input to preventfiltering of high-frequency components. The membrane voltage is summedto the negative of the steady membrane potential desired (the holdingpotential HP) plus the pulses or waveforms to be imposed on the axon(the imposed waveform lzo) at the summing junction of operational amplif ierAo. The combination C, and Ru act as a lead compensator and C. and R.as a stabil izing network. We have also (at the suggestion of R. Levis) foundthat the use of a lead compensator in the current feedback for series-resistancecompensation (Section 10.3.5) provided by C, is helpful. The output of theoperational amplif ier Ao is connected directly to the axial wire when thevoltage clamp is on or connected to an auxil iary feedback when the clampis off. The voltage clamp can be turned on remotely by replacing the switchS, by two field-effect transistors connected as switches or using two bipolartransistors.a' This electronic switch makes it possible to interrupt the freecourse of the action potential to study the ionic conductances during the

ao M. Wester l ie ld, personal communicat ion.or I . L lano and F. Bezani l la. Prot . Nut l . Atod. Sci . L/ .5.A.77. l2 (1980).4 t " R . E . Tay lo r r nd F . t s czan i l l : r . unpuh l i shcd obsc rva t rons .a 2 F . B e z a n i l l a , E . R o j a s , a n d R . E . T a y l o r . . / . P h y s i o l . ( L o n d o n l 2 l l , 7 2 9 ( 1 9 7 0 1 .

io R* I ' Ko(ror ' )' ' | , : .

' . c o s ( r ' r - - ) r 1 r , r .I t Jo 0^ t | ( uu )

r

462

_r-L

vorrstr

Ftc. 7. Schemal ic d iagram of a squid-axot t vol tage c lamp. A, and A, are very h igh- input-

impedance, low- input-capaci tance. wide-bandwidth operat ional ampl i l iers wired in a vol tagc

fol lower conf igurat ion to measurc the potent ia l o[ the internal e lectrode E and external e lec-

trode l-". A. is an operational amplifier connected as a drfferential amplilier giving the difference

between the outputs of Ar and Ar as 12, , . the membrane vol tage. Zoorr , , , is used to compensate

for e lectrode and junct ion potent ia ls. Ampl i f ier Ao is the contro l ampl i f ier , and i ts summingjunct ion I surns the negat ive of the desired potcnt ia l ( / .o") , which consists ofa steady potent ia l

(HP) and a waveform ( l/r) such as a pulse pattern, the membrane voltage ( lz.), and a iraction of

the negative of membrane current ( a1.) f or series resistance compensation. Note that tr/. is

connected to I through Rn in parallel, with (',.. acting as a lead compensator. Cr- and R. are a

stabi l iz ing network. S, is a swi tch that connects the output to e i ther the axia l wire (AW): c lamp

in posi t ion ON. or to an auxi l iary fcedback loop: c lamp in posi t ion OFF. Membrane current

der ived by the centra l p late (P.) is connected to the negat ive input ofoperat ional ampl i f ier ,4r ,

whose output wi l l be equal to the negat ive of the current der ived by Pc t imes the feedback

resistor R, . . The high gain of ,4. guarantees that P. is held at the same ground potent ia l as the

guard plates P1,, which are connected di rect ly to ground; however, at h igh f requencies the gain

clecreases and P. may not be at ground potent ia l Exar lp les of commercia l ly avai lable ampl i f iers

A, and A, are Nat ional Serniconductors LF356: of , , \ .1 , r \a and A. arc Nat ional Semiconductors

LF357 .

spike. Also, it can be used to protect the axon from electrical transients thatcould drive the membrane potential beyond safe l imits. This is accomplishedby measuring the membrane potential at all t imes and activating a bistablefl ip-flop whenever the membranc potential reaches unsafc l imits. The output ofthe fl ip-flop is used to control the clcctronic switch to open the clamp loop.

10.3 . a ,x ra l -wrRE voLTA( ;E cLAMP 463

The current amplif ier A, is connected as a current-to-voltage converterwith the input connected directly to the measuring plate (central). This makesthe central plate potential equal to ground potential when the amplif ier has avery large gain. However, it must be remembered that the open-loop gain ofoperational amplifiers decreases with frequency, and therefore at highfrequencies the central plate wil l not be at ground potential, making a gradientof potent ia l in thc external so lut ion a long the axon (scc Sect ion 10.4. 1) . I t isimportant then to select an amplif ier with a large gain-bandwidth productto maintain space clamp over the entire frequency range of interest duringvoltage clamping.

Unless partit ions are used, the resistance between the central plates andthe guards is quite low because it is given by the external solution. This low-resistance pathway between the summing junction and ground drains alarge current from the input voltage noise generator of the operationalamplif ier, producing significant noise at the output of the current amplif ier.Frequently, this is the predominant source of noise in the voltage clampsystem, and it can be minimized using current amplif iers with low inputvoltage noise.

Levisa3 has analyzed the noise performance for a clamp and concludedthat the major source o[ noise is the voltage measuring electrodes if theresistance between the center chamber and the guard chambers is high.If this resistance is low (10 Q. say), this may become the major source ofnoise. This problem may be avoided by not using guards44'45 or by the useof insulating partit ions.a6

1 0.3.5. Ser ies-Resistance Compensat ion

As mentioned above, the electrodes should sense the potential as close aspossible to the membrane. In the squid axon, the Schwann cell layer con-stitutes a barrier that cannot be penetrated by the electrodes, and a significantresistance is included in series with the axolemma. The origins of this resis-tance are the narrow clefts between Schwann cells. which amounts to about4 O cm2,r8 '47 's6 but i t var ics considcrably wi th the type of external so lut ionand possibly with the state of the axon.

Hodgkin et al.t8 introduced in their clamp circuitry positive feedbackthat subtracted the voltage drop across the series resistance (compensated

ar R. Levis, Doctoral Disser lat ion Department of Physio logy. Univers i ty of Cal i fbrnia.Los Angeles, l98l .

oo E. Wanke, L. J. DeFel ice, and F. Cont i . Pf lue4ers Ar<'h.347,63 (1974).45 F. Cont i , L. J . DeFel ice, and E. Wanke, .1. Ph.r 's io l (Londonl248.45 (19751.an F. Bczani l la, R. E. Taylor . and J. M. Fcrni indcz. J. Gen. Phl ' : io \ .79. 2 l (19u2).+ ' L. Binstock, W. J. Adelman, Jr . , P. Senf l . and H. Lecar. J. Membr. Bio l .2 l ,25 (1975).

10. volr,qcE c'LAMptNG oF EXCITABLE MEMBRANES

J, l-*""

ri

J(,-l

feedbr r r ' l ) l r r I ' . . r \ l l n i la r a r r : rn l l ( . rn , r r i r . r \ l ) cen incorpora ted . A vo l tagep l ' t t p 1 ' 1 1 r , , r , r l r r , l r l i a C t i O n p f t l r , , ' , , r r r l ) r i r l e C g f f e n t e q U a l t O - A I . R , i S, r , l r l , , l r , ' r l ) e n r e a s u r c < l l j , , . r r r 1 , , r r n n l i r - l g j u n c t i o n o l a m p l i f i e r A o . I f a i s, r , l l r r r r r ' t l t o m a k c l l r r r r , ' 1 r . . ' r ' e c l u i l l t o t h c / n , R . r . o l t a g e , t h e c o n t r o l, r r r l ) l l f i e r Ao w i l l ln l , ( , . , \ . l l re command vo l tagc p lus ho ld ing po ten t ia l onv^- In ,R, . r r l r r , l r rs the rea l membrane po ten t ia l I z " [see F ig .2 and Eq.( 1 0 . 2 , 2 ) l

10 3 5 1 . Effects of Series Resistance. If the electronics yielded a per_[cc( clrr 'rp, i.e., the measured potential faithfully followed the commandp.tential, there would sti l l be errors introduced by uncompensated seriesreslstance. These errors have been discussed by many people includingHodgkin et al.t8 (pp. 430, 435), Taytor et e1.,23 anj ginstoct eial.u, W" muidistinguish three cases: (i) passive membranes which may be described bymodels containing elements which do not vary with voltage, current, ortime; (i i) active membranes where the region of interest

-contains only

current-voltage curves at any given time (isochronal curves). which arestraight l ines with positive slope; and (i i i) systems in which the isochronalcurrent-voltage curves in some region are either curved and varying withtime or there is a transient or steady-state negative resistance region. In thefirst two cases, the data as recorded could be corrected with the use of a loadline for a known series resistance. In case (i i i), i t may not be possible to makecorrections. Not much can be added to the discussion of these points inTaylor et a1.,23 and the computer plots of current versus time for solutions ofthe equations of Hodgkin and Huxrey in Binstock et al.ai For more detailedstudies of stability in the negative-resistance region as affected by serresresistance. and the stabil ity of the second patch in the model of raylor, seeChandler et al.ag

we may summarize here by saying that in studies of the squid giant axon involtage clamp, the additron of series resistance producei chinges in theshape of the current-time curve in response to an appried voltag; step andchanges in the peak inward and steady-state outward current-voltagecurves. Load-line corrections, as predicted, correct for the l inear portionsof the peak inward and steady-state outward curves, but not lor the maxi-mum peak inward current or the shape of the peak inward current curves inthe negative-resistance region. The time to peak of the transient inwardcurrent was corrected by the use o[ a load l ine over most of the curve. Tayloret al-23 also demonstrated that, for their system, addition of negative-resistance compensation did correct, to a large degree, for the effects ofremoving the external reference electrode, which was, in fact, an addition ofseries resistance.

48 W. K. Chandler , F. Fi tzHugh, ancl K. S. Cole, Biophvs. J. 2, 105 ( 1962).

46510.3 . ,A ,x laL-w lRE voLTAGE cLAMP

It is then advisable to perform experiments using series-T:l::i::" *t-

pensation in order to measure the reat current-voltaee c.haracllistics of the

membrane. The first 't"fio "o*pensate

for series.resistance i1::^l"tu'u"

J"t.rrninution of its value; we address this problem in the next sectlon

1 0.3.5.2Oeterml t 'a- t - ion of S" ' i " ' Rei is tanceThe measr ' r rement of thc

resistance in series *lrrr-irr. membrane and between the voltage measuring

elecrrodes would b" "

i;i;i; ,,n10r" marter if (i) the membrane capacitances

were loss free, (i i) the seiies elements were'pure resistance" and (i i i) the

applied pulses and t"u'ut"-"nts were not distorted by electrode impedances

and amplif ier delays. ln this case' the most accurate way would probably be

tomeasu re the impedanceandex t rapo la te to in f i n i t e f requency 'Themos trapid would U. to uppfvl '"t iuogutot pulse of current and extrapolate the

measured voltage a J"-1i-.. tt iould also be possible toapply a voltage

clamp pulse and fit the measured current' All of these methods have been

rt"O, *,t variable and confusing results'

10.3.5.2.1. Lossv C'qp'AclrrNcs' Curtis and Cole'b using transverse

currents with external electrodes' measured the impedance of the axon

membrane and concluded that the capacitance was lossy and could be

approximated over ;;;;;;"y 'ung" by a constant-phase-angle impe-

clarrcc of the form 2"" : z* (i<rn)-' ' ,wherc-l : r l -1' <'t : 2nl. dld I is the

frequency. They reported un uu"ruge a ofO'SS' representing a constant phase

angle of 41 : a (90') : lO ' U'inglnternal and eiternal electrodes' Hodgkin

et eil.ts conclurjed that their cafacity transient for a voltage clamp pulse

was roughly "on,i't"lt

*iift "

phase angle of 80" (but see FitzHugh and

Coleae;. Taylor ana Cnandler, 'd using internal and external electrodes with

a bridge, found ttratf";"; l0 and 70 kHz the impedance closely fit the

constant-phase-angle expression above' *i ' ft lotpurable -values for a'5t

Taylors2 later showJ<litut tf. '" same.ata could be fi ir ly well approximated

by a single ."tu*utio].-ii*. b.uy.-type dielectric (and fit very well for a

moderate distributi ln'. i ' ."f"-i i"n't imes). FitzHugh and Coleae have

computed voltage and current transients for the constant-phase-angle

capacitance with parallel leakage conf1l-tance and series resistance' Their

resu l t scou ldbecomparedw i thexpe r rmen ta l resu l t s i | suchwereava i l ab le .A treatment of th" itunsient response to an applied' current of the form

I(r): Io(1 - e-'t ');;;;" founi in Binstock it al 'o' for a capacitative

element with a single-relaxation Debye-type dielectric with parallel leakage

and series resistance'

4e R. Fi tzHugh an<l K S' Cole ' Biophvs' J l3 ' I l2-5 (1971) '

. , , R. E. Taylor and w. r . cr lu"al . r , Biophts. Sor ' . ( .4b.srr . ) TDt 1 l?9?). . ̂ -^

, I N. Matsutnoto, l ' Inoue, and U' Kishimo to, Jpn. J. Pht ,s i t l l . 20, 5 l6 ( l 970).

t ' R. E. Taylor ' J . L 'e l l ' Comp' Ph'r 's iot 66' 2 l (1965) '

T

466 10. volrecg cLAMptNG oF EXCITABLE MEMBRANES

10.3.5.2.2. SEnrns l lrpEprNcn. The squid giant axon is surrounded by aSchwann cell layer which would not have a pure-resistance impedance.This is probably not serious in most cases because it approximates a pureresistance for frequencies below 100 kHz.53

10.3.5.2.3. FrNrrE RespoNsn Trnar . Hodgkin et u l . t8 repor ted t r rc resul tsof two experiments in which they determined the membrane capacitanceand series resistance for the squid giant axon membrane using rectangularpulses of current. They considered that the amplif iers for measuring currentand voltage had equal response times and that the effect would cancel.This is an important concept so we shall elaborate somewhat. considerFig.7. The membrane potent ia l is g iven by Vr :1_R, + ( l /C_)o [ ,o t^at , .Denoting the Laplace transform with argument s by an overbar, tor f.(O; : 0we find

2"1.t; : /_{s)R. * ( l/C_)/-(.s)/s.

Say the electrodes plus amplifiers distort the measurement o[ v, and I ̂into V *and 1-, where the transler functions are given by y,,(r) : V,^6)lV*\r)and yr(.9 : f;1s;7f-1s). If Y"(s) : y,(s), rhey wilt indeed cancet, and we mayuse the equation we started with, but using the measured V*(t) and I,^(t).In principle, for any applied current we may fit the measured curves andobtain values for C- and R..

one way to reduce the effects of amplifier delays would be to slow down thetime course of the applied current. Using a known finite-rise-time currentpulse, Binstock et al.al derived the time course of the voltage for a capacitywith parallel conductance and series resistance and presented results ofmeasurements on squid axons and Myxicola central nerve cords. If the leakconductance is small, extrapolation of the l inear portion of the voltagecurve [o zero t ime g ives an in tercept equal to /oR. : Io t lc^ . where r isthe time constant of the rising phase of the current pulse.

A somewhat more elegant, but to our knowledge untried, scheme ispresented by cole and Lecar.5a one may apply almost any shaped currentpulse asymptotic to a constant and extract the series resistance in the follow-ing way: draw a straight l ine asymptotic to the late l inear portion of thevoltage record, pick a time to, integrate the difference between the straightline and the recorded voltage from t : 0 to , : ,o, and subtract this fromthe integral of the difference from I : to to t : cx,; vary ro unti l the result iszero. The value of the voltage given by the straight l ine at / : fo is thenequal to 1oR., where 1o is the value for a rectangular pulse giving ih. tu-.voltage at long times.

5r K. S. Cole, Bioph. t 's . J . 16, t t4 (1976).t ' K. S. Cole and H. Lecar, J. Menhr. g iot .25,209 (1975).

10.3. rxr,qr--wtRE voLTAGE cLAMP 467

For any ofthese approaches it is necessary to carefully consider the distor-

tions produced by electrode impedances and ampli{ier delays for each

particular case.A very novel approach has been triedss'56 in which voltage-sensitive dyes

are employed. Application of a voltage clamp pulse which is rectangular as

measured electrically results in a light response which, in the presence of a

series resistance. wil l reflect the time course of the membrane current due to

the 1-R. drop. For a nerve membrane where the sodium current component

is large, the amount of negative-resistance compensation needed to make the

response of the dye rectangular gives an estimate for the series resistance.

fO.a.S.S. Vol tage Clamp Stabi l i ty and Ser ies-Resistance Compensa-

tion. A thorough analysis of thc properties of a voltage clamp system as

actually employed would be very complicated, and most voltage clamp

systcms in use today are constructed on the basis of in tu i t ion and cut and t ry .

