Nonlinear Cable Equations for Axons H ...
Nonlinear Cable Equations for Axons H. Computations and Experiments with External Current Electrodes J . w . M O O R E and N. J . A R I S P E From the Department of Physiology and Pharmacology, Duke University Medical Center, Durham, North Carolina 27710 and the Marine Biological Laboratory, Woods Hole, Massachusetts 02543. Dr. Arispe's present address is Escuela de Biologia, Facuhad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela.
A B S T R A C T We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold. INTRODUCTION
In the p r e c e d i n g p a p e r (Arispe a n d Moore, 1979), we e x a m i n e d the steady-state spatial distribution o f c u r r e n t a n d voltage in an a x o n into which c u r r e n t was injected. We d e m o n s t r a t e d the validity a n d applicability o f the Cole equation to extract the m e m b r a n e c u r r e n t - v o l t a g e relation f r o m the cable i n p u t characteristic. In this p a p e r we take u p the m o r e f r e q u e n t l y e n c o u n t e r e d but analytically difficult case o f an a x o n b a t h e d in a restricted m e d i u m t h r o u g h which c u r r e n t is passed between e x t e r n a l electrodes. In 1965, M o o r e a n d G r e e n m a d e a p r e l i m i n a r y r e p o r t for solutions o f this case, but we were unable to c o m p a r e o u r solutions with the e x p e r i m e n t a l results obtained by Cole a n d Curtis (1941) because the s e p a r a t i o n b e t w e e n their electrodes was not k n o w n . It was not given in the original p a p e r , a n d Cole told us that the c h a m b e r h a d b e e n disposed o f so that it could not be m e a s u r e d . F u r t h e r m o r e they stated that " t h e r e were a n u m b e r o f unsatisfactory aspects to the potential m e a s u r e m e n t . . . a n d a n u m b e r o f c o m p r o m i s e s between theory a n d e x p e r i m e n t which are not easily e v a l u a t e d . " I n particular, their value o f resting m e m b r a n e resistance was 23 l~cm 2, s o m e 50-fold lower t h a n the generally accepted value o f =1,000 l)cm 2. T h e r e f o r e , the p r e s e n t a u t h o r s decided to r e p e a t a n d e x t e n d the e x p e r i m e n t s o f Cole a n d Curtis. We also wanted to have data in the i n t e r p o l a r a n d e x t r a p o l a r regions as well as at the electrodes for c o m p a r i s o n with o u r calculated potential distributions. F u r t h e r m o r e , we wanted to see h o w m u c h the well-known d e t e r i o r a t i o n o f a x o n s a f t e r dissection would c h a n g e the distributions o f c u r r e n t a n d voltage a l o n g the cable. J. GEN. PHYSIOL. 9 The Rockefeller University Press 9 0022-1295/79/06/0737/09 $1.00 Volume 73 June 1979 737-745
737
738
THE JOURNAL OF GENERAL PHYSIOLOGY ' VOLUME 73 9 1979 CABLE
EQUATIONS
FOR
EXTERNAL
ELECTRODES
T h e e x p e r i m e n t a l case to be simulated was that described by Cole and Curtis (1941), a n d consists o f an a x o n lying in a n a r r o w c h a n n e l in which n a r r o w c u r r e n t passing electrodes were placed in the wall o f the c h a n n e l as shown in Fig. 1. Because o f the restricted outside v o l u m e of solution b a t h i n g the a x o n , the e x t e r n a l resistance r e is finite a n d the potential d r o p outside the a x o n b e c o m e s very i m p o r t a n t . Insight into the c u r r e n t a n d voltage distributions u n d e r these conditions comes f r o m the physical principles used in cable equations a u g m e n t e d by r u n n i n g the p r o b l e m o n a h y b r i d c o m p u t e r which gave very r a p i d solutions. C u r r e n t flows t h r o u g h the a x o n as illustrated in Fig. 1. T h e c u r r e n t I0 is applied between electrode E1 a n d E2. I f there were no a x o n in the channel, all
EXTRAPOLAR
_1_
INTRAPOLAR
Ef
_l-
EXTRAPOLAR
Ee
.... ........
......
(Z
.--'-
._\
--. ; _.:
.......
