Electric field distributions in neuronal membranes

Electric Field Distributions Neuronal

in

Membranes

DOUGLAS K. McILROY Department of Applied Mathemaiics University of the Witwatersrand Johannesburg,

South Africa.

Communicated

by D. Perkel

ABSTRACT The “instantaneous” current-voltage relationship of a neuronal membrane is examined in the light of the Planckian and Goldmanian limits of the membrane’s electric field distribution and a detailed model of facilitated ionic transport through the membrane. The model assumes that ionic conduction occurs through pores defined by chains of dipoles which in effect provide ion-selective negatively charged carriers for the current-carrying species. It is suggested that, provided the mean distance between conducting pores is greater than the membrane Debye length, the current-voltage relationship will be linear as in the case of the natural squid giant axon; theoretical expressions are deduced for the membrane conductances in terms of identifiable membrane parameters. The theory also suggests that if the foregoing inequality is reversed, the current-voltage relationship will be non-linear, the case of the frog node of Ranvier being cited as an example. Alternatively non-linearity may arise if more than one mobile ionic species is present in the conducting pores, and the current-voltage relationship of the squid giant axon in choline bathing solutions is discussed as an instance of this behavior.

1.

THE STRUCTURE OF THE AXONAL RESTING ELECTRIC FIELD

The observed disparity conductances g, and g,,

between the magnitudes of the axonal membrane in the resting state as well as at the peak of an

action potential permits us to estimate the corresponding membrane potential differences (p.d.s) as a first approximation from the Nernst equilibrium potential for respectively K and Na ions alone. Higher approximations must take into account the simultaneous finite values of g, and gNa, i.e., the essentially non-equilibrium character of the actual p.d. But even if the calculated non-equilibrium corrections of the p.d. were small, or if present knowledge of the ionic diffusion coefficients were insufficient to evaluate MATHEMATICAL

BIOSCIENCES

26, 191-206 (1975)

0 American Elsevier Publishing Company, Inc., 1975

191

192

DOUGLAS

K. McILROY

the corrections with numerical accuracy, the structure of the resting p.d. and its corresponding field would be of fundamental importance in the construction of microscopic models of the initiation of action potentials by depolarization, because of the following observations. The resting p.d. may be decreased by reducing the internal potassium concentration through perfusion of the axon. If the resting p.d. is substantially lowered by reducing only the internal potassium concentration under conditions of constant internal ionic strength, action potentials cannot be initiated. But if the internal ionic strength is reduced in proportion to the potassium concentration, action potentials can be initiated by depolarization from resting p.d.s even in the range 30 mV to zero (outside of the axon taken as positive) [ 1, 151. Now the average membrane field strength E is related to the membrane p.d. + and membrane thickness 6 by

EC

-

;

=

5 yE(x)dx 0

and becomes small as C$ does. If E(x) were uniform, E(x)= E, action potentials could then be initiated by changes of a low field (e.g., at low internal ionic strengths), and all microscopic mechanisms of the initiation of the nerve impulse which relied on the effects of the reduction of a high resting field would be inadequate. These include the Wien dissociation effect on ionic dissociation, the rotation of dipoles, and the displacement of ions. But if, on the other hand, the resting field E(x) is so distributed within the membrane that it remains high loca& despite a low average value, then the high-field mechanisms remain tenable. Is the resting field uniform or distributed in the membrane? It is perhaps curious that informed opinion (e.g., [3, 12, 181) cannot agree on such a basic question. However, by theoretical analysis the consensus does agree to narrow down the problem to a more specific point of uncertainty. We begin such theoretical considerations with Planck’s formulation of electrodiffusion as applied to neuronal membranes. The resting membrane is in a steady (non-equilibrium) state maintained in the long term by active transport. Since the membrane is thin as compared with the radius of the axon, we may regard it as an infinite plane region of dielectric constant E between x = 0 (inner surface) and x = 6 (outer surface) within which ionic species k have concentrations ck(x), charges zke, and diffusion coefficients D,, where e >0 is the elementary charge. The ions move through the membrane (or, on closer examination, through its permeable parts such as channels) under the action of the electric field superimposed on diffusion. In the steady state the flux Fk of each species is independent of time and

FIELD

DISTRIBUTIONS

IN NERVE

position: Fk= -L&z

+ j$.~,ec,E=constant,

where T denotes absolute temperature and the non-subscript k Boltzmann’s constant. We therefore obtain as many differential equations as there are ionic species. However, the field is at first unknown and must be determined from the collective effect of the ions (to which any fixed charge may be added) from Poisson’s equation: dE -=dx

he t.

