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Postsynaptic potential and postsynaptic current in muscle fibres with large time-constant. Epsp amplitude is independent of membrane resistance
Gen. Pharmac. Vol. 13. pp. 477 to 484, 1982 Printed in Great Britain. All rights reserved
0306-3623/82/060477-08503.00/0 Copyright © 1982 Pergamon Press Ltd
POSTSYNAPTIC POTENTIAL AND POSTSYNAPTIC C U R R E N T IN M U S C L E FIBRES WITH LARGE T I M E - C O N S T A N T . EPSP A M P L I T U D E IS I N D E P E N D E N T OF M E M B R A N E RESISTANCE* L. FISCHER and E. FLOREY Fakult~it fiJr Biologie, Universit~it Konstanz, D-7750 Konstanz, FRG (Received 19 April 1982) Abstract--1. Equations have been developed for a computer programmed calculation of synaptic current from synaptic potentials (epsps). 2. The method permits separation of presynaptic and postsynaptic effects in experiments involving drugs which affect synaptic transmission. It is particularly applicable where the postsynaptic cells are large as in the case of commonly employed crustacean muscle fibres. 3. Contrary to a widely held view the theoretical approach used predicts that epsp-amplitude is relatively independent of membrane resistance. 4. Confirmation is provided by experiments involving the application of barium which increases, and of GABA which decreases membrane resistance.
INTRODUCTION
The determination of synaptic current in crustacean muscle fibres is hindered by their large size and high capacity. In order to achieve an approximation of synaptic current our intention was to design a model system adequate to the electrical properties of multiterminally innervated crustacean muscle fibres which permits the calculation of the synaptic current from the time course of the epsp. At a time when techniques for recording synaptic current with microelectrodes were not yet developed Eccles et al. (1941) made similar assumptions to calculate the transmitter action from the endplate potential in frog muscle fibres. The method was occasionally applied later (e.g. Curtis & Eccles, 1959; Gerschenfeld & Stefani, 1968; Martin & Pilar, 1963; Bywater & Taylor, 1980) to different preparations. A test for the applicability of this method, however, is wanting. Calculations based on the responses of the model to a simulated synaptic current yielded equations which allow a description of the relationship between epsp-amplitude and membrane resistance in a system characterized by the high membrane capacity typical for crustacean muscle fibres. These equations, which are simple enough to be readily applicable to experimental results, reveal the astonishing fact that the epsp-amplitude is relatively independent of membrane resistance changes. The validity of this result was demonstrated in experiments on crayfish muscle fibres. In the literature (e.g. Hironaka & Otsuka, 1972; Sarne & Parnas, 1977; Ishida & Shinozaki, 1980) it has generally been implied that changes in membrane resistance would be reflected in a propor* This investigation was supported by the Sonderforschungsbereich 138 of the Deutsche Forschunggemeinschaft.
tional change of epsp-amplitude. It is the aim of this paper to show that such an assumption can be grossly misleading. MATERIALS A N D M E T H O D S
The experiments were done on the opener muscle of the third walking leg of mature male crayfish (Astacus leptodaetylus). The dissection and handling of the nerve-muscle preparation and the experimental set-up are the same as described earlier (Fischer & Florey, 1981). Extracellular epsps were recorded with low resistance (1 2 Mr2) NaCl-filled microelectrodes connected to a high impedance d.c. amplifier. The saline used had the following composition (mM/l): Na205, K 5.4, Ca 13.5, Mg 2.6 and C1242. In some experiments, BaC12 was substituted for equi-osmolar amounts of NaCI. The preparation was always perfused at a rate of 2-3 ml/min (bath vol. 1.5 ml): The temperature was held at 10°C thermostatically. For computation and averaging, a Nicolet MED 80 computer was used. For application of the developed mathematical algorithms the system was programmed in assembler language. RESULTS
The model The passive electrical properties of the sarcolemma of crustacean muscle fibres can be conventionally represented by a simple equivalent circuit consisting of one resistor, R, and one capacitor, C, connected in parallel (see Fig. 1A). In the following equations which describe the synaptic activity of muscle cells, R represents the total membrane resistance (f~) and C the total capacity (F). Simulated synaptic currents (instant rise followed by exponential decay) injected into an actual circuit of this type yield potentials of a time-course closely resembling actual epsp's, provided resistance and capacity values are chosen which conform to those of actual opener muscle fibers. The 477
478
L. FISCHER and E. FLOREY
is
The potential V(t), which can be measured across the RC-circuit, when a (synaptic) current is tlowing satisfies the differential equation of the RC-circuit
Is
dV(t) !,'(t) dt + R. C
\
(4)
The time-constant r of the system is defined by the product R. C
i,I \ ~
T = R.C.
u
A
B
C
Fig. I(A). The RC-equivalent-circuit of the model, into which the assumed current, i+, is injected. In Figs I(B) and I(C) two examples of the potential response. V, of this circuit to an applied current, i,, are presented. The time constant of current decay is 5 msec in example (B) and 100 msec in example (C). The resulting potential is identical in (B) and (C), because R is adjusted so that r = R-C is 100 msec in (B) and 5 msec in (C) (represented by the horizontal bars).
more complex equivalent circuit of Falk & F a n (1964) which involves a second capacity in series with a resistance, does not simulate actual current-voltage relations nearly as well• Their model has, therefore, not been used. In the case of nerve evoked potential changes, cable theory can be neglected and isopotentiality can be assumed because (1) the length-constant is large (about fibre length) and (2) n u m e r o u s synapses of the multiterminal innervation are distributed over the whole fibre length. The synaptic conductance, (j,(t), as measured in a wide variety of cells, has a very short rise time relative to the decay time. It can approximately be described by a m o m e n t a r y rise at t = 0 to the peak amplitude, ,q~0, and by an exponential decay with the time-conslant co
g+(t)
i~(t) C
=
g+o.exp
(')
- ~o "
(1)
In general, it is true that the current, i,(t), responsible for the synaptic potential, changes as the memb r a n e potential approaches the reversal potential, V,+,., according to the relation
i~(t) = g+(t). [V~+~ - V(t)]
(2)
where V(t) represents the actual m e m b r a n e potential at the time t. V,~v and V(t) are defined as potential difference relative to the resting potential. However, since epsp-amplitudes are very small in most crustacean muscle fibres it is legitimate to omit corrections for a change in driving force, V,.c+- V(t), during the course of the epsp and to describe the synaptic current by the relation
i+(t) = i,o.exp
- ~
(3)
where i~o represents the peak amplitude of synaptic current. The omission of the driving force is even necessary if one wishes to obtain the explicit relationships between epsp-amplitude and m e m b r a n e resistance as given below.
(5)
Assuming the current i+(t) to be of the form given in (3) and V a t t = 0 to be zero (which means that the m e m b r a n e potential is at its resting level), equation (4) can be solved for the time-variant potential V(t)
V(t)
(t) =
• T
ira. C.(r
x
~,))
-
[(:) exp
-
- exp
()1 - c,J
"
(6)
\
Considering the extreme case where the model consists only of a capacitor, the resulting potential, V(t), reaches the maximal amplitude i,0. +o/C from which it never decays. In the opposite extreme case where the model consists only of a resistor, the resulting potential parallels the time-course of the current, idt). The normal, intermediate case representative of the behavior of crayfish muscle fibres is approximated by the ratio r/+o of 210 as shown in Fig. I B, which also shows the time-course of the applied current. It is an i m p o r t a n t consequence of this model that even if the values of z and (o are exchanged, the time-course as well as the amplitude of the resultant epsp remain the same as demonstrated in Fig. 1C. In the case represented in Fig. 1B the time-course of the epsp-decay reflects the passive repolarization of the membrane, in the other case (Fig, IC) it reflects the time-course of the synaptic current. The relationships pointed out here make it clear that it cannot be decided from the time-course of the epsp alone whether its decay is governed by the time-course of the synaptic current or by the passive repolarization of the membrane. With this simple model, it is indeed possible to generate "'artificial epsp's" according to equation (6) which match the epsp's recorded from muscle fibres during nerve stimulation. Figure 2 shows an example of such a fit of artificial and recorded epsp's. The differences are almost undetectable. This already warrants confidence in the validity of the model. In the following it will be shown that the model permits
lo,~s
A
B
C
Fig. 2. Comparison of a recorded and averaged epsp (A) with a model-epsp (B) calculated according to equation (6); the values of r, e) and im/C used in this calculation are 37msec, 3msec and 1.38/IA//~F, respectively. In (C) measured and fitted epsp's are superimposed proving the quality by which the model can fit the epsp's in crustacean muscle fibres.
