physical principles and formalisms of electrical excitability

7/25/2019 Physical Principles and Formalisms of Electrical Excitability.

1/53

C H A P T E R 6

A A * I I

A

A

A

Physical principles and formalisms

Departments of Physiology and Neurology, Albert Einstein

College of Medicine, New Yo rk , Ne w YoFk

Departmen t of Biophysics, T he Rockefeller Unive rsity,

New York , New York

of electrical excitability

C H A P T E R C O N T E N T S

Membrane Potentials and Ionic Fluxes

Fundamental concepts

Equilibrium state

Nonequilibrium state

Summary

The Donnan equilibrium

Phase-boundary potentials

Two important, examples of equilibr ium situ ations

Electrodes

-

he measurement of potential differences

Quasi-equilibrium systems

A membrane with a very large fixed-charge density

An oil membrane

Homogeneous uncharged membrane

Homogeneous membranes with special properties

Mosaic membranes

Formal Consequences of Voltage-dependent Conductances

Ion transport (the Nernst-Planck flux equations)

The nature of electrical excitability

Reasons for believing th at electrical excitability does not

Hodgkin-Huxley equivalent circuit

Current-voltage (I-V) characteristics

Negative-slope conductance

Changing the I - V characteristic without change of the g-V

result from the shifting of ionic profiles

characteristic

Voltage-dependent Conductance in Thin Lipid Membranes

The unmodified t hin lipid membrane

Formation

Permeability and electrical properties

Carriers

Channel formers

A mosaic membrane formed with two modifiers

Summary

Monazomycin

Alamethicin

Excitability-inducing material

Nonvoltage-dependent modifiers

Voltage-dependent modifiers

system, the modified thin lipid (or bilayer) mem-

brane, which illustrates most of the relevant phe-

nomena associated with nerve excitation. Before dis-

cussing this model, however, we develop more or less

from first principles the concepts of membrane poten-

tials and ionic fluxes. This forms a ra ther large pa rt

of the article and may be superfluous for the more

sophisticated reader. Nevertheless we have included

this material because, despite its importance for un-

derstanding nerve excitation, it is rather inaccessible

to students and investigators attempting to make

initial contact with the neurophysiological literature.

In the second section we discuss some of the formal

aspects of the behavior of systems containing ele-

ments whose conductances are profoundly affected by

the voltage across them. With thi s as background, we

then discuss the fascinating voltage-dependent phe-

nomena that can arise in suitably doped thin lipid

membranes.

MEMBRANE

POTENTIALS

A N D IONIC FLUXES

Fundamental Concepts

Here we consider two general situations: equilib-

rium and nonequilibrium states. We analyze the

equilibrium sta te from both th e thermodynamic and

the statistical mechanical viewpoint; the nonequili-

brium state is handled by the Nernst-Planck flux

equations.

EQUILIBRIUM STATE.

hermodynamic approach.

The

fundamental relation that we need from thermody-

Single channels

Summary and conclusion

_ _ _ _ ~

namics is that at thermal equilibrium the electro-

chemical potential, pi, f any species i is the same in

all phases to which the species has access. Thus, for

phases 1 and 2 we can write

pi (1) =

pi

(2) (1)

provided that species i can move between the two

phases. Equation

1

is the starting point of all our

IN

THIS CHAPTER

we intend to elucidate the presently

understood physicochemical principles and formal-

isms underlying the electrical excitability of biologi-

cal membranes. The primary analysis is of a model

161


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162

HA N D BOOK O F PHYSIOLOGY HE N ER V OU S SY STEM I

thermodynamic treatments. For a n ideal solution, p,

is given by

p , = p , + RT In

X ,

+ PV, + z , F

$

+ any other

relevant work terms (2)

where p , is the standard chemical potential (in the

particular phase being considered) of the ith species,

X ,

is mole fraction of ith species,

V ,

s partial molar

volume of ith species, z, is valence of ith species (0,

k l , +2,

.

. .), P

is

hydrostatic pressure of the phase, 4

is

electrostatic potential of the phase,

R is

the gas

constant, T is temperature in degrees Kelvin, and F

is the Faraday. (If the solution is nonideal, an activ-

ity coefficient, y , , is included with the mole fraction

term. In this article we deal only with ideal solu-

tions.) If species i is dilute, then Equation 2 can be

rewritten as

p, = p ,

+ RT

I n c, +

PV, + z,

F 4 +

. . . (2a)

where

c,

is the concentration of i.

( p l ( 0 )

n Eq. 2a is

different from p , n Eq. 2 because the standard state

must be newly defined in going from mole fraction

units to concentration units.)

A s

indicated in Equation 2 , other work terms must

be included if they contribute

to

the potential energy

of the species i. For example, if the system is in a

gravitational field (or ultracentrifuge), there will be a

term for the gravitational potential energy. For the

systems we consider, these terms do not arise. In fact,

even the

PV

term will generally be trivial and there-

fore not ente r into our treatments. Thus, for our

purposes,

p,

can be written as

p, =

p, (

+ RT

In c,

+

z , F

$

(2b)

Stat i s t ical mechanical approach. The basic rela-

tion tha t we need from statistical mechanics is tha t at

thermal equilibrium the particles satisfy the Boltz-

mann distribution a t all points in the system to which

they have access. That is, a t any point x (considering

only a one-dimensional situation)

c , x ) =

C,,e-%(x)/h7

(3)

where,

w ,

XI is the potential energy per particle of

the ith species a t

x ,

c,, is its concentration a t th e point

defined as zero potential energy, and

k

is th e Boltz-

mann constant. Multiplying the numerator and de-

nominator of the exponent by

N A ,

Avogadros num-

ber, we can write Equation 3 in the form

c , x ) = c , , e - , s ) lRT (3a)

where

W,

is the potential energy per mole of the ith

species. If W,

is

purely electrostatic energy, then

Equation 3a becomes

c , ( x ) =

~ ~ e - 2 ~W X ) l R T

(3a)

For example, consider particles in a gravitational

field. Then Equation 3 becomes

=

c e -l l l v s : k ? (4)

where m is the mass of the particle and g is the

acceleration due to gravity. If there were no thermal

energy, all the particles would sit a t x = 0, he point

of minimum potential energy. On the other hand if

there were no gravitational field and only thermal

energy, the particles would be homogeneously dis-

tributed throughout space. Equation 4 is the compro-

mise when both terms operate.

It is important to realize th at Equation 3a is equiv-

alen t to Equation

1.

(Take the logarithm of both sides

of Equation 3a and identify W, as all the terms on the

right in Equation

2a

except

RT

In

ci.)

Thus the ther-

modynamic statement that the electrochemical po-

tential of a species is the same in all phases is equiva-

lent to the statistical mechanical statemen t that the

molecules of the species satisfy the Boltzmann distri-

bution.