(2)

where Q(x) is the membrane potential. Normalizing Q(O)=0 we obtain (P(6)=+ for the membrane p.d. In the presence of an external e.m.f. a conduction current Z is maintained (taking inward current to be positive) Z= - ex z,F,,

(3)

k

ck(O+) and ck(8 -) being prescribed through the partition coefficients Uk which depend on the membrane structure determining not only the phaseboundary potentials but the ionic solubilities at the membrane boundaries as well: Ck(o+)=~TSck(o-),

ck(&)=7Jkck(s+).

(4)

For N ionic species, Eqs. (3) and (4) supply the 2N + 1 conditions needed to determine the N fluxes and N + 1 integration constants of the set of N + 1 first order differential equations [(l) and (2)]. It will suffice for the present purpose to consider only monovalent ions /I~(= 1. We (K+, Na+, Cl-, etc.) as being possibly potential determining, introduce the total concentration of all such ions &,,= xkck, c+ the total concentration of all positives, and 8_ the total concentration of all negavariables, all of order zero to tives, Z_ = Z,,, - Z, and the dimensionless unity in neuronal membranes,

whence Poisson’s equation

becomes (5)

194 where

DOUGLAS

X is the Debye

Two limiting membranes:

length

cases

K. McILROY

in the membrane.

of

(5)

have

usually

been

discerned

for

nerve

(a) if 6*
uniformity

of field

gradient of the by E(x)=constant,

limit.

This

case

field

remains negligible; i.e., the Goldman [4]

is considered

by some

notably Katz [12], to be appropriate for neuronal (b) if h*<
membranes. multiplication

pmn,

is now

even

if d2+/dq2+0.

Poisson’s

equation

authors,

by h*/S*)

replaced

by

the

condition of electrical neutrality, 2 k c~(x)=O. This corresponds to the thick-membrane, high-solubility limit and is in effect Planck’s argument [ 161 in modern form. Though the field calculated from the simplified set given in (I) and (2) is not uniform, the field gradient is multiplied by the small parameter AZ/S*, and the corresponding charge density remains negligible. Some authors (e.g., Tasaki [ 181) consider this case to be the relevant one for some neuronal membranes. If we suppose that the electrical characteristics are deducible in terms of simple electrodiffusion geneous membrane [which is effectively assumed (a) and (b) above], then we may results of the above two methods Z&O + ) = Z&S

-),

in which

of neuronal membranes of ions across a homoin the discussion of limits

make the following of solution coincide

case it is easy to show

observations: the if (and only if)

that Planck’s

field too

is uniform. Making the assumption that the Uk are equal (see also Sec. 3). this is the situation in the natural axon (since the internal and external ionic strengths are equal to within a few percent, the concentration of polyvalent ions being slight), and it is only in perfusion experiments with changed ionic strength that the field structure is different in the two limits. In this case it is also easy to demonstrate that only Planck’s limit can yield a distribution of the field E(x) which arises as a result of the need for the individual fluxes to be independent of time and position in the steady state-where ionic strength is reduced, the electric field must be increased at the expense of the electric field elsewhere. Hence, in the Planckian limit, decreasing the axoplasmic ionic strength (by perfusion) increases the field near the inner surface of the membrane (n = 0+) which renders the high-field models of the excitable membrane still tenable. On these considerations, if limit (a) above were the actual situation in the nerve membrane, a high resting field could exist only if the measured membrane p.d. were not the p.d. across the membrane alone. Thus, e.g., if