479
Calculation of synaptic current important conclusions concerning the relationship of membrane resistance and epsp-amplitude.
Relationship between epsp-amplitude and membrane resistance Theory. The amplitude, A, of the artificial epsp is defined as the maximum of V(t). The mathematical condition of a maximum is met if the first derivative of V(t) with respect to time is equal to zero dV(tm.x)
dt
0.
(7)
r . (z
co~. In
.
(8)
The equation for the amplitude, A, can be obtained by evaluating V(t) [equation (6)] at the time tmax A
ts0.~.
.
Corr =
(10)
The property of this factor can be clarified by its behavior in the two extreme cases: (1) if the timeconstant of current decay, co, is large compared to the time-constant of the membrane, z, this correction factor, Corr, approximates z/co. Hence, the epspamplitude, A, is proportional to z and to R. (2) If, on the other hand, the current is short compared to the time-constant of the membrane, Corr approximates unity. It follows, therefore, that the amplitude of the epsp is independent of z and of the membrane resistance R! Alterations of R have almost no influence on the epsp-amplitude in the extreme case co < z. In order to obtain convenient estimates of the changes of the epsp-amplitude, dA/A, which result from changes of the membrane resistance, dR~R, at a given value of z/co, an alternate dimensionless correction factor, Corr', can be used
A
dR - Corr'.--. R
(11)
F r o m the equations (5) and (9), the relationship between A and R is known; Corr' is calculated as
dA R Corr' -
Uco
~ ln('r/o.~)
dR" A - (1 Z ~/~o)" k(i Z ~ )
c o y
1.o 08
0.6
06
Q4
0.4
0.2
02
; ~ ; ' , ; ~ 001
; 01
;; ;;:'.:', 1
: ;
~;',',',~ . . . . . . . . . . . 10
o
100
z/co Fig. 3. Relationship of magnitude of correction factors Corr and Corr' on the ratio of membrane time constant z and of the time constant of current oJ. Corr represents the deviation of the epsp-amplitude from the amplitude reached in a purely capacitive system (without R, r--* :~). Corr' describes the deviation from proportionality in the relationship between relative changes of epsp-amplitude and relative changes of membrane resistance.
(9)
The first factor in this equation, iso.u)/C, is the maximal amplitude reached in a purely capacitative system. The second dimensionless correction factor represents the contribution of the membrane resistance and will be referred to as Corr
dA
-~-.~... c~'
Q8
o
-
It follows that the time tm~x needed to reach this point is tma X =
1.0
~1 +
(12)
Corr' depends only on z/co. While Corr [equation (10)] represents the deviation of the epsp-amplitude from the amplitude reached in a purely capacitive system (without R; z ~ 0o), Corr' describes the deviation from the proportional case in the relationship between relative changes of the epspamplitude and relative changes of membrane resist-
ance found in a purely resistive system (without C; z--* 0). Hence, Corr' approximates unity in this case where the current is long compared to the membrane time-constant. In the opposite case where the current is short compared to the membrane time-constant, Corr' approximates nearly Uco. This "inverse" relationship between Corr and Corr' can be seen in Fig. 3 where both correction factors are presented. An application of the correction factor, Corr', in the computation of epsp-amplitudes is illustrated by the following example: assuming a value of 40 for Uco, a 10Vo variation of R results in a 0.TVo variation of the epsp-amplitude, since Corr'(40). 10~o = 0.07.10~o = 0.7~o. This result is in sharp contrast to the nearly proportional variation of A at small z/co-values, e.g. at z/co = 1/40 a 10~o change of R alters the epsp-amplitude by 0.93.10~o = 9.3~o. Experimental testing. According to the argument presented in the preceding section, epsp-amplitude should be relatively independent of membrane resistance in muscle fibres whose epsp's last very much longer than the synaptic currents. Calculations based on the equations to be given below indicate that in crayfish opener muscle fibres the time constant of epsp decay is about 1(~80 times greater than that of the synaptic current. In order to verify the deduced relationship between epsp-amplitude and membrane resistance, experiments were designed in which membrane resistance was altered by application of Ba 2 + and GABA. The application of relatively low concentrations of Ba2+-ions (1-10 mM) is known to increase the membrane resistance in crustacean muscle fibres considerably (Fatt & Ginsborg, 1958; Werman & Grundfest, 1961). We substituted equi-osmolar amounts of Ba z+ for Na + and found that z, which is equivalent to R in these muscle fibres (C can be expected to be constant; Fischer and Florey 1981) increases by a factor of 2.8 within 5 min after application of 10mM Ba 2+ (see Fig. 4). In spite of the large increase of the membrane resistance, the epsp-amplitude did not increase proportionally, in fact, it sometimes decreased slightly. In
4S0
L. FISCHER and E. FI.ORIiY
';
, -
250ms
I
t
2mV
I
t ! lOmin
Ba~"
Fig. 4. Effect of increased membrane resistance on the time constant of epsp-decay (upper trace) and on the epspamplitude (lower trace). An increase of membrane resistance of 2.8-fold was accomplished by the application of 10raM Ba 2+ in the saline. Both parameters were determined every 5 sec and presented as single dots in the computer printout. Arrows indicate the application and washout of the Ba 2 + saline. The decay time constant of synaptic current calculated from extracellularly recorded epsp's was 5.9 msec.
a series of 6 experiments of this type, the application of 5 m M Ba 2 + induced an increase of m e m b r a n e resistance by a factor of 2.41 4- 0.61, while the epspamplitude increased only by a factor of 1.07 4- 0.08 (mean 4- SD). The predicted mean increase of epspamplitude which can be calculated from the measured change of m e m b r a n e resistance in each individual experiment by means of equation (9) is 1.065 4- 0.03 (mean 4- SD; n = 6). This fits very well with the measured mean increase of 1.07. These experiments with Ba 2+ illustrate the relative independence of the epsp-amplitude from m e m b r a n e resistance changes in crustacean muscle fibres and confirm the theory inherent in the model. While Ba 2 +-application increases the m e m b r a n e resistance, application of G A B A decreases it. In addition to its postsynaptic effect, G A B A is also k n o w n for its powerful presynaptic, epsp-depressing action when applied to crustacean operner musle fibres (Dudel & Kuffler, 1961). Hence a much s,ronger reduction of the epsp-amplitude must be expected after application of G A B A than that predicted from the decrease in m e m b r a n e resistance alone. For this reason natural synaptic currents produced by nerve stimulation could not be used to test the theory. Instead, the synaptic current was simulated by rapidly increasing and exponentially decreasing currents (time-constant ¢9) injected by a microelectrode into the muscle fibre. Although a single electrode c a n n o t imitate the effect of multiterminal innervation, the effect of spatial decrement is evidently not too large to invalidate the result of such an experimental arrangement (fibre length ~ length-constant). In the typical experiment presented in Fig. 5, nerve evoked epsp's, a single simulated epsp and an electrotonus were recorded from the same muscle fibre. After application of 2 . 1 0 4 M G A B A (Fig. 5B), the epsp's nearly vanish while the simulated epsp is only
reduced to 0.5 of its control amplitude. This is indeed predicted by equation (9): using the value r/o) = 101 msec/9.4 msec = 10.7 {Fig. 5A) and r/"(o = 1.47 (T was estimated from the reduction of the electrotonus) (Fig. 5B) the correction factors, Corr, are 0.78 and 0.39, respectively; hence the reduction of the epsp-amplitude resulting from m e m b r a n e resistance changes is 0.39/0.78 = 0.5. The wdue of the m e m b r a n e time-constant, r, cannot be calculated directly from the decay of the simulated epsp in Fig. 5B, because due to the low m e m b r a n e resistance produced by G A B A the wdue of ~ is in the range of the value of ~:); compared to a merely passive repolarization phase of the epsp the actual, now rapidly decaying epsp in Fig. 5B is prolonged by the decay phase of the applied current. Calculation ~71the synaptic current The amplitude of the epsp as described by equation (9) is determined by four parameters: i~o, co, C and ~. Alterations of any one of these parameters may result in a variation of the epsp-amplitude detected in experiments in which, for example, temperature or the composition of the solution is changed. The time-constant ~ and its variation can easily be determined from the epsp-decay because it has been shown for crayfish muscle fibres (Fischer & Florey, 1981), that the decay is determined only by r (not by
L_
B GABA f i !