NONEQUILIBRIUM STATE.

The bases of our discussion of

ion transport a re the Nernst-Planck flux equations

where $,+and

$k-

represent the flux (in mol/s)

of

the

jth and kth ion, respectively, across unit area at any

point x in the system,

u ,

s th e molar mobility of the

jth cation,

uk

is the molar mobility of the kth

anion,

c,+

and

ck-

represent the concentration of the

jth and kth ions, respectively, a t any point I, nd

is the electrical potential

at

any point x . (Again we

are considering a one-dimensional situation with gra-

dients of concentration and electrical potential occur-

ring only in the x direction.)

Whereas our basic equilibrium equations (Eqs. 1

and 3) have

a

solid foundation in thermodynamics

and statistical mechanics, the flux equations are less

firmly grounded in theory, and in some instances

they are even grossly incorrect. We therefore wish to

make a few points concerning them tha t will give the

reader some feel for their meaning.

Equations 5 can be written i n the form

where

D = uRT

(7)

The Boltzmann distribution iS

a

quantitative state-

merit of the balanced but incessant competition be-

tween potential energy

( w )

nd thermal energy ( k T ) .

I some

heoretical problems associated with justifying the use

of the Boltzmann distribution in electrolyte theory do not concern

us

67).


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CHAPTER 6: PHYSICAL PRINCIPLES

A N D

FORMALISMS

OF ELECTRICAL EXCITABILITY 163

For

a

nonelectrolyte (z

=

O), Equation 6 simply be-

comes

dc

dx

4 = D-

which is Ficks

first

law of diffusion. [Equation

7

is

the famous Nernst-Einstein relation between the dif-

fusion constant an d th e frictional coefficient (13,

14);

the mobility (u)

s

simply the reciprocal of the fric-

tional coefficient.] Thus th e flux of a species is propor-

tional

to

the negative gradient of its concentration.

On the other hand for an ion

at

uniform concentra-

tion throughout the system dc/dx = 01, Equation

6

becomes

which is simply the equation for electrophoresis.

That is, the ionic flux

is

proportional to the electric

field (-d+/dx). Thus Equation 6 says that if there is

both a concentration gradient and an electric field,

the ionic flux is a l inear sum of the fluxes that would

arise from each effect alone.

Another way of looking at Equations

5

is to say

that

flux = concentration x velocity

and

velocity = mobility x driving force,2

so that combining we have

flux = mobility

x

concentration

x

driving

Z F9)

T dc

4 = u x c x

dx

forc

Substituting Equation 2b, the thermodynamic

expression for the electrochemical potential, into

Equation 8 we obtain Equation 5, the Nernst-Planck

flux equations (if p,( is constant). Note that Equa-

tion

8

reduces to the thermodynamic equilibrium

condition, Equation

1,

when

4i = 0

at all points; or

Equation

5

reduces to the BoItzmann distribution,

Equation 3ar, when Cbi

=

0.

[The relationship of the

Boltzmann distribution (equilibrium state ) to the

flux equations (nonequilibrium state) is clearly seen

upon differentiating Equation 3a

d%

T

dci

CI dx du

(which is the same relation a s obtained by se tting di

= 0 in Equation 5). Thus a t equilibrium the sum of

the electrical force

( -

z,F d$ldx) and the diffusional

force

( -

RT/ci dc,/dx) cting on an ion is zero at

every point in the system, and hence there is no flux

of matter. When these two forces do not balance,

there is a net driving force on the ion and hence a

flux.

For our purposes the Nernst-Planck flux equations

(Eqs. 5) are perfectly adequate. Nevertheless the

reader should be aware th at there exist situations for

ziF 0

The flux equation states that the force on an ion is the

sum of two terms: -RTlc

dcidx

and -zF d$ldx. The

second term is th e electrical force familiar from ele-

mentary electrostatic theory, but the first term is

more subtle.

It is

a phenomenological force resulting

from the random (Brownian) motion of each individ-

ual ion. Though the random movements of each ion

are equally likely to be in the positive or negative

direction, statistically it appears as if there is a force

operating in the direction of the concentration gra-

dient (4 1).

Equation

5

is

a

special case of the more general

expression

which st ates tha t the driving force on the ith species

is the negative gradient of its chemical potential.

In a condensed phase such as water, an ion subjected to a

force very quickly accelerates to its terminal velocity. In practice,

therefore (unless we are dealing with very high frequencies),

force gives rise to velocity rather than to acceleration.

which these equations are insufficient. In particular,

Equations

5

(or more explicitly Eq.

8)

say that the

only driving force on a species i is the negative gra-

dient of chemical potential for that species alone; it

neglects the coupling of the gradients of chemical

potentials for other species j to th e flux of species i. (A

familiar example of such coupling is solvent drag,

where the gradient of the chemical potential of water

gives rise not only to a flow of water, but also to a flow

of solute dissolved in the water.) Though these cou-

plings, expressed as cross coefficients to forces acting

:e

(5)

on other species, are extensively used by those who

deal with the formalism of irreversible thermody-

namics (36), they do not concern us here. For the

systems we consider (which are

of

neurophysiological

interest), they introduce corrections tha t are a t best

second order.

SUMMARY. We consider dilute, ideal solutions of ions

in which gradients of concentration, potential, or

solvent composition occur in only the x di re~ t ion .~

The sta rting point for the t reatment of these systems

is one of the following equations

4.

-

dc, d%

I uiRT ziuiFci 5) Transport

dx dx

For the general case, d/dz is replaced by

C

n all equations.


4/53

164 HA N D BOOK OF PHYSIOLOGY THE NERVOUS SYSTEM I

m e m b r a n e

C I -

F I G .

1.

The Donnan system.

Two Impor tan t Examples of Equi l ibr ium S i tua tions

THE DONNAN

E Q U I L I B R I U M .

Aside from its intrinsic

physiological intere st, the Donnan equilibrium illus-

trates almost all the important concepts and difficul-

ties associated with membrane potentials; we there-

fore analyze it in some detail. The Donnan equilib-

rium arises when a membrane separates two solu-

tions containing both permeant and impermeant

ions. For simplicity we restrict our trea tment to the

case of an infinitesimally thin membrane separating

two infinite solutions; both solutions contain a single

permeant univalent positive (Na+) nd negative (Cl-)

ion species, and, in addition, one solution contains a

univalent impermeant ion, zN, where

z = + 1

(Fig.

1). (The mechanism of membrane semipermeability

is irrelevant to our treatm ent. In practice, the Don-

nan situation commonly arises when a porous mem-

brane, such as dialysis tubing, holds back a macro-

ion (e.g., protein) which cannot

fit

through the pores.