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195

adsorbed negative charges on the inner surface of the membrane together with their counterions outside the membrane are assumed to form an additional double layer partially masking the true membrane p.d. and increasing with decreasing internal ionic strength, then a large range of even uniform membrane fields may be postulated, depending on the magnitude chosen for the adsorbed negative charge. This assumption, however, seems to permit, in the present state of observations, such a multiplicity of permissible theoretical constructions even for the actual natural membrane, that it is clearly an uneconomic way of interpreting the results of perfusion experiments at low ionic strengths-at least until the requisite adsorbed surface charge (my2 PC cm-* [9]) is observed, or unless (a) and not (b) is shown to be the appropriate limit for neuronal membranes. However the non-aqueous non-ionizing medium of the membrane interior [3] makes extremely unlikely the unassisted passage of inorganic ions across the membrane (upon which the foregoing theories and deductions depend). Indeed, in the subsequent sections, we will show that the conclusions and implications of the preceding two paragraphs are weakened, if not altogether invalidated, when facilitated passive ion transport occurs through well-defined pores exhibiting selectivity towards the current-carrying species. Subject to molecularly interpretable a priori assumptions about the mode of such transport of these ions through the channels, we will show that the various linear and non-linear behavior of the“instantaneous” current-voltage relationships exhibited by axonal membranes is deducible from our theory. 2.

A MODEL OF THE LINEAR SQUID GIANT AXON

The “instantaneous” current-voltage relationship of the squid giant axon in its natural ionic environment is linear [lo, 111. We first propose a model of facilitated ion transport across such a membrane, and then show that it leads to a detailed mathematical agreement with the Hodgkin-Huxley formulation of the membrane’s electrical properties. We suppose that I,, the jth component of the total axonal membrane conduction current I, is carried through the membrane via well-defined pores which conduct the ionic species j only (for simplicity’s sake we will here understand j to refer to either K+ or Na+, though the theory may obviously be extended to include other conducting ions of either positive or negative charge). In the steady state, from Eq. (l), dc,+ - dx + &_Ejcj+ = - Aj,

(6)

DOUGLAS K. McILROY

196

where A, is independent of time and position, and E,(x) is the electric field in aj pore. Furthermore, we suppose that the motion of thesej ions through the membrane is partially or totally compensated by a mobile negatively charged ion of charge z,-e and concentration c,-(x) which is confined to the membrane and does not contribute to either I, or I:

where B, is independent

dx

+

m

E,c,= - B,,

of time and position

c_+ = I Equations

z/-e

dc,-

-

c-

c,(x)

=

I

(7)

and

for 0 Q x < S.

(8)

(6) (7) and (8) give B, - z,-A,

dc, -_= dx

E

/

=

1 - z,-

’

Bj- Aj

g

(10)

e (1-z,-)c,’

and

where cojrc,(O+), cs,-~,(a-). It follows from Eqs. (3) and (IO)

E

so that the membrane

/

=

‘1

g

ecJ

eD,

I

‘OJ

csJ -

8

p.d. is

whence

9, = eD, (co, - c,,)/64,>

(12)