Katz and Schwartz22 have considered a very simplif ied circuit in which

the resistance (R. in Fig.2) in series with the membrane and between the

voltage measuring electrodes is ignored and only the time constants of the

membrane and the control amplif ier are considered. This gives a second-order

system which is probably insufficient to say very much about an actual

system. They analyze the effect of a feedback arrangement to reduce the effects

of the access resistance R"" between the output of the current-supplying

amplif ier and the preparation. This is important when using microelectrodes

(Chapter 10.4) to supply current, and in some of the cases that they considet

the series resistance R, is small. For squid axon membrane clamping the

access resistance is small, and R. is a major problem.

Levisar,sT has done many voltage clamp analyses and has concluded that

an important considerat ion for s tabi l i ty and a fast c lamp. us ing negat ive-

resistance compensation (for R.), is that the frequency response characteris-

tics of the voltage and current measuring systems be comparable and

combined before feeding back to the control amplif ier through a compensa-

ting network.ionsider Fig. 2 with the additional feature that the current is measured

as shown in trig. 7 and a voltage -eI-Rs proportional to this current is fed

back to the control amphfier (trig. 7). We have analyzed this system in some

detail with the simplifying assumptions that the amplif iers are ideal opera-

tional amplif iers with zero output and infinite input impedance and neglecting

the effects of the electrodes. We shall only make a few general comments

here. Say that the control amplif ier A" (Fig 2) has a transfer function Y.o"(s) :

s5 L. B. Cohen and B. M. Salzberg, Rer. PhJ's io! . , Biothcm. Pharmacol . l t3,35 (1978).

56 B. M. Salzbcrg, F. Bczani l la, and H. V. Davi la ' l i iophls. J 3 l .90a (1981) 't ' R . Lev i s . pc r sona l comtnun i ca t i t r n .

-

468 10. volr lcr cLAMprNc oF EXCITABLE MEMBRANES

u.',(s)/u'.(.s) wherc u(s) : [; v1t7e "'r lr is the Laplace transform of v(t).Similarly. the transfer functions of amplif iers Ao (Fig. 2) and A. (Fig. 7)ar-e Y, and Yr, respectively, and the admittance of the membrane is y"(s).Also let Rl : R, * R"" (Fig.2). It is important, even in simple analysis, toco'sider the effects of the resistance from the input of the curient measurrngampli{ier A. to ground through the guard eleclrodes. cail this Ro, and retR i : R 1 R 6 / ( R 1 + R c ) .

with these definit ions we can say that the transfer runction ror the relationbetween the membrane potential v* and the command potentiar z.o" inFig. 2 or Fig. 4 is a function of the complex variable s which has two zerosand four poles. The behavior of this system is completely determined by thevafues of the zeros and the poles (see. e.g. .D,Azzo and Houpiss8; . We havefound that there are always two real pores. The other two poles may be realor complex conjugates. The response to a step in 260" wil l be a constant(the step introduces another pole at s : 0) plus the sunioffour exponentidls.For absolute stabil ity the real part of any pole must be negative, so the res-ponse wil l be either the sum of a constant and four decreasing exponentials,or the sum of a constant prus two decreasing exponentials-ptu, on

"^po-nentially decaying sinusoid.There are several ways to approach the question of the behavior of the

clamp system. one is to consider the frequency dependence of the transferfunction and plot the rrragnitude of r,*(ir,t), ir,.o*(ftrr) versus to : 2nl,where/is frequency, on double-logarithmic paper. This is often referred to asa Bode diagram, and is particularly useful when one does not have ananalytic expression for the system function. For stabil ity the Nyquistapproach is particularly useful when it is possible to readiiy measure theopen-loop transfer function frequency characteristics [Hfro)G(iar), seebelow]. For an analytical expression, stabil ity can be determined with theuse of the Routh criterion for the presence of positive real parts of the rootsof the denominator. we have done this, but the most usef;l for us seems robe the so-called root locus procedure. (Ail of the above are well discussed inD'Azzo and Houpis.s8)

If we look at the distribution of the poles and zeros of the transfer functionon the s plane and how they move when some parameter of the system isvaried, we can observe the complete behavior of the system and obtain ciuesas to what one can do by adding more poles and zeros to improve it ( i.e.,with compensating networks). The major diff iculty with this approach hasalways been the necessity of obtaining the complex roots of the denominatorof the transfer function. For the simplif ied version we are considerins there

58 J. J. D'Azz. and c. H. Houpis. "Feedback contror System Analysis and Synthesis."McGraw-H i l l . Ncw Yo rk . 1966 .

10.3. nxrel-wrRE voLrAGE cLAMp 469

are four such roots. If the electrode impedances are included, as well as thecompensation introduced by the elements in the feedback across and theinput to the control amplif ier (Fig.7), there are seven roots to obtain. Theavailabil ity of modern computers has changed this picture drastically,and such analyses are becoming feasible.

The term "stabil ity" is used in more than one sense. Absolute stabil ityrefers to the situation in which there are no positive real parts of any poleand the system does not go to infinity with time. If the poles are all real andnegative. the response is a sum of exponentials and there are no dampedoscil lations. This is a relative stabil ity often referred to as crit ical damping.The stability of the system we are copsidering here depends very much onthe bandwidths of the amplif iers employed, relative to the time constant ofthe membrane being clamped. For very wide-bandwidth amplifiers thesystem is unlikely to be stable for R. : 0. For infinitely wide bandwidths withR. > 0 the system may be stable, but no negative-resistance compensationis possible. On the other hand, it is quite possible to arrange things so thatone can compensate for many times the series resistance. It is intuit ivelyreasonable (and analytically true) that if the clamp system is slow, then onlythe low frequencies are involved and the presence of the capacitor in themembrane equivalent circuit wil l have very l itt le effect; one could thencompensate for R- + R". If the time-constant of the amplif iers is ro : 11,ro,one can compensate up to near R, + R./(1 + ofinlCz1.se For a usualcase with the dc gain of the amplif iers equal to l0s and tuo equal to 2 x 107;the sys tem i s uns tab le f o r R . :0 . l f , 4 i s on l y 103 and R , :5Ocm2. onecan compensate for about twice R. without instabil ity.

One reason for the complexity of the analyses of clamp behavior is that themembrane for which one is atlempting to control the potential is part of thefeedback and not just a load as one finds in most textbooks. With the abovedefinit ions for our simplif ied system, it can be shown that the transferfunction can be put into the standard form

if we let

and

u"(s)/u66"(s) : G(.s)/[1 + G(s)H(s)]

G(.s): -JX,n"(s)/{ l + Y"[Ri + RKl - v,)]]

H(s ) : Yy ( l + R .Y" ) - Y1eRrY" .

This is shown diagrammatically in Fig. 8. The open-loop transfer functionis H(s)G(.s) and is the function that would be used for the Nyquist diagramapproach to stabil ity.

5e R. E. Taylor and F. Bezani l la. Prognnn .1hstr . Soc. Nruro.vt i . . p. 306 (1973).

l l

471470 10. volracp cLAMpING oF EXCITABLE MEMBRANES

Ftc. 8. Equivalent c i rcui t for the t ransf 'er funct ion r ' " ( . r ) / r , . . r r ( . r ) lbr the system shown inF ig .2w i t h theadd i t i ono l ' nega t i ve - res i s t ancecomper ) sa t i onasshown inF ig .T .He reV . - r ' cou lY,r" . is the t ransfer funct ion of ampl i f ier A" (Fig. 2) ; ) / " is the admit tance of the membrane wi thpotent ia l V6 Y, and ) , , are thc tota l t ransfer funct ions of the vol tage and current nreasur ingarrangements: Ri is the sum of the ser ies resistance R. and the access resistance R.. (Fig. 2) ;aR, is the amount of negat ive-resistance feedback (Fig. 7) , where R, is the feedback resistance ofthe current measur ing ampl i f ier Ar. Ri - RrRc/(Rr * Ro) where Ro is the resistance f rom theinput o i , {5 to ground (not shown). The compensat ing networks Ru, ( 'u. and Ry, ( '1 shown inFig. 7 are not inc luded here.

10 .3 .6 . Pu l se Genera t i on and Da ta Acqu i s i t i on

We now describe the basics of an electrophysiological setup used to studyionic and gating currents.

The setup contains (i) a unit to produce the pulse patterns to drive themembrane potential, (i i) voltage clamp electronics to control the membranepotential, (i i i) a current measuring ampli{ier, and (iv) a data acquisit iondevice to store currents obtained for the imposed membrane potentials.The different units of the setup can be easily assembled with conventionalpulse generators, oscil loscope, and cameras, but digital computers areobtained at such reasonable prices today that the setup can best be assembledwith a computer as the core block.

The description is then that of a particular computer-based system we haveused successfully in our laboratory, although many similar systems havebeen used by many other investigators for many years.6o 62

Basically, the computer is used to produce the pulse patterns used to drivethc membrane potcntial and also to acquirc and storc thc mcmbrane currents.The computer, a Data General Nova 3 (Southboro, Ma) is programmed ina combination ol'FoRTRaN rv and ASSEMBLER languages. ASSEMBLER is usedto handle a l l the per ipherals or input /output inst ruct ions and a lso in por t ionsof the program when the ponrn,q,N execution times are excessively long.

u " B . H i l l c . Ph .D . Thes i s . Rockc l i ' l l c r Un i ve rs i t y , New Yo rk . I 9 ( r 7 , I t Jn i vc l s i t y M i c ro f i lms(No. 61t-9. -5t . l ,4) . Ann Arbor. Michiganl .

' ' ( ' . M . A rms t rong anc l F . Bezan i l l L t . / 1 t n t . N . l ' - Acu l . 5c i . 264 .26 -5 (1975 ) .

"r W. Nonner. E. Ro. jas. ancl R. Str inpf l i . Pt ' luuL'rs Arth.351 I (1975).

10.3. nxru-wrRE voLTAGE cLAMp

10.3.6.1. Pulse Generat ion. The opcrator assembles the pulse pat tern,and the program stores the sequence of amplitudes and durations in a bankof random-access memories. The random-access memories are communi-cated to a digital-to-analog (D/A) converter. Communication between thememorics and the D/A convcrter takes place via optical isolators to preventground loops and, consequently, to decrease the pickup of digital noise inthe analog side of the setup. The output of the D/A converter contains thepulse patterns, as decided by the investigator, and is applied as the commandsignal of the voltage clamp circuit. Most often the pattern is a series ofrectangular pulses, but rampsrs'61 and other waveforms are sometimesemployed.ba

10.3.6.2. Data Acquis i t ion. The membranc currsnt is ampl i f ied andfiltered conveniently to cover the dynamic range and speed of the sampleand hold amplif ier preceding the analog-to-digital (A/D) converter thatdigit izes the signal to be entered into the computer memory. Data from theA/D convertcr arc transferred by way of optical isolators to the computermemory v ia the "data channel" or d i rcct memory acccss (DMA); datat ransfer is not under d i rect program contro l . having the h ighcst pr ior i ty inthc computer cyc lc operat ion. Thc stored current s ignal may or may not beprocessed beforc it is stored in hard or f lexiblc rnagnetic disk.

An important design consideration in this setup is a clear separationbetween the analog and digital sections of the instruments. We have fouqdthat unless the computer is physically separated from the experimentaltable, significant digital noise is picked up by the analog circuitry, withadverse effect on the signal-to-noise ratio. In practice, the voltage clamp andcurrent amplif ier are set on or very near the experimental chamber. The D/Aconverter used to generate the voltage pulses is fed with its own powersupply, and the same is done with the multiplexer, sample and hold, andA/D converter used to enter data into the computer. Both A/D and D/Aconverters are connected to the computer with multiple twisted pair cables.The analog signals, voltage, and current, are displayed in an oscil loscopedifferent from the oscil loscope used to display data stored in the computer.

10.3.6.3. Inst rumentat ion and Recording of Gat ing Currents. The useof a computcr to generatc ths command pulscs in a voltage clamp syslcm iscspecially uscful in the dctcction of gating currents. Gating currcnts aredisplaccment currcnts produced by the movcmcnt of charge inside Ihemembrarne that are thought to bc re lated to the opcning and c los ing ofthe ionic channels. r8b 'os l for a rev icw, scc Almers66). Thc currents are

n r F I . M . F i shn t : u r . Nu ru r t , ( L tn l on \221 , I l l 6 ( 1969 )

"+ Y. Paf t i ancl W. .1. At lc lntan. Jr . . .1. l l l r ,nthr . l ] io l . l . .1.11 ( f 969).u5 R. I ) . Kcvncs arr t l E. Ro. ias. J. Ph), t i r t l . ( l t tnthnl239. 391 (1974).

"" W. Alrncrs, Rer. Phr.s io l . . Biotht ,n. Phurntutol .82.96 (1978).

Yv { '1 + RsYm) - YI a RrYm

472 10. volrncr cLAMpTNG oF EXCTTABLE MEMBRANES

voltage and time dcpendent, and the method used to separate them fromthe larger capaci t ive currents is based on thei r nonl inear dependence onvoltage. The first method (*P procedure) used to detect them consistedof recording the membrane current for equal-magnitude but opposite-polarity pulses and adding the resultant currents. With the computer,it is an easy task tcl generate a scquence of positive (test) and negative(subtracting) pulses very well matched in amplitude and time course if theD/A converter has a resolution of I part in 4096 ( l2 bits). (The main problemis, however, to reduce the "glitches" to an acceptable minimum.) It was soonfound that to measure gating currents more accurately, the subtractingpulse had to be sttrrted from a more hyperpolarized potential. Becausestrongly negative potentials can have deleterious effects on the membrane.the "+P" procedure was devised.6r '67 1n th is technique a test pulse ofamplitude P is given, followed by four pulses of anrplitude {P riding on avery negat ivc or vcry posi t ivc potent ia l . Again, t l - re cornpr-r ter can gcneralc thescquencc and ampl i tude of the pLr lses wi th a h igh degrec of accuracy. Besides,thc program can bc rnade f lcx ib le cnough to accommoclatc the " + P," " }P"proccclures, or any othcr combinat ion. When the * P proccdure is used. thectl rreltts are addcd : in t lrc jP prrrc.'d u rc t he cLrrrents from t hc four smallpulsesare added, and thc rcsul t is subtractcd f rom the current pr t tduced by thc tcstpulse. With the computer it is quite simple to process the currents recorded,because under program control the current produced by each pulse can bestored and later added to of subtracted from the others. Furthermore, it is anormal procedure to signal-average the currents recorded to improve thesignal-to-noise ratio; again, this procedure can be implernented with thecomputer with the same program that generates pulse amplitudes, durations,and sequences, and records the membrane currents.

The high gain required to record gating currents may produce saturationof the amplif iers. This problem has been solved by adding to the currentsobtained, for both test and subtracting pulses, the current produced by apassive network which mimics the l inear part of the membrane response.6?If this transient generator is itself highly l inear, the exact form produced is ofno importance because it wil l be eliminated by the subtraction procedure.

10.3.6.4. Inst rumentat ion for Studying Noise and Single Channels.The studies of membrane channel noise and single-channel current f luc-tuations have contributed precise values for the conductances of trans-membrane ionic channels. As well as providing the most tangible proofof the existance of discrcte molecular channels. thc noise and channel-Jump experiments provide unique insights into the kinetics of gating.Membrane noise analysis has by now become a large enterprise which has

67 F. Bezani l la and C. M. Armstrong, J. Gen. Physio l .7O.549 (19' r -7\ .

10.4. volrncE cLAMp WITH MICRoELECTRoDES 473

been reviewed extensively.68 osc Two of the main considerations in thisimportant area of research are how to extract the noise spectrum gen-

erated by the activation of ionic channels from other extraneous sources ofnoise generated in a voltage-clamped preparation, and how to interpret the

observed noise spectrum in terms of appropriate stochastic models of thegating process.

