I. Computations and Experiments with Internal

Nonlinear Cable Equations for Axons

I. Computations and Experiments with Internal Current Injection N. J . A R I S P E

a n d J . W. M O O R 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, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela.

A B S T a A C T Steady-state potential and current distributions resulting from internal injection o f current in the squid giant axon have been measured experimentally and also computed from nonlinear membrane cable equation models by numerical methods, using the Hodgkin-Huxley equations to give the membrane current density. T h e solutions obtained by this method satisfactorily reproduce experimental measurements o f the steady-state distribution o f membrane potential. Computations o f the input current-voltage characteristic for a nonlinear cable were in excellent agreement with measurements on axons. Our results demonstrate the power o f Cole's equation to extract the nonlinear membrane characteristics simply from measurement of the input resistance. INTRODUCTION

For a long time axons have been considered analogous to a leaky electrical cable and the differential equations describing it derived in many papers (e.g., Cole and Curtis, 1941; Hodgkin and Rushton, 1946; Taylor, 1963). Extensive and excellent reviews of the analytic solutions of these equations for passive membranes have been provided by Cole (1968) and Jack et al. (1976). However the marked nonlinearities of the squid axon membrane (Cole and Curtis, 1939, 1941; Cole, 1941, 1949; Hodgkin et al., 1952) limit analytic solutions for the potential and current distributions generally to small potential purturbations. For large potential changes in an axon having a nonlinear membrane, numerical integrations are necessary to obtain solutions of the cable equations? We have obtained such solutions using the 1952 equations of Hodgkin and Huxley (HH). Moore and Green (1965) gave a preliminary report on steady-state solutions of the nonlinear cable equations by both analog and digital methods. Two different commonly used experimental situations for axons were simulated: When this work was done in 1970, no analytic methods were available. In their book, Jack et al. (1976) showed that some qualitative solutions could be obtained by using a polynomial approximation for the membrane's nonlinear characteristics. J. GEN. PHYSIOL. • The Rockefeller University Press 9 0022-1295/79/06/0725/11 $1.00 Volume 73 June 1979 725--735

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(a) injection o f c u r r e n t at a point inside a n e r v e in a large volume o f electrolyte solution containing the r e t u r n c u r r e n t electrode; (b) b o t h c u r r e n t electrodes on the e x t e r i o r o f the n e r v e m e m b r a n e b a t h e d in a small v o l u m e o f solution whose longitudinal resistance is c o m p a r a b l e to that o f the cell interior. At that time e x p e r i m e n t a l data was either not available or entirely satisfactory for c o m p a r i s o n with the calculations. For case a , Doctors J. Brinley a n d L. Mullins, University o f M a r y l a n d School o f Medicine, Baltimore, supplied us with data relating the potential inside an a x o n as a function o f the distance f r o m a cut e n d . A l t h o u g h curves could be fitted to the data points, r a t h e r large (or small) values o f a x o p l a s m resistance h a d to be a s s u m e d . Cole a n d Curtis (1941) m a d e m e a s u r e m e n t s o n squid axons u n d e r the conditions o f case b. At first we t h o u g h t that their data was a d e q u a t e but we f o u n d that an essential point o f i n f o r m a t i o n , the distance b e t w e e n the electrodes was not given. T o o u r inquiry, Cole replied that he h a d no r e c o r d o f the value o f this p a r a m e t e r a n d that the a x o n c h a m b e r was not available for m e a s u r e m e n t . F u r t h e r m o r e , Cole wrote (1968, p. 155) that the c o n d u c t a n c e s which the), m e a s u r e d were " f a r higher t h a n f o u n d b e f o r e a n d the action potential e x t r e m e l y low for what still a p p e a r to be quite u n k n o w n reasons . . . . It m a y only be g o o d luck that the results have b e e n f o u n d at least qualitatively correct." T h e r e f o r e , it s e e m e d necessary a n d reasonable to r e p e a t their e x p e r i m e n t s a n d to c o m p a r e t h e m with c o m p u t e r simulations o f the cable equations. In the process we p l a n n e d to obtain a n d fit data for e x p e r i m e n t a l changes, including natural d e t e r i o r a t i o n of the axon's excitability. T h e m e t h o d s used in the e x p e r i m e n t s a n d c o m p u t a t i o n s are r a t h e r d i f f e r e n t for the two cases. T h e r e f o r e , we have decided to treat the steady-state case o f internal injection o f c u r r e n t at a point inside an a x o n in a large b a t h in this p a p e r . T h e following p a p e r will deal with c u r r e n t flows f r o m external c u r r e n t electrodes situated in a n a r r o w channel h o l d i n g the a x o n (Moore a n d Arispe, 1979). COMPUTATIONAL