O3 Io FIGURE 1. Schematic representation of current flows in and around an axon in narrow channel in which thin current electrodes are embedded in the sides. o f the c u r r e n t flow would be i n t r a p o l a r . I n s e r t i o n o f the a x o n causes s o m e c u r r e n t to flow e x t r a p o l a r to obtain access to the interior o f the a x o n . All o f the extracellular e x t r a p o l a r c u r r e n t in the c h a n n e l to the right o f Ee eventually crosses the m e m b r a n e t h e n flows towards the left inside the a x o n . T o the left o f E 2 , c u r r e n t also enters the a x o n for p a r t o f the distance to El, b r i n g i n g the axial c u r r e n t i a to a p e a k , a n d t h e n i a declines as c u r r e n t leaves t h r o u g h the m e m b r a n e . T h e axial c u r r e n t r e m a i n i n g at the position o f E a, continues to the left, is gradually lost t h r o u g h the m e m b r a n e a n d r e t u r n s to E1 extracellularly. For an o h m i c m e m b r a n e , the p e a k o f i a occurs at the m i d p o i n t b e t w e e n E 1 a n d E2, but f o r a n o n l i n e a r a x o n m e m b r a n e , it is shifted towards the depolarizing electrode. In the i n t e r p o l a r region, the s u m o f the internal, ia, a n d e x t e r n a l c u r r e n t s , i e, equal the injected c u r r e n t , I0 Io = ie + ia.
(I)
MOOREANDAR:Se~ Nonlinear CableEquations: External Current Electrodes
739
Most of the current remains external to the axon, although a small part flows in or out of it depending upon the membrane potential at any particular point. At any extrapolar cross section, I0 is zero and the current flowing external to the axon is exactly equal to and opposite in direction to the internal current. The cable equations of the first paper are modified to include Va and V e, the internal and external potentials, respectively. The membrane potential difference Vm equals Va - V e so that dVm
dVa
dVe
dx
dx
-
dx
(2)
= -raia + reie,
where r a and r e are the axial and external resistances per unit length. Again, from conservation of charge, the exit of current per unit length of membrane must equal the loss in the axial current and the gain in the external current, i,. COMPUTER
SOLUTIONS
FOR
dia
die
dx
dx"
THE
NONLINEAR
(3) MEMBRANE
Because we must consider two current sources in this case, the integration of these equations again presents the problem of instability associated with integration proceeding away from current sources. It is not possible to generate the full solution by reflection of the current and voltage distributions about the point of current injection, as was done in case a of the preceding paper, because the intrapolar patterns must be different from the extrapolar distributions. For speed and convenience in solving this problem, we used a hybrid computer system, in which the 580 analog computer (Electronic Associates, Inc., Westland Branch, N.J.) solved the cable equations and a PDP-15 (Digital Equipment Corp., Marlboro, Mass.) computed the membrane current density. Although integration on an analog computer is carried out with respect to time, we followed conventional practice and substituted the space variable x for time and thus simulated integration over distance. A digital program was written in FOCAL language for a PDP-15 computer to operate and control the analog computer and provide the membrane characteristics from the Hodgkin and Huxley (1952) equations. A table of the nonlinear current density-membrane voltage characteristic was calculated for increments in voltage of 0.1 mV and stored in the buffer memory of the computer. Simultaneous computation of Eqs. 2 and 3 were carried out on a hybrid system shown in Fig. 2. The digital computer samples the membrane potential from the output of amplifier 18, looks in the stored table for the corresponding value for the membrane current density in mA/cm 2, and feeds back this value to the analog computer. Potentiometer 29 is set to convert the digital output of membrane current density Im to membrane current per unit length ira. This is integrated to give a voltage proportional to the axial current i~ (Eq. 3). The values of the potential inside (V~) and outside (Ve) the axon were found as a function of distance by integration of +i ~r~ and - i ere. The membrane potential was obtained by summation of -V~ and V e sign inversion. Initial conditions were established as follows:
740
THE J O U R N A L OF G E N E R A L P H Y S I O L O G Y " V O L U M E 7 3 "
1979
(a) T h e value o f the e x t r a p o l a r V m was chosen as 1 mV or less (as in Arispe and Moore, 1979). H e r e V J V a = r e / r a because i ~ = - i a. (b) T h e initial value for Va was calculated f r o m the relation V~ = V m / ( 1 + r e / r ~ ) .