II
I
2mV
[
'\
i
C
1
2
3
Fig. 5. The effect of GABA on epsp's (1), simulated epsp (2) and electrotonus (3). (A) shows the control, (B) the maximal effect of 2.10 4M GABA and (C) the response after 15 min washout. A second electrode was inserted 120 ktm from the recording site to produce electrotonus (injected current 20nA) and simulated epsp (time constant of injected current 9.4 msec). The nerve evoked epsp's nearly vanish after GABA-application due to presynaptic inhibition. The amplitude of the electrotonus (membrane resistance) is reduced to ~o of its control value, while the amplitude of the single simulated epsp is only reduced to ½ of its control amplitude. This reduction is predicted by the model.
Calculation of synaptic current e~). Should the membrane capacity, C, not be constant it can be evaluated from C = T/R, where R is determined from electrotonic measurements. The synaptic parameters, i.e. the amplitude of the synaptic current, i~o, and its time-constant, ~o, cannot be evaluated explicitly from equation (9). In order to determine them separately, the synaptic current must be known. The method described below permits the calculation of the current from recorded epsp's alone. Theory. For a cell to which the RC-circuit of our model is applicable, the corresponding differential equation (4) can be used to determine the synaptic current, i~(t), if the epsp V(t) and the time-constant = R. C is given by measurement
is(t)
V(t)
dV(t) -
+
C
- -
dt
For the calculation reversal potential of account as described equation (4) can thus
(13)
T
of the synaptic conductance, the the epsp's can be taken into in equation (2). The differential be rewritten
V(t)
dV(t) _
gs(t)
_
-}-
_ _
dt
T
-
Kov- v(t)
c
(14)
The reversal potential in opener muscle fibres is around + 15 mV (Onodera & Takeuchi, 1975). Hence VreV is 90 mV with reference to a resting potential of 75 mV. In order to make use of these equations for appropriate programming of a computer the following points must be taken into consideration: the usual way for digitizing potentials is to take samples at a fixed interval, called dwell time and referred to as DW. According to the definition of the zero levels of the potentials used in the equations the digitized epsp in the computer must be shifted so that the resting potential is at the zero level. Then the membrane time-constant T can be determined by a least square fit applied to the logarithmized decaying phase (z can be assumed to be large relative to co in crayfish muscle fibres). If Vand V' are two adjacent potential samples taken at the time t, and t' = t + DW, respectively, equation (13), for instance, can be transformed for programming into the following form -
V
i,(t)
C
-
V' DW
+
V + V' 2.T
-
05)
The units of this current are ~A//~F, if V and V' are given in mV and D W in msec. The current amplitude is obtained from comparison with a calibration pulse of known amplitude: it is e.g. 1/~A//~F, if it is of the same magnitude as a 1 mV potential pulse. Application and testin.q. An example of the application of current calculation [-equation (15)] to an epsp, simulated by injecting a rapidly rising and exponentially decreasing current into an actual RC-circuit, is shown in Fig. 6. The artificial epsp was recorded with a 1 0 M ~ resistor simulating a microelectrode and fed through the amplifiers and tape recorder into the computer. The calculated time-course of the current fits that of the actually "injected" current very well. The noise in the epsp recorded and hence in the
481
2"* I I
0.5pA
I
,,
i\,
20 ms
Fig. 6. Comparison of injected and calculated current in a simulation of an epsp by injecting current into and recording epsp from an RC-circuit (100k~, lffF) via 10M~ resistors. Upper trace: artificial epsp, middle trace: calculated current (dots), middle and lower trace: injected current (continuous curvet.
computer current is due to the thermal noise of the resistors and recording apparatus and can be reduced by averaging. The calculation of the current according to equation (13) includes the membrane capacity. If the membrane capacity is known (it can be determined approximately from C = T/R by electrotonic measurements) the absolute amplitude of the current can be calculated iso = i~°.C - is° z C C 'R"
(16)
This is a kind of scaling which was employed in the simulation experiment of Fig. 6. The close correlation of calculated and injected current amplitude is obvious. An estimation of the absolute current amplitude in a muscle fibre can be obtained from epsp's of the type presented in Fig. 5: the electrotonus yields an approximation of R, z can be determined from the epspdecay and i~o/C can be calculated according to equation (15); thus the absolute value of the maximal current corresponding to the unfacilitated epsp in Fig. 5A (1 mV amplitude) is estimated as follows iso = 0.28/~A. 101 msec _ 0.132/~A. /~F 216kD In order to test the validity of the calculation of synaptic currents from epsp's recorded from crayfish muscle fibres the result of the computation should be compared with a genuine synaptic current actually measured in the same fibre. As measuring method we chose the extracellular recording technique, since the voltage clamp technique with microelectrodes is not applicable because of the large size of crayfish muscle fibres. An example of extracellularly recorded synaptic current and of intracellularly recorded epsp is shown in Fig. 7 together with the computed synaptic current. It must be emphasized that the computed current is the whole current flowing in this muscle fibre in contrast to the extracellular recording which
4~2
L. FISCHERand E. FLOREY
2 mV
20 ms
Fig. 7. Extracellular record of synaptic current at one syuaptic spot (middle trace) in parallel with intracellular record of epsp's (upper trace). The current calculated from the epsp (lower trace) shows a time constant of 2.8 msec which is only 15"~i shorter than the time constant of the synaptic current recorded extracellularly at one synaptic spot.
represents only the synaptic current at one of a large number of synaptic spots. For this and other reasons, the amplitudes of calculated and recorded epsp's are not equal. The time-constants of current decay can be compared, however. The duration of the extracellularly recorded current is very similar to the duration of the calculated current. Evaluation of the time constants by a least square fit revealed a small difference of about 15'~,~,. Possible explanations for this difference will be discussed later. Additional extracellular recordings from different preparations have confirmed the close correlation between calculated and measured time constants. DISCUSSION
Calculation ~f the synaptic current As already stated, the technique of extracellular recording cannot provide absolute values of current amplitude and the voltage clamp technique is not applicable to crustacean fibres of large size in the high frequency range needed for clamping the synaptic currents. The advantage of the mathematical method lies in the simplicity of the experimental arrangement necessary to provide the experimental data. The major effort is needed for the programming of a computer. Once this is accomplished, fast on-line computations of the synaptic current from the recorded epsp's can be performed• The calculation is not affected by artefacts. The noise amplification which is a result of the differential form of the algorithm used can easily be reduced by averaging. The method is well suited for application not only to single epsp's but also to facilitating and summating trains of epsp's. For the calculation method a value of the membrane time constant, 3, must be fed into the computer.