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197

From Eq. (13) we note the following cases: (i) when c,,, > csj, +, > 0 (e.g., for the K current-carrying system) and I, < 0 for I# < +,, i.e., the current is outwardly directed as observed for I k; (ii) when cs, > coj, +j < 0 (e.g., for the Na current-carrying system), 4 > 0 for I$>+,, and the current is inwardly directed as observed for I,,; (iii) when either co, or cS,+O, g,-+O though I, is finite for finite 9, approaching csjeDj/S and -c,,,eD,/S, respectively; (iv) and I, = c,e2Dj+/SkT, also non-zero when coj = cs,, g, = co,e2Dj/SkT(#0), for non-zero 9. The foregoing treatment is clearly not invalid in these important cases. In particular our definition of the chord conductance [Eq. (13)] holds in an external bathing solution free of the current-conducting species (e.g., see Hodgkin and Huxley’s remark [ 10, p. 4801). In Eq. (13) we have recovered the famous Hodgkin-Huxley (hereafter HH) linear current-voltage relationship for the natural squid giant axon and have given an interpretation of the chord conductances gj in terms of identifiable membrane parameters. This has been achieved by applying three a priori constraints: I. complete specificity of the j conducting pores towards ionic species j, II. partial or total compensation of the charge of the transported ion for all 0 < x g 6, and III. such “electroneutrality” to be achieved with the aid of a mobile negative ion which makes no effective contribution to the measured conduction-current component I,. The question of how these conditions can be implemented involves basically the question of how simple inorganic ions can enter and traverse with relative ease the hostile non-polar low-dielectric-constant medium which constitutes the interior of a neuronal membrane. One model which satisfies the above three criteria depends on the existence of dipole chains traversing the membrane parallel to the x direction. A single approximately cylindrical pore could then be defined by a group of such chains. If the dipoles of the chains are perpendicular to the direction of the chains with the negative poles all pointing in towards the center of the pore, a core of polarizable negative charge is provided to compensate the positive charge of a positively charged ion moving through the pore. We suppose that on the approach of such an ion to the mouth of a pore, the negative core becomes polarized, providing an effective negative charge zj- to offset the entrance of the ion to the pore, whereupon the ion is able to traverse the membrane accompanied by its negative “carrier” according to the equations of electrodiffusion as set out above. On the transference of the ion to the external medium the negative core relaxes to its equilibrium configuration, so that the net effect is the transportation of one positively charged ion across the membrane. If in addition the dipole core (e.g., by means of a

198

DOUGLAS K. McILROY

specialized shape or size) exhibits ion-selective properties towards the transported ion and to no other species, the above three a priori criteria are satisfied. The magnitude of zj- would of course depend upon the reduction needed in the (Born) solvation or immersion energy of the transported ion. The well-known autonomy of the axonal membrane conductances is an obvious feature of the model both because of the physical separation of the pores and because of the proposed ion-selective properties of the dipole carriers. The current-conducting ions do not co-determine the local electric field strength as in the case of the homogeneous membrane discussed in Sec. 1, so that we now see the results of the previously described perfusion experiments in a new light. Our model predicts that changes in the ionic environment of the nerve membrane could affect its excitability through changing its g’s [Eq. (13)]. This arises directly through the factors depending on concentration or indirectly through the D’s, which may themselves be functions of the local electric field strength redistributed by changes in the extramembrane media. It is therefore possible that, for high-field mechanisms (e.g., [13, 141) for the control of ionic permeability at least, membrane excitability could be maintained in the face of a low internal potassium concentration (reduced +) provided E,.,, remains locally high despite a low average value. Thus, e.g., the value of E&O+) [large for the natural nerve, since c,,~~<
FIELD

DISTRIBUTIONS

operate quite independently 3.

199

IN NERVE

THE GOLDMAN

of any such gating system.

RELATION

FOR AXONAL

CURRENT

Unlike the membrane of the natural squid giant axon, other axonal membranes (e.g., that of the node of Ranvier of frog nerve) exhibit a non-linear “instantaneous” current-voltage relationship. In this section we show how our model of facilitated ion transport can result in such a non-linearity and how the distinction between linearity and non-linearity can be interpreted in terms of two limiting cases of the model. We suppose here that “electroneutrality” again constrains the passage of current-carrying ions through the membrane, but in a form different to that of the preceding section: we replace Eq. (8) by the new condition Fc/=

Cc,-=Z(x) j

forO
(14)

where the first summation is over all the positive ions contributing to the total conduction current, and the second summation is over their corresponding negative carriers of valence zj- whose properties were discussed in Sec. 2. We now have in the steady state Eqs. (6) and (7) replaced by dc,+ dx

+zEc+=-A’ kT ’

J’

dc,zj-e dx + kT EC,- = - B,‘,

where E(x) and (15),

is the electric field corresponding

(Isa) (15b)

to Eq. (14). From Eqs. (14)

-$$+&Ex=-CA;=-A. 1 dZ - dx+$$EZ:=-xB;=-B,

if

j =z-,

qy=z&=...

with A,!, B,’ (and therefore Hence

A and B) independent

d2 _=_ dx

B-z-A

l-z-

’

z=z,+ $(Z,-C,),

of time and position.