Usually the solvent, water, is also permeant.)

Thermodynamic ana lys i s . The conditions that

must be satisfied are, from Equation

1

/- .I

( l ) ~ N . I 2 ) (9a)

k l

(1 )

= P(I

( 2 ) (9b)

which become, upon substituting from Equation 2b

&,+ +

RT In a+], -t F ,

(10a)

= &;;+ + RT In [Na+In+

F I / J ~

Note that there is no such equation for N, since we

have specified that it cannot move between the two

aqueous phases. (We have excluded a third equation

equat ing the chemical potential

of

water on the two

sides of the membrane. Although the osmotic effect

accompanying the Dorinan equilibrium is of consider-

able general physiological interest, it is not signifi-

can t for electrophysiology.) Adding Equation 10a and

10b we obtain

or

This is known a s the Gibbs-Do nnan condi t ion, and

r

is called the Don nan ratio .

From Equations 10a or lob we can directly obtain

the potential difference across the membrane

RT a+],

v =

($, - J J ~ )=

~

In

~

F

a+],

RT

[Cl-ll RT

F

[Cl-]? F

-

- -

In ~=

-

~ In r (13)

To

calculate r, and hence the membrane potential,

we invoke the familiar electroneutrality condition

on both sides of the membrane

[Na+I1= [Cl-1, =

c I

a+],

+

z[N]

=

LCl-1, (14b)

(I t is to be understood tha t the electroneutrality con-

dition holds only for remote regions on eith er side

of

the membrane, that is, for macroscopic regions. As

will be seen presently, the al terna tive treatment by

means of the Poisson-Boltzmann equation establishes

the concentrations and potential as continuous func-

tions ofx, and indeed regions very close to the mem-

brane are not electrically neutral

.)

Combining Equa-

tions 14 and 12 we obtain

(14a)

(electroneutrality condition)

Figure 2 is a sketch of r as a function of the imper-

meant ion (N) concentration for

z =

+1. As [N+]

increases,

r

decreases from

1;

that is, permeant posi-

tive ion concentration decreases on side 2 and per-

meant anion concentration increases. In the limit of

large +I, there is virtually no Na+ on side 2 and

[Cl-1,

=

+I. The converse occurs for large -1.

Note also from Equation 13 th at t he membrane po-

tential, \Ir, will be positive or negative depending on

whether

z

is

+

1 or -1 and th at the absolute value of

\I

increases as [N] increases. Figure 3 is a diagram of

r

-

~

= r

Na+;l,

-

[Cl-1,

a+:[, [Cl-1,

FI G . 2 .

Qualitative plots of the Donnan ratio, r, as a function

of the impermeant ion concentration,

IN],

for an impermeant ion

of

valence

+1

or - 1 . (See Eq.

1 5 . )


5/53


6/53

166

HANDBOOK OF PHYSIOLOGY THE NERVOUS SYSTEM I

and

[Cl-]+,

=

[Cl~]~,e+~YH

FIG.

5.

The two-phase system.

generated from a given initial condition. Thus, al-

though the impermeability of N is the ultimate cause

for the membrane potential and ion asymmetries,4

the $ and concentration functions are inexorably

coupled. That is, the

$

profile affects the concentra-

tion profiles which in turn affect the

J,

profile, and

which is the Donnan potential as given in Equation

13. Furthermore by equa ting the two expressions for

q

we see that

lCI-I+,lNal,

=

[Cl-lL,[Na+l-, so on.

We have dealt in some detail with the Donnan

equilibrium because the general nature of the

results

which we shal l discuss shortly. Basically the thermo-

which is the Gibbs-Donnan condition a s in Equation

we have the more general statement

2. (Note tha t bY

Equations 16a and 16b

applies to many other equilibrium systems, one of

lC1-l[Natl = [C-J-,[Na+]_,

=

constant

dynamic approach gives the-concentrations and po-

at every point x . )

Second, we now finally see the specific charge sepa-

ration that gives rise to the electrostatic potential.

There exist narrow space-charge regions on the two

sides of th e membrane; in solution 1 he space charge

is negative and in solution 2 it is positive, with of

course

.r:& = -

J1:pbr

(24)

The potential, $, and the ion concentrations, a+]

and [Cl-I, vary continuously in these regions to their

final values in the remote regions.

A

comparison of

Figure 3 with Figure 4 clearly illustrates the differ-

ences between the thermodynamic and th e statistical

mechanical point of view. In the former case, the

$

and concentration functions are constant a t thei r re-

mote values on the two sides of the membrane, with

a discontinuity in these values occurring at the plane

of the membrane; in the latter case these functions

are seen to vary continuously from one remote region

to the other.

The ex ten t of the space-charge regions is deter-

mined by the Debye length, L,) Eq. 23). Roughly

speaking, p falls about e-fold for every Debye length.

For 0.1

M

salt solution in water

( E =

801,

L,, =

10

A;

thus the region where electroneutrality is signifi-

cantly violated extends about 40-50

A,

too small for

direct sampling with a pipette. Note from Equation

23 th at L,, is directly proportional to 4 2 nd inversely

proportional to f l a a Thus in low dielectric con-

stant media, L,, will tend to be smaller, whereas in

media of low ionic strength,

L,,

will be larger. We

shall refer back to this point later when discussing

ion distributions in a lipid, or hydrocarbon, phase.

Finally it should be realized that the system is a

typically nonlinear one, which makes it difficult to

describe how the J, and concentration functions are

tentials far from the membrane, or interface,

whereas the Poisson-Boltzmann treatment explicitly

describes how the membrane potential arises from

space-charge regions near the membrane; electroneu-

trality holds only a t remote (many Debye lengths)

regions.

PHASE-BOUNDARY POTENTIALS. In th e Donnan equilib-

rium, the asymmetry of permeable-ion distributions

(the Donnan condition) and the membrane poten-

tia l ar ise because a macro-ion in one of the aqueous

solutions

is

impermeant. Here we consider the case in

which no impermeant ion is present, but the two

phases are different (for simplicity we consider them

immiscible); for example, one phase

is

water and the

other is oil (Fig. 5).

Thermodynam i c anal y s i s . From Equations

1

and 2b

we have

pi;;,+,,

+ RT In a+], + F$,

= p l&t )2

RT In [Nail2

+

F$2

(25a)

+ RT In [Cl-1,

-

F $ ,

= p;: )l-)s

RT In [ClV], - F$2

(25b)

These are the same equations employed for the Don-

nan equilibrium (Eqs. 10a and lob); this time, how-

ever, the p s are not the same on sides l and 2,

because the solvents are different. The conditions of

electroneutrality for remote regions also give

a+], = [Cl-I, = [NaCIl, (264

[Na+12

=

[Cl-I,

=

[NaCll, (26b)

Substituting Equation 26 into Equation 25 and add-

ing we obtain

For the t rue Donnan case, where N

is

not fixed

at

a constant

value to th e right of the membrane, we would h ave

an

Equation

16c, stating th at N satisfies the Boltzmann dis tributio n for x

>

0.