This initial value also equals i~0 times the i n p u t resistance (set by p o t e n t i o m e t e r 27) for a linear cable. P o t e n t i o m e t e r 0! is adjusted to give the calculated value for Va, also setting the a p p r o p r i a t e initial value o f ia in the process. This initial value o f i~ is less than the value chosen (by
-IO-(~ao
I-IO~ Im
"~176v't
-J
FIGURE 2. Hybrid computer method of solution of equations for axon in Fig 1. Voltage-summing (and sign-inverting) operational amplifiers are represented by triangles. A rectangle to the left of a triangle represents a summing integrator whose initial condition is set by the voltage at the top. Attenuating potentiometers are represented by circles. p o t e n t i o m e t e r 04) for the axial c u r r e n t between the electrodes. T h e r e fore, the c o m p a r a t o r o p e n s the electronic switch. T h u s , the initial value o f i e ( o u t p u t o f amplifier 08) is the negative o f the initial value o f i a. (c) P o t e n t i o m e t e r 10 was set to the ratio o f r J r a to establish the initial value for V~. (d) T h e initial value o f V,, (output o f amplifier 18) is automatically established as the difference, V a - V e. Integration is initiated in the e x t r a p o l a r region to the left o f E1 (see Fig. 1) and proceeds towards this electrode. W h e n i a has r e a c h e d the level selected by p o t e n t i o m e t e r 04, the electronic c o m p a r a t o r closes the electronic switch (SW), a n d the electrode c u r r e n t I0 is a d d e d to the external c u r r e n t . T h e integration p r o c e e d s in this condition while i a first increases and then r e t u r n s to the chosen switching value. T h u s , the c o m p a r a t o r senses when electrode E2 is reached and opens the switch to r e m o v e Io. As the integration proceeds, Vm and i m both r e t u r n to zero b e y o n d the second electrode.
MOORE ANDA g l s ~ Nonlmear Cable Equattons: External Current Electrodes EXPERIMENTAL
741
METHODS
All e x p e r i m e n t s were p e r f o r m e d at the Marine Biological Laboratory, Woods Hole, Mass., in a chamber similar to that used by Cole and Curtis (1941). A cleaned squid giant axon was placed in a narrow slot (600 p,m wide and 680/.Lm deep) between two side pools cut in the top o f a lucite block. External stimulating c u r r e n t was applied between a pair o f thin (50 /zm wide) electrodes e m b e d d e d in the c h a m b e r on opposite sides of the channel and a large indifferent electrode in an end pool o f the chamber. Such a chamber allows a reduction in the length o f axon required because the polarity of the thin -6( -4( Depolonzohon
- 20
o
40
AV m p~rl~fion
"~la
"-~ 401 30
~'te
20
I0
,,,
50
m
H
-I
0
I
2
3
X,cm
FIGURE 3. C o m p u t e d spatial steady-state distribution o f currents and voltages resulting from external application of current to an axon in a narrow channel.
electrodes can be reversed, providing two data points for each location of measuring electrodes. Axial potentials were recorded with a low impedance micropipette which had a long stem (several centimeters) o f uniform outer diameter making it possible to slide the micropipette axially back and forth into the extrapolar and the intrapolar regions without touching the inner side o f the axon m e m b r a n e . T h e external potentials were recorded with low impedance micropipettes placed just outside the axon. Simultaneous r e c o r d i n g o f the potential inside a n d outside the m e m b r a n e as well as their difference (Vm) were m a d e at several locations on both sides o f the stimulating electrode pair by moving the tips o f both r e c o r d i n g electrodes together.
742
T H E J O U R N A L OF G E N E R A L P H Y S I O L O G Y 9 V O L U M E 7 3 '
1979
A
E2
I I
X, cm
2 I
i
-I0 -20 Vm, mV
\\,I/./S,,
-30
.o
,o.
-40 -50 60
-20 B
depolarization
-I0 E2
I
X , cm
2
0
El I0 I0
- - 15
hyperpolarization
30 vm , mv 4o
I o , ~A
5o
X \ I
\T/ 70
80
INTRAPOLAR
/
/'--z5
...n,.....r EXTRAPOLAR
"
FIGURE 4. (A), C o m p u t e d family of m e m b r a n e voltage distributions for a range of currents. (B) Family of experimental observations for same conditions.
MOORE AND ARISPE Nonlinear Cable Equations: External Current Electrodes
743
RESULTS
T h e c o m p u t e d spatial steady-state distribution o f the c u r r e n t s and voltages associated with a H o d g k i n - H u x l e y axon in a n a r r o w channel is shown for external c u r r e n t injection in Fig. 3. T h e external resistivity o f sea water was taken as 20 flcm 2 and that o f the axoplasm as 35 l~cm ~. Note the relatively small ratio o f axial to external c u r r e n t . A family o f c o m p u t e d distributions o f t r a n s m e m b r a n e potential is shown in the u p p e r part o f Fig. 4 and a family o f e x p e r i m e n t a l observations below. A l t h o u g h these figures a p p e a r satisfactorily similar, a m o r e direct c o m p a r i s o n o f the model and e x p e r i m e n t can be m a d e by plotting the i n p u t current-voltage relationship.