Single-channel currents observable with an isolated small (of the order ofI pm2 in area) patch of membrane are the ultimate in resolution for mem-

brane currents. The major considerations for this technique are how tofabricate electrodes which can be pressed up against the cell surface to

make a seal and how to construct sufficiently low-noise virtual-ground

current detectors for the picoampere-level currents involved. These tech-niques are presented in three excellent reviews by Neher and his colla-borators.6sr 68h

10.4. Vol tage Clamp with Microelectrodes

1 0.4.1. Two Microelectrodes

In principle, it is possible to control the membrane potential using two

intracellular electrodes, one to measure potential and the other to inject

current. The technique, however, has two serious drawbacks: (i) it does notprovide space clamp, and (i i) the high resistance of the microelectrodes make

the system inherently slow.The high resistance of the microelectrodes is in some cases not a serious

problern. For example, the two-microelectrode technique Iras been used tovoltage-clamp the end-plate potential in the frog neuromuscular junction.6e

Since in this preparation the end plate is usually activated by stimulation of

the presynaptic terminal or by direct microiontophoresis of transmitter

o8 L . J . DeFe l i ce . l n r . Re r ' . N (u rob io l . 20 , 169 (1977 ) .68 ' L . . 1 . I ) eFe l i ce . " l n t r o t l uc t i on t o Membrane No i se . " P lenum. New Yo rk . 1981 .n8b V. E. [ ) ionnc, i r r " lechnic]ucs in Ccl lu lar Physio logy" (P. F. Bakcr, ed.) , in press. Elscvicr .

No r t h -Ho l l and , New Yo rk , 1981 .

"n 'H. Lecar ancl F. Sachs, i l "Exci table Cel ls in Tissue Cul ture" (P. G. Nelson and M.L iebe rman . c t l s . ) , p . 137 . P lenum. New Yo rk , I 981 .

nn " E . Nehc r and C . I - . S t cvcns . . 1 r l r . R t , r . . B l op i r . s . B i o t ' nq .6 .345 (1977 ) .

" n " F . Con t i and E . Wanke . Q . Re r . B i oph . v . r . 8 , 4 , s I ( 1975 ) .n t IE . Nehe r , B . Sakmann . and J . H . S te i nbach . P f l ueqc r . s A r t h .3 ' 15 . l l 9 ( 1978 ) .nne E . Nehc r . i r r "Tcchn rqucs i n ( ' e l l u l a r Phys io l ogy " (P . F . Bakc r . c t l . ) . i n p ress . E l sev ie r

No r th -Ho l l : r nd . Neu Yo rk . I 9 l { 1 .unuO. P . Ham i l l . A . Ma r t y . E . Nchc r , B . Sakmann , and F . J . S i gwo r th , I ' f l ue1 le r s .4 r t l r . 39 l ,

8 5 ( 1 9 8 1 ) .o ' A. Takeuchi and N. Takeuchi . J . Neurophl 's io l .22.195 (1959).

474 10. volracp cLAMpTNG oF EXCITABLE MEMBRANES

substance at constant potential, there is no need to charge the membranecapacity at high speed. In some other applications, when it is necessary tocontrol the membrane potential in a small region but the recording of thecurrent is not important, two-microelectrode voltage clamps have beenused successfully (see, for example, Adrian et al.1o).

However, the usual application of the voltage clamp to study the current-voltage characteristics of the membrane is a problem which requires specialattention and varies with the type of cell considered because the potentialdistribution depends on the cable properties of the cell, which are partiallyrelated to its geometr;.

To consider the problems in probably one of the worst cases, let us assumethat we wish to voltage clamp a long cylindrical cell with two microelec-trodes as was briefly discussed at the beginning of this article. The membranecurrent wil l be measured as the total current injected by the current micro-electrodes. The potential at the tip of the voltage microelectrode wil l becontrolled by the feedback circuitry, but the potential in other regions of thefiber wil l be given by the potentialdistribution in a long cylinder. In particular,in the steady state and assuming that the conductances are independent ofvoltage (not a very interesting case but useful for i l lustrative purposes), thevoltage wil l be distributed according to voe ' i1, where zo is the potential atthe current injection point and ,1 is the space constant. It is easy to see that inthe case of length long compared to ,tr, thc current measured wil l not cor-respond to the membrane current in the controlled patch of membrane, butwil l be a mixture of the currents of different membrane patches all at differentmembrane potentials. For example. the length of a frog sartorius musclefiber is about 20 times the space constant at rest, making it inappropriatefor two-microelectrode voltage clamping. This situation is acceptable ifthe variation of the membrane potential with distance is negligible, a condi-tion that can be approximated in two common cases: the short cylindricalcell and the spherical cell.

10.4.1 .1. The Short Cyl indr ica l Cel l . I t is in tu i t ive ly c lear thar a shor tfiber wil l show less potential variation along its length the shorter it is incomparison to its length constant ,t. weidmannTr has presented solutionsof the distribution of potential in a short cable for the steady-state case.It can be easily shown that the potential distribution wil l be more homo-geneous when the control of membrane potential is done in the centerrather than at the end ofthe fiber.

To get an idea of the homogeneity of potential control, we show the pre-diction made by the solution of the cable equation for the case of a short

to R. H- Adr ian, W. K. Chandler , ar . rd A. L. Hodgkin. J. ph_v.s io! .2O4.207 ( l96gJ? t S . We idmann . J . Ph l . s i o l . ( London ) l l 8 , 34u (19 ,52 ) .

10.4. volr,qcE cLAMp wtrH MI('RoELECTRoDES 475

cable of length 2/ with both ends open circuited and with potential control althe center of the fiber.

The equation to be solved is (e.g., Taylor5)

( 10 .4 .1 )

where x is the distance measured from the center to either end: Z is the

membrane voltage, t is t ime, ,t is the space constant equal t" ,/aJr,; r is themembrane time constant equal to rn,('-, and r;, r-, and C- are the internalresistance, membrane resistance, and membrane capacitance for unitlength, respectively.

The boundary conditions at the ends require no current circulation:

^ , o tv ?vA ' ^ , - r ^ - - V : 0 ,

(.,-x- 0t

tdv li ( x : / ) | : 0 .

ri (.,.x lr_l(t0.4.2)

At the center, the membrane potential is set equal to 2.o", and the totalcurrent injected /o is divided equally into the two segments of length /(total l iber length is 2/). The solution of the above equations for Z.o*,an arbitrary function of voltage Vcou -- / (r), was obtained by the use o[ theLaplace transform, the complex inversion formula, and the convolutiontheorem. g iv ing

v(x , t ) :4 i ( -1) , (2n- t )cos Q!-_ lyU - )t -T n= t Z l

f t f - - / n r (2n - l ) r , i r \ - l ,I f O - z l e x p l - - | ^ r 2 1 , 1 : . ( 1 0 . 4 . 3 )

Jo L t r r r / l

Several driving functions can be analyzed with this equation. The mostobvious case is the step funct ion: . / - :0 for r < 0 and f : Vo for r > 0.In this case the distribution is given by (see also WaltmanT2)

l (x , r ) : , ' , ,1 toth l t { - x t " t ]

" \ coshl /7,1;

( - l f n ' ( 2 n - l ) e x p { - ( t l t ) l n 2 ( 2 n - l ) 2 1 2 1 4 t 2 + t l l \' ' .. _ cos[(2n - t)n(t -f!?!l

_ a "L 4 ! ) ,A2 + x212 t r _ 112 t .

(10.4.4)

Figure 9 shows an example of Z(x, t ) when I l l :0 .44. I t is c lear that thepotential rises at the ends more slowly than at the center. and at long times

t 2 B . Wa f t n ran , A t t u Ph . t ' s i o l . S tun l . 66 . Supp l - 264 (1966 ) .

If one analyzes the ideal case in which r, : 0 (perfect space clamp) the totalcurrent is given by

I o : 2 V o l l r ^ . t > 0 . (r0.4.6)

7r F. Bezani l la. C. Caputo, and P. Horowicz. Act t t Cient . Venez.22, Suppl .2, 72(19'r -1) .' * P. Heistracher and C. C. Hunt. J . Phr, .s io l . (Lon&tn) 201, 589 (1969).'4" F. Bezani l la, C. Caputo. and P. Horowicz, pcrsonal communicat ion.

1[

, 2Vo frann(t,t) 4n' i Qn - 112'o :

[ i , - 1 , 4 ,4 t ] 1Az+nz12n -112

t t ln212n - l ) t , t t \ l l' .^oL- , \ 4r2 )lJ ( r0.4.5)

416 10. volrncg cLAMptNG oF EXCITABLE MEMBRANES

t/r m

Ftt ; . 9. Normal ized membrane potent ia l as a funct ion ol t ime in a vol tage-clamped shortliber'. At .y - 0 (center of the fiber) a step function of voltage of magnitude I/o is applied (curve a)and at d is l ances x/ / of 0.2 (curve b) , 0.4 (curve c) , and I .0 (curve d) ; the potent ia l is p lot ted as afunct ion ol ' t inre. Abscissa is normal ized t ime / / r , wi th r the menrbrane t ime constant . Ordinate isnormal ized membrane potent ia l def ined as the membrane potent ia l d iv ided by the contro l ledpo ten t i a l l " , , . l . i s t hespacecons tan t , and / / iwas takenas0 .44 .C r .uvescompu tedw i t hEqua t i on(10 .4 .4 ) .

the potential at the ends is different from the potential at the center. Theparameters used in Fig. 9 were obtained from experimental measurementsperformed in the lumbricalis muscle of digit i IV o[the frog.73 When a thirdmicroelectrode was inserted at the end of a voltage-clamped fiber, a verysimilar voltage waveform as observed in Fig.9 was recorded.'74.'74^

It is interesting to compare the current predicted by the above modelwith the current expected for the case in which there is no decrement alongthe fiber. The total current 1o injected by the current microelectrode is givenby

10.4. volr,qcE cLAMp wrrH MTcRoELECTRoDES 417

I o - cn a t r : 0 , and 1o :0 a t r < 0 . I t shou ld be no ted tha t Eq . (10 .4 .5 )approaches (10.4.6) when ri - 0. The current measured in a typical two-microelectrode clamp is the total current /o, and from the above discussionit is clear that it wil l not represent the properties of a single patch of themembrane under study, but instead the charging of the distributed membranecapacitance. The case approaches ideality when the space constant is verylarge compared to the length of the fiber. The error can be computed withEqs.(10.4.5) and (10.4.6). lf the length and diameter of the cell are known,it is possible to roughly estimate whether two-microelectrode clampingcould be used. The value of ,1 is given as

1 : ",GJ;,

: utn^"FJz""n, : J R^,t2R,,

where a is the radius, R- is the specific membrane resistance 1Qcm2);and R, is the resistivity of the internal medium (Q cm). We can approximateAto

S" = ulZ1u (cm)

for typical average values of R- and R,. For an appropriate space clampwith two microelectrodes, //,1 should be less than or equal to 1, and using the

above approximation the length of the fiber should be

r < 16;.It should be noted, however, that the theoretical expressions presented

above have been obtained with a simple core-conductor model which doesnot take into account current spread in three dimensions. Three-dimensionalconsiderations are important near the site of current injection, and this maylead to error in the controlled potential at the tip of the voltage electrodewhen both electrodes are very close together.ts It is also known that whenthe radius a of the fiber approaches ,tr, the three-dimensional equationshould be used instead.Ts lt is probably not appropriate to discuss thatcase in any detail because when a = 2 the space clamp will be a completefailure" as can be seen lrom the previous equations, unless I = a, in whichcase we are approaching the case of the spherical cell.

10.4.1.2. The Spher ica l Cel l . In the approximat ion of the core-con-ductor model made in the previous paragraph, the spherical cell would beconsidered an idealcase. Wectn test the val id i ty o l the assumpt ion of potent ia luniformity by approaching the problem in the three-dimensional case.The solution to the problem of a spherical cell with two microelectrodes,one to inject current and the other to record potential, has been presented

rs R. S. Eiscnberg and E. A. Jolrnson. Proq. Ri t tphts. Mol . ts io| .20, I (1970).

418 10. volrecE cLAMptNG oF EXCITABLE MEMBRANES

in detail by Eisenberg and Engel,76 Peskoff and Eisenberg,TT and Peskoffand Ramirez.Ts Although their solution has not been produced for voltageclamp, we can assume that their voltage electrode is in our case the site ofmembrane potential control, and then ask for the potential in other placesof the cell, without moving the current electrode and without changingthe injected current, that wil l maintain the controlled potential in theoriginal position. It should be noted that in the derivation by Eisenberg andEngel,76 lhe current source was assumed ideal, that is, of infinite impedance.This is true in the case of " current clamping " because the electronics in thatcasc are clesigned to havc very high-impcdancc current clectrodc. Whenthe objective is voltage clamping, the electrode resistance is made lowpurposely to be able to inject as much current as needed to control themembrane potential in the shortest possible time. However, a very lowsource resistance cannot be obtained because in practice the minimumresistance is given by the current microelectrode, which typically wil l varybetween 5 and 100 MQ. In any case, the problem discussed by Eisenbergand EngelT6 can be applied without modifications for the steady statebecause the impedance of the current source is not important provided theamount of current i t suppl ies is known. In the t ransient so lut ion, the s i tuat ionis more involved, and it has not been treatec'l in a lorm convenient for thediscussion of the voltage clamp.

Let us consider the steady-state situation. Using Eqs.(1) and (2) ofE,isenberg and EngelT6 we find that the potential distribution tr2.,((/) is givenby

V,,,(0) : (l o R,,l4nbz) | (0, b I N,

where 0 is the angular separation between the current and voltage electrode;b is the radius; A is the space constant , equal to R," /R,wi th R, . , membraneresis tance per uni t area (Ocm2) and R, internal res is t iv i ty (Ocm); and 1o isthe total injected current. If the potential is homogeneous in the cell surface,it is given by

Vo: IoR, , . l4nh2,

whence it is clear that the factor / '(0, b/A) is defined as

.f (0, h I A) : v,,,(o)l v 6.The situation sought in voltage-clamping a spherical cell is a minimum

variation of potential with angle separation. It must be remembered thatduring voltage clamping the potential wil l be controlled at the voltageelectrodc; thercfore, we must ask what is the deviation from a consranrmembrane potential with any angle between current and voltage electrode.

' ( ' R. S. Eiscnberg and E. Engel , . , / . Gq. Ph. t 's io l .55, 736 (1970).

'7 A. Peskof fand R. S. I l isenberg, J. t l ' larh. Biot .2.21l (1975\.r8 A. Pesktr{Tand D. M. Ramirez. J. l l turh. Biot .2- 301 (1915\.

10.4. vclr-rncE cLAMp wrrH MIcRoI.LEC'TRoDES 419

Eisenberg and Engel (Ref . 76, Tablc I I ) havc publ ishcd a tablc of / (0, b lL)as a function of angle 0 and the ratio blA.. It must be emphasized that in thevicinity of the current electrode there is a steep rise in membrirne potential,but for most of the cell (for example, angles larger that 20", and b/A < 0.01 ),the variation in membrane potential wil l be less than 5",,. This is not verydiff icult to achieve because a cell 100 lm in diameter with a membraneresis tunce of 5 kQcm2 and an internal res is t iv i ty of lO0Qcrrr wi l l haveblL:0.0001. This value could be reduced 100 t imes ( for example, dur ingexcitation) and sti l l result in a potential distribution with less than 5 ",,

variation for angles larger than 20'. It should be noted, however, that avariation of 5",, could produce local circuit currents near the thresholdregion for a very steep conductance versus voltage curve and, consequently,produce a total current that would be practically impossible to interpret interms of the properties of an elementary membrane patch.

The above analysis assumes that the electrodes are inserted just under themembrane surface and that the external medium is isopotential. The firstassumption is reasonable in the majority of the cases because it is notdesirable to introduce the tip of the electrode too deep to prevent cell damage.In very small cells. however, this assumption may not be valid. The externalisopotentiality has been demonstrated to be a very good approximation byPeskoff and Eisenberg,TT who examined the difference it would make intheir solution by including a finite conductivity of the external solution.