The final result is not much different from th at shown in Fig. 4 ,

except that

N

is perturbed upward near the membrane.


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CHAPTER

6:

PHYSICAL

PRINCIPLES

AND

FORMALISMS

OF

ELECTRICAL

EXCITABILITY 167

Upon substituting Equation 26 into Equation 25 and

subtracting we also obtain

*O/\V =_ ( 2 - 1 )

(28)

(dk+,] P$L)J - (PL:%+)y PII.l-),)

-

2F

Equation 27 is an expression for the oil-water (o/w)

partition coefficient

of

NaCl in terms of the standard

chemical potentials of Na+and C1- in the two phases.

Equation 28 gives the phase-boundary potential in

term s of these same quantities.

It is instructive to transform these expressions into

ones containing the intrinsic partit ion coefficient of

each ion. At the boundary between the two phases

there is a discontinuity in standard chemical poten-

tial and hence a discontinuity in ion concentrations.

Of course, in (classical) reality there are never actual

discontinuities; all functions are continuous. Never-

theless, because th e boundary between two phases is

established through short-range van der Waals inter-

actions, the phase transition occurs over molecular

distances (-2 A) and hence can be practically treated

as a discontinuity, compared to the space-charge

re-

gions that can extend over tens or even hundreds of

angstroms. Thus the p*.))s,nd hence t he ion concen-

trations, can change precipitously over a distance

where the electrostatic potential,

$,

remains un-

changed.s With this in mind, and assuming that

there is not a layer

of

dipoles at th e interface, we can

write for th e conditions at the boundary

Equations 30a and 30b are expressions for the intrin-

sic partition coefficients

G.1

of Na+ and C1-, respec-

a Image forces, which make a major contribution to the intrin-

sic partition coefficient, extend over distances longer than the

phase-transition region, but these are still in general much

shorter than the space-charge regions.

tively. That is, they a re th e partition coefficients tha t

would be observed in remote regions

i f

somehow elec-

trostatic interaction among ions did not occur. Sub-

stituting Equation 30 into Equation 27, we have

p o / w = d(PNa+)(PCI-) (31)

and substituting Equation

30

into Equation 28 gives

,

(32)

Equation 31 states that the macroscopically ob-

served partition coefficient for NaCl is the geometric

mean of the individual partition coefficients

@ )

for

Na+ and C1-. Note th at if one of these approaches

zero, the parti tion coefficient for the sal t approaches

zero. Equation 32 states that the phase-boundary

potential is determined by the rat io of the individual

partition coefficients. Note that if these are equal,

the phase-boundary potential is zero. Also note that

in contrast to the Donnan potential, the phase-bound-

ary potential is not a function of the NaCl concentra-

tion (provided, of course, the partition coefficients

are

not concentration dependent).

Stat is t ical mechanical analys is .

Had we stopped

our thermodynamic analysis with Equations 27 and

28, we would have been in much the same position we

were in with our thermodynamic treatment of the

Donnan equilibrium; the concentrations of NaCl are

different in the two phases, and there is a potential

difference between th e two phases (Fig. 6). Again th e

formal expressions are perfectly correct, and again

the origin of the boundary potential

is

obscure. By

extending our thermodynamic analysis to the parti-

tioning occurring a t the boundary, we have given a

fairly strong hint as to the source

of

the boundary

potential. Clearly, applying the Poisson-Boltzmann

analysis with these boundary conditions will show

that space-charge regions exist in both the aqueous

and oil phases. Figure 6 will then be transformed into

the more complete Figure 7. We shall not go through

the formal treatment since, aside from the mathe-

matical details, there are no new physical principles

that have not already been considered for the Donnan

case. The extent of the space-charge regions in the

two phases will depend on their respective Debye

I

ql=O

I

FIG. 6 . Concentrations and potentials in the water and oil

phases. The potential, 2, in the oil phase i s positive, because we

have assumed that the partition coefficient for sodium, pNa+,

between oil and water

is

greater than that for chloride,

p c , - .


8/53

168

HANDB OOK OF PHYSIOLOGY -

THE

NERVOUS

SYSTEM

I

lengths. Since

L D =

Jmhere tends to be a

compensating effect between dielectric constant and

salt concentration. Thus a low dielectric constant by

itself would lead to a smaller Debye length, but in

general a low dielectric constant

is

accompanied by a

small partition coefficient of the salt between water

and the low dielectric constant phase, which leads

to

a larger Debye length. In the cases we consider later

of membranes with an essentially hydrocarbon inte-

rior, the concentration term strongly predominates,

so that the space-charge region in the membrane

interior is much more extended than the one in the

aqueous phase. Of course, regardless of the extent of

the space-charge regions, overall charge conservation

must always obtain

pdx = - 1: p d x

(24)

We might also note that a truly continuous treat-

ment would not show the discontinuities in ion con-

centrations at = 0, as is depicted in Figure

7,

but

rather would show

a

continuous transition over a

distance of a few angstroms. Such a t reatment would

require a "van der Waals, image force,

. . .

-Boltz-

mann" analysis of this region, in analogy to the Pois-

son-Boltzmann treatment. Needless to say, the much

more complex nature

of

these forces makes such an

analysis extraordinarily difficult (if not impossible).

il ,

Electrodes - he Measurement

of Potential Difference

Up to this point in our discussion of potentials

associated with ionic systems, we have not described

how one goes about measuring these potentials. Be-

fore considering other examples

of

membrane poten-

tials, we must discuss this problem. We are con-

I

*O/

w

N a t , q

*--- - -

N o t l $/'

I /

A

r

I

I S o l u t i o n

1

I S o l u t i o n 2 I

FIG.

8. Method for measuring the potential difference across a

membrane.

cerned not with the technical questions of which am-

plifiers to use

o r

what brand of oscilloscope is best,

but rather with an important theoretical question

(which also happens to have important practical im-

plications). The basic problem is the following: in

order to measure the potential difference across a

membrane, we must insert a pair of electrodes into

the system- ne electrode on each side of the mem-

brane (Fig. 8). By necessity there will exist

at

each

solution-electrode (soln/elec) interface a potential,

generally called an

electrode pot entia l .