Io,/~A axon diometer : 432/,Lrn
-30
-I0
Vm, rnV 60
[
50
40
30
I I I hyperpolorizotion
20
I
~
I0
I
~"
I~
-I0
-20
I I depolorizot,on
20
FIGum~ 5. Experimental steady-state I0 - Vm relations showing changes with the duration of the experiment. The points on curve 1 were determined as quickly as possible after placing the axon in the chamber. The a• quickly settled into a stable condition (curves 2 and 3) up to nearly 1 h. Then the resting potential began to drop, and curve 4 was taken. T h e e x p e r i m e n t a l I0 - V,n input relation varied somewhat with time a n d an example is shown in Fig 5. T h e first curve (curve 1) was taken immediately after m o u n t i n g the axon in the c h a m b e r . A few minutes later it had c h a n g e d to that labeled curve 2 (without a resting potential change) and was r a t h e r stable for nearly 1 h (curve 3). Later the resting potential declined and the I0 - V relation c h a n g e d to (curve 4). T h e stable I0 - V relation (points in Fig. 6) could not be fitted by the cable model with H o d g k i n - H u x l e y equations shown as a heavy line. H o w e v e r , the cable model fit can be i m p r o v e d greatly for hyperpolarizations by r e d u c i n g the leakage c o n d u c t a n c e in the H o d g k i n - H u x l e y equations fivefold,
744
THE
axon
JOURNALOF GENERALPHYSIOLOGY9 VOLUME73 9 1979 -30-
diameter:400Fro
I~
//
/
-20- / -IOVm , mV 60 [
50 I
40 [
30 I
hyoerpolarization
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J ~
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20 I
j
I0 I
~
9/
-I0 I
-20 I
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~
-I0
/ _ Hodgk,n ~ Huxley equations
-20
-30
Fmuaz 6. Fitting the stable I0 - V mrelation with the Hodgkin-Huxley equations for the cable membrane with the leakage conductance reduced fivefotd. c o r r e s p o n d i n g to a better physiological condition. It is possible that the p o o r e r fit this gives for depolarizations could be partly o v e r c o m e by also changing the leakage equilibrium potential but the m e a n i n g o f such an additional change is not clear.
DISCUSSION We f o u n d , as did Cole and Curtis (1941), that the input c u r r e n t voltage relation o f squid axons c h a n g e d with time. H o w e v e r , it was usually in a stable condition for m a n y minutes, long e n o u g h for at least a few full families o f data. O u r e x p e r i m e n t a l observations o f the current-voltage relation for axons in a n a r r o w channel with external electrodes were in reasonable a g r e e m e n t with calculations for a cable with a H o d g k i n - H u x l e y m e m b r a n e . A better fit was obtained w h e n the leakage was r e d u c e d fivefold. This means that o u r axons were in as good or better condition as those o f H o d g k i n and Huxley. T h e y were in m u c h bettea-condition ( - 6 0 - m V resting potential, 100-mV spike) than those o f Cole and Curtis who r e p o r t e d an average resting potential o f - 5 0 inV. If we estimate a - 6 0 - to - 6 5 - m V resting potential for the H o d g k i n - H u x l e y axon and calculate the m e m b r a n e conductance at - 5 0 mV we find - 2 5 0 D.cm z. This is still 10 times larger than the 23 f~cm2 which they r e p o r t . Because o f the stated problems with the Cole and Curtis (1941) experiments, we still are not able to d e t e r m i n e the reason for their extraordinarily low m e m b r a n e resistance. H o w e v e r , we see no reason for c o n t i n u e d c o n c e r n because o u r new observations are in such good a g r e e m e n t with the HodgkinH u x l e y equations.
We appreciate the technical assistance of Mr. E. Harris and the secretarial assistance of" Mrs. D.
MOOREANDA m s ~ Nonlinear CableEquations: External Current Electrodes
745
Munday and Mrs. D. Crutchfield. We are grateful for invaluable discussions with Doctors F. Ram6n and R. Joyner, and Dr. K. S. Cole. This work was supported by grant NS-03437 from the National Institutes of Health.
Receivedfor publication 3 April 1978. REFERENCES
ARISPE, N. J., and J. W. MOORE. 1979. Nonlinear cable equations for axons. I. Computations a n d experiments with internal current injection.J. Gen. Physiol. 73:725735. COLE, K. S., and H. J. Curtis. 1941. Membrane potential of the squid giant axon d u r i n g c u r r e n t flow.J. Gen. Physiol. 24:551-563. H'ODC~N, A. L., a n d A. F. HUXLEY. 1952. A quantitative description of m e m b r a n e current a n d its application to conduction and excitation in nerve. J. Physiol. (Lond.). 117:500-544. MOORE, J. W., and J. E. GREEN. 1965. Integration of the cable equation for membranes with variable conductances. Biophys. Soc. Annu. Meet. Abstr. 12.