The discussions about space clamping in two-microelectrode arrange-ments have been limited to the case where the system is l inear. In practice.the most interesting preparations wil l have conductances which are depen-dent on membrane potential ; consequently, the above equations are notdirectly applicable. However, some insight can be obtained by changing themembrane parameters in tlre steady-state situation. The situation is evenmore diltrcult when regenerative phenomena are present because voltage-dependent negative conductances are involved and, in general, numericalsolutions are required to analyze each particular case. Complications arisewhen the spherical cell has dendrites or axons. because current wil l bedrained into cablelike structures, sometimes excitable, which could resultin very large nonuni formi t ies.

Having examined the problem of space clamp when voltage clampingwith two microelectrodes, let us consider now the l imitations imposed bythe use of micropipettes. Microelectrodes are made by pull ing glass tubing,leaving tips less than I trrm in diameter, which are then ll l led with a concen-trated electrolyte solution. Normally, electrodes to record membranepotential are fi l led with 3 M KCl. Electrodes to pass current are normally notfi l led with KCI because they cannot pass enough current for most appli-cations; it has been found that electrodes fi l led with 2 M K-Citrate exhibitlower resistance to current iniection. Because of the small diameter near

' f '

480 10. voI-r,q,cr cLAMptNG oF EXCITABLE MEMBRANES

the tip, the resistance of these electrodes ranges between 5 and 500 Modepending on tip diameter, shank shape, and the electrolyte fi l l ing them.The high resistance of microelectrodes imposes serious l imitations on theimplementation of a voltage clamp system. voltage recording must bedone with very high-input-impedance amplif iers so that the electrodeamplif ier combination does not drain current from the cell. The distributedcapacitance of the wall of the microelectrode, the electrical shield. and thernput capacitance of the amplif ier, together with the high rcsistance of thcpipette, constitute a low-pass fi l ter that gives an erroneous measurementof the actual membrane potential when varying with time. The approachcommonly used to solve this problem is to electronically compensate for theinput capacitance using positive feedback through a capacitor (e.g., BakTe).The adjustment of the compensat ion is nol a s imple rnat ter ( for a d iscussionof methods, see Moore2' ) , and as some of the capaci tance is d is t r ibuted,it cannot be compensated. Although the result is a faster response than thecase without capacity compensation, the transfer function of the amplif ierintroduces phase lags that are important in the overall performance andstabil ity of the voltage clamp feedback circuit. To decrease the need fornegative capacity compensation, "driven shields" have been used whichconsist of connecting the shields at the input to the output or the amplif ier.As the output is a low-impedance point with respect to ground, it shieldseffectively. and being at the same potential as the shielded conductor (theinput), the capacity of the shield does not need to be charged or dischargedwhen the potential of the microelectrode changes.

A very ingenious way to decrease the influence of the electrode capacitancehas been devised by Eisenberg and Gage8o based on the idea of bringingthe interior of the cell to virtual ground.sr In trris arrangement a high-input-impedance operational amplif ier is used, the microelectrode is con-nected directly to the inverting input. the output of the amplif ier is connectedto the bath. As the positive input of the amplif ier is grounded, the negariveinput wil l be at "virtual " ground, and the oulput wil l be at minus the mem-brane potential. That is, measuring the potential in the bath with respecr roground gives the cell membrane potential. Meanwhile, all the effects ofcapacity to ground have been eliminated because the electrode is at groundpotential, and only the capacitance across that portion of the wall of themicroelectrode in the bathing solution remains, which can be reducedsignificantly, lowering the level o[ the external solutions. This configurationis recommended for measuring membrane potentiar, especially when usedas part of the feedback loop in a voltage clamp system.

tn A. F. Bak, Eletrroent'epholoor. Clin. Neurophyslol. lll. 745 ( l95g).8 ' ' R . S . E i senbe rgund p . W. Gage , J . G . , n . p l t r i i n t 53 . 27s (1969 ) .8r B. Frankenhauser, /. Ph.y,.yi<tl. (London) 135, 550 (1957).

10.4. volre.clr cLAMp wrrH MICRoELECTRoDES 4ul

The current electrode also has a high resistance which imposes problemsin the design o[ the voltage clamp. It has been found that to inject enoughcurrent to charge the membrane capacitance following a step change involtage, a high-voltage amplif ier is required because the high resistance of thepipette effectively l imits the amount of current to be passed. Some investi-gators have used the internal amplif ier of the oscil loscope,82 and others haveused operational amplif iers with large output-voltage range.ta

However, even the use of high-voltage control amplif iers does not solvethe problem of speed encountered when the membrane capacitance ischarged through a large resistor. Katz and Schwartz22 have proposed acompensating network for reducing the effect of the access resistance asdiscussed in Section 10.3.5.3. If the amplif ier had infinite gain at all fre-quencies, it would be possible to impose a fast rise in potential across themembrane, but the l imited bandwidth and slow rate of practical amplif ierslimit even further the speed of the voltage clamp system.

The interaction ("cross talk") between the voltage and current electrodesdue to capacitative coupling between them is important during fast transients,because it introduces undesired transients in the voltage clamp feedbackcircuit which may render it unstable. The coupling can be effectively reducedby positioning a shield between the electrodes and impaling the cell with awide angle between them.

1 0.4.2. Three Microelectrodes

Adrian et a1.83 devised a technique to control the membrane potential ofa muscle fiber and measure the current in the neighborhood of the controlledregion. Their technique is based on the fact that membrane current is roughlyproportional to the voltage gradient along the fiber. They impaled threemicroelectrodes near the natural end of a fiber, and, controll ing the potentialat the electrode closer to the end (control electrode) by passing currentthrough the electrode farthest from the end, they measured the membranecurrent as the potential difference between the central electrode and thecontrol electrode. Using the l inear cable equation, they derived the exactexpression for the current as a function of the potential difference betweenthe two electrodes, and they found that if l lTis less than 2, the error is lessthan 51, if the current is measured just as the potential difference betweenthe electrodes (l is the distance between the end of the fiber and the controlelectrode and between the two voltage electrodes, 2 is the space constant).The method works very well to measure delayed rectif ication, but fails inthe range of potentials needed to measure the sodium current. The reader is

8r L. L. Costanr in. . l . Pht ,s io l . (London) 195, I l9 ( l96U).8 r R . H . Ad r ran . W. K . Chand l c r ' . and A . L . Hodsk in . . 1 . Phys io l . ( London \ 208 ,607 (1970 ) .

482 10. volrace cLAMptNG oF EXCITABLE MEMBRANES

relerred to the original paper for details of the sources of the errors of thistechnique.8s Kootsey8a has calculated the errors of this technique usingnumerical procedures for the case o[negative conductance and has found thatUI:2 produces unacceptable errors. The error decreases to small butstill noticeable values when Il7 : 0.86.

Recently, the technique of making the interior of the fiber virtual groundhas been applied to the three-microelectrode voltage clamp.sau

10.5. Vol tage Clamp of an lsolated Patch UsingExternal Pipettes

The smaller the region in which the membrane current is measured, thecloser one gets to the ideal situation of recording current from an isopotentialregion. In this chapter we review briefly some of the attempts that have beenmade to isolate a small patch of surface membrane using external pipettesto record the current through it.

10 .5 .1 . Ex te rna l Pa tch l so la t i on

strickholm8s'86 manufactured fire-polished pipettes and applied them tothe surface of skeletal muscle fibers of the frog. He relied on the fact that theleakage conductance produced by imperfect contact with the membrane wasmuch larger than the membrane conductance of the patch, and with this thepatch of external surface membrane may be considered to be "clamped"at a constant potential. However, the lack of a feedback circuit to hold thepatch voltage controlled when the patch conductance changed made thisarrangement of l imited value.

A significant improvement was made by Frank and rauc.87 These authorsused two internal microelectrodes to voltage-clamp the membrane potentialofa mollusk neuronal body and brought an external pipette near the externalsurface of the soma. The interior of this pipette was connected to the input ofan operational amplif ier connected in the current-to-voltage configuration.The other input of the amplif ier was connected to the bath, making theoutput of the amplif ier proportional to the current f lowing through the patch

uoJ. M. Kootsey, Fcd. Prot . Ft ' t l . An.5b<. t rp. Bio l .34, l34l(197_5).84" P. c. vau{ l ran, J. G. Mclarnan. and D. D. [ " . L.o. ca, . J . pht 's io l . phurmutol . sg.999

( I 980).8 s A. Str ickholnt , J . Gan. Pb' .s io1.44. l07l ( 196 I ) ." " A . S t r i c kho lm . J . Ce l l . Con ry . Ph t s i o l . 60 . 149 (1962 )u7 K. Frank and L. Tauc. i , "Cel lu lar Funct ion of Membrane Transport" (S. Hol l rnan,

(ed.) . Prent icc-Hal l . Enslewood Cl i l rs . New Jcrsey. 1963.

10.5. volracE cLAMp oF AN TSoLATED pArt 'H -ll{.1

under the pipette tip. The comparison between total current and currentthrough the patch revealed that the total current had contributions from theaxonal membrane which was not under voltage control. The patch revealedcurrent pat terns that increased smoothly wi th mernbrane depolar izat ion aswas expected from a well-controlled membrane patch.

A further improvement was made by Neher and Lux88 when they incor-porated a feedback loop in the extracellular pipette using a partit ionedpipette, one half to measure the pipette potential and the other to injectcurrent. In their arrangement, the potential inside the pipette was maintainedat bath potential by nreans of a feedback circuit. The current needed tocontrol the potential of the pipette was measured as the current through thepatch, the cell membrane potential sti l l being controlled by an independentfeedback circuit with two microelectrodes. Using feedback in the extra-cellular pipette effectively decreases the access resistance (pipette resistance)to the input of the amplif ier, decreasing significantly the effect of the shuntresistance produced by the leakage pathway between the pipette wall andthe cell surface.

The original approach of Strickholm using the extracellular pipetteto control the membrane potential of the patch of membrane under thepipette was notably improved by Fishman.se He made a double-barreledconcentric pipette, the internal to measure patch voltage and to injectcurrent, and the external to introduce sucrose to improve the seal o[ thepipette against the membrane. As the preparation used was the squid giantaxon. which is surrounded by a coat of Schwann cells and connectivetissue, the sucrose was essential to decrease the leakage under the walls ofthe central pipette. Even with the sucrose flow the seal resistance was of theorder of 2 MQ, which is about ten times smaller than the membrane resistanceunder the pipette. The voltage clamp was established as a feedback circuitto control the potential inside the pipette at a predetermined voltage, andthe current needed to maintain this potential was considered to be the currentof the membrane patch under the pipette plus the leakage current due tothe imperfect seal. This latter current could be subtracted off electronicallybecause it was found that the leakage resistance was ol.rmic. The membranecurrents recorded by Fishman resemble the current recorded with theconventional axial-wire technique, and he provided tests for isopotentialityinside the axon by inserting a voltage measuring pipette. This techniquehas been used to record membrane noise taking advantage of the fact t l"atthe currents are measured from a small natch of membrane.eo

"E E. Noher and H. D. Lnx, Pf luct tL ' r . ; Ar th.3 l l ,272 ( 1969).n" H. M. Fis l . rnran. .1. I Iamhr. 8 io1.24.265 (1975).' " H . M . F i shman . D . J . M . Poussa r t . and L . E . Moo re . . , / . Mcm l , r . B i o l . 21 .2 l t l ( 1975 ) .

f {

I

484 10. volr l ,cr cLAMplNc oF EXCITABLE MEMBRANITS

The isolation of a patch of membrane with an external pipette has beenperfected by Neher and Sakmannao and Neher et al.68r By using collagenase-treated denervated muscle fibers, the seal resistance was increased up to50 MQ, and they were able to record the opening and closing of singlepostsynaptic channels. In their arrangement an extremely low-noise opera-tional amplif ier was used as a current-to-voltage converter with its inputconnected to the pipette. Using 500 MO as a feedback resistor, the mainsource of noise in that configuration is the input voltage noise of the opera-tional amplif ier that appears as a current in the leakage resistance under theseal between the pipette and the cell surface. With 50-MO seal resistancethe noise was low enough to observe a single acetylcholine-activated channelwi th u good s ignal - to-noise rat io .

Recently, Sigworth and Neherer and Horn and Patlakero have achievedseal resistances above 1 G O using tissue culture cells. Sigworth and Neherhave reported fluctuations of single sodium channels. Horn and Patlak havebeen able to excise a membrane patch attached to the pipette, enabling themto readily change the medium on the inner side of the membrane. Thesematters. and others, are d iscussed in recent rev iews.68* '68h

1 O.5.2. In ternal Access

Several groups of investigatorse2-e4 have used the external pipette togain access to the interior of the cell (Fig. 10). Although these techniques arenot strictly regarded as patch isolation, we have included them here becausethey uti l ize external pipettes (or perforated plastic partit ions) that have tobe pressed against the membrane to obtain a good seal.

Once the seal is in contact with the pipette, suction is applied to increaseseal resistance and if the suction is increased even more, it is possible tobreak the cell membrane (or break it with a wire inside the pipette) and,consequently, have access to the internal medium (Fig. 10a). Under theseconditions when the interior of the pipette is voltage clamped, the whole cellwil l be voltage clamped. The accuracy in the current recordings wil l dependon membrane parameters and geometry of the cells as discussed in the sectionon microelectrode clamping. This technique has the added advantage ofmaking possible the exchange of the internal solution by known solutionsloaded and circulated through the pipette shank.

' ' F. J . Sigworth and E. Neher, Nclr r re \London\ 287,447 (1980).e ' ' R. Horn and J. B. Pat lak. Proc. Nur l . Acad. Sci . U.S. ,1. 77. 6930 ( I 980).e2 P. G. Ktrstyuk and O. A. Kr ishtal , J . Ph. t 's io l . (London) 270, 545 (1977\.' r K. Takahashi and M. Yoshi i , J . Phl ,s io! . (London)279,519 (1978).n4 K. S. Lr .e, N. Akaike, and A. M. Brown, J. Gen. Phl ' .s ioL 7l ,489 (1978).

10.5. volrecE cLAMp oF AN ISoLATED pATCH 485

Ftc. 10. Schcmu(ic d iugnrnrs of patch c lamps wi th intcrnal access. (a) Sinrplc p ipct tcarrangement. The inte l jor o l 'prpct te P, is in d i rect contact wi th the cel l contents. The vol tageclamp ci rcui t (Ar. A:) contro ls the internal nrcrnbrane potent ia l and the tota l cel l current ismcusu rcd by t hc cu r reD l t o vo l t agc convc r t c r A .3 . ( b ) Vo l t agc c l amp r s i n (A ) bu t cu r ren t i s on l ymeasured out o la smal l mcmbrane patch wi th the smal ler p ipet te P, that is against Ihe e. \ t?rnalsur l i tce of thc cel l .e5 (c)

- lwo pipct tcs havc acccss to thc intcr ior of thc ccl l . P, is uscd [o mcasurc

the membrane potent ia l wi th ampl i f ier A, and then compared to the command vol tage atampl i f ier Ar that in jects current through pipet te P2. Current 1" is measured as the tota l currentin jected into the cel l .eo

Kostyuk et al.e5 have carried this technique one step further (Fig. 10b).They have manufactured plastic pipettes that, wlren applied with suctionagainst a molluscan neuronal soma, resulted in seal resistances of up tolOe Q. They use one pipette to gain access to the cell interior by applying

'5 P. G. Kostyuk.O. A. Kr ishtal , and V. I . Pidopl ichco. Dokl . Akat l . Narrk S^SSR 238,471'J( l 97u ) .

vc-o"

g

vc-orta

486 10. volrecp CLAMPING oF EXCITABLE MEMBRANES

enough suction to break the cell membrane and to voltage clamp the soma.

Another pipette with smaller t ip diameter is applied to the intact part of

the cell soma; enough suction is applied to obtain a good seal, but not

enough to rupture the membrane. This second pipette is connected to a

current-to-voltage transducer to record the membrane current of the patch

isolated under the pipette. With this technique they have been able to

record current noise related to calcium channels.Krishtale6 has used the same type of pipettes to gain access to the cell

interior with two pipettes in the same cell (trig. l0c). One pipette is used to

measure voltage and the other to pass current, approaching an almost

ideal situation when the cells are small and have high membrane resistance.