Thus the po-

tential th at we measure,

Y,,, , ,~i5u,. , ,c, ,

s in principle the

algebraic sum of three potentials: the membrane po-

tential (the quantity

of

interest) plus two electrode

potentials

-

q'measured

-

Vrrnernbrane

4-

qeleclsoln I +

q s o i n

2leier

(33)

How then do we make contact with the solutions so

that the sum of the last two terms in Equation 33 is

negligible? This

is

crucial, for when we go to measure

a membrane potential, we want to measure a quan-

tity tha t is a unique property of the ionic system and

not a quantity that is dependent on the particular

pair

of

electrodes we happen to choose.

To illustrate more concretely the problem

of

meas-

urement, consider again the Donnan system of Fig-

ure 1. With two theoretical formulations we have

shown that

RT [Cl-1,

In~

[Cl-1,

- --

* ,,.ml,r;,nC.

-

( 1 3 )

and the question now is, can we measure it? Suppose

we use reversible Ag-AgC1 electrodes. [The electrode

reaction i s AgCl + e + Ag")'+ C1- (soln).] What then

is

~ , , , ( . ; l S U l . ( . t , ?

The answer turns out t o be zero. Let us

see why. The "electrode potential" of a Ag-AgC1 elec-

trode in contact with a solution containing C1- is

where qo s the so-called standard potential. (In thi s

case it is the potential

of

the half cell when the

solution is one molar in chloride.9 If we keep our sign

FIG.

7. Sketch of concentration and potential profiles for a

phase-boundary equi librium a s determined from the Poisson-

Boltzmann analysis (cf. Fig. 6, which is the result

of

the thermo-

dynamic analysis; as in

Fig. 6, P , ~ . >

p c i - ) . The space-charge

density is shown in th e lower par t

of

the figure.

It is interesting to note that Eq. 34 can be derived by exactly

the Same methods we have employed in treating our pure ionic

systems. Thus, since the system is in equilibrium and

C 1 ~

an

move between the two phases (solid and solution), we have from

our thermodynamic relation


9/53

CHAPTER 6: PHYSICAL PRINCIPLES AND FORMALISMS O F ELECTRICAL EXCITABILITY

169

convention straight we then have

RT

+ (*(,

- p

n [Cl-I,) = 0

r

* i * Iv r / w1 n

2

The electrode potentials exactly cancel the mem-

brane potential, and operationally we measure no

potential difference

at

all. We certainly made a poor

choice of electrodes

We could have predicted th is result purely from the

second law of thermodynamics without going

through the above algebra. Since each electrode

is

in

equilibrium with its solution and the solutions are in

equilibrium (Donnan) with each other, the entire

system must be in equilibrium. If there indeed were a

potential difference between the electrodes, we could

construct a perpetual motion machine of the second

kind. That is, we could connect a load between the

two electrodes and do work. At one electrode, C1-

would go into th e solution, and at the other electrode,

C1- would come out; the overall chemical composition

of the solutions would not change. If the Donnan

condition became perturbed by the transport of NaCl

from one solution to the other, we could merely pause

for a while and let the Donnan condition reestablish

itself (utilizing, of course, only thermal energy).

When one electrode becomes almost depleted of AgC1,

we merely switch the electrodes from one solution to

the other, a process tha t, in principle, requires negli-

gible work. Thus we could indefinitely convert ther -

mal energy into work without any other change in

the universe -a clear violation of the second law of

thermodynamics.

Since a reversible pair of electrodes will always

give =

0,

we must try something different.

Why not use a pair of stainless-steel wires? We might

indeed measure the correct Donnan potential, but

then aga in we might not. The problem is that there is

not a well-defined process dominating the potential of

k I (solid) = k 1 soh)

&? solid) - FJlplrc

&','

(sold

+

R T In

ICI-I

- F&,

or

and rearranging, we obtain

Eq.

34 where

and

~ ' , 4 , . , h < , l ( elPC - l W d

1

Po = j

(d'i) solid) -

i' (soh))

Similarly a Poisson-Boltzmann analysis would show an extended

space-charge region in the solution near the electrode and a very

narrow one in the electrode itself.

the steel wires. What is the dependence of thi s poten-

tial on Na+ and C1- (or in th e more general case of the

Donnan equilibrium, on any other permeant ions

present in the system) concentration? What effect

does the macro-ion N have on this potential? It is

possible that none of these have significant effect on

the electrode potential, and therefore the two elec-

trode potentials will be equal and cancel each other

out, leaving the Donnan potential a s the only quan-

tity measured. But we cannot be sure , since we have

no theory to work from.

It turns out that we

can

measure the membrane

potential by introducing appropriate sa lt bridges. In-

stead of putt ing t he Ag-AgC1 electrodes directly into

the solutions, we place them into 3

M

(or saturated)

KC1 and make contact to the solutions through t he 3

M

KCl (Fig.

9) .

Now at

first

glance it might appear

that things are even worse, because the measured

potential is now the sum of five potentials instead of

three:

In addition to the two electrode potentials, there are

now two liquid junction potentials -one between 3 M

KCl and solution 1 and one between 3 M KCl and

solution 2. On reflection, however, it is clear that

things are not worse than before, for since the Ag-

AgCl electrodes are in identical solutions (i.e., 3 M

KCl), the electrode potentials must be equal and

hence cancel. This then leaves the two liquid junction

potentials with 3

M

KC1, and it t urn s out that these

are small, both because K+ and C1- have the same

mobility and because their concentration is large

(40).

Given a sal t bridge with a large concentration of

a salt whose ions have equal mobilities, it can be

shown tha t the liquid junction potential between this

bridge and any "reasonable" solution is small

(40).

(KCl happens to be the most convenient salt to use.)

Thus the only term left in Equation 35 hat is either

not negligible or does not cancel out is 9m ml ri nhe

quantity we wish to measure.

Note t hat , although solutions 1 and 2 in our Don-

nan example contain C1-, the presence of thi s partic-

ular ion is not relevant to our reason for using 3 M

KCl; these bridges would be equally effective with

chloride-free solutions. Also note th at since the metal

Ag/AgCI, ,Ag/AgCI

rne m i

one

FIG. 9. Method of measuring membrane potential by making

contact with the solutions through

3

M

KCI

junctions.


10/53

170

HANDBOOK

O F

PHYSIOLOGY HE NERVOUS SYSTEM I

electrodes are in identical solutions, any identical

pair of electrodes in theory could be used (even stain-

less-steel wires). For practical reasons of stability and

convenience, the most commonly used electrodes ar e

either Ag-AgC1 or calomel (Hg-Hg,Cl,).

It should also be clear that the measurement, by

use of the arrangement shown in Figure

9,

of a finite

potential difference for a Donnan equilibrium is not a

violation of the second law, since with the salt

bridges present, the complete system of membrane

and

electrodes is not in equilibrium.