A B S T R A C T We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold. INTRODUCTION
In the p r e c e d i n g p a p e r (Arispe a n d Moore, 1979), we e x a m i n e d the steady-state spatial distribution o f c u r r e n t a n d voltage in an a x o n into which c u r r e n t was injected. We d e m o n s t r a t e d the validity a n d applicability o f the Cole equation to extract the m e m b r a n e c u r r e n t - v o l t a g e relation f r o m the cable i n p u t characteristic. In this p a p e r we take u p the m o r e f r e q u e n t l y e n c o u n t e r e d but analytically difficult case o f an a x o n b a t h e d in a restricted m e d i u m t h r o u g h which c u r r e n t is passed between e x t e r n a l electrodes. In 1965, M o o r e a n d G r e e n m a d e a p r e l i m i n a r y r e p o r t for solutions o f this case, but we were unable to c o m p a r e o u r solutions with the e x p e r i m e n t a l results obtained by Cole a n d Curtis (1941) because the s e p a r a t i o n b e t w e e n their electrodes was not k n o w n . It was not given in the original p a p e r , a n d Cole told us that the c h a m b e r h a d b e e n disposed o f so that it could not be m e a s u r e d . F u r t h e r m o r e they stated that " t h e r e were a n u m b e r o f unsatisfactory aspects to the potential m e a s u r e m e n t . . . a n d a n u m b e r o f c o m p r o m i s e s between theory a n d e x p e r i m e n t which are not easily e v a l u a t e d . " I n particular, their value o f resting m e m b r a n e resistance was 23 l~cm 2, s o m e 50-fold lower t h a n the generally accepted value o f =1,000 l)cm 2. T h e r e f o r e , the p r e s e n t a u t h o r s decided to r e p e a t a n d e x t e n d the e x p e r i m e n t s o f Cole a n d Curtis. We also wanted to have data in the i n t e r p o l a r a n d e x t r a p o l a r regions as well as at the electrodes for c o m p a r i s o n with o u r calculated potential distributions. F u r t h e r m o r e , we wanted to see h o w m u c h the well-known d e t e r i o r a t i o n o f a x o n s a f t e r dissection would c h a n g e the distributions o f c u r r e n t a n d voltage a l o n g the cable. J. GEN. PHYSIOL. 9 The Rockefeller University Press 9 0022-1295/79/06/0737/09 $1.00 Volume 73 June 1979 737-745
737
738
THE JOURNAL OF GENERAL PHYSIOLOGY ' VOLUME 73 9 1979 CABLE
EQUATIONS
FOR
EXTERNAL
ELECTRODES
T h e e x p e r i m e n t a l case to be simulated was that described by Cole and Curtis (1941), a n d consists o f an a x o n lying in a n a r r o w c h a n n e l in which n a r r o w c u r r e n t passing electrodes were placed in the wall o f the c h a n n e l as shown in Fig. 1. Because o f the restricted outside v o l u m e of solution b a t h i n g the a x o n , the e x t e r n a l resistance r e is finite a n d the potential d r o p outside the a x o n b e c o m e s very i m p o r t a n t . Insight into the c u r r e n t a n d voltage distributions u n d e r these conditions comes f r o m the physical principles used in cable equations a u g m e n t e d by r u n n i n g the p r o b l e m o n a h y b r i d c o m p u t e r which gave very r a p i d solutions. C u r r e n t flows t h r o u g h the a x o n as illustrated in Fig. 1. T h e c u r r e n t I0 is applied between electrode E1 a n d E2. I f there were no a x o n in the channel, all
EXTRAPOLAR
_1_
INTRAPOLAR
Ef
_l-
EXTRAPOLAR
Ee
.... ........
......
(Z
.--'-
._\
--. ; _.:
.......