METHODS

AND

RESULTS

Cable Equations and Solution f o r a Linear Membrane

When current is injected inside the axon at a point as in Fig. I, it divides into two symmetrical, longitudinal flows which decrease rapidly with distance. For an axon in a large bath, the external longitudinal resistance is negligible and the external voltage gradient may be neglected. The transmembrane potential, V m, will equal the internal potential. The internal longitudinal potential gradient is given by the product of the longitudinal current, ia, and resistance per unit length of axoplasm, r~ d Vm dx -

raia.

(1)

The sign is negative because the flow of current away from the point of injection causes the displacement in membrane potential to decrease with an increasing x. For conservation of charge, the exit of current per unit length of membrane, ira, must equal the longitudinal gradient of the axial current, i a, or di, -dx

= -im.

(2)

ARISPEANDMOOmE Nonlinear Cable Equations: Internal Current Injectzon

727

Elimination o f i a between Eqs. 1 and 2 leads to the usual form of the cable equation d 2Vm dx 2 = ra'im

=

Vm ra--,

(3)

~+ra

where r r, is the m e m b r a n e resistance per unit length. For a point source of c u r r e n t in an infinite cable with an ohmic m e m b r a n e , the analytic solution is Vm = Vmoe-x/x, where h = ~

(4)

in centimeters and is called the length constant.

"m*]~

'\

~

v

"-+X

I 0

,,, +

I0

,,";,

la.~v

ro

ra

++++t

FIGURE 1. Schematic drawing of current injection into an axon and measurement of potential as a function of the distance x from the point of injection, In this case the axon is in a large bath and the current returns to a large electrode at ground. T h e electrical equivalent circuit is shown below. T h e resistance external to the m e m b r a n e is taken as zero.

Computer Solutions f o r the Nonlinear Membrane For the cable with a n o n l i n e a r m e m b r a n e such as the squid axon, where r m is a strong function of the m e m b r a n e potential, the value of potential for a given i n p u t c u r r e n t (or vice versa) cannot be f o u n d analytically and must be solved by machine methods. T h e current crossing the m e m b r a n e may be calculated from the H H equations (or other models) for any potential and inserted on the right side of Eqs. 2 and 3. O u r original attempts to integrate these equations iteratively on an analog or digital computer, assuming better values of the voltage at the origin for the given c u r r e n t injected proved to be impractical. T h e m e m b r a n e potential was unstable with distance from the origin, flying off to -+- infinity because of error in the value of the voltage assumed at the origin. T h e higher resolution in this value possible in the digital computer allowed the integration to proceed to a few length constants.

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We found that it was much more satisfactory2 to start the integration far enough from the origin so that the membrane potential is small enough for the membrane currentvoltage relation to be linear. For a given small voltage perturbation the axial current flowing at this point can be calculated. With these initial conditions, integration can proceed toward the origin without instability under any conditions. A single integration sufficed to provide solutions for any input current value up to the maximum. This method is also applicable to short cables. The terminating conditions are readily determined and entered as the initial conditions for the integration toward a current source. Furthermore, it is most convenient for generating other characteristic curves such as the cable input conductance-voltage relationships.

Digital Program A digital program was written in the FOCAL language and computation carried out on either a LINC-8 or PDP-15 computer (Digital Equipment Corp., Marlboro, Mass.). For the longitudinal axoplasm resistivity, we used a value of 35 ~km (used by Hodgkin and Huxley, 1952; also see Cole and Moore, 1960). The axon diameter was taken as 500 ~m for most computations. For the initial voltage deflection (0.1 mV or less), the initial axial current distal to the starting point was found for the initial value of V m from the characteristic resistance for a semi-infinite cable (r~-~-~ra). Integration of Eqs. 1 and 2 (with signs reversed ) proceeded toward the point of current injection until the desired current or voltage maximum was attained. With this technique, the computation is almost trivial and the axial currents and membrane potentials may be immediately plotted or stored as a function of the distance for later plotting.