This technique makes possible the internal perfusion of the soma, as solution

is passed from one pipette to the other removing the contents of the cell.

Cont i and Nehere6u have used an L-shaped p ipet te on the ins ide of thc

squid giant axon to isolate a small patch of membrane and record the

fluctuations of single potassium channels.Recently a new preparation has been described in which a squid axon

had been cut open and the resulting membrane sheet positioned in a chamber

separat ing two compartments. Normal- looking ionic and gat ing currents

have becn rcpor tcd f rom th is cut axorr ,ar and f luctuat ious f rom a few sodium

channels havs bccn observed by apply ing patch p ipct tcs to the in ternal

surface membranc.e6b

10.6 . Vo l tage C lamp wi th GaPlsolat ion Techniques

Some of the most rewarding voltage clamp techniques can be gathered

under the heading gap isolation techniques. In this case, cylindrical cells are

mounted across partit ions and patches of membrane are isolated by means of

Vaseline, sucrose, Vaseline-sucrose, or Vaseline air gaps. Voltage clamping

of excitable cells using gap isolation techniques has provided sustained

experimental evidence about electrophysiological mechanisms or pre-

parations in which microelectrodes, pipette patch isolation. and axial-wire

techniques cannot be used. The methods in general were init ially developed

for electrical recordings from the naturally isolated patch of excitable

membrane in the node of Ranvier, but they have been modil ied to artif icially

define a small patch of mernbrane in unmyelinated nerve axons and muscle

fibers.

' )u O. A. Kr ishtal . Dokt . Akud. NauA' S^S.SR 2- l l t , 4 lJ2 (1978).

""" F. C'( ) l l t i i | | r t i F- . Ncl tct . N( l t l r ,c (Lr) , , ( / r ) , ; ) 285. l '10 ( l r ) l l0 t .' " ' ' L L l un , r an r l F . Bczan i l l a . B i op l r t ' s . J . 37 . l 0 l a ( l 9 l i l ) .

10.6. vol lncg cLAMp wrrFl GAp ISoLATIoN TECHNIeUES 481

1 0 . 6 . 1 . N o d e o f R a n v i e r

Myelinated nerve fibers propagate electrical impulses (action potentials)not as a continuous wave of depolarization as the unmyelinated axons do,but in a saltatory fashion in which action potentials are regenerated insmall patches o[ bare membrane called nodes of Ranvier.eT We will discussfirst the l imitations imposed by the characteristics of this preparation onthe electrical techniques used to record resting potentials, action potentials,and later in voltage clamping. As said above, essentially all the patch isola-tion techniques were derived from those developed for the node of Ranvier,and thus the natural node wil l be described rather extensively. Voltageclamping of the node of Ranvier has become a standard technique in whichlow-noise current records, fast settl ing times, and accurate membranepotential control can be obtained without the use of extremely sophisticatedelectronics. Although studies of the node of Ranvier began in order to seei f the ins ights devclopcd by Hodgkin and Huxley were val id for the node,this preparation has become so popular in the past ten years that many ofthe recent developments in electrophysiology have been made on thispreparation.eT

10.6.1.1. General Character is t ics. The techniques for d issect ion andmounting of single myelinated axons from the frog and toad have beendiscussed extensively by St i impf l i and Hi l lee? and wi l l not be covered here.The design of the chamber used to mount the nerve fibers and to make'electrical recordings from the node of Ranvier depends on the techniqueemployed, and its characteristics are described in connection with each olthe available methods.

In order to orient the reader to the technical rcquirements of the node ofRanvier as an electrophysiological preparation, we have included in Table Ithe typical electrical parameters of a frog's myelinated fiber. From Table Iwe see that the node represents a very small patch of excitable membrane(22 pmz) with a very high resting resistance (40-80 MO).

10.6.1 .2. Membrane Potent ia l Measurement in a Node of Ranvier .We recall that t l-re rcquircments of a good voltage clamp technique arethat the potential measurement must be accurate, /d.st, sampled at anisopotential pat(h o.l memhrure. We discuss now to what extent this has beenachieved in myelinated nerve fibers.

A straightforward method for measuring transmembrane potentialin a biological preparation is the use of micropipettes. However, as suggestedby the work of Woodbury,q?u they cannot be successfully used in the node ofRanvier.

e 'R . S temp f l i and t s . H i l l e . i r "F rog Neu rob io l ogy " (R . L l i nas and W. P rech t , eds . ) .

Spr inger-Ver lag. Ber l in and New York, 1976.

"-" J. W. Woodbury, J. Ccl l . ( 'omp. Pht 's io l .39, 323 (19-s2).

,-

488 10. volr ,q.ce CLAMpING oF EXCTTABLE MEMBRANES

Tasls L Electr ical Character is t ics of a Fros 's Mvel inated Fibers

Fiber d iameterThickness of myel inDistance between nodesArea of nodal membrane (not measured di rect ly)Resistance per uni t length of axis cy l inderSpeci f ic resistance of axoplasmCapaci ty per uni t length of myel in sheathCapaci ty per uni t area of myel in sheathResistance t i rnes uni t area of mycl in sheathSpeci f ic resistance of myel in sheathCiapaci ty of node of RanvierResistance of rest ing nodeAct ion potent ia l arnpl i tudeRest ing potent ia lPeak inward current densi ty

14 pm

2 p m2 rnm

zl ltm-140 MQ/cml l 0 Q c ml0 l 6 pF / cm

2 .5 5 nF / cm20 . I 0 . 16 MQ cm2500 800 MQ cmz0 . 6 . 1 . 5 p F40 80 MO

I 16 rnV- 7 1 m V

20 mAlcm2

( o )

(b )

An alternative method for measuring the potentials in this preparationwas introduced by Huxley and StAmpfli.e8 Figure 11a shows a shematicdrawing o[ the preparation and their electronic arrangement. The rationaleof the Huxley Stdmpfli method is that ol a simple potentiometric recording.The membrane potential Z- of node 0 in Fig. I I b can be measured withoutattenuation if the current source H supplies current through the partit ion ABuntil the meter G shows no deflection, indicating no current f low across theresis tance Ro. . Sincc nodc - I is not contr ibut ing to the membrane potent ia l(because i t is depolar ized by KCI) , thcn V^s : V^. I f there is current f lowingthrough tl.rs cxternal resistor R"., the potential at C does not represent ym,

but is an attenuated mcmbrane potential value. If there is no currcnt f low,though, Vc: Vs: Vo: I / - . Huxley and St i impf l ie8 were able to measureaccurate ly the rest ing potcnt ia l and the peak value of the act ion potcnt ia l innodes of Rarr a esurlentu by setting the current at H manually by trial and errorto a vtrlue which blocked currcnt f low. Their method has been called staric'potentiometrfu' since the zero-current condition was only met at spccificpoints of the e lect r ica l cyc le.eT

In the experiments of Huxley and Stampfli the gap BC, measuring ap-proximately 600 pm, was fi l led with paraffin oil and had a resistance ofabout l0 MQ. The resistance of the whole loop DCBAD was between 60and 90MQ; the voltage drop across the gap BC is only a small fraction ofV^, and an action potential wil l seem very small because of this voltagedividing effect (attenuation) unless the potentiometric method is used.In order to rccord action potentials, this attenuation has to be reduceddramatically, and two approaches have been followed in order to achieve

e8 A . F . Hux leyand R . S tamp f l i , J . Ph1 , s i o l . ( London ) 112 ,476 (1951 ) .

B A

( c )

a

JL

zco ilr " l lil

B

I Zro

l *';":" I ;;^

----+^\7et

J - 7=

l i tc ; . l l . Potcnt iomctr ic mct l tot ls o[ rccording mcrnbrane potcnt ia l . (a) Schernat ic d iagramof the Huxley Stampf l ie8 stat ic potent iometr ic methods. Node 0 has a rest ing potent ia l Z"and node - I is depolar ized by isotonic KCl. Shaded regions represent part r t ions. The BCpart i t ion is f i l led wi th paraf l in o i l and has a resistance of about l0MQ. The AB part i t ion is asmal l t rough wi th a resistance of about 20 KQ. H is a current source, G is a galvanometerindicat ing the zero-current condi t ion, and A is a d i f ferent ia l ampl i f ier measur ing the vol taged rop t c ross R r r ( . . ( b ) Schcn ta t i c d i ag ram o f F runkcnhausc r8 I dynam ic po ten t i onc t r i c mc thod .Node 0 has a rest ing potentra l / " and nodes I and + I are depolar ized by isotonic KCl.St ippled regions represent petro leum je l ly seals wi th resistance of about 5 l0MO. Importantpoints are labclcd and corrcspond to thosc in thc c i rcui t d iagranr shown in (c) . (c) Equivalcntc i rcui t of Frankenhauser 's method. The electrode impedance and balancing c i rcui ts are notconsidcrccl : Z inc l icates gcneral izct l impcdances as uscd in Sect ion l0.A. l . They are drawn anddcscr ibcd in thc tcxt as rcs istanccs l i r r s impl ic i ty only. R. is thc output impcdancc of thc st imulator(abou t500Q) .A rnp l i f i c rA ' i sah igh - i npu t - impcduncc .w idc -handw id tha rnp l i f i e r .Fo rcon rp l c t ed i scuss ion scc t cx t and Scc t i on l 0 .A . l .

490 10. volr,ccl cLAMprNc oF EXCTTABLE MEMBRANES

this goal: (i) improved gap insulation and (i i) the dynamic potentiometricmethod.

10.6.1.2.1 . GnpINsuI-nr lox. The res is tance of the gap across whichthe potential is measured has to be made very high. In one method, Sti impflieereplaced the conductive solution in pool B of the Huxley Stiimpfli record-ing apparatus by deionized isotonic sucrose solution. In another method,Tasaki and Frankroo e l in- r inated the pool B and made a large ar i r gap inthe in ternodal region. Wi th these methods the problem of at tenuat ionhas been largely reduced even though the use of atr air gap, for example,requires very extensive drying of the preparation to obtain less than l0'l/"attenuation. The disadvantages of gap insulation methods for voltageclamp techniques reside not only in the problem of the static or dc attenua-tion factors that violate our requirement of accuracy of potential

measurement, but also in the dynamic attenuation factors introduced bystray capacitances. In the gap insulation techniques. the pool at whichnode - I is located (pool C) is effectively insulated from the measuring pool(poolA) , which is usual ly grounded. This means that pool C is a very h igh-impedance pool with respect to ground and that i iny measurement ofpotential made there wil l be afl 'ected by stray capacitances to ground, slowingdown the recording system. Action potentials at room and even low temp-eratures are fast processes that can be seriously distorted if the frequencyresponse of the recording apparatus is below 5 kHz. With high-impedancegaps o[ the order of 100MQ, only 2pF are necessary to cut the f requencyresponse below I kHz. ln these cases it is necessary to introduce negative-capacity compensation techniques or driven-shield techniques that introduceinstabil ity in the voltage clamp circuits and/or complicate the electroniccircuitry.

10.6.1 .2.2. PornNrronnrRlc MErHoDs. We have already discussed theHuxley Stiimpfli method as a static-potentiometric method in which theaction potential t ime course cannot be continuously displayed because theamount of current suppl ied by H is manual ly changed. Their method,though, did not require a very high gap resistance Rs. since the currentsource H was adjusted unti l no current f lowed through the partit ion BC.From the diagram shown in Fig. 11a it can be seen that if the potential dropacross Ro. is dif l 'erentially measured by an amplif ier that in the case ofHuxley and Stiimpfli was used to inject current into pool B in a negativefeedback configuration, then the potential drop across Rss wil l be dynamicallymaintained at zero. This condition has been successfully used by Derksenrorto record action potentials potentiometrically. The method sti l l has the

o' R. Sternpf l i , E. tptr i t 'n t iu 10. 5() l t (195.+)r " " I . Tasak i and K . F rank , . { n . J . Ph rs i o l . l 82 , 572 ( 1955 ) .

"" H. E. Derkscn. . l t tu Pht .s io l . Phurnrut t ; l . Naer l . 13.173 (1965).

10.6. volracE cLAMp wrrH GAp rsoLATroN TECHNTeUES 491

limitation that there is a relatively high impedance between the voltagemeasuring circuit and ground. This l imits the bandwidth of the recordingsystem and negative-capacitance compensation is required to avoid attenua-tion of fast signals such as the action potential. A related but new method ofpotential measurement in the node of Ranvier that has overcome manyof these diff iculties has been designed by Frankenhauser.8'

The schematic diagram of the experimental arrangement of Franken-hauser's Vaseline gap technique is shown in Fig. l lb, and the equivalentcircuit is shown in Fig. l lc.t It can be seen in Fig. l lb that there are threenodes ( - l, 0, and + 1) involved in the circuit instead ol the two nodes usedin the Huxley Stiimpfli method. The third node (in pool E, Fig. 11b) is usedto pass current and stimulate the preparation. The circuit equations ofFrankenhauser's method can be deduced from Fig. 11c.

I t was shown by Frankenhauser8r [see Sect ion 10.A.1, Eq.( l0 .A. l ) ]that the potent ia l in poolA is g iven by:

Va: -V^ARBcl lRBc(A + 1) + RcDl , ( 10 .6 .1 )

where Z- is the membrane potential of the node, ,4 is the galn of amplif ier,4,, and R". and R.o are the resistances between the respective points in theci rcui t d iagranr{ (1. ' ig . l lc ) . This resul t dsmonstratcs the ef fect iveness of thepotentiometric recording of Z- since from Eq.(10.6. 1), when the open-loopgain of the amplif ier ,{, bccomes large, Zo approximertcs the membranepotential,

V a : - V ^ A - t . ( 10.6.2)

In this situation it can bc seen that /o and I/. are bclth cqual to zero. Thisis the same as saying that /6 is virtual ground, and since B is at groundpotential, no current can flow through R66, in which case, no current f lowsthrough the resistance Rco. verifying that the recording of /- is potentio-rnetric. In fact, the elTective loop resistance defined as (l/o - Vr)l lro isvery high, given approximately by

R r o o p : R B c ( , 4 + 1 ) . ( r0.6.3)

In the der ivat ion of Eqs.(10.6.1)- (10.6.3) , we d id not consider any currentdrain at the input of the amplif ier A,, for which purpose a high-impedanceamplif ier has to be used. Field-eflect-transistor input amplif iers are currentlyavai lable wi th impedances of l0 t2 10r3Q; th is ef fect ive ly prevents any

t In th is d iugrarn and thc fo l lo$ ' ing oncs. c lcctrodc potcnt ia ls and baluncir rg c i rcul ts arc no1inc l udcd . l n t hc ac tu r l imp l cn rc r r t r r t i on o f t he l , r ankcnhausc r mc lhod . : r t r l l anc i ng c i r eu i l r srequ i r c t l t oc l ncc l c l c c t r odcpo t cn t i : r l s . l I t h i sba lancepo t cn t i u l i sm isad . j us t cd . l . . r v i l l b cd i f l c ren tl rom zcro: consequent ly. I i , wi l l : r lso c le ' r ' ia tc f rom zcro potcnt ia l . but in l largcr l ] ropr)r t ion thunl . { l i r r a no rn ra l nodc 1 , , : 20 t , . . H i l l c ( " ' .

; ln Scct ion l ( ) .A. I and in l " igs. I I l l thc l incar c i r -cui t c le mcnts urc l , rbclcd us gcncnr l izct iimpc t l l n ccs . F i r r s imp l r c i t l , * c r : s c l l r c s i r r nc r cs i s t ancc no t i r l r ( ) n i r r t hc t c x t .