(16a) (16b)

200

DOUGLAS

K. McILROY

where Z,=C(S-).

C,=C(O+), It follows

that

(B-A)

E,g e

and so the membrane

p.d. is, from

(l-z-)8

Eqs. (16) and (17)

kTS(A-B) += Now if we suppose, axonal

conduction

-T

e(1 -z-)(Z,-2,)

as before,

(17)

’

(18)

q.

that only positive

ions contribute

to the total

Eqs. (15). (16) and ( 17) yield

current,

(19)

where

U= c,D,c,’

and I = ec,

D,A,‘. Setting (A - BM/(l -rC)Cs,

=CJ

(20) integration

of Eq. (19) gives Z~~,-eUO(l+v)(~,-ZO)

8(A-B)

’

[= 16X,-eU,(l+v)(Z,-Z,)

In the natural it follows

axon x,c,+(O-)x

that Z o=&

x,c,‘(S+)

and so, from

,_ * -

kT6

v= (l-z-)(&-8,)~

(21)

[8]. so that if we set UK=uNa,’

Eqs. (18) and (21) U,eP+/kr-

ee+/kT_

U, 1

’

(22)

In Eqs. (13) and (22) we have two distinct relationships for membrane current as a function of voltage, and we now seek an interpretation of this difference in terms of the two limits discussed in Sec. 1 and the two “electroneutrality” conditions defined by Eqs. (8) and (14). Now most estimates would set the mean distance d between current-carrying channels (of both types) at w 1000 A [5], so that in all probability d>S=70 A. If the ‘If we suppose that the capture of K+ by the carriers of a potassium pore is as efficient a process as that of Na+ by the carriers of a sodium pore, the simplest way of satisfying this injunction on the distribution coefficients is to take the area1 density of potassium pores equal to that of sodium pores.

FIELD

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201

Planck domain of validity applied in the neuronal membrane under consideration, A would certainly be much less than d. The passage of K ions through the membrane is then quite independent of the passage of Na ions through it, since no interaction is then possible between the two currents; the measured j component of Z from a test patch of membrane (of dimension >d) is then simply the sum of the identical contributions from the j pores in this patch; the pertinent “electroneutrality” condition is evidently that given by Eq. (8) and the relationship between current and voltage is that described by Eq. (13). But if, on the other hand, the Goldmanian domain of validity applied, h could be greater than d, and interaction between currents in channels transporting different ionic species would then occur. In this case Eq. (8) is inappropriate because the current from the j pores is not observable microscopically (i.e., over patches the size of a pore cross section) as before; because of interference from neighboring pores, the measurement of membrane current is now possible only at the macroscopic level (i.e., over areas the size of the entire membrane test patch) where the relevant “electroneutrality” condition is clearly that given by Eq. (14), which therefore replaces Eq. (8) in our a priori criteria whenever h > d. Thus the Ei of Eq. (10) is written with the subscriptj because it is microscopic in character, and the E of Eq. (17) has no such subscript, in view of its essentially macroscopic construction. Notwithstanding this amendment of our a priori constraints, however, we still of course assume complete specificity of j channels towards the jth ionic species as before. Thus Djcj is taken to be the only non-vanishing current term inj channels, whence from Eq. (22) e*+ (c,j/coj)e’*/krrj = DjcOjkTs

ee+/kT_ 1

1 ’

(23)

the so-called Goldmanian expression which has been found to describe the data for the “instantaneous” I,-+ relationship for the frog node of Ranvier

PI* We therefore suggest that the linearity property exhibited by the natural squid giant axon is a consequence of the independence of the Q--of the non-interaction between current-carrying channels. Though the Planckian domain may still apply in this case, it seems unnecessarily severe in the light of the foregoing argument, which merely requires that the linearity property of a neuronal membrane applies for X < d (which of course includes the Planckian domain). The non-linearity as exhibited by the “instantaneous”* $-+ relationship of the frog node of Ranvier we interpret as resulting from h > d, which could arise by a decrease of ionic solubility (increase in A) or ‘Measurements made before the physicochemical processes which control ionic permeability in nerve can react to a sudden change in membrane p.d.