With ratios of z/co encountered in large crustacean muscle fibres the calculated current is rather insensitive to relatively large variations of z. It is possible that there are fibres whose properties are not described by our simple model (Fig. 1). In this case more than one time-constant and special cable properties appropriate to the spatial decrement of the imposed potential changes would have to be taken into account. From impedance measurements in crayfish muscle fibres Falk & Fatt (1964) deduced an equivalent circuit which consists of a resistance and a capacitance connected in parallel with another path composed of a resistance in series with a capacitance. Simulation experiments with such an equivalent circuit showing that a ~'splitting" of the total capacity, C, of Fig. 1 into the two components used in the Falk/Fatt-model by introducing the series resistance results in a "sharpening" of the rise and initial decay of the epsp while its later decay is barely affected. If the Falk/Fatt-model were indeed applicable the calculations based on the simple model would yield current durations which are too short. However, epsp's recorded from crayfish opener muscle fibres usually do not show the shape predicted by the Falk/Fatt-model. The influence of cable properties on the calculated currents is of general interest in application to innervated cells which are depolarized by synal~tic currents in a non-uniform manner. Such a non-uniform depolarization simulating an "epsp" in a muscle fibre can be achieved by injecting into the fibre a brief exponentially decaying current. The current computed from the configuration of such an "epsp" by means of equation (15) is significantly shorter than the actually injected one. Crayfish muscle fibres are short cables and the nerve evoked epsp's can be expected to depolarize the fibres relatively uniformly: the modifying effects of cable properties can thus be expected to be very weak. It is indeed found (see Fig. 7) that the currents calculated according to the equations based on the simple model are slightly shorter than the extracellularly measured synaptic currents. It is possible that this discrepancy between calculated and measured current is based on the morphology of the muscle fibres. Crustacean muscle fibres are known for their large clefts and invaginations of their surface. Axonal branches may run in and form synapses within such a cleft. Due to the restricted connection with the surrounding fluid, the cleft can be assumed to behave like a leaky cell (the extracellular space represents the interior of the cell and the sarcoplasma the "'extracellular" space). The synaptic potential recorded with a microelectrode placed at the opening of the cleft which is generated by the synaptic currents inside of this "invaginated" cell can be expected to be prolonged by the capacity of the surrounding muscle fibre membrane.
Relationship between epsp-amplitude and membrane resistance Different fibres of one crustacean muscle are known to differ in presynaptic and postsynaptic properties. As shown by equations (9) and (ll) the epsp-amplitude is relatively independent of the postsynaptic
Calculation of synaptic current membrane resistance and is mainly determined by the synaptic current (transmitter action), if T is much greater than ~o as it is in crustacean muscle fibres. It must be concluded, that there is no obvious correlation between epsp-amplitude and membrane resistance, since the variation of other factors from fibre to fibre, for example the amplitude of the synaptic current, is much larger than the variation of a few percent due to the observed membrane resistance changes. This lack of correlation between epsp-amplitude and membrane resistance was indeed found by Meiss & Govind (1979) who scanned the fibres of the accessory flexor muscle in the lobster. They concluded from their experimental data that 'tit appears, that epsp-amplitude is independent of the membrane resistance". This is a rather indirect, but interesting confirmation of the relationships deduced from the model. Their explanation, however, that classical cable theory is not applicable to crustacean muscle fibres appears to be inadequate. Cable properties have a definite, modifying influence on the relationship of the epsp-amplitude to membrane resistance as derived from the simple RC-circuit which is also the fundamental circuit used in cable theory. The experiment with Ba 2 + as shown in Fig. 4 was presented to provide evidence for the deduced equations which reveals the relative independence of epspamplitude from membrane changes in crayfish muscle fibres. If, in contrast, a proportional effect of membrane resistance on epsp-amplitude is assumed, the constancy of the epsp-amplitude can only be explained if it is assumed that Ba 2+ depresses (by a presynaptic or subsynaptic mechanism) the epsp's which otherwise would have to be increased in proportion to the resistance increase produced by Ba 2 +. It seems very unlikely, however, that in all the experiments with Ba2+-application the time-course as well as the magnitude of such a depression just compensate for the large, Ba 2 +-induced change of the membrane resistance. A presynaptic depression can be excluded, because the quantum content, calculated from the variance of the epsp-amplitude according to equations given by del Castillo & Katz (1953) before and during Ba2+-application, does not change significantly; in a series of 6 experiments with 5 mM Ba 2+ the quantum content even shows a tendency to increase by a factor of 1.21 _+ 0.35 (mean + SD). The non-proportionality of epsp-amplitude and membrane resistance is perhaps unexpected and has not been taken account of in previous publications: Hironaka & Otsuka (1971), for instance, describing drug effects at crayfish neuromuscular junction conclude that "'augmentation of ipsp size is attributed to the increase in membrane resistance of the muscle fibre". Ishida & Shinozaki (1980) deduce that "the increase in amplitude of the epsp's caused by diltiazem was due to the increase in membrane resistance". S a r n e & Parnas (1977) attempt to separate presynaptic from postsynaptic inhibition in crab muscle fibres by attributing the reduction of the epsp to the postsynaptic effect assuming that this is proportional to the reduction of the input resistance after GABA application. Although the values of z/e~ in the investigated crab muscle fibres are less than those observed in crayfish muscle fibres, the relationship is still far from proportionality. The experiment presented in
483
Fig. 6 demonstrates that the effect of GABA on the simulated epsp (which is exclusively of post-synaptic origin) is much less than the GABA effect on the input resistance. This can be predicted by equation (9). By utilizing the presented relationships in the evaluation of such experiments, a much more realistic separation of pre- and postsynaptic GABA-effects and a better differentiation of drug effects on epsp's can be derived. Furthermore, by applying current-calculation the "presynaptic" effects can be separated into effects on the amplitude and effects on the decay time constant of synaptic current [both parameters contribute to the epsp-amplitude in a manner described by equation (9)].
SUMMARY Direct measurement of synaptic currents (voltage clamp technique) in big crustacean muscle fibres is impractical because of the large membrane capacity. On the basis of a simple equivalent circuit, equations have been developed for a computer programmed calculation of synaptic current from intracellularly recorded synaptic potentials. Confirmation of the adequacy of the method has been obtained in experiments on crayfish opener muscle fibres with a fibre length about equal to the space constant. The theoretical approach used here predicts a relative independence of epsp-amplitude and membrane resistance in the case of muscle fibres where membrane time-constant is long compared to the time-constant of the synaptic current. Confirmation of this independence is provided by experiments involving application of Ba 2 + (increases membrane resistance) and GABA (decreases membrane resistance). The new method permits separation of presynaptic and postsynaptic effects in experiments involving procedures which affect synaptic transmission; the equations furthermore allow a differentiation of effects on amplitude and effects on decay time-constant of synaptic current.
Acknowledgement We would like to thank Ms Birgit Bremer for excellent technical assistance.
REFERENCES
BYWATER R. A. R. & TAYLOR G. S. (1980) The passive membrane properties and excitatory junction potentials of the guinea-pig vas deferens, d. Physiol. 300, 303 316. CURTIS D. R. & ECCLESJ. C. (1959) The time courses of excitatory and inhibitory synaptic actions. J. Physiol. 145, 529 546. DEL CASTILLOJ. & KATZ B. (1954) Quantal contents of the end-plate potential. J. Physiol. 124, 560 573. DUDELJ. & KUFFLERS. W. (1961) Presynaptic inhibition at the crayfish neuromuscular junction. J. Physiol. 155, 543-562. ECCLES J. C., KATZ B. & KUFFLER S. W. (1941) Nature of the endplate potential in curarized muscle. J. Neurophysiol. 4, 362 387. FALK G. & FATT P. (1964) Linear electrical properties of striated muscle fibres observed with intracellular electrodes. Proc. R. Sot'. Lond. B Biol. Sci. 160, 69 123.