10. volrecr cLAMptNG oF ExclrABLE MEMBRANES

signi f icant current f rom leaking in to the ampl i f ier i tse l f . (Frankenhauser8lused a gr id pentode that dra ined less than l0- t t A.) I t is in terest ing to notethat the point D inside the node is held at virtual ground by the amplif ierA, and that the external solution at pool A is at - I/-, which is the inverseof the situation in conventional methods of recording membrane potential.In the three Vaseline gap techniques of Frankenhauser, node * I (pool E)is used to stimulate node 0. Vaseline insulation is necessury to inject currentinto the fiber, preventing current leaks between pools E and A. It is clear,however, by inspection of Fig. 1 1c that in this case the membrane potentialZ. would be shunted by the resistance given by the series combinationReo * Ruo. This would produce current f low through the membrane,defeating the whole purpose of the potentiometric method unless thepotential at pool E is forced to be close to zero, preventing current f lowacross the Ruo resistance. In the actual implementation of the Frankenhausertechnique. a stimulator is connected between E and ground, providing alow-res is tance pathway g iven by the output impedance of the uni t . [n th iscase current wil l circulate through Ruo only during the stimulus when trzuis madc d i f ferent f rom zero. FrankenhausersI pointed t rut th is problem andused a -500-Q-output irrrpedance stirnulator (R,) to correct it. It is cor.rcluded,then. that the st imulator used in the Frankenhauser method should bea voltage rather than a current generator.

Probably the most i rnpor tant inrprovement of Frankenhauser 's method ofnreasur ing the potent ia l o f the node of Ranvier over prev ious methods is thefact that pool C, a high-impedancc pool, is electronically at virtual ground.This means that stray capacitances to ground would be insignificant alter-native current pathways, thus improving the frequerrcy response of therecording c i rcu i t wi thout the need of negat ive-capaci ty compensat ion andother speeding-up techniques. Frankenhauser8 I repclrted a bandwidth forhis apparatus of 30 kHz which seems satisfactory for most electrophysiolo-g ical s tudies.

1 0.6.1 .3. Vol tage Clamp of the Node of Ranvier . Scvcra l arrangemenrsto contro l thc mcmbrane potent ia l in a s ingle node of Ranvier have bccnreported. l02 r08 Of these, the most commonly used today is that in t roducedby Dodge and Frankenhauser los wi th modi f icat ions descr ibed by Nonnerr08

r" ' J . del Cast i l lo . J . Y. Let tv in, W. S. McCul loch, and W. Pi t ts . Nuture (Lont luu) l t t0, 1290( I 957 ) .

r ( ' r B. Frankenhauser and A. Pearson. . l t ru Phl ' .s io l . Stund. 42, Suppl . 145, 45 ( 1957).r"o I . T i rs i rk i and A- I - - . Bak. J. Neuntphrs io l .2 l , I l . l ( 1958).r"5 F. A. Dodge and B. Frankenhauscr. .1. Phv.s io l (Londonl 143,76 ( l 95U).

" 'n F. A. Dodge and B. Frankcnhauser, J. P/r_r ' . i io l . (London) 148, l t l | (1959).r" t C. Bcrgman and R. St inpf l i , Hclr . Pht .s io l . Phurntatol . A( te24,211 (1966\.roE W. Nonner. Pf luurcr .s .4rch. 309. I76 ( 1969).

10.6. volrncE cLAMp wtrH GAp ISoLATtoN TECHNIQUES 193

and Hil le.roe Del Casti l lo and Moore (see Moore and Cole2o) cut thenerve fibers in pool E to be able to inject current more efficiently into thenode 0. Nonnerro8 inc luded as standard in h is methods the cut of thef iber at the in ternodes: Hi l leroe a lso cut the f ibers at the in ternodes. butused the Dodge Frankenhauser electronic arrangement using two amp-lif iers for voltage clamping instead of one as described by Nonner.ro8

Figure l2a shows Hi l le 'sroe modi f icat ion of the Dodge-Frankenhauser

system. The circuit diagram oi the Dodge Frankenhauser voltage clamp is

shown in F ig. l2b.We have previously shown that if ,4 (the voltage gain of amplif ier Ar)

is large, the fo l lowing statements are approximately val id :

(a) Vc : lir, : 0,( b ) v r : - v ^ .(c) no current f lows ir.r circuit ADCB; therefore. current iniected through

Rpp l iows only through c i rcu i t EDA.

It is easy to see how Frankenhauser's potentiometric voltage measuring

circuit can be used in a voltage clarnp configuration if we connect a second

ampl i f ier A, in the c i rcu i t as shown in F igs. l2a and 12b. The fo l lowing

cqr . rat ion can bc wr i t tcn when the second ampl i f icr A, of gai r t A ' is contrected:

V p : A ' ( V * - V - ) : A ' ( V o + 4 . - ) .

From th is equat ion i t fo l lows that /o : - l lcor" r i s i l rce I /n is mainta ined at- V^6y amp l i f i e r A t ,

V^ : Vcou '

The conclusion is that amplif ier A, passes current through the membrane

to keep V^: Vcou. In order to calculate the value ol the current I- we

can observe that point D is at zero potential, and since no current f lows

through DCB, then

/E : 1-R6p, I - : Vel Rro.

In Section l0.A.l a general derivation is presented with Laplace-transformed

var iables consider ing A and A'as funct ions of f requency. The above equat ions

are only valid in the hmiting case when A and A' - ' t for all frequencies.

One of the l imitations of the Dodge Frankenhauser voltage clamp is

related to the attenuation factor. When we discussed Frankenhauser'spotentiomctric rrrethod for measuring potential from the node of Ranvier,

we found tht t the loop res is tance wi th ampl i l ier gain.4 was Rs6(,4 * 1)

" ' u B . H f l l e , J . Gen . Ph t ' s i o l . 5 l l . 599 (1971 ) .

( o )

HIi fl

10.6. volracE cLAMp wtrH GAp rsoLATroN IECHNTeuES 495

IEq. (1U.6.3) ] , which min imized the at tenuat ion factor . A l imi tat ion in th ismethod arises becausc pool B cannot be made wide enough to prevent theat tenuat ion; a wide pool leads to a large phase lag in the potent ia l recordingsystem. This phase lag was acceptable for recording the membrane actionpotential but seriously l imits the stabil ity characteristics of the closed-loopcorrfiguration when the voltage clamp anrpli l)er (Ar) is incorporated in theloop. A very simple rule dcfining the stabil ity characteristics of a feedbackcontrol circuit is that the feedback should never become a positive feedbackfor gains larger than l. Phasc lags introduced by the loop network and theamplif ier itself may transform the negative feedback into positive feedbackif t lre signal frequency becomes higher than certain values. A phase lag of90' between the input and output signals of a network is produced by each"pole" in a transfer function equation. A system with two poles, then, canproduce a phase lag of 180., and if suclr a network is in the leedback loop.the closed-loop system may become unstable for gains larger than l. Whenthe second amplif ier (A:) is added in order to voltage-clamp the preparationthe situation becomes more crit ical. Wc can observe that the input signal toA, is /a, which already has a phase lag introduced at least by the productA(s)Zur$) . Ampl i f ier A, wi l l add at least another pole to the system, makingIrecessary a careful analysis of thc network closcd-loop equations in order to

Fr< ; . 12 . H i l l e ' sn rod i f i ca t i ono f t heDodge -F rankenhause r ro5vo l t agec lampo f t henodeo f

Ranv i c r and Nonnc r vo l t agc c l amp . ( a ) Schc rna t i c d i ag ra rn : Myc l i na t c t l t i b c r s o f ab t l u t l 6 pn r

internodal d iameter and 1.5 mm rnternodal length are used. The nerve is mounted ln an acry l icchamber in which the s ize of thc pools is ad. justable. to8 Four pools (shaded areas) are lbrmedbl luy ing three str ings ul Vascl ine (b lack arcas). Thc t ibcr is cut in pools E ant l ( . Pools A and Bcontain Ringer 's solut ion and pools E, 'and C contain isolonic KCI or CsF solut ion. The s izesol ' the gaps are 500 (BC), 200 (AB), and 200 l im (EA); the s izes of the pools are 250 (pool B)and 150 pm (pool A). Seal resistances are about 5 MO. The electr ical connect ions are madethrough lM KCI agar br idges to calomel e lectrodes to which the electronic equipment isc t r n n c c t e d . I . . , , n , i n c l u t l c s t h c p u l s c a n d h o l d i n g p o t c n t i a l . ( b ) E q u i v u l c n t c i r c L r i l o f l h c n r c t h o d :Ampl i f ier A, has gain / (s) and ampl i f ier A, has gain,4 ' ( , r ) . Ampl i f iers A, and A, need to bephase- lag compensated. Membrane current ( /M) is measured as VulZuoand membrane potent ia las f^ . To measure the menrbrane potent ia l under current-c lamp condi t ions, ampl i f ier A, isdisconnectcd. and E is connected to a pulse gcnerator to st i rnulate the preparat ion as shown inI - ' i g . l l c . l - o r dc ta i l s sce t cx t anc l Scc t i on l 0 .A . l . ( c ) Equ i r a l en t c i r cu i t o f Nonnc r ' s me thod :S i r r ne l r s ( b ; r : x ccp t t ha t on l y r l nc lmp l i l i c r i s uscd t o vo l t agc -c l an rp l hc p r cpu lu t i on . Thc vo l t agcgenerat() r used to impose the membrane potent ia l ( 1, . . . " ) had an output impedance of 200 O andt l r e f c c d b a e k a m p l i f i c r A , h a d l r i g h i n p t r t i m p c t l a n c c ( l ( X ) M O , 7 p F ) . l o i l b i a s c u r r c n t ( 1 0 ' r A ) .and lc tw t ' rutput impedance (100 O) in addi t ion to wide bandwidth and high ampl i l icat ion (upto t i6 dB). The ampl ihel was cernpensated based on meas'urements of lhe open- loop character-i s t i c s o l ' t h c c i r cu i t us i ng t hc NyqL r i s t s t ab i l i t y c r i t c r i on . ' n

- f hc mcn rb rane cu r ren t i s mcasu rcc l

indirect ly ( l 'n) and the membrane potent ia l is measured as the negat ive of the potent ia l at pool A.In order to lneasure the vol tage in a current-c lamp conl igurat ion, the oulput of ampl i f ier A,has to be connected to A ( instead of E), d isconnect ing the st imulator V.o*. Vu is measured as- /^ under those condi t ions.

( b )z - ^zco

Vcou

," l[zec B Zae

l u 'A zea

I

-+^L -+o>.'F4 ,7

lf

496 10. volrnc;p cLAMpTNG oF ExctrABLE MEMBRANES

evaluate i ts s tabi l i ty . Such a study has been done by Nonner, r08 who int ro-duced a new voltage clamp system for the node preparation.

Nonner's method represents an improvement of the Dodge Franken-hauser method mainly in two respects: (i) the attenuation artifact waseliminated even though the size of pool B was made very small, and (i i)by studying carefully the loop impedances and eliminating one of the ampli-f iers of the loop, the frequency response of the voltage clamp could beincreased to at least l0kHz, which allows one to observe the rapid ionicconductance transients occurring at room temperature. A circuit diagramof Nonner 's technique" '3 is shown in F ig. 12c, and a general der ivat ion ofthe circuit equations is presented in Section 10.A. l. The air gap included inhis technique helped to eliminate the attenuation artifact described byDodge and Frankenhauser (discussed above), probably because the leakthrough the Schwann cell space disappeared with the drying of a certainportion of the internode. To record membrane potential under currentclanrp condition, Nonnerlos used directly Frankenhauser's potentiometricmethod, but to voltage-clamp tlre nodal membrane he switched the outputof the potentiometric amplif ier fronr pool A (current clamp) to pool E(voltage clamp). The voltage clamp method is also based on the fact that cbecomes virtual ground when the gain A of the amplif ier is large. In thatcondi t ion the current through R.o is zero, and point D is a lso v i r tual ground.The potential at pool A becomes -v^. lf a low-impedance voltage sourceis used to set the value of voto - vcoM,itfollows from the equations discussedin Sect ion 10.4.1 that V^: Vcou. In order to measure the membranecurrent 1- we demonstrate a lso in Sect ion 10.A. 1 that / - : Zu/Rup. I t isinteresting to observe that the final voltage clamp conclit ion is achievedwith Nonner's technique without several of the stabil ity complicationspresent by the use of two amplif iers in the Dodge Frankenhauser technique.The second ampl i f ier (Ar) of the la t ter technique is redundant in the vol tageclamp s i tuat ion;A, can bc replaced by connect ing the output o[ampl i f ierA r to E and adding a low-impedance source to set the value of I/o as Nonnerdid. The equations can be compared with the Dodge-Frankenhauserloop equations to verily that the elimination of one of the arnplif iers simplif iesthe equations for the loop and thus makes easier the design of a fast andstable voltage clamp. For an analysis of the stabil ity characteristics, it ispossible to set values for the terms included in the loop equations and studysystem performance using the stabil ity criteria commonly used in feedbackcontrol systems (see, for example, D'Azzo and Houpiss8 arrd Nyquisttl0).Nonnerrt '8 analyzed his voltage clamp system extensively and the reader isreferred to his paper for an explicit discussion.

r r " H . Nyc lu i s t . B t , / / S l s l . 7 ( , ( ' 11 . . / . ( 1912 ) .

10.6. volracE cLAMP wlrH GAP lsoLATIoN TECHNIQUES 491

Instrumental and thermal noise associated with Nonner's voltage clamp

technique were analyzed by Conti et al.tt I when they used this technique

to measure Na current f luctuations in the node of Ranvier. They found that

the background thermal noise contributions from the partit ions and passive

membrane patch may be reduced by increasinE Zeo, Zu., and membrane

impedance Z^. and by decreasinE Zco. Besides, the amplif ier's voltage

(e,2) and current (i,2) input noise are multiplied by factors (containing the

inrpedances Zuo, Zuc, Zrr. and Z^) that are minimized by the same proce-

dure.

10 .6 .2 . Vase l i ne -Gap Techn iques i n S ing le Musc le F ibe rs

Frankenhauser et al.t r2 demonstrated that the principles involved in

potentiometric recording of membrane potential at the node of Ranvier

car be applied to single muscle fibers. The error in the potential measure-

ment in this latter preparation is about 1\ ar low frequencies, but can

reach about lO')t, at frequencies of 50 kHz. Since a 50-kHz bandwidth isadequate for most purposes. their method has been attractive enough to

€ncourage muscle investigators to use it for voltage clamp studies. Moorer r3

appl icd for thc f i rs t t ime the potent iomctr ic ntc t l tod for t r ip le-Vascl ine-gtpvol tage c lamp stuc l ics in s inglc musclc f ibers.

A significant improvement on the three-Vaseline-gap voltage clamp, and

on muscle voltage clamps in general resulted from the work of Hil le and

Campbell. lra These authors applied the Dodge Frankenhauserlos tech-

niques to short segments of muscle fibers cut at both ends instead of oneend as Moorer l3 d id. They a lso cut the f ibers in a solut ion of isotonic CsF,

which helped to keep the cut ends unswollen and left the muscle membranein pool B depolarized but with high resistance. l-he diagram and schematic

c i rcu i t o f the Hi l le Campbel l vo l tage c lamp technique is shown in F ig. 13.

The circuit equations for that configuration are developed in Section 10.A.1

In order to lrave a last and stab]e voltage clamp circuit, Hil le and

Campbellr ra employed the discrete amplif iers prcviously used by Nonnerro8

that can be externally compensated. The compensation network for ampli-

f iers A, and A, in the Hil le-Campbell voltage clamp can be calculated

theoretically from the open-loop equations and/or determined by trial and

error unti l the open-loop Bode plot has a rolloff of less than 20 dB/decade.

r r r F. Cont i , ts . Hi l le , B. Neunrcke, W. Nonner, and R. St impf l i . . l . Phl 's io l . (London)262,

699 ( I 976).f f 2 B . F r a n k e n t r a u s e r . B . D . L i n d l e y , a n d R . S . S n r i t h , J - P h - t s i o l . ( L o n d o n ) 1 8 3 ' l 5 l ( 1 9 6 6 ) .t t t L . E . Moo re . . / . Gcn . Ph . r ' , s i o | . 60 , I ( 1972 ) .I to B. Hi l le and D. T. Canrrrbcl l , J . Gcn. Physio l . 6 '1,265 (1976).