Finally, we must warn the reader who wishes to

pursue th is subject

of

the measurement of membrane

potentials fur ther tha t he will come across the view of

certain purists who claim that, since one can only

measure ~ l , l , , l , , , ~ , . , l l , , . through some such artifice as the

introduction of liquid junctions, it is not sensible to

even talk about membrane potentials. That is, it

is

impossible to assign values to the individual te rms in

Equation

35,

and hence one can only talk about

*,)

),,,,,

In fact, there are some ingenious argu-

ments that prove that most of the measured Don-

nan potential occurs not a t the membrane but a t the

contact between salt bridges and solution. It is not

our purpose to contribute to these polemics. We hope

that the Poisson-Boltzmann analysis of the Donnan

equilibrium will convince anyone th at indeed there is

a membrane potential intrinsic to that system and

that our treatment of the phase-boundary potential

(and other systems we shal l discuss shortly) will also

convince the skeptic of the reality of membrane po-

tentials. It might also be germane to point out that if

electrophysiologists had taken the purists critique of

membrane potentials seriously, the subjects of elec-

trical excitability, receptor potentials, and postsyn-

aptic potentials would never have gotten

off

the

ground, and most neurophysiologists today would be

out of business.

Quasi-equilibrium S,ystems

Before we take up the problem of diffusion poten-

tials involving the flux equations, we consider two

nonequilibrium systems t ha t a re sufficiently close to

equilibrium that they can be treated with good accu-

racy by the methods already employed. The consider-

ations developed here a re particularly relevant to our

future discussions of bilayer membranes.

SITY. uppose tha t we modify the Donnan system in

Figure

1 so

that the solution with the macro-ion

separates

two

solutions of NaCl (Fig. 10). Further-

more, let the macro-ion, N, be uniformly distributed

and immobile (as in the ion-exchange resin we dis-

cussed previously (see subsection

Statistical mechan-

ical analysis (Poisson-Boltzmann equation).

Then

the middle compartment (m) of Figure

10

is a mem-

brane (in point

of

fact, an ion-exchange membrane)

separating solutions 1and 2 . Let us also assume that

A

MEMBRANE

WITH A

V E R Y LARGE FIXED-CHARGE DEN-

the concentration of fixed charge is large compared to

the concentration of NaCl in the two compartments;

that

is,

[Nl

+

[NaCI],, [NaCll,. If the membrane

thickness is large compared to the Debye length in m,

then with solutions of equal concentration on the two

sides, there exist two Donnan potentials of the same

magnitude,

as

shown in Figure

1 l A

(where for the

sake of concreteness we take z = + l ) . igure 1 l A is

simply a symmetrical duplication of Figure 3 . At each

interface, there is a large Donnan potential jump

between solution and membrane, but there is no

potential difference across the membrane, because

these two jumps are equal.

Now consider the case in which th e concentrations

of NaCl in solutions

1

and 2 are not equal (e.g.,

[NaCll, < [NaCl],). This system is no longer in equi-

librium, and NaCl will diffuse slowly from solution

1

to 2; within the membrane there is a concentration

1 ) m ( 2 )

FIG. 10. A n ion-exchange membrane separating two solu-

tions.

Concentra t ion

I /

/i

P o t e n t i a l

p r o f i l e

63

\ C o n c e n t r a t i o n

I I p r o f i l e s

w

Pot en i a I

o r o f i l e

= O i

1)

( m )

2

FIG. 11.

A:

concentration and potential profiles for an ion-

exchange membrane of large positive fixed-charge density sepa-

rati ng two solutions of equal concentration of NaCI. B : same as

A , except tha t NaCl concentrations in compartments 1 and

2

a r e

unequal. (T he Na and C1- profiles within the m embrane have a

small negative slope tha t is not clearly seen in the figure.)


11/53

CHAPTER

6: PHYSICAL PRINCIPLES AND

FORMALISMS

OF ELECTRICAL

EXCITABILITY

171

gradient of Na+ an d C1- -the concentration profiles

within the membrane have a finite slope. However,

the high concentration of fixed charge, N +, permits

only a very small concentration of Na+ in th e mem-

brane; thus the salt concentration gradient will be

quite shallow, and diffusion of NaCl across the mem-

brane will be very slow indeed. To a

first

approxima-

tion we can therefore neglect th e finite slopes within

the membrane and the slow flux of NaCl across the

membrane, and tr eat t he system by our equilibrium

methods. Thus, as in the case when the concentra-

tions of NaCl in solutions

1

and

2

were equal, we

again have two Donnan equilibria. This time, how-

ever, they are not identical, and the re exists across

the membrane a potential difference (Fig. 11B).

For highly charged membranes, the counter ion is

virtually the only mobile ion present within the

membrane. Thus the membrane is "permselective"

for C1- (if z = -1, it is permselective for Na+). Note

th at this permselectivity arises purely from the Don-

nan effect and is not dependent on an y steric factors.

On the basis of th e permselectivity for C1-, we can

immediately calculate the membrane potential from

our thermodynamic equilibrium criterion

k T ( 1 )

= k T ( 2 )

pi ;' +

RT

In [Cl-I, - FICII

= & ;)-

+

R T In [Cl-I, - FI, I~

If instead the membrane contained a negative fixed

charge,

it

would

be

permselective for Na+,and by the

same argument as above we would obtain

R T "af],

Y = + - l n -

F "a+],

In general, for a membrane that is permselective for

an ion, i, of valence z (regardless

of

the mechanism

for the permselectivity), we obtain, by equating the

electrochemical potential of the ion on the two sides

of the membrane

(37)

Equation

37

is often called the

Nerns t

equation, and

v'

is called the Nerns t potent ia l .

It is instructive to see how the membrane potential

can also be derived from the algebraic sum of the two

Donnan potentials. Since N+

s

very large, within the

membrane [Cl-1,

--

"+I. The Donnan potential be-

tween solution

1

and th e membrane

(m)

is

and between solution 2 and the membrane

Combining we have

which is Equation 36. Thus the transmembrane po-

tential is made up of the difference between two large

Donnan potentials.

We have been assuming that the membrane thick-

ness is large compared to the Debye length within it ,

and we have therefore been able to draw the concen-

tration and potential profiles as in Figure

11,

without

worrying about the very th in space-charge regions. If

the membrane thickness and the Debye length were

comparable, the continuous profiles in the space-

charge regions would have to be explicitly calculated.

Also it would now be meaningless to speak of the

transmembrane potential as the sum of two Donnan

potentials

at

each interface, since the distinction be-

tween interface and electroneutral membrane inte-

rior no longer exists; the space-charge region extends

throughout the enti re membrane. To explicitly calcu-

late th e membrane potential would require t he solu-

tion of the Poisson-Boltzmann equation. It turns out,

however, that even in such a thin membrane, vir-

tually the only mobile ion present is C1-, for which

the thin membrane is still permselective. Thus our

equilibrium results are still applicable, and the mem-

brane potential will still be the Nernst potential for

c1-.