DOUGLAS K. McILROY

202

by a decrease of d as compared to the squid giant axon. At a node of Ranvier the peak inward current density during excitation is some 10 times that of the squid giant axon [8]. Such an increase could be explained most simply by a greater density of current carrying pores giving rise to a smaller value of d and a possible transition to the case X> d. In view of the estimated value of d, this strongly suggests that the Planckian domain of validity may be inappropriate even for the natural squid giant axon. The postulated method of ionic transport through neuronal membranes requires that the electric field should be a function of position [Eqs. (10) and (17)], so that our theory bears resemblance to both the Planck and the Goldman treatment. The preceding analysis assumes that aj channel conducts onlyj ions. If however, mobile species other than j ions are present in j channels, this analysis is modified in a novel way. In the next section we consider an example of this in relation to the choline-seawater experiments of HH [lo]. 4.

A MODEL OF THE NON-LINEAR SQUID GIANT AXON

Choline has been employed in the separation of the axonal membrane conduction current into its potassium and sodium components on the assumptions that, when it replaces sodium in the bathing solution, the resting potential is little affected and choline makes no contribution to I, [lo]. The HH choline-seawater experiments show that the squid giant axon exhibits the striking property change from linearity in sodium (natural) sea water to non-linearity in sodium-deficient bathing solutions. Now, in terms of our model of the membrane conduction process, this result could be interpreted in two ways: either (1) the substitution of choline for sodium in the extracellular medium is accompanied by a transition to the case X > d, and our treatment of Sec. 3 applies, or (2) though X < d, Eq. (8) is replaced by Eq. (14) in a single pore because of the presence in the pore of more than one mobile ionic species, viz.,j ions and choline ions in what is aj pore in the natural axon. (For example, Hille [8, 91 demonstrates that mobile ions other than sodium can penetrate sodium channels when the axon is placed in an unnatural ionic environment.) In other words, at the expense of a relaxation of apriori constraint I, Eq. (14) and the theory of Sec. 3 have a microscopic interpretation. In the first case h could presumably be increased by the removal of most of one of the conducting species, i.e. sodium, from the membrane. However, this interpretation would mean that the value of the membrane h is determined largely by the concentration of the permeable species, which is unlikely in view of the apparent sparcity of pores [5]. The alternative case

FIELD

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203

IN NERVE

seems to be more attractive because of the large concentration of extracellular choline (~z460 mM) used in these experiments. Linearity of the “instantaneous” Z,-+ relationship is preserved in choline solutions [lo], so we simply assume that choline is excluded from K pores but not from Na pores; this ensures that neither the resting potential nor the potassium current is much affected by the replacement process. If borh choline and sodium ions are present in sodium pores, then from Eqs. (20) and (21),

+e kTln(Z,/Z,)

(24)

+ ’

where in this case

uo=

DN&oN~,

h = %Na+ %Ch?

zO= CONa.

In Fig. 1 we show the theoretical “sodium” current curves versus depolarization from the natural resting potential, drawn for different values of the extracellular sodium concentration. Exact comparison with experiment is precluded by the uncertainty in the experimental values of cNa(8+) (since measurements, beginning at data point 1, were started before all the sodium had diffused away from the nerve [lo]), but the theory is clearly in tolerable agreement with experiment. In the curve fitting DNa/DCh= 100 much different to 100 would lead and %Ch/ CONa -6.64. Values of D,,/D,, to either unacceptably large values of DNacONa(see below) or cOCh/cONa.The values actually chosen for DNa/ D,, and csCh/cONa would mean that though the affinity of the sodium carriers for choline is rather high (comparable to that for sodium itself, since in normal seawater cGNa/cONa=9.2), the mobility of choline in a sodium pore is negligibly small in comparison with that of Na+ . This would mean in turn that choline does not contribute directly to ZNa but affects g,, through distortion of ENa( Furthermore we are able to determine the important membrane parameters Djcoj from our theory: expressing the co’s in moles per cm3, (i) from Eq. (13) and Fig. 7 of [lo], DNacONa= 1.47X lo-l5

cm-’

set-’

204

DOUGLAS K. McILROY Al

a ’

I

b ------IT--

4----

I ,___* P..