498 10. volrncl cLAMptNc oF EXCTTABLE MEMBRANES

Ftc; . l3 l . Vol t : rgc c l i tn lp of s inglc rnusclc f ibers wi t l r Vuscl i r rc-gup lcchniqrrc. Expelmcntalchamber: The chanrber (CH) is bui l t of Luci te and rcsts on an aluminur l b lock 14B) cooled byPel t ier coolers (PC) connected to a leedback c i rcui t to contro l the temperature measured by athcrnistor (T) insta l led in pool A (see inset) . HD is the heat t l iss ipator for the pel t rer coolers.The f t rurelectrodes (E. , En, En, and Er) are bui l t wi th pel lets of s intered Ag/Agcl in l M KCIconnected to rhe chamber wi th agar.br idges (BR) f i l led wi th I M KCI and contain ing a f loat ingplat inum wire to decrease the high-f requency impedance. The electrodes are mounted in anelectrode holder (EH) that can be removed for storage of the electrodes. Pools are separated byvasel ine str ings (vS), and solut ions are changed by supply ing solut ion at I and suckrng rr awayat O. Inset : detai lof a muscle f iber segment (MF)instal led in the charnber. The pools are marked(A, B, C. E). VS are the Vasel ine seals, and T is the thermistor

The reason trial and error may be necessary is that the phase lags introducedby the preparation arise from a cable structure and not a lumped RC network.

one of the l imitations of t lre vaseline-gap voltage clamp (common toany voltage clamp system in which the rength constant of the preparation isnot modified by impalement with an axial wire) is that the membrane currentis collected from a patch of membrane of f inite length, while tlre potentialcontro l is rest r ic ted to a membrane r ing at the AB par t i t ion. we shai l d iscussthis problem extensively below in Section 10.6.4 and Section 10.A.2 as ageneral l imitation of gap voltage clamps with special reference to a discussionby Hil le and Campbell I ra of the muscle-fiber voltage clamp.

The Hil le campbell technique for muscle-fibers voltage clamps hasbeen used successfully to record sodium channel gating currents and excita-t ion-contract ion (EC) coupl ing charge .nou",n"ntr . ' t r I t hu, fur ther been

t ' ' . f . Vergara and M. Ca hala n. Bioph1r. J. 21, | 67a (197$.

10.6. volracE cLAMp wrrH GAp rsoLATroN TECHNTeUES 499

Ftc; . l3b. Vol tagc c lanrp of s inglc musclc f ihcrs u, i th Vascl ine-gup tcchnique. Equiv l lcntc i r cu i t : Thed iag ra rn i ss i r r r i l a r t o th t t o l ' F i g . I 2b ( t hcDodge F ranke r rhause rvo l t agec lamp) . l nth is casc there is an acld i t ional impcdance T, t t , lhLr t reprcsents thc membrane patch in pool B-A r cs i s t ancc i n sc r i c s w i t h t he ou tpu l o famp l i f i c r A1 (R . ) i s i nc l udcd 1o measu re t hc t o ta l cu r r cn tl . . b u t i s s e l e c t e d o f a s r n a l l v a l u c ( l K O ) a n d i s n o t i n c l u d e d i n t h e d e r i v a t i o n l i r u n d i n S c c t i o n10 .A . l . Ano the r r es i s t a r r ce t ha t h i r s r r o t been i nc l uded i n ( he c l e r i va ( i ons o f Sec t i on l 0 .A . l i s ascr ics rcs istance Ro wi th thc mcmbnrnc in pool A that should bc compcnsatcd for by posi t ivcf ccdbuck . r r a I t c rn bc r c l d i l y i nco rpon r t ec l bv r cp l t c i ng t " , by 1 , i , , + 1 -RA i n Eq . ( 10 .A .14 ) o fS e c t i o n l 0 . A . l . A t l i s c u s s i r r n o f s c r i c s r e s i s t a r r c e c o r . n p L ' u s a t i o r . l i s p r c s e n t e d r n S e c t i o n l 0 l . 5 . S .

used to study optical events related to the EC coupling in muscle fiber.1 I t" I r 7

Vergara et al.tt6 report a nrodification of t lre Dodge-Frankenhauserroschamber to include an optical f iber in pool A to i l luminate the musclefiber and verif ied that it can sti l l be used for voltage clamp experiments.They also improved the Hil le-Campbell technique in three aspects.

(l) The fibers were cut in relaxing solutions containing K-Aspartateinstead of CsF or KF. Under these conditions the fibers are able to contractin the region of pool A (and only there) when they are depolarized. Themovement associated with the depolarization is blocked by adding 2-4 mMEGTA in pool E and C and waiting about 20 min for diffusion.

(2) The fibers were depolarized for the first t ime, after dissection. in ahigh-K solut ion kecping the f ibers st rc tched. thus prevent ing the over-shortening that occurs when they are depolarized without being held.Under those conditions the fiber contracts and relaxes as described byHodgkin and Horowicz.r t8 The sarcomere length of the {ibers, mounted after

' r6 J. Vergara, F. Bezani l la, and B. M. Salzberg, .1. Gen. Ph. t 's io1.72,115 \1978).I r7 P. Palade, Bioph. t ' .s . J . 25, l42a ( 1979).' rE A . L . Hodgk in and P . Ho row icz , . l . Ph rs i r t l . ( London ) 153 , 3 t t 6 ( 1960 ) .

zec ! zot

500 10. volrecE cLAMptNG oF EXCTTABLE MEMBRANES

this precaut ion is taken, is about 2.4 + 0.2,umttu instead of 1.6 * 0.2pmreported by Hi l le and Campbel l . r ra

(3) When the fibers were treated with TTX, normal-looking delayed Kcurrents were described, whereas one of the major dif l jcult ies in the Hil leCampbell 'ra report was that they were not able to record normal-looking Kcurrents.

It should be pointed out f inally that the fact that both ends of the fibers arecut means that both can be used to diffuse substances to the fiber interiorand eventually replace the normal ionic content of the sarcoplasm. Un-doubtedly, a good blockage ofthe K current can be obtained in this prepara-t ion by soaking the cut ends in a solut ion conta in ing CsF. I ra

The Hil le -Campbell technique and subsequent modifications of itconstitute a reasonably accurate voltage clamp system for muscle fibers.We discuss bc low in Scct ion 10.6.4 thc l imi tat ions of gap iso lat ion vol tageclamps and give there error criteria for judgment of the longitudinal disper-s ion of the membrane potent ia l o f the l ibcr a long thc contro l pool . One con-c lus ion to be drawn f rom that analys is is that i f thc gap is made smal l wi threspect to the radius of the fiber and the conductancc of the membrane doesnot becomc too large, the membrane in the gap can be considered almostisopotential in the longitudinal direction.

1 0 .6 .3 . Suc rose -Gap Me thods

Stdmpfliee introduced the use of deionized sucrose to increase the impe-dance of the recording gap. This technique was extended to voltage clampby Jul ian et u l . t te ' t20 for nonmyel inated g iant axons of the lobstcr and byMoore et a l . t2 t for the squid g iant axon.

Two sucrose gaps and three pools (two lateral and one central) aredefined for voltage-clamping a cylindrical cell. Julian et al.tte't20 calledthe lateral pools 1 pool (at which current was injected to the fiber) and Zpool (at which voltage was recorded). The central pool was used for currentrecording and was kept at virtual ground by a current-to-voltage converter.The length of their sucrose gaps was about 600 pm, and the central poolcould be made as narrow as 50 pm. The gap resistances were at least 25 MQ,and under these conditions the potential measurements are accurate withless than 5') '"error. The sucrose-gap technique has the problem that there is a20 60mV hyperpolarization in the recorded potentialr20 that has beensuggested to arise from a l iquid junction potential between sucrose and

r r e F . J . Ju l i an . J . W . Moo re , and D . E . Go ld rnan , J . Gcn . Phy . s i o l . 45 , l 2 l 7 ( 1962 ) .r 2o F . J . Ju l i an , J . W . Moo re , and D . E . Go ld rnan , J . Gcn . Ph t . sk t l . 45 , I 195 (1962 ) .r2r J. W. Mcrore, T. Narahashi , and W. Ulbr icht , . l . Phr,s io l . (London\ 112. 163 (1964).

10.6. volrncE cLAMp wrrH GAp rsoLATroN TECHNTeUES 501

sea water.r22 Despite this problem, the currents recorded with the sucrose-gap technique in giant axon suggest that good potential control is achievedprovided that the central gap length is not longer than the fiber diameter(see discussion below).

The sucrose-gap technique described by Julian et al.r re'r20 has beenused almost without modil ication in skeletal muscle fibers by Nakajimaand Bast ian. t23 These authors found, though, that the Jul ian-MooreGoldman techniques could not be used directly with single muscle fibersof the frog but could be applied to Xenopus muscle fibers. Nakajima andBastianr2s also found, as expected and discussed above, that an importantbandwidth l imitation (in the voltage clamp) is induced by stray capacitancesto ground; their potential recording pool V is a high-impedance pool.

A modification of the double-sucrose-gap technique was made by Ildefonseand Rougiert2o to voltage-clamp single muscle fibers of the frog. Theirsystem is not only a sucrose-gap technique but also uses Vaseline andsucrose.

Recently, Duval and Leotyr2s have used a double-sucrose-gap techniquein mammalian muscle fibers in which they cut the ends of the fibers, thusimproving the performance of the voltage clamp. Duval and Leotyt2s alsomeasured the potential in the test gap lengthwise with a microelectrode,verifying that there was a partial lack of longitudinal control. They decidedthat gap lengths of 100 pm for f ibers ol50 70 1rm were safe values to preventthis lack of control. In the next section we discuss this problem extensively.

10.6.4. Errors Int roduced by the Fin i te Length of the Gap

The measurement of current in the gap voltage clamp will only be exact ifthe length of the gap is made infinitesimally small. We consider in thissection the errors introduced by making the gap of f inite length.

This problem has been considered theoret ica l ly by Cole (Ref .7, p.a l8)and is analyzed in detail in Section 10.4.2. We show there that in mostexperimental situations the core-conductor approximation is valid.f Wegive formulas to calculate both the maximum error in the voltage at the EA

t 22 M . P . B laus te i n and D . E . Go ld rnan , B iophvs . " / . 6 , 453 (1966 ) .'2r S. Naka. l inra and J. Bast ian, J. Gen. Ph. t , .s io l .63, 235 (1974).' t t M. I ldefonse and O. Rougier , .1. Ph.r ,s io l . (Londonl222.373 (1972\.'2s A. Duval and C. Leoty. J. Phv.s io l (London)278,403 (197U).

tOur three-dimensional model in Sect ion 10.A.2 docs r tot inc lude a f -system nctwork:

consequenl . ly , the corc-conductor c l rb lc moclc l may not bc a goot l descr ipt ion of a musclc I ihct

i n a san .

502 10. volrecr CLAMptNG oF EXCITABLE MEMBRANES

Frc. 14. Errors produced by f in i te gap length: Maximum error in membrane vol tage 11" andin membrane current ry, produced by the length of the gap as computed by Eq. (10.A.2g) and(10.A.29) der ivet i f rom rhe core-conductor model in Sect ion 10.A.2.

partit ion with respect to the controlled voltage and the error in the measuredcurrent with respect to the current acloss a controlled patch of membrarre.These errors are plotted in Fig. I 4 as a function of a single parameter U \[b1 ,in which / is the length of the gap at pool A, b is the frber diameter, and A isthe generalized space constant defined in Section 10.A.2. The experimentalistcan est imate f rom Fig. l4 the maximum gap length that r r r r r . , h im vol tageand current control, within a selected tolerance, when the fiber parameters band A are known. Hi l le and Campbei l r ra calculated the lonsi tudinalvariations of the potential inside the fiber assuming that the rnimbranecurrent is constant and that the fiber in pool A can be considered as a semi-infinite cable. They derived

V ( l ) : I M R t l 2 l A . (10.6.4)

In the potentiometric method, the internal potential voe) can be defined as

V p ( z ) - V ( z ) * V r ,

in which I, '(:) is the membrane potential as a function of the distance fromthe AB par t i t ion and vo is the potent ia l a t the AB par t i t ion (z :0) . I f wemake th is change of var iable in Eq.(10.A.27) of Secr ion 10.A.2. we obta in(see a lso ColeT)

I p{ : ; : Zolcosh(-- r r /2 / , hAy - 1y. ( 10.6.5)

10.6. volrncE cLAMp wrrH GAp rsoLATroN TECHNTeUES

that the membritne current Iy: VslR^ is constant over the entire length of

the gap. It is interesting to observe in Fig. 14, though, that only at I l l t ismaller than 0.05 can the membrane current be assumed constant. but at

larger and sti l l realistic values of I1J-bA(for an activc membrane for example)this approximation no longer holds and the exact formula should be used.Within the restrictions imposed by the first-order approximation, Hil le andCampbel l ' ra showed that lzo( / ) can be roughly cst imated f rom thc potent ia lat pool E (Ve.) by the formula

V " ( l ) : v ' E l l ( + 2 k ) .

in which l i is the length of the fiber segment from z : I to the cut end at poolE. An exact formula relating Vo() and Vu can be derived from Eq. (10.6.5)and other equations for the core-conductor model included in Section 10.A. l.The resul t

sr

;r

l

0.3I

vrun

0.2

V"(l) : Yucosh(/ . Zru/nl ' l - t

cosh(1 , 2 t ! /hA) + $J2 , hA)s inh(1 . " z , rFL , t t l - I

A f i rs t -orde. appr .x imat ion of thc Taylor-ser ies cxpansi .n of Eq. (10.6.5)evafuated ar z : / g ives the Hi l le campbel l resul t [Eq.(10.6.4) ] prov ided

shows that, in the exact case, it is necessary to know the membrane parametersto estimate fzo(/) as a function of V.. The above equations may give thewrong impression that the potential lzo(D diminishes as k is made larger.This is not the case because the feedback amplif ier (Ar) makes lzu larger if kincreases, compensating for the change. The deviations of Zo(/) are onlyfunct ions of the error ryy in F ig. l4 and the imposed potent ia l /o .

The error criteria developed above can be used to roughly estimate theerrors in voltage-clamping an active membrane. For example, when the Kconductance increases upon depolarization A decreases, and if the currenthas reached a steady stlte (that is, when there is no K inactivation), a fairlygood estimate of the errors in the records can be made from Fig. 14. In anon-steady-state s i tuat ion Fig. l4 may st i l l be used to obta in a roughestimate of the errors. For the case of negative conductance we have in-cfuded in Sect ion 10.A.2 Eqs.(10.A.31) and (10.A.32) to est imate 4 i andr;f , respectively. lt can be noticed lhal 4i and 4i in this case have oppositesigns with respect to those calculated in the positive conductance case.

Numer ical computat ions wi th Eqs.(10.A.31) and (10.4.32) show that for

valucs o[ tt/,r[ahAin the range used in Fig. 14, 4i. and 4f do not differ signifi-cantly, in absolute value, from 4u and ryr. These equations could be used inthe steady statc for cases ir.r which Na inactivation is absent, but care shouldbe exerciscd in using them whcn normal sodiLrm condustance is present.For both positive and ncgative conductances in the non-steady-statesituation. a better estimation of the errors can be obtained with numerical

10. j . Concluding Remarks

Thc conccpt that ail celrs are bounded by a superficiar rayer with veryspecial properties was derived rrom measurements of electrical and osnroticproperties. These ceil membranes are highry permeabre ro water and tolipid soluble substances, and they transpJrt lons, sugars, and amino acidsup and down electrochemical gradients. In this articre we have been con-cerned with certain techniques wrrich are used to study the erectricar prop-erties ol some cell membranes which are excitable, i.e., ttrey respond toelectrical or chemicar stimuri and produce electrical efrects. In order tostudy these electrical properties it i i necessary to measure or control trrevoltage across and the current through the membrane over an area in whichthey are relatively uniform. This has been done most successfuily with thegiant axon of the squid, the node of Ranvier in myerinated axons, and by thedevelopment of gap techniques and patch isolation with external electrodes.Some very good work has been done using microerectrodes in both cyrin-drical and spheroidal cells, but the high rJsistance of such electrodes andproblems of uniformity make quantitatlve studies diff icult.

Much of this paper wourd be relevant to attempts to control current(current clamp)' but the voltage cramp is emphasized because it i .s thenatural quantity to contror in a system with large capacitance and vortage-dependent elements.