AN

OIL MEMBRANE.

Let us now, in the same way th at

we extended the Donnan system of Figure

1

o make

an ion-exchange membrane in Figure 10, extend the

water-oil system of Figure 5 to make an oil mem-

brane bounded by water phases

as

in Figure 12.

Suppose, to start with, that the membrane

is

thick

compared to

the

Debye length within it and that

the

NaCl concentrations on the two sides differ. If we

assume tha t the partition coefficients of one or both of

the ions is very small, then again we can neglect the

small gradients of NaCl within the membrane and

the slow flux of NaCl across the membrane and

treat

the system

as

being

at

equilibrium. The concentra-

tion and potential profiles are shown in Figure 13,

where for the sake of concreteness we have made the

Na+ partition coefficient considerably larger tha n

th at of C1-. We see that there are two large positive

(H,O)

; (o i l ) ;

H,O)

1) m ) ( 2 )

FIG. 12.

An oil membrane separating two NaCl aqueous solu-

tions,


12/53

172 H ANDBO O K O F

PHYSIOLOGY

- THE NERVOUS SYSTEM I

Not

1 - 1 -

I

2

I

I

Concentration

:

( o i l )

I

Na+.CI- prof I

les

I +m I

I I p r o f i l e

-

Potential

I

1

I I

I I

I

F IG . 13 .

Concentration and potential profiles for a thick oil

membrane separating two NaCl aqueous solutions. The potential

within the membrane

($,)

is positive, because we have assumed

that N a+ partitions better into the membrane than C1-.

phase-boundary potentials, but they are equal and

given by Equation

32

As we pointed out earli er, the phase-boundary poten-

tial is not

a

function of the NaCl concentration in the

aqueous phase. Even though N a+ is much more fa-

vored in the oil phase than C1- and consequently

there are large phase-boundary potentials, the total

membrane potential is zero. This result contrasts

sharply with that for the high-density fixed-charge

membrane we discussed earlier,

across which the

Nernst potential appears.

The above analysis was predicated on the mem-

brane being thick compared to the Debye length

within it. Consider now a membrane of thickness

comparable with t he Debye length. Instead of Figure

13,

we must now draw the complete ionic profiles

including the space-charge regions, since there is no

electroneutral region within the membrane. The pro-

files in Figure 14 ar e an extension of those in Figure

7 . For comparison we have redrawn Figure 13 in

Figure

14A,

exaggerating the space-charge regions

for

a

thick membrane; in Figure 14B we have ex-

panded the scale, since the space-charge regions ex-

tend through the enti re membrane.

To calculate the membrane potential we cannot use

Equation

32,

but must solve the Poisson-Boltzmann

equation. By inspection of the profiles, however, we

see

that Na+

is

virtually the only ion in the mem-

brane; that is, the membrane is permselective for

Na+. Consequently the membrane potential mus t be

given by the Nernst potential for Na+

This is true provided the mobilities of Na+ and C1- in the

membrane are equal. If they are not, then,

as

we shall see in a

later section, there will be a diffusion potential due to the asym-

metry in mobility. Nevertheless our main conclusion continues to

hold; namely, the phase-boundary potentials do not contribute to

the membrane potential.

Thus for the thick membrane (membrane thickness

* LJ

the membrane potential is zero, whereas for

the th in membrane (membrane thickness c , ;

if

c ,

> c p ,V,, the potential a t

I =

0 )

would change sign. T hat is, th e plots for

u

>

u

and

u < u

would

be interchanged.

current-voltage characteristic of thi s circuit for u

=

u,

u

> u, and u oes

change during

the

tran-

FIG.

21. Concentration profiles within the membrane for the

bi-ionic case of a membrane separating 0.1

M

NaCl from 0.1

M

KCI. A , I = 0; B , I is large and positive; C, I is large and negative.

10.1M NaCl

? I

FIG. 22. Sketch of the steady-state current-voltage character-

istic for the bionic case of Fig. 21. Note the characteristics(dashed

l ines) for a membrane separating either symmetrical 0.1 M NaCl

solutions or symmetrical 0.1 M KCl solutions, which the bi-ionic

characteristic approaches asymptotically for large positive and

negative potentials, respectively. Also shown are the chord and

slope resistance lines at point P .


18/53

178 HANDBOOK OF

PHYSIOLOGY

- THE

NERVOUS

SYSTEM I

T

t I

F I G . 2 3 .

Nonlinear steady-state current-voltage characteristic

with chords drawn at

I , ,

9,)nd

I x ,

qJ. f current is stepped

fro ml, toZ,, the voltage originally attains the value a t P and then

increases with ti me along the vertical line from

P

until it reaches

the value q 2 . imilarly, if curre nt is stepped from I , to I,, the

voltage originally attains the value at Q and then moves along

the vertical line from Q until it reaches the value 9,.

sient from one steady state to another (81.1

The current-voltage characteristic in Figure 22 also

illustrates several concepts tha t will be important for

our later discussion of excitable systems: the system

showsrectification; tha t is, for voltages (i.e., values of

q

-

of the same magnitude but opposite sign, the

(absolute values of the) currents a re

not

equal. Also,

as for any circuit element with a nonlinear current-

voltage characteristic, the term

resistance

is ambigu-

ous and has a t least two reasonable meanings. One is

the chord resistance given by

(58)

In this case, since

\If,,

s not voltage dependent, the

chord resistance is identical to the integral resist-

ance. Another equally valid meaning for resistance is

the

slope resistance

given by

(59)

Both of these terms are illustrated in Figure

22

at a

given value

of

q.

n discussing membrane resistance

for a nonlinear system, one must always specify the

resistance referred to.

Of

course,

for

linear current-

voltage characteristics, such as in Figure 19, the two

resistances are identical, and no distinction need be

made.

It is very instructive to consider how the system

passes i n t im e from one steady-state point to another

(Fig. 23, which is a redrawing of the characteristic in

Fig. 22). Suppose we are sitting at the point ( I I ,PI)

and suddenly step the current up to I,. The ins tanta-

neous voltage will be given by I , times the chord

resistance at

I , ,

since the ionic profiles have not had

time

to

change. As they rearrange themselves, the

resistance rises (Na+

is

replacing K + in the mem-

brane) and the voltage increases until it attains the

steady-state value for I,. Similarly, if the current is

suddenly stepped from

I ,

to

14,

he instantaneous

voltage will be given by

I ,

times the chord resistance

at

I,,

and then the (absolute value of the ) voltage will

decrease

as the ionic profiles rearrange. The re-

sponses are schematized in Figure 24.

We see that the membrane resistance is not only

nonlinear, it is also

t ime uariant .