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L. FISCHER and E. FLORliY
FATT P. & GINSBORG B. L. (1958) The ionic requirements for the production of action potentials in crustacean muscle fibres. J. Physiol. 142, 516 543. FISCHER L. & FLOREY E. (1981) Temperature effects on neuromuscular transmission (opener muscle of crayfish, Astacus leptodactylus), d. exp. Biol. 94, 251 268. GERSCHENFELD H. M. & STEFANI E. (1968) Evidence for an excitatory transmitter role of serotonin in molluscan central synapses. Adv. Pharmac. 6, 369 392. HIRONAKA T. & OTSUKA M. (1972) The effect of dimorpholamine on crayfish neuromuscular junction. Neuropharmacolo:ly I1, 573 584. ISHIDA M. & SHINOZAKI H. (1980) Differential effects of diltiazem on glutamate potentials and excitatory junctional potentials at the crayfish neuromuscular junction. J. Physiol. 298, 301 319.
MARTIN A. R. & PILAR O. (1963) Dual mode of synaptic transmission in the avian ciliary ganglion. J. Physiol. 168. 443 463. MEISS D. E. & GOVIND C. K. (1979) Regional differentiation of neuromuscular synapses in a lobster receptor muscle. J. exp. Biol. 79, 99 114. ONODt!RA K. & TAKfUCHI A. (1975} Ionic mechanism of the excitatory synaptic membrane of the crayfish neuromuscular junction. J. Physiol. 252, 295 318. SARNE Y. & PARNAS I. (1977) Differential effects of GABA on three muscles innervated by a c o m m o n inhibitory axon of Ocypode. J. comp. Physiol. 114, 327 333. W~RMAN R. & GRUNDFEST H. (1961) Graded and all-ornone electrogenesis in arthropod muscle. 11. The effects of alkali-earth and onium ions on lobster muscle fibers. J. Gen Physiol. 44, 997 1027.
0306-3623/82/060477-08503.00/0 Copyright © 1982 Pergamon Press Ltd
POSTSYNAPTIC POTENTIAL AND POSTSYNAPTIC C U R R E N T IN M U S C L E FIBRES WITH LARGE T I M E - C O N S T A N T . EPSP A M P L I T U D E IS I N D E P E N D E N T OF M E M B R A N E RESISTANCE* L. FISCHER and E. FLOREY Fakult~it fiJr Biologie, Universit~it Konstanz, D-7750 Konstanz, FRG (Received 19 April 1982) Abstract--1. Equations have been developed for a computer programmed calculation of synaptic current from synaptic potentials (epsps). 2. The method permits separation of presynaptic and postsynaptic effects in experiments involving drugs which affect synaptic transmission. It is particularly applicable where the postsynaptic cells are large as in the case of commonly employed crustacean muscle fibres. 3. Contrary to a widely held view the theoretical approach used predicts that epsp-amplitude is relatively independent of membrane resistance. 4. Confirmation is provided by experiments involving the application of barium which increases, and of GABA which decreases membrane resistance.
INTRODUCTION
The determination of synaptic current in crustacean muscle fibres is hindered by their large size and high capacity. In order to achieve an approximation of synaptic current our intention was to design a model system adequate to the electrical properties of multiterminally innervated crustacean muscle fibres which permits the calculation of the synaptic current from the time course of the epsp. At a time when techniques for recording synaptic current with microelectrodes were not yet developed Eccles et al. (1941) made similar assumptions to calculate the transmitter action from the endplate potential in frog muscle fibres. The method was occasionally applied later (e.g. Curtis & Eccles, 1959; Gerschenfeld & Stefani, 1968; Martin & Pilar, 1963; Bywater & Taylor, 1980) to different preparations. A test for the applicability of this method, however, is wanting. Calculations based on the responses of the model to a simulated synaptic current yielded equations which allow a description of the relationship between epsp-amplitude and membrane resistance in a system characterized by the high membrane capacity typical for crustacean muscle fibres. These equations, which are simple enough to be readily applicable to experimental results, reveal the astonishing fact that the epsp-amplitude is relatively independent of membrane resistance changes. The validity of this result was demonstrated in experiments on crayfish muscle fibres. In the literature (e.g. Hironaka & Otsuka, 1972; Sarne & Parnas, 1977; Ishida & Shinozaki, 1980) it has generally been implied that changes in membrane resistance would be reflected in a propor* This investigation was supported by the Sonderforschungsbereich 138 of the Deutsche Forschunggemeinschaft.
tional change of epsp-amplitude. It is the aim of this paper to show that such an assumption can be grossly misleading. MATERIALS A N D M E T H O D S
The experiments were done on the opener muscle of the third walking leg of mature male crayfish (Astacus leptodaetylus). The dissection and handling of the nerve-muscle preparation and the experimental set-up are the same as described earlier (Fischer & Florey, 1981). Extracellular epsps were recorded with low resistance (1 2 Mr2) NaCl-filled microelectrodes connected to a high impedance d.c. amplifier. The saline used had the following composition (mM/l): Na205, K 5.4, Ca 13.5, Mg 2.6 and C1242. In some experiments, BaC12 was substituted for equi-osmolar amounts of NaCI. The preparation was always perfused at a rate of 2-3 ml/min (bath vol. 1.5 ml): The temperature was held at 10°C thermostatically. For computation and averaging, a Nicolet MED 80 computer was used. For application of the developed mathematical algorithms the system was programmed in assembler language. RESULTS
The model The passive electrical properties of the sarcolemma of crustacean muscle fibres can be conventionally represented by a simple equivalent circuit consisting of one resistor, R, and one capacitor, C, connected in parallel (see Fig. 1A). In the following equations which describe the synaptic activity of muscle cells, R represents the total membrane resistance (f~) and C the total capacity (F). Simulated synaptic currents (instant rise followed by exponential decay) injected into an actual circuit of this type yield potentials of a time-course closely resembling actual epsp's, provided resistance and capacity values are chosen which conform to those of actual opener muscle fibers. The 477
478
L. FISCHER and E. FLOREY
is
The potential V(t), which can be measured across the RC-circuit, when a (synaptic) current is tlowing satisfies the differential equation of the RC-circuit
Is
dV(t) !,'(t) dt + R. C
\
(4)
The time-constant r of the system is defined by the product R. C
i,I \ ~
T = R.C.
u
A
B
C
Fig. I(A). The RC-equivalent-circuit of the model, into which the assumed current, i+, is injected. In Figs I(B) and I(C) two examples of the potential response. V, of this circuit to an applied current, i,, are presented. The time constant of current decay is 5 msec in example (B) and 100 msec in example (C). The resulting potential is identical in (B) and (C), because R is adjusted so that r = R-C is 100 msec in (B) and 5 msec in (C) (represented by the horizontal bars).
more complex equivalent circuit of Falk & F a n (1964) which involves a second capacity in series with a resistance, does not simulate actual current-voltage relations nearly as well• Their model has, therefore, not been used. In the case of nerve evoked potential changes, cable theory can be neglected and isopotentiality can be assumed because (1) the length-constant is large (about fibre length) and (2) n u m e r o u s synapses of the multiterminal innervation are distributed over the whole fibre length. The synaptic conductance, (j,(t), as measured in a wide variety of cells, has a very short rise time relative to the decay time. It can approximately be described by a m o m e n t a r y rise at t = 0 to the peak amplitude, ,q~0, and by an exponential decay with the time-conslant co
g+(t)
i~(t) C
=
g+o.exp
(')
- ~o "
(1)
In general, it is true that the current, i,(t), responsible for the synaptic potential, changes as the memb r a n e potential approaches the reversal potential, V,+,., according to the relation
i~(t) = g+(t). [V~+~ - V(t)]
(2)
where V(t) represents the actual m e m b r a n e potential at the time t. V,~v and V(t) are defined as potential difference relative to the resting potential. However, since epsp-amplitudes are very small in most crustacean muscle fibres it is legitimate to omit corrections for a change in driving force, V,.c+- V(t), during the course of the epsp and to describe the synaptic current by the relation
i+(t) = i,o.exp
- ~
(3)
where i~o represents the peak amplitude of synaptic current. The omission of the driving force is even necessary if one wishes to obtain the explicit relationships between epsp-amplitude and m e m b r a n e resistance as given below.