504 10. volrecE ctLAMprNc oF ExcrlABLE MEMBRANES

computations of the cable equations using the Hodgkin and Huxrey equations.This has been done for the sucrose gup iur" by Moore et al.t26.t '2-l

10.A. Append ix

10 .A .1 . C i r cu i t Equa t i ons o f Vase l i ne_Gap Vo l t age C lampThe purpose of this section is to deverop the circuit equations of thepotentlometric method of Frankenhauser. and three commonry usedvol ta-ee c lamp systems: ( i ) the Dodge-Frankenh.user vol tage cramp o 'amyelinated-fiber node of Ranvier, 1ii) tne Nonner vortage clamp of a node

of Ranvier' and (i i i) the Hii le and Campbeil voltage c-ramp oi u ,t "t"tutmuscle l iber. The equations deveroped here can be adapted to modifications

of these methods and we think tl iey wii l herp in unierstanding the basic

' to J. W. Moore. F. Ramon, and R. W. Joyner. Bioph. t .s . J . 15, I I (1975).' t t J . W. Moore. F. Rarnon. and R. W. Joyner, g i r ry;hr , . r . . / . 15.25 i l975).

10.A. a,ppsNnrx 505

principles involved in their application. voltage ancl currents are Laplace-transform variables, and the circuit elements are defined as generalizedimpedances Z,r , that may behave as pure res is tors or a combinat ion of l inearelements. The potent ia l a t a point w of the c i rcu i t is ca l red u* . The t rans-formed membrane potcnt iar and current are cal red l , - and i - , respectrvery.The d iag rams shown i n F igs . l l , l 3 a re used i n t he de r i va t i on o f t he c i r cu i tequat io 's . Nei ther potent ia ls associated wi th the c lect rodes nor balanci 'gpotent ia ls wi l l be considered here for two reasons: ( i ) the resut t i r rg equat iorrsare simpler. and (i i) every electro<Je pote-ntial should be balanced ele-ctronicallyto reach the configuration analyzcd irr the circuit diagr.ms that correspo'dsto the ideal case.

10.A.1.1. Potent iometr ic Method of Frankenhauser. Theexper imenta larrangcment and c i rcu i t d iagrarns are shown in F ig. l lc wi th Lo :0. Usingelementary c i rcu i t analys is and the re lat ion

we find

and

aP' : - Auc '

ue : -u_AZsrf(Zsc(,{ + l) * Z.r)

i* : -r ,_( l + K)l lZBcU t l ) * Zcol,

l - + -u^R , fZnnZur ,

w i l h

K : l(Zsc + Zc;(Rs * Ze) + ARsZBC),,LRs(Zue t Zuo) + ZEDZE^].

( r0 .A .3 )

ln order to record the.actuar membrane pote. t iar wi thout at tenuat ion, i -must be zero; th is condi t ion is only , . , . , . t *h"n both ,4 _ :c and R. _ 0.

In practice when ,4 + trt we get. for R, small,

( 1 0 . A . 1 )

( r0 .A .2)

( r0 .A.4)

from which we can estimate how rarge R. can be to keep i- under a specifiedto lerance. For the node of Ranvier R, r i . /o- x l0ra i l . " io dra in ress thanl0 - " A f rom the node a t I z - r 100n rV l "R . : lOke . I n t he case o l - t hepotentiometric method appried to muscle fibers, Zuo is at least r00 timessmaller than in myelinated nerve fibers and one can allow a maximumcLlr rent dra in f rom the membrane patch o[about r0 roA. This requrres areslstance R. of at most I kC).

506 10. volrnct cLAMpTNG oF EXCnAITLE MEMBRANES

10 .A .1 .2 . Dodge -F rankenhause r Vo l t age C lamp o f t he Node o fRanvier. We consider the expcrimental arrangements and circuit diagramsshown in Figs. 12a and l2b, respectivcly. The final equations wil l con-tain the terms,4 and A', but only after the final derivation is made wil lthe l imits A and A' - ;c, be taken. Ideally. A and.4'are complex gains of theform ,4(s) : Aol0 f sr), and the reader can use the equations derived heremaking that replacement to study thc performance of his particular voltageclamp giving the appropriate form to every impedance element as well.In this case. we can analyze the circuit equations with the following equationslor the ampl iher conf igurat ions:

10.A. eppaxorx

We can calculate u-:

D- : ucov - i*,zFD(zBC + zc)ll(A + l)zBC I Zco * Zuof

and

(10.A.r2)

i^ -- upl l lZuo - (Zro * Zec + Z')IAZB.Zurf. (10.A.13)

f f rve take the l imi t A + ' r , ,wc get Eqs. ( 10.A.9) and ( I 0 .A. l0) der ived for rheDodge-Frankenhauser voltage clamp. It can be observed though that inNonner's clarnp, the general expressions for tr/- and 1- are simpler than inthe Dodge and Frankenhauser method. Because a single amplif ier is usedin the vol tage c lamp c i rcu i t the product of AA'present in Eqs.(10.A.7)and (10.A.8) does not appear in Eqs. (10.A.12) and (10.A.13) .

10 .A .1 .4 . H i l l e -Campbe l l Vo l t age C lamp o f S ing le Musc le F ibe rs .Thc differencc between the Hil le Campbell and Dodge Frankenhausermethods is that in the former there is a patch o[ membrane in pool B notex is t ing in the la t ter . This is shown in F ig. l3b as the impedarrce Zou. Thistcrm s l ight ly changcs the equat ions der ived in par t (b) , g iv ing

t 'm : ( trcoM A' - i" ,Zo,o)ul l l , ( r 0 .A. l4)

wi th

u : ZsgZps(A + l) + ZFD(ZBC -l Zc) + Z:DZFB

and

$ : Zee(Zp1, I Zsc t Zr) -l (Zes + ZF)(Z)D * Zod + AA',ZBCZFR.

I f rve take the l imi t Zps- t - i l a and / , Eq. (10.A. l4) becomcs idcnt ica l toEq. (10.A.7) . The equat ion for in , is

i,, : L'n l l I(.4 A' Z EDZ BC Z t) - r ' .or[[/, '1 Z B('Z ̂BZ ED) - A' i Z vo]. ( I0.A. I 5)

I f we take the l imi ts A + , r ' and A' + 7: ws get Eqs. (10.A.9) and (10.A.10) ,which are then common lor the three voltage clanrp techniques discussedin th is sect ion. l t should be noted that in Eqs.(10.A.14) and (10.A.15) thegain ,4 of amplif ier A, appears always multiplied by the impedance termZrrrZrrc @.g. . .42t , , Zo.) instead of Zr* as is the case for the node of Ranvier .This d i f fercucc is important in dcs igning a conlpL ' r ' lsat ing netwt l rk forstabi l izat ion of thc vol tase c lamD of musclc f ibcrs.

507

u n : A ' ( L t t * r , 1 . e y )

Ds - - Auc .

We obtain for u- the fol lowing equation:

t m - U C O M

; - , , / 7t m - u E / L F D .

(10 .A.5)

( r 0.A.6)

t , - : ( u . o " , 4 ' - i ^ Z r o )

x lZrr (A + l ) + Zr") l l (Zur r Zro * Zr) + AA'ZBC). (10.A.7)

For stabil ity analysis of the type described earlier it is necessary to corllputeu^/ucou which can readily be obtained lrom Eq. (10.A.7) and to assumea linear membrane impedance to relate i- and u-. Thus i- is given by

i ^ : \ r o lAA 'Zvc * (Zsc I ZcD + ZE i )

- A 'u.or(Zuc - t Zco + ZEJI IAA'Z.oZ"r . (10.A.8)

I f we take the l imi ts 1+ t r ) and A' - : r , in Eqs.(10.4.7) and (10. , { .8) weobtain respectively

and

( r0 .A.9)

(10.A.10)

Equations (10.A.9) and (10.A.10) have always been considered the fun-damental equations of the Dodge Frankenhauser voltage clamp, but itshould be kept in mind that Eqs.(10.A.7) and (10.A.8) are the more generalequations because they consider A and A' f inite and frequency dependent.

10.A.1.3. Nonner 's Vol tage Clamp. We shal l fo l low the same schemeas in part (a) but the circuit equations wil l be solved for the diagram ofFig. l2c. The gain of amplif ier A, is connected in such a way that

I 'n : - ADc ( 1 0 . A . 1 1 )

,l-l

50U 10. volr,q.cp cLAMpTNG oF EXCTTABLE MEMBRANES

10.A.2. Potent ia l Dist r ibut ion for a F iber in aGap Vo l t age C lamp

In this section we first consider models that describe the membranepotential distribution at the control pool (pool A) in nonmyelinated fibersvoltage clamped using a Vaseline- (or sucrose-) gap technique. We thenstudy the influence ofthe gap length and fiber characteristics (space constantand diameter) on the accuracy of the voltage control and current measure-ments when the Vaseline- (or sucrose-) gap voltage clamp is used.

The first model used is a three-dimensional cylinder extending frompartit ion AB (z : 0) to partit ion EA (z : /) in the case of a Vaseline-gaparrangement.

The potential V(r,:) anywhere in the fiber can be obtained by solving theLaplace equation in cylindrical coordinates:

l o .A. nPPrNolx 509

We solved this boundary value problem applying the Hankel transform

with the kernel (Ozis ik , ' 'z8 P. 135)

J5J o( [ ]^r ) l {b [ (Rt /R- l - ) ' + l ] | i2 J o( lJ^h)1,

where .,/o is the Bessel function of zeroth order and $,' are the roots of the

equat ion

fJ J l th) A- 'Jo(11,) : 0 .

where J, is the Bessel function of order one and A is the generalized space

constant (see. e.g. , E isenberg and Johnson?s) g iven by

A : R - /R i .

The resultant transformed ordinary dif l 'erentral equation

- pt^V + d2V l(tz2 -- o,

in which Z indicates thc Hankel t ransform ol V(r , : ) , was in tegrated, g iv ing

V : (I 8,1 ll ̂ )[cosh(/- ;)/sinh(/- /)]' ( r0.A.20)

in wl r ich l , is thc Hankel t ransform of 1, . Thc potent ia l V ( r , z) can be obta ined

by i nvc rs ion o f Eq . (10 .A .20 ) (Rc f ' l 2 t t ' p . 135 ) ' g i v i ng

V( r . : ) : 2 I ,R .h , , , ' ' . ! ,V . y \ . J ' , : ( f , , r ) cosh (p " , : ),",*., L iut,,,h)l(h t L)2 T'o' i;-:s;nit*l1 ( I0'A'21 )

Tlrc membrnnc potcnt ia l V(b, : ) is g ivcn by

a2v I dv a2v n^ t I a I r )

- v r

or- r ( ) r ( ' : -( 1 0 . A . 1 6 )

( r0 .A . r7 )

( 10 .A.18)

(10.A.1e)

where r is the radial coordinate from the axis of the cylinder and z is thelongitudinal coordinate along the axis. There is no circular dependencebecause there is circular symmetry imposed by the following boundaryconditions at the ends:

0 V l A z : 0 , z : 0 .

AVIAz : RtI t , z -- l ,tll = \t

V ( h . : ) : 2 1 r R i I, t : I

with the boundary condit ions

d V l t l z : 0 , ; : 0 '

The solut ion is

bl1. cosh(/- :)

/ l , , f rh A)2 + f t l l , )21sinh11t, , / j '( 10 .A.22)

where R, is the internal resistivity (in O cm) and 1, is the density of current(in A/cm2) injected at pool E which has no radial or angular dependence.

The membrane boundary condition is given by As a sccond modelT wc consider the s implc core conductor (see' e.9. ,

Taylors) ,

dzv idzz : 2v lb \ ( 1 0 . A . 2 3 )

z : I . (10.A.24)t lV l t l z : I rR t .

I d V V ^- l - : O r : hR , 0 r ' R - - " '

where we have assumed that the potential distribution has reached a steadystate and that the external solution is isopotential at a potential zero.tR- is the membrane resistance (in Q cm2) and b is the fiber radius.

f The condi t ion ofexternal potent ia l equal to zero has been used to s impl i fy the mathemat ical

t reatment and is d i rect ly appl icable to the case of sucrose gap when using electrometr ic polent ia l

measurement. When using potent iometr ic potent ia l measurement lhe internal potent ia l is made

vir tual ground (as in the Dodgc Frankcnhauser or Nonncr mct l rodsi to use th is t rcatntent thc

intcrnal potent ia l /o( : , r ) can be calculatcd as l "o( : . r ) : V( : . r ) - Vo.

( r0 .A.25)

' . * M . N . O z i s i k . " B r t u n t l a r y V l r l u c P r o b l e m s o l ' H c a t ( ' o n d t t c t i ( ) 1 1 . " l l l t ! ' r l l i l t l ( ) l l l l l ' l ' c x 1 -

book ( 'o . . Scrar t to t t . l ) cnnsy lvan ia . I961 i .

v(z):1rR,v/jb^Hil##*

l r l ltf

5 1 0 10. volracp cLAMpTNG oF EXCTTABLE MEMBRANES

L'o : I r R, x/-Lzbilsinh1/y,716A ). ( t 0.A.26)The voltage cramp circui t is imposing vo at: :0 and the potent iar as a[unction of the longitudinal pu.urn.,". i i,t !n

V(:) : Z, cosh(:. i in[. (10.A.27)

I 0 . n . , \ t , l , t , \ t ) t x

and replacing in Eq. (10.A.26),

5 l II t can be demonsrrated argebraicaty that when b/A is very sma', Eq. (r0.A.22)becomes Eq'(r0A'25). we have nu'n..r.uiiy computed and compared theresults of Eqs. (r0.4.22).and (10.A.25t unJ'rouna thar the core-conductormodel [Eq.(10.A.22)l does not aeuiat. more than 3\ f rom the three_dimensional model

-tEq- 1f 0.A_2Slj, p-",."d that two conditions aremet simultaneously:

l ! Ul .n,< O.t 'ana <;; i t t t < 5. These condirrons arefulfil led in most experimental cases; therefo.., we shall discuss the accuracy

:jJj;"i;lJ"trage ctamp only considering the simpler

"ur. oirhe core_From Eq'(r0'A'25) we can see that the membrane potent iar at ; : (J,defined as t'o, is equal to

4 r : l o o 1 1 1 1 t ( / t J [ r \ )

1.,i 2,'irA( r0 .A.2e)

( 10 .A.31)

(10 .A.32)This equat ion atows us to quant i fy the deviatron of the membrane potentrarfrom the control led potent ial lzo, ieingrnu^irnof ar z: / . We can est imatethis nraximar deviatlon of the;.";;;; porenriar (given as percenrageerror' ryr) as a function of the,gap rength /, the uber'iiu..t., i]ano thespace constant A. This is given by

4, is plotted as a function of Uuibi in Fig. t4.The above discussion is valid lor steady-state positivc rrrcprhr.irqe c91-

ductance. In order to roughly est imate the error dur ing lhc.ct iv i r l i . ' . fa negative conductance? we may use the same derivation ancl repr.cc. inEq. (10.A.25), A by A", defined as

n o _ - - R - R , .

When this replacement is done we obtain

t ' (z t : i rRi \ jbn" t . , t j : . r : ,T , , ( ru .A. lo)

srn(tJ 2lhA")The errors 4i and ra1 cdn be carcurated in an equivalent way as done above:

ti i ' : 100[cos(1.ri 2bA,) - l],

ryi : r00[sin(tJ2tL))l tJ2h "

Acknowledgments

wc t l rank Dr ' Richard Fi tzHugh f t r r chccking thc dcr i !at i ( )n of the cxprcssion l i r r thc, c l i f fcrcn t i a l c l ec t r o t l c s . Th i s n ,o r k was suppo r (ed by UpHS AMl520 l .

t hen .

, / r ' ( '1.) : l (V( l) - Vi lVol x t00;

? r : 1 0 0 [ c o s h ( / . ] b A ) _ l l : ( r0.A.28)t1r has been plotted as-a function of l lu/ 'bAin Fig. 14. We can also estimatethe deviation o[ the measured .r.."n't frorn the current . ir*i i t ing ut ucontrolled membrane patch (z : 0). At itr i, point the membrane currentdensity is given by

I 6 : V 6 l R ^ .

The measured membrane current density is experrmentary defined as

I - : I , ( xh2 ) /2xb l : 1 ,11121 .

Now we can define the percentage error in the current (r7r) as

r / r : [ (1 - - I i l to l x 100,