O3 Io FIGURE 1. Schematic representation of current flows in and around an axon in narrow channel in which thin current electrodes are embedded in the sides. o f the c u r r e n t flow would be i n t r a p o l a r . I n s e r t i o n o f the a x o n causes s o m e c u r r e n t to flow e x t r a p o l a r to obtain access to the interior o f the a x o n . All o f the extracellular e x t r a p o l a r c u r r e n t in the c h a n n e l to the right o f Ee eventually crosses the m e m b r a n e t h e n flows towards the left inside the a x o n . T o the left o f E 2 , c u r r e n t also enters the a x o n for p a r t o f the distance to El, b r i n g i n g the axial c u r r e n t i a to a p e a k , a n d t h e n i a declines as c u r r e n t leaves t h r o u g h the m e m b r a n e . T h e axial c u r r e n t r e m a i n i n g at the position o f E a, continues to the left, is gradually lost t h r o u g h the m e m b r a n e a n d r e t u r n s to E1 extracellularly. For an o h m i c m e m b r a n e , the p e a k o f i a occurs at the m i d p o i n t b e t w e e n E 1 a n d E2, but f o r a n o n l i n e a r a x o n m e m b r a n e , it is shifted towards the depolarizing electrode. In the i n t e r p o l a r region, the s u m o f the internal, ia, a n d e x t e r n a l c u r r e n t s , i e, equal the injected c u r r e n t , I0 Io = ie + ia.
(I)
MOOREANDAR:Se~ Nonlinear CableEquations: External Current Electrodes
739
Most of the current remains external to the axon, although a small part flows in or out of it depending upon the membrane potential at any particular point. At any extrapolar cross section, I0 is zero and the current flowing external to the axon is exactly equal to and opposite in direction to the internal current. The cable equations of the first paper are modified to include Va and V e, the internal and external potentials, respectively. The membrane potential difference Vm equals Va - V e so that dVm
dVa
dVe
dx
dx
-
dx
(2)
= -raia + reie,
where r a and r e are the axial and external resistances per unit length. Again, from conservation of charge, the exit of current per unit length of membrane must equal the loss in the axial current and the gain in the external current, i,. COMPUTER
SOLUTIONS
FOR
dia
die
dx
dx"
THE
NONLINEAR
(3) MEMBRANE
Because we must consider two current sources in this case, the integration of these equations again presents the problem of instability associated with integration proceeding away from current sources. It is not possible to generate the full solution by reflection of the current and voltage distributions about the point of current injection, as was done in case a of the preceding paper, because the intrapolar patterns must be different from the extrapolar distributions. For speed and convenience in solving this problem, we used a hybrid computer system, in which the 580 analog computer (Electronic Associates, Inc., Westland Branch, N.J.) solved the cable equations and a PDP-15 (Digital Equipment Corp., Marlboro, Mass.) computed the membrane current density. Although integration on an analog computer is carried out with respect to time, we followed conventional practice and substituted the space variable x for time and thus simulated integration over distance. A digital program was written in FOCAL language for a PDP-15 computer to operate and control the analog computer and provide the membrane characteristics from the Hodgkin and Huxley (1952) equations. A table of the nonlinear current density-membrane voltage characteristic was calculated for increments in voltage of 0.1 mV and stored in the buffer memory of the computer. Simultaneous computation of Eqs. 2 and 3 were carried out on a hybrid system shown in Fig. 2. The digital computer samples the membrane potential from the output of amplifier 18, looks in the stored table for the corresponding value for the membrane current density in mA/cm 2, and feeds back this value to the analog computer. Potentiometer 29 is set to convert the digital output of membrane current density Im to membrane current per unit length ira. This is integrated to give a voltage proportional to the axial current i~ (Eq. 3). The values of the potential inside (V~) and outside (Ve) the axon were found as a function of distance by integration of +i ~r~ and - i ere. The membrane potential was obtained by summation of -V~ and V e sign inversion. Initial conditions were established as follows:
740
THE J O U R N A L OF G E N E R A L P H Y S I O L O G Y " V O L U M E 7 3 "
1979
(a) T h e value o f the e x t r a p o l a r V m was chosen as 1 mV or less (as in Arispe and Moore, 1979). H e r e V J V a = r e / r a because i ~ = - i a. (b) T h e initial value for Va was calculated f r o m the relation V~ = V m / ( 1 + r e / r ~ ) .