(5)
Assuming the current i+(t) to be of the form given in (3) and V a t t = 0 to be zero (which means that the m e m b r a n e potential is at its resting level), equation (4) can be solved for the time-variant potential V(t)
V(t)
(t) =
• T
ira. C.(r
x
~,))
-
[(:) exp
-
- exp
()1 - c,J
"
(6)
\
Considering the extreme case where the model consists only of a capacitor, the resulting potential, V(t), reaches the maximal amplitude i,0. +o/C from which it never decays. In the opposite extreme case where the model consists only of a resistor, the resulting potential parallels the time-course of the current, idt). The normal, intermediate case representative of the behavior of crayfish muscle fibres is approximated by the ratio r/+o of 210 as shown in Fig. I B, which also shows the time-course of the applied current. It is an i m p o r t a n t consequence of this model that even if the values of z and (o are exchanged, the time-course as well as the amplitude of the resultant epsp remain the same as demonstrated in Fig. 1C. In the case represented in Fig. 1B the time-course of the epsp-decay reflects the passive repolarization of the membrane, in the other case (Fig, IC) it reflects the time-course of the synaptic current. The relationships pointed out here make it clear that it cannot be decided from the time-course of the epsp alone whether its decay is governed by the time-course of the synaptic current or by the passive repolarization of the membrane. With this simple model, it is indeed possible to generate "'artificial epsp's" according to equation (6) which match the epsp's recorded from muscle fibres during nerve stimulation. Figure 2 shows an example of such a fit of artificial and recorded epsp's. The differences are almost undetectable. This already warrants confidence in the validity of the model. In the following it will be shown that the model permits
lo,~s
A
B
C
Fig. 2. Comparison of a recorded and averaged epsp (A) with a model-epsp (B) calculated according to equation (6); the values of r, e) and im/C used in this calculation are 37msec, 3msec and 1.38/IA//~F, respectively. In (C) measured and fitted epsp's are superimposed proving the quality by which the model can fit the epsp's in crustacean muscle fibres.
479
Calculation of synaptic current important conclusions concerning the relationship of membrane resistance and epsp-amplitude.
Relationship between epsp-amplitude and membrane resistance Theory. The amplitude, A, of the artificial epsp is defined as the maximum of V(t). The mathematical condition of a maximum is met if the first derivative of V(t) with respect to time is equal to zero dV(tm.x)
dt
0.
(7)
r . (z
co~. In
.
(8)
The equation for the amplitude, A, can be obtained by evaluating V(t) [equation (6)] at the time tmax A
ts0.~.
.
Corr =
(10)
The property of this factor can be clarified by its behavior in the two extreme cases: (1) if the timeconstant of current decay, co, is large compared to the time-constant of the membrane, z, this correction factor, Corr, approximates z/co. Hence, the epspamplitude, A, is proportional to z and to R. (2) If, on the other hand, the current is short compared to the time-constant of the membrane, Corr approximates unity. It follows, therefore, that the amplitude of the epsp is independent of z and of the membrane resistance R! Alterations of R have almost no influence on the epsp-amplitude in the extreme case co < z. In order to obtain convenient estimates of the changes of the epsp-amplitude, dA/A, which result from changes of the membrane resistance, dR~R, at a given value of z/co, an alternate dimensionless correction factor, Corr', can be used
A
dR - Corr'.--. R
(11)
F r o m the equations (5) and (9), the relationship between A and R is known; Corr' is calculated as
dA R Corr' -
Uco
~ ln('r/o.~)
dR" A - (1 Z ~/~o)" k(i Z ~ )
c o y
1.o 08
0.6
06
Q4
0.4
0.2
02
; ~ ; ' , ; ~ 001
; 01
;; ;;:'.:', 1
: ;
~;',',',~ . . . . . . . . . . . 10
o
100
z/co Fig. 3. Relationship of magnitude of correction factors Corr and Corr' on the ratio of membrane time constant z and of the time constant of current oJ. Corr represents the deviation of the epsp-amplitude from the amplitude reached in a purely capacitive system (without R, r--* :~). Corr' describes the deviation from proportionality in the relationship between relative changes of epsp-amplitude and relative changes of membrane resistance.
(9)
The first factor in this equation, iso.u)/C, is the maximal amplitude reached in a purely capacitative system. The second dimensionless correction factor represents the contribution of the membrane resistance and will be referred to as Corr
dA
-~-.~... c~'
Q8
o
-
It follows that the time tm~x needed to reach this point is tma X =
1.0
~1 +
(12)
Corr' depends only on z/co. While Corr [equation (10)] represents the deviation of the epsp-amplitude from the amplitude reached in a purely capacitive system (without R; z ~ 0o), Corr' describes the deviation from the proportional case in the relationship between relative changes of the epspamplitude and relative changes of membrane resist-
ance found in a purely resistive system (without C; z--* 0). Hence, Corr' approximates unity in this case where the current is long compared to the membrane time-constant. In the opposite case where the current is short compared to the membrane time-constant, Corr' approximates nearly Uco. This "inverse" relationship between Corr and Corr' can be seen in Fig. 3 where both correction factors are presented. An application of the correction factor, Corr', in the computation of epsp-amplitudes is illustrated by the following example: assuming a value of 40 for Uco, a 10Vo variation of R results in a 0.TVo variation of the epsp-amplitude, since Corr'(40). 10~o = 0.07.10~o = 0.7~o. This result is in sharp contrast to the nearly proportional variation of A at small z/co-values, e.g. at z/co = 1/40 a 10~o change of R alters the epsp-amplitude by 0.93.10~o = 9.3~o. Experimental testing. According to the argument presented in the preceding section, epsp-amplitude should be relatively independent of membrane resistance in muscle fibres whose epsp's last very much longer than the synaptic currents. Calculations based on the equations to be given below indicate that in crayfish opener muscle fibres the time constant of epsp decay is about 1(~80 times greater than that of the synaptic current. In order to verify the deduced relationship between epsp-amplitude and membrane resistance, experiments were designed in which membrane resistance was altered by application of Ba 2 + and GABA. The application of relatively low concentrations of Ba2+-ions (1-10 mM) is known to increase the membrane resistance in crustacean muscle fibres considerably (Fatt & Ginsborg, 1958; Werman & Grundfest, 1961). We substituted equi-osmolar amounts of Ba z+ for Na + and found that z, which is equivalent to R in these muscle fibres (C can be expected to be constant; Fischer and Florey 1981) increases by a factor of 2.8 within 5 min after application of 10mM Ba 2+ (see Fig. 4). In spite of the large increase of the membrane resistance, the epsp-amplitude did not increase proportionally, in fact, it sometimes decreased slightly. In
4S0
L. FISCHER and E. FI.ORIiY
';
, -
250ms
I
t
2mV
I
t ! lOmin
Ba~"
Fig. 4. Effect of increased membrane resistance on the time constant of epsp-decay (upper trace) and on the epspamplitude (lower trace). An increase of membrane resistance of 2.8-fold was accomplished by the application of 10raM Ba 2+ in the saline. Both parameters were determined every 5 sec and presented as single dots in the computer printout. Arrows indicate the application and washout of the Ba 2 + saline. The decay time constant of synaptic current calculated from extracellularly recorded epsp's was 5.9 msec.