Computation Results INTEGRATION METHOD TESTS Solutions using a single-step Euler integration were f o u n d to converge to an invariant plot as the x i n c r e m e n t was r e d u c e d to 1 m m or less. This solution was indistinguishable f r o m one using a R u n g e Kutta m e t h o d a n d all subsequent calculations were m a d e using the Euler m e t h o d with steps o f 0.25 or 0.5 ram. We also investigated the effect o f the choice o f the initial m e m b r a n e voltage on the shape o f the steady-state m e m b r a n e voltage distribution. We f o u n d that the solutions for a 500 g m (diameter) axon for initial voltages o f 0.01,0.1, and 1 mV in the depolarizing direction overlaid each o t h e r to within the width o f the plotted lines. Similar superpositions o f solutions were also obtained with equivalent initial h y p e r p o l a r i z i n g displacements. T h u s , we were assured that no appreciable e r r o r was m a d e by choosing an initial m e m b r a n e potential displacem e n t o f 1 m V o r less. NONLINEARITIES Fig. 2 A gives a c o m p a r i s o n o f the relative steepness o f the potential distribution in an o h m i c and a H H m e m b r a n e . Identical initial conditions were used and the c o m p u t a t i o n s were t e r m i n a t e d when a depolarization o f 100 mV h a d been achieved. For convenience in c o m p a r i n g distributions associated with different parameters, the x origins will be relocated at the point of c u r r e n t injection in subsequent figures. Fig. 2 B shows the m e m b r a n e c u r r e n t per unit length, i 0,, and the axial c u r r e n t , ia, c o r r e s p o n d i n g to the H H potential distribution shown in Fig. 2 A. z Jack et al. (1976, p. 435) also suggest this procedure.

AmsvE ANDMOORE Nonlznear Cable Equations: Internal Current Injectwn

729

Rectification o f the H H m e m b r a n e is reflected in the very d i f f e r e n t potential distribution f o r injection o f h y p e r p o l a r i z i n g a n d d e p o l a r i z i n g c u r r e n t s (Fig. 3 A). T h e h i g h e r m e m b r a n e c o n d u c t a n c e associated with depolarization causes the decay o f the m e m b r a n e potential with the distance to be m u c h s t e e p e r a n d r e q u i r e d injection o f m u c h l a r g e r c u r r e n t s t h a n f o r an equivalent h y p e r p o l a r i zation. T h e deviation o f the a x o n f r o m an o h m i c cable is most readily seen in semi-log plots w h e r e a linear m e m b r a n e p r o d u c e s a straight line relation between the log V m a n d x. Fig. 3 B shows such semilog relations f o r absolute values o f potential d i s p l a c e m e n t arising f r o m d e p o l a r i z i n g c u r r e n t s (right) ,.. I00

,-

8 0

,-60 mV 9-

40

,-20

I

ii

!

cm

B

I ' " JO0

HH MEMBRANE

It v

J(]

-2

-J

0

0

cm

FIGURE 2. (A) A comparison of the potential distribution for a depolarization of 100 mV in an axon with an ohmic (linear) membrane and one with a H H membrane. (B) The distributions of H H membrane potential and current density along with the axial current. The units for full scale are: V = 100 mV, I,, = 5 mA/ cm 2, i a = 50/~A.

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, tO0 m V

HH MEMBRANE

0(POLARIZING

A i

! -3

HYPERI=OLARIZING

~OmV

lOOmV

B

L.. (V)

IO

I'

I:"

' -'

I

FIGURE 3. (A) T h e voltage distribution for a d e p o l a r i z i n g c u r r e n t (above) a n d for a h y p e r p o l a r i z i n g c u r r e n t (below) in an a x o n with an H H m e m b r a n e . For p u r p o s e s o f c o m p a r i s o n , the c u r r e n t s injected at the origin w e r e a d j u s t e d to give equal and o p p o s i t e m a x i m u m m e m b r a n e potential displacements. (B) T h e l o g a r i t h m s o f the absolute values o f the same potential distributions plotted a l o n g with those for a linear m e m b r a n e for p u r p o s e s o f c o m p a r i s o n .

ARISPE AND MOORE

NonlinearCableEquations: lnternal Currentlnjection

731

applied to H H a n d linear cables a n d f r o m hyperpolarizing c u r r e n t s (left). For purposes o f comparison, the c u r r e n t s injected at the origin were adjusted to give equal and opposite m e m b r a n e potential displacements. It is clear that the H H cable is essentially ohmic only if the m e m b r a n e voltage displacement is < 1 o r 2 mV. DIAMETER EFFECTS In a cable with an ohmic m e m b r a n e , the f o r m o f the voltage distribution (Eq. 4, an exponential decay with distance) is d i a m e t e r i n d e p e n d e n t . T h e d i a m e t e r affects the length constant k which varies with the square root o f the d i a m e t e r , D, i.e.,

r~-_ , l Rm / 4R"

=/