This initial value also equals i~0 times the i n p u t resistance (set by p o t e n t i o m e t e r 27) for a linear cable. P o t e n t i o m e t e r 0! is adjusted to give the calculated value for Va, also setting the a p p r o p r i a t e initial value o f ia in the process. This initial value o f i~ is less than the value chosen (by
-IO-(~ao
I-IO~ Im
"~176v't
-J
FIGURE 2. Hybrid computer method of solution of equations for axon in Fig 1. Voltage-summing (and sign-inverting) operational amplifiers are represented by triangles. A rectangle to the left of a triangle represents a summing integrator whose initial condition is set by the voltage at the top. Attenuating potentiometers are represented by circles. p o t e n t i o m e t e r 04) for the axial c u r r e n t between the electrodes. T h e r e fore, the c o m p a r a t o r o p e n s the electronic switch. T h u s , the initial value o f i e ( o u t p u t o f amplifier 08) is the negative o f the initial value o f i a. (c) P o t e n t i o m e t e r 10 was set to the ratio o f r J r a to establish the initial value for V~. (d) T h e initial value o f V,, (output o f amplifier 18) is automatically established as the difference, V a - V e. Integration is initiated in the e x t r a p o l a r region to the left o f E1 (see Fig. 1) and proceeds towards this electrode. W h e n i a has r e a c h e d the level selected by p o t e n t i o m e t e r 04, the electronic c o m p a r a t o r closes the electronic switch (SW), a n d the electrode c u r r e n t I0 is a d d e d to the external c u r r e n t . T h e integration p r o c e e d s in this condition while i a first increases and then r e t u r n s to the chosen switching value. T h u s , the c o m p a r a t o r senses when electrode E2 is reached and opens the switch to r e m o v e Io. As the integration proceeds, Vm and i m both r e t u r n to zero b e y o n d the second electrode.
MOORE ANDA g l s ~ Nonlmear Cable Equattons: External Current Electrodes EXPERIMENTAL
741
METHODS
All e x p e r i m e n t s were p e r f o r m e d at the Marine Biological Laboratory, Woods Hole, Mass., in a chamber similar to that used by Cole and Curtis (1941). A cleaned squid giant axon was placed in a narrow slot (600 p,m wide and 680/.Lm deep) between two side pools cut in the top o f a lucite block. External stimulating c u r r e n t was applied between a pair o f thin (50 /zm wide) electrodes e m b e d d e d in the c h a m b e r on opposite sides of the channel and a large indifferent electrode in an end pool o f the chamber. Such a chamber allows a reduction in the length o f axon required because the polarity of the thin -6( -4( Depolonzohon
- 20
o
40
AV m p~rl~fion
"~la
"-~ 401 30
~'te
20
I0
,,,
50
m
H
-I
0
I
2
3
X,cm
FIGURE 3. C o m p u t e d spatial steady-state distribution o f currents and voltages resulting from external application of current to an axon in a narrow channel.
electrodes can be reversed, providing two data points for each location of measuring electrodes. Axial potentials were recorded with a low impedance micropipette which had a long stem (several centimeters) o f uniform outer diameter making it possible to slide the micropipette axially back and forth into the extrapolar and the intrapolar regions without touching the inner side o f the axon m e m b r a n e . T h e external potentials were recorded with low impedance micropipettes placed just outside the axon. Simultaneous r e c o r d i n g o f the potential inside a n d outside the m e m b r a n e as well as their difference (Vm) were m a d e at several locations on both sides o f the stimulating electrode pair by moving the tips o f both r e c o r d i n g electrodes together.
742
T H E J O U R N A L OF G E N E R A L P H Y S I O L O G Y 9 V O L U M E 7 3 '
1979
A
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X, cm
2 I
i
-I0 -20 Vm, mV
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-30
.o
,o.
-40 -50 60
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depolarization
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X , cm
2
0
El I0 I0
- - 15
hyperpolarization
30 vm , mv 4o
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X \ I
\T/ 70
80
INTRAPOLAR
/
/'--z5
...n,.....r EXTRAPOLAR
"
FIGURE 4. (A), C o m p u t e d family of m e m b r a n e voltage distributions for a range of currents. (B) Family of experimental observations for same conditions.
MOORE AND ARISPE Nonlinear Cable Equations: External Current Electrodes
743
RESULTS
T h e c o m p u t e d spatial steady-state distribution o f the c u r r e n t s and voltages associated with a H o d g k i n - H u x l e y axon in a n a r r o w channel is shown for external c u r r e n t injection in Fig. 3. T h e external resistivity o f sea water was taken as 20 flcm 2 and that o f the axoplasm as 35 l~cm ~. Note the relatively small ratio o f axial to external c u r r e n t . A family o f c o m p u t e d distributions o f t r a n s m e m b r a n e potential is shown in the u p p e r part o f Fig. 4 and a family o f e x p e r i m e n t a l observations below. A l t h o u g h these figures a p p e a r satisfactorily similar, a m o r e direct c o m p a r i s o n o f the model and e x p e r i m e n t can be m a d e by plotting the i n p u t current-voltage relationship.