a series of 6 experiments of this type, the application of 5 m M Ba 2 + induced an increase of m e m b r a n e resistance by a factor of 2.41 4- 0.61, while the epspamplitude increased only by a factor of 1.07 4- 0.08 (mean 4- SD). The predicted mean increase of epspamplitude which can be calculated from the measured change of m e m b r a n e resistance in each individual experiment by means of equation (9) is 1.065 4- 0.03 (mean 4- SD; n = 6). This fits very well with the measured mean increase of 1.07. These experiments with Ba 2+ illustrate the relative independence of the epsp-amplitude from m e m b r a n e resistance changes in crustacean muscle fibres and confirm the theory inherent in the model. While Ba 2 +-application increases the m e m b r a n e resistance, application of G A B A decreases it. In addition to its postsynaptic effect, G A B A is also k n o w n for its powerful presynaptic, epsp-depressing action when applied to crustacean operner musle fibres (Dudel & Kuffler, 1961). Hence a much s,ronger reduction of the epsp-amplitude must be expected after application of G A B A than that predicted from the decrease in m e m b r a n e resistance alone. For this reason natural synaptic currents produced by nerve stimulation could not be used to test the theory. Instead, the synaptic current was simulated by rapidly increasing and exponentially decreasing currents (time-constant ¢9) injected by a microelectrode into the muscle fibre. Although a single electrode c a n n o t imitate the effect of multiterminal innervation, the effect of spatial decrement is evidently not too large to invalidate the result of such an experimental arrangement (fibre length ~ length-constant). In the typical experiment presented in Fig. 5, nerve evoked epsp's, a single simulated epsp and an electrotonus were recorded from the same muscle fibre. After application of 2 . 1 0 4 M G A B A (Fig. 5B), the epsp's nearly vanish while the simulated epsp is only
reduced to 0.5 of its control amplitude. This is indeed predicted by equation (9): using the value r/o) = 101 msec/9.4 msec = 10.7 {Fig. 5A) and r/"(o = 1.47 (T was estimated from the reduction of the electrotonus) (Fig. 5B) the correction factors, Corr, are 0.78 and 0.39, respectively; hence the reduction of the epsp-amplitude resulting from m e m b r a n e resistance changes is 0.39/0.78 = 0.5. The wdue of the m e m b r a n e time-constant, r, cannot be calculated directly from the decay of the simulated epsp in Fig. 5B, because due to the low m e m b r a n e resistance produced by G A B A the wdue of ~ is in the range of the value of ~:); compared to a merely passive repolarization phase of the epsp the actual, now rapidly decaying epsp in Fig. 5B is prolonged by the decay phase of the applied current. Calculation ~71the synaptic current The amplitude of the epsp as described by equation (9) is determined by four parameters: i~o, co, C and ~. Alterations of any one of these parameters may result in a variation of the epsp-amplitude detected in experiments in which, for example, temperature or the composition of the solution is changed. The time-constant ~ and its variation can easily be determined from the epsp-decay because it has been shown for crayfish muscle fibres (Fischer & Florey, 1981), that the decay is determined only by r (not by
L_
B GABA f i !
II
I
2mV
[
'\
i
C
1
2
3
Fig. 5. The effect of GABA on epsp's (1), simulated epsp (2) and electrotonus (3). (A) shows the control, (B) the maximal effect of 2.10 4M GABA and (C) the response after 15 min washout. A second electrode was inserted 120 ktm from the recording site to produce electrotonus (injected current 20nA) and simulated epsp (time constant of injected current 9.4 msec). The nerve evoked epsp's nearly vanish after GABA-application due to presynaptic inhibition. The amplitude of the electrotonus (membrane resistance) is reduced to ~o of its control value, while the amplitude of the single simulated epsp is only reduced to ½ of its control amplitude. This reduction is predicted by the model.
Calculation of synaptic current e~). Should the membrane capacity, C, not be constant it can be evaluated from C = T/R, where R is determined from electrotonic measurements. The synaptic parameters, i.e. the amplitude of the synaptic current, i~o, and its time-constant, ~o, cannot be evaluated explicitly from equation (9). In order to determine them separately, the synaptic current must be known. The method described below permits the calculation of the current from recorded epsp's alone. Theory. For a cell to which the RC-circuit of our model is applicable, the corresponding differential equation (4) can be used to determine the synaptic current, i~(t), if the epsp V(t) and the time-constant = R. C is given by measurement
is(t)
V(t)
dV(t) -
+
C
- -
dt
For the calculation reversal potential of account as described equation (4) can thus
(13)
T
of the synaptic conductance, the the epsp's can be taken into in equation (2). The differential be rewritten
V(t)
dV(t) _
gs(t)
_
-}-
_ _
dt
T
-
Kov- v(t)
c
(14)
The reversal potential in opener muscle fibres is around + 15 mV (Onodera & Takeuchi, 1975). Hence VreV is 90 mV with reference to a resting potential of 75 mV. In order to make use of these equations for appropriate programming of a computer the following points must be taken into consideration: the usual way for digitizing potentials is to take samples at a fixed interval, called dwell time and referred to as DW. According to the definition of the zero levels of the potentials used in the equations the digitized epsp in the computer must be shifted so that the resting potential is at the zero level. Then the membrane time-constant T can be determined by a least square fit applied to the logarithmized decaying phase (z can be assumed to be large relative to co in crayfish muscle fibres). If Vand V' are two adjacent potential samples taken at the time t, and t' = t + DW, respectively, equation (13), for instance, can be transformed for programming into the following form -
V
i,(t)
C
-
V' DW
+
V + V' 2.T
-
05)
The units of this current are ~A//~F, if V and V' are given in mV and D W in msec. The current amplitude is obtained from comparison with a calibration pulse of known amplitude: it is e.g. 1/~A//~F, if it is of the same magnitude as a 1 mV potential pulse. Application and testin.q. An example of the application of current calculation [-equation (15)] to an epsp, simulated by injecting a rapidly rising and exponentially decreasing current into an actual RC-circuit, is shown in Fig. 6. The artificial epsp was recorded with a 1 0 M ~ resistor simulating a microelectrode and fed through the amplifiers and tape recorder into the computer. The calculated time-course of the current fits that of the actually "injected" current very well. The noise in the epsp recorded and hence in the
481
2"* I I
0.5pA
I
,,
i\,
20 ms
Fig. 6. Comparison of injected and calculated current in a simulation of an epsp by injecting current into and recording epsp from an RC-circuit (100k~, lffF) via 10M~ resistors. Upper trace: artificial epsp, middle trace: calculated current (dots), middle and lower trace: injected current (continuous curvet.
computer current is due to the thermal noise of the resistors and recording apparatus and can be reduced by averaging. The calculation of the current according to equation (13) includes the membrane capacity. If the membrane capacity is known (it can be determined approximately from C = T/R by electrotonic measurements) the absolute amplitude of the current can be calculated iso = i~°.C - is° z C C 'R"
(16)
This is a kind of scaling which was employed in the simulation experiment of Fig. 6. The close correlation of calculated and injected current amplitude is obvious. An estimation of the absolute current amplitude in a muscle fibre can be obtained from epsp's of the type presented in Fig. 5: the electrotonus yields an approximation of R, z can be determined from the epspdecay and i~o/C can be calculated according to equation (15); thus the absolute value of the maximal current corresponding to the unfacilitated epsp in Fig. 5A (1 mV amplitude) is estimated as follows iso = 0.28/~A. 101 msec _ 0.132/~A. /~F 216kD In order to test the validity of the calculation of synaptic currents from epsp's recorded from crayfish muscle fibres the result of the computation should be compared with a genuine synaptic current actually measured in the same fibre. As measuring method we chose the extracellular recording technique, since the voltage clamp technique with microelectrodes is not applicable because of the large size of crayfish muscle fibres. An example of extracellularly recorded synaptic current and of intracellularly recorded epsp is shown in Fig. 7 together with the computed synaptic current. It must be emphasized that the computed current is the whole current flowing in this muscle fibre in contrast to the extracellular recording which
4~2
L. FISCHERand E. FLOREY
2 mV
20 ms
Fig. 7. Extracellular record of synaptic current at one syuaptic spot (middle trace) in parallel with intracellular record of epsp's (upper trace). The current calculated from the epsp (lower trace) shows a time constant of 2.8 msec which is only 15"~i shorter than the time constant of the synaptic current recorded extracellularly at one synaptic spot.
represents only the synaptic current at one of a large number of synaptic spots. For this and other reasons, the amplitudes of calculated and recorded epsp's are not equal. The time-constants of current decay can be compared, however. The duration of the extracellularly recorded current is very similar to the duration of the calculated current. Evaluation of the time constants by a least square fit revealed a small difference of about 15'~,~,. Possible explanations for this difference will be discussed later. Additional extracellular recordings from different preparations have confirmed the close correlation between calculated and measured time constants. DISCUSSION
Calculation ~f the synaptic current As already stated, the technique of extracellular recording cannot provide absolute values of current amplitude and the voltage clamp technique is not applicable to crustacean fibres of large size in the high frequency range needed for clamping the synaptic currents. The advantage of the mathematical method lies in the simplicity of the experimental arrangement necessary to provide the experimental data. The major effort is needed for the programming of a computer. Once this is accomplished, fast on-line computations of the synaptic current from the recorded epsp's can be performed• The calculation is not affected by artefacts. The noise amplification which is a result of the differential form of the algorithm used can easily be reduced by averaging. The method is well suited for application not only to single epsp's but also to facilitating and summating trains of epsp's. For the calculation method a value of the membrane time constant, 3, must be fed into the computer.