II’ I!5 12 13

5bmv

4

-1

FIG. 1. Sodium current-voltage relationship in sodium-choline seawater bathing solutions. The curves are plotted from Eq. (24); for curve (a) the concentration of external sodium is 10% of that in normal seawater, for curve (b) it is I%, and for curve (c) choline seawater is the external medium. The crosses are experimental results for the squid giant axon in choline seawater, with the numbers indicating the order in which the measurements were taken (Fig. 7 of [lo]).

(S’C, initial (ii) from

depolarization 110 mV, in natural Fig. 1 of this article, DNacoNa= 1.73X lo-l5

(5”C, initial (iii) from

cm-’

depolarization 110 mV, in choline Eq. (13) and Fig. 12 of [lo], D,q,,=

(2O”C, initial depolarization

13.4~ lo-l5

cm-’

seawater),

see-’ seawater),

set-’

84 mV, in choline seawater).

Bearing in mind that with equal distribution coefficients (fJ,=&J, cOK/ cONa= 8 and that under the experimental conditions the sodium and potassium conducting systems are in similar states of excitation, results (i), (ii) and (iii) are at least plausible. Exact comparison would depend on construction of a detailed model of the ionic permeability controlling systems. In

FIELD

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205

addition, if we suppose that in the experiment associated with result (ii) g, is increased by a factor of about 75 relative to its resting value [ 1l] and if we take Cole’s estimate [2] of the resting value of D, as ~2.5 X lo-” cm* set-‘, then (iii) yields cOKx0.72 m&f/liter and an average value of conducting ion in the natural squid giant axonal membrane of x0.4 mM/liter and &(=&a)~l.8~ 10-3. 5.

CONCLUSION

To the questions posed in the introduction, the foregoing model of ionic transport through nerve membranes provides answers which may be summarized as follows: (1) Electric field is distributed in the membrane. In the case of “linear” membranes (A < d) the distribution in sodium pores is different to that in potassium pores in a way determined by the concentrations of Na+ and K+ in the external media. For “non-linear” membranes (A> d) the field distribution also depends on the ionic constitution of the surrounding solutions; the distribution is, however, observable (with the electrical techniques normally employed) only over membrane patches large compared with the cross section of a pore. (2) In general the Goldmanian domain of validity is more likely to apply than the Planckian one, so that the above theory of neuronal membranes bears (superficial) resemblance to both the Goldmanian and the Planckian treatment. (3) Investigations of the excitability of “linear” neuronal membranes by perfusion at low intracellular ionic strength seem to support the idea of a masking double layer at the inner membrane surface, but do little else to elucidate the molecular mechanism underlying membrane excitability. This latter goal might be better approached in such experiments by exploiting the autonomy of the sodium and potassium current-carrying systems as quantitatively defined in this article; e.g., by simultaneous extracellular and intracellular perfusion of the squid giant axon, the level of excitability might be maintained near its natural value by offsetting an unfavorable redistribution of &(x) against a favorable redistribution of ENa( REFERENCES 1

2 3

W. K. Chandler and H. Meves, Sodium inactivation in internally perfused squid giant axons, Arch. Ges. Physiol. 281, 25-26 (1964). K. S. Cole, Electrodiffusion models for the membrane of squid giant axon, Physiol. Rev. 45, 346379 (1965). K. S. Cole, Membranes, ions and impulses, University of California Press, Berkeley, 1968.

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DOUGLAS

K. McILROY

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