Io,/~A axon diometer : 432/,Lrn
-30
-I0
Vm, rnV 60
[
50
40
30
I I I hyperpolorizotion
20
I
~
I0
I
~"
I~
-I0
-20
I I depolorizot,on
20
FIGum~ 5. Experimental steady-state I0 - Vm relations showing changes with the duration of the experiment. The points on curve 1 were determined as quickly as possible after placing the axon in the chamber. The a• quickly settled into a stable condition (curves 2 and 3) up to nearly 1 h. Then the resting potential began to drop, and curve 4 was taken. T h e e x p e r i m e n t a l I0 - V,n input relation varied somewhat with time a n d an example is shown in Fig 5. T h e first curve (curve 1) was taken immediately after m o u n t i n g the axon in the c h a m b e r . A few minutes later it had c h a n g e d to that labeled curve 2 (without a resting potential change) and was r a t h e r stable for nearly 1 h (curve 3). Later the resting potential declined and the I0 - V relation c h a n g e d to (curve 4). T h e stable I0 - V relation (points in Fig. 6) could not be fitted by the cable model with H o d g k i n - H u x l e y equations shown as a heavy line. H o w e v e r , the cable model fit can be i m p r o v e d greatly for hyperpolarizations by r e d u c i n g the leakage c o n d u c t a n c e in the H o d g k i n - H u x l e y equations fivefold,
744
THE
axon
JOURNALOF GENERALPHYSIOLOGY9 VOLUME73 9 1979 -30-
diameter:400Fro
I~
//
/
-20- / -IOVm , mV 60 [
50 I
40 [
30 I
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J ~
o J
20 I
j
I0 I
~
9/
-I0 I
-20 I
depolarization
~
-I0
/ _ Hodgk,n ~ Huxley equations
-20
-30
Fmuaz 6. Fitting the stable I0 - V mrelation with the Hodgkin-Huxley equations for the cable membrane with the leakage conductance reduced fivefotd. c o r r e s p o n d i n g to a better physiological condition. It is possible that the p o o r e r fit this gives for depolarizations could be partly o v e r c o m e by also changing the leakage equilibrium potential but the m e a n i n g o f such an additional change is not clear.
DISCUSSION We f o u n d , as did Cole and Curtis (1941), that the input c u r r e n t voltage relation o f squid axons c h a n g e d with time. H o w e v e r , it was usually in a stable condition for m a n y minutes, long e n o u g h for at least a few full families o f data. O u r e x p e r i m e n t a l observations o f the current-voltage relation for axons in a n a r r o w channel with external electrodes were in reasonable a g r e e m e n t with calculations for a cable with a H o d g k i n - H u x l e y m e m b r a n e . A better fit was obtained w h e n the leakage was r e d u c e d fivefold. This means that o u r axons were in as good or better condition as those o f H o d g k i n and Huxley. T h e y were in m u c h bettea-condition ( - 6 0 - m V resting potential, 100-mV spike) than those o f Cole and Curtis who r e p o r t e d an average resting potential o f - 5 0 inV. If we estimate a - 6 0 - to - 6 5 - m V resting potential for the H o d g k i n - H u x l e y axon and calculate the m e m b r a n e conductance at - 5 0 mV we find - 2 5 0 D.cm z. This is still 10 times larger than the 23 f~cm2 which they r e p o r t . Because o f the stated problems with the Cole and Curtis (1941) experiments, we still are not able to d e t e r m i n e the reason for their extraordinarily low m e m b r a n e resistance. H o w e v e r , we see no reason for c o n t i n u e d c o n c e r n because o u r new observations are in such good a g r e e m e n t with the HodgkinH u x l e y equations.
We appreciate the technical assistance of Mr. E. Harris and the secretarial assistance of" Mrs. D.
MOOREANDA m s ~ Nonlinear CableEquations: External Current Electrodes
745
Munday and Mrs. D. Crutchfield. We are grateful for invaluable discussions with Doctors F. Ram6n and R. Joyner, and Dr. K. S. Cole. This work was supported by grant NS-03437 from the National Institutes of Health.
Receivedfor publication 3 April 1978. REFERENCES
ARISPE, N. J., and J. W. MOORE. 1979. Nonlinear cable equations for axons. I. Computations a n d experiments with internal current injection.J. Gen. Physiol. 73:725735. COLE, K. S., and H. J. Curtis. 1941. Membrane potential of the squid giant axon d u r i n g c u r r e n t flow.J. Gen. Physiol. 24:551-563. H'ODC~N, A. L., a n d A. F. HUXLEY. 1952. A quantitative description of m e m b r a n e current a n d its application to conduction and excitation in nerve. J. Physiol. (Lond.). 117:500-544. MOORE, J. W., and J. E. GREEN. 1965. Integration of the cable equation for membranes with variable conductances. Biophys. Soc. Annu. Meet. Abstr. 12.