With ratios of z/co encountered in large crustacean muscle fibres the calculated current is rather insensitive to relatively large variations of z. It is possible that there are fibres whose properties are not described by our simple model (Fig. 1). In this case more than one time-constant and special cable properties appropriate to the spatial decrement of the imposed potential changes would have to be taken into account. From impedance measurements in crayfish muscle fibres Falk & Fatt (1964) deduced an equivalent circuit which consists of a resistance and a capacitance connected in parallel with another path composed of a resistance in series with a capacitance. Simulation experiments with such an equivalent circuit showing that a ~'splitting" of the total capacity, C, of Fig. 1 into the two components used in the Falk/Fatt-model by introducing the series resistance results in a "sharpening" of the rise and initial decay of the epsp while its later decay is barely affected. If the Falk/Fatt-model were indeed applicable the calculations based on the simple model would yield current durations which are too short. However, epsp's recorded from crayfish opener muscle fibres usually do not show the shape predicted by the Falk/Fatt-model. The influence of cable properties on the calculated currents is of general interest in application to innervated cells which are depolarized by synal~tic currents in a non-uniform manner. Such a non-uniform depolarization simulating an "epsp" in a muscle fibre can be achieved by injecting into the fibre a brief exponentially decaying current. The current computed from the configuration of such an "epsp" by means of equation (15) is significantly shorter than the actually injected one. Crayfish muscle fibres are short cables and the nerve evoked epsp's can be expected to depolarize the fibres relatively uniformly: the modifying effects of cable properties can thus be expected to be very weak. It is indeed found (see Fig. 7) that the currents calculated according to the equations based on the simple model are slightly shorter than the extracellularly measured synaptic currents. It is possible that this discrepancy between calculated and measured current is based on the morphology of the muscle fibres. Crustacean muscle fibres are known for their large clefts and invaginations of their surface. Axonal branches may run in and form synapses within such a cleft. Due to the restricted connection with the surrounding fluid, the cleft can be assumed to behave like a leaky cell (the extracellular space represents the interior of the cell and the sarcoplasma the "'extracellular" space). The synaptic potential recorded with a microelectrode placed at the opening of the cleft which is generated by the synaptic currents inside of this "invaginated" cell can be expected to be prolonged by the capacity of the surrounding muscle fibre membrane.
Relationship between epsp-amplitude and membrane resistance Different fibres of one crustacean muscle are known to differ in presynaptic and postsynaptic properties. As shown by equations (9) and (ll) the epsp-amplitude is relatively independent of the postsynaptic
Calculation of synaptic current membrane resistance and is mainly determined by the synaptic current (transmitter action), if T is much greater than ~o as it is in crustacean muscle fibres. It must be concluded, that there is no obvious correlation between epsp-amplitude and membrane resistance, since the variation of other factors from fibre to fibre, for example the amplitude of the synaptic current, is much larger than the variation of a few percent due to the observed membrane resistance changes. This lack of correlation between epsp-amplitude and membrane resistance was indeed found by Meiss & Govind (1979) who scanned the fibres of the accessory flexor muscle in the lobster. They concluded from their experimental data that 'tit appears, that epsp-amplitude is independent of the membrane resistance". This is a rather indirect, but interesting confirmation of the relationships deduced from the model. Their explanation, however, that classical cable theory is not applicable to crustacean muscle fibres appears to be inadequate. Cable properties have a definite, modifying influence on the relationship of the epsp-amplitude to membrane resistance as derived from the simple RC-circuit which is also the fundamental circuit used in cable theory. The experiment with Ba 2 + as shown in Fig. 4 was presented to provide evidence for the deduced equations which reveals the relative independence of epspamplitude from membrane changes in crayfish muscle fibres. If, in contrast, a proportional effect of membrane resistance on epsp-amplitude is assumed, the constancy of the epsp-amplitude can only be explained if it is assumed that Ba 2+ depresses (by a presynaptic or subsynaptic mechanism) the epsp's which otherwise would have to be increased in proportion to the resistance increase produced by Ba 2 +. It seems very unlikely, however, that in all the experiments with Ba2+-application the time-course as well as the magnitude of such a depression just compensate for the large, Ba 2 +-induced change of the membrane resistance. A presynaptic depression can be excluded, because the quantum content, calculated from the variance of the epsp-amplitude according to equations given by del Castillo & Katz (1953) before and during Ba2+-application, does not change significantly; in a series of 6 experiments with 5 mM Ba 2+ the quantum content even shows a tendency to increase by a factor of 1.21 _+ 0.35 (mean + SD). The non-proportionality of epsp-amplitude and membrane resistance is perhaps unexpected and has not been taken account of in previous publications: Hironaka & Otsuka (1971), for instance, describing drug effects at crayfish neuromuscular junction conclude that "'augmentation of ipsp size is attributed to the increase in membrane resistance of the muscle fibre". Ishida & Shinozaki (1980) deduce that "the increase in amplitude of the epsp's caused by diltiazem was due to the increase in membrane resistance". S a r n e & Parnas (1977) attempt to separate presynaptic from postsynaptic inhibition in crab muscle fibres by attributing the reduction of the epsp to the postsynaptic effect assuming that this is proportional to the reduction of the input resistance after GABA application. Although the values of z/e~ in the investigated crab muscle fibres are less than those observed in crayfish muscle fibres, the relationship is still far from proportionality. The experiment presented in
483
Fig. 6 demonstrates that the effect of GABA on the simulated epsp (which is exclusively of post-synaptic origin) is much less than the GABA effect on the input resistance. This can be predicted by equation (9). By utilizing the presented relationships in the evaluation of such experiments, a much more realistic separation of pre- and postsynaptic GABA-effects and a better differentiation of drug effects on epsp's can be derived. Furthermore, by applying current-calculation the "presynaptic" effects can be separated into effects on the amplitude and effects on the decay time constant of synaptic current [both parameters contribute to the epsp-amplitude in a manner described by equation (9)].
SUMMARY Direct measurement of synaptic currents (voltage clamp technique) in big crustacean muscle fibres is impractical because of the large membrane capacity. On the basis of a simple equivalent circuit, equations have been developed for a computer programmed calculation of synaptic current from intracellularly recorded synaptic potentials. Confirmation of the adequacy of the method has been obtained in experiments on crayfish opener muscle fibres with a fibre length about equal to the space constant. The theoretical approach used here predicts a relative independence of epsp-amplitude and membrane resistance in the case of muscle fibres where membrane time-constant is long compared to the time-constant of the synaptic current. Confirmation of this independence is provided by experiments involving application of Ba 2 + (increases membrane resistance) and GABA (decreases membrane resistance). The new method permits separation of presynaptic and postsynaptic effects in experiments involving procedures which affect synaptic transmission; the equations furthermore allow a differentiation of effects on amplitude and effects on decay time-constant of synaptic current.
Acknowledgement We would like to thank Ms Birgit Bremer for excellent technical assistance.
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