The time depend-

ence enters because of the time required for the ionic

profiles

to

shift from one steady-state configuration

to

another.

As

the profiles are shifting, the integral

resistance is continuously changing. (Y,, is also con-

tinuously changing, but we are neglecting that in

this discussion.) Note also th at the response in Figure

24A is phenomenologically similar to the response of

the RC network in Figure 25A, whereas the re-

sponse in Figure

24B

is phenomenologically similar

to

the response of the RL network in Figure 2%. It

can be shown formally tha t the AC-impedance char-

acteristics of time-variant resistances can display

phenomenological capacitances and inductances (42).

To

summarize the time-variant aspect of the system:

the instantaneous response to a change in current (or

voltage) is along the chord resistance; the voltage (or

1

A

I

F I G . 24. The change of voltage with time in response to steps

of cur ren t. These responses follow from the a nalysis of Fig. 23 as

given in the legend and text.

P ?

FIG. 25. RC

( A )

and RL ( B ) networks that would give re-

sponses similar to those shown in Fig. 24.


19/53

CHAPTER 6: PHYSICAL PRINCIPLES AND FORMALISMS O F

ELECTRICAL EXCITABILITY

179

current) then relaxes to the point on the steady-state

current-voltage characteristic appropriate for the

new current (or voltage).

We have been discussing th e behavior of a membrane

whose only property is tha t

it

allows stir ring on both

sides for maintenance of boundary conditions; other-

wise the membrane imposes no new restrictions on

ion movement. Our real example has been coarse

filter paper. We now wish to consider two examples

in which the membrane phase is not merely a con-

fined salt solution of composition grading between

that of the two aqueous phases bathing the mem-

brane. After discussing the two examples, we shall

comment on the general situation of membranes with

special properties.

Fixed-charge membrane. We have already consid-

ered the quasi-equilibrium situation th at arises when

the fixed-charge density of

a

membrane is very large

and the membrane separates a single salt at two

different concentrations. We now extend the treat-

ment of fixed-charge membranes to the case of lower

charge densities. Since the co-ion is no longer ex-

cluded, transport (when

Z

= 0) of ions and sa lt cannot

be neglected. (Even in membranes of large fixed-

charge density, significant fluxes of counter ions oc-

cur; e.g., for the bi-ionic case of NaCl vs. KCl sepa-

rated by

a

high-density negatively charged mem-

brane, there is, a t

Z

= 0, a flux of Na+ from compart-

ment

1

o compartment 2, and an equal and opposite

flux of K+ from

2

to

1.

Our treatment will of course

also apply to these cases.) We do not go through the

details of the analysis, but merely point out the high-

lights of the principles involved.

The general analysis of fixed-charge membranes is

a straightforward combination of the Donnan equilib-

rium and the flux equations

(71). It

is assumed that

the interfacial processes are rapid compared to trans-

port through the membrane interior; hence one as-

sumes that equilibrium is maintained a t each inter-

face a t al l times (even in the face of curren t flow). For

the fixed-charge membrane, this means that the

Donnan equilibrium is satisfied at each interface.

Thus th e concentrations of ions just within the mem-

brane satisfy the Donnan condition with respect to

the ions just outside the membrane. Subject to these

new boundary concentrations, the ions then move

according to the flux equations. Essentially then the

fixed charge has established new boundary condi-

tions for the Nernst-Planck regime of ions. Current

flow will shift the concentration profiles within the

interior of the membrane, but

it

is assumed th at th e

concentrations Ijust inside the membrane are not

perturbed by the current. Figure 26A illustrates the

concentration profiles for the single-salt case in a

positive fixed-charge membrane. If +I = 0, the

system reduces to the single-salt case for an un-

charged membrane, and if +I % [NaCll,,,, the sys-

HOMOGENEOUS MEMBRANES WITH SPECIAL PROPERTIES.

I

I

( 1 ) ( m )

( 2 )

FIG.

26.

Concentration profiles (A ) and potential profile ( B ) or

a positive fixed-charge membrane separating NaCl solutions at

different concentrations. Note the Donnan

jumps

in concentra-

tions and potential at the interfaces.

tem approaches the quasi-equilibrium case of a mem-

brane with a very large fixed-charge density, de-

scribed earlier. Note tha t in th is lat ter case the mem-

brane is almost exclusively permeable to C1- not

because the mobility of chloride in the membrane is

so much larger tha n th at of sodium, but because the

concentration of chloride is so much greater than that

of sodium. This illustrates a n important general prin-

ciple applicable to both electrolytes and nonelectro-

lytes: the permeability of a membrane for

a

species is

generally dependent on two factors. One is the mobil-

ity of the species in the membrane phase, and the

other is the ability of the species to enter the mem-

brane in the first place. Unless one has other infor-

mation, i t is impossible to tell which is responsible for

a

given molecule being poorly permeable.

For a fixed-charge membrane, the total membrane

potential

is

the algebraic sum of three terms: two

Donnan potentials at the interfaces, generally called

rr, and

rr2,

and a Nernst-Planck diffusion potential

within the membrane, called (GI, - 2m). Thus

(

60)

(The potential profile is illustrated for the single-salt

case in Fig. 26B.) To find - 2m) one must solve

the flux equations subject to the new boundary condi-

tions established by the Donnan equilibria at the

interfaces

(71).

The electroneutrality condition

within the membrane now takes the form

C

cj+ +

ZN -

X C ~ -

0 (61)

Note tha t for the bi-ionic case, the Donnan potentials

ml nd

rr2

are equal an d hence cancel, so that the

membrane potential in Equation 60 is given just by

the internal diffusion potential -

Oil membrane. The analysis of th is system is com-

pletely analogous to that of the fixed-charge mem-

brane, but the equilibria a t the boundaries are now

partition equilibria, not Donnan equilibria. Subject

to the new concentrations established just within th e

membrane, the ions once again diffuse according to

the Nernst-Planck flux equations. (The concentration

profile for the single-salt case is illustrated in Fig.

27.) Once again the membrane potential is the alge-

braic sum of three potentials: two phase-boundary

potentials and an inte rnal diffusion potential. Thus

9 = T(o/u)a +

T( u /o) z

+ ( I m - 2m)

(62)

= T +

~2

+ ( 1,

-

2m)


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180

HANDBOOK

O F PHYSIOLOGY

THE

NERVOUS SYSTEM I

As we pointed out earlier (see subsection

Quasi-egui-

Zibrium Systems.

AN OIL

MEMBRANE),

the phase-

boundary potentials are equal for the singe-salt case

and hence cancel. Thus, despite the relative solubili-

ties of the anion and cation in the oil phase, the

membrane potential will only be a function of their

relative mobilities in this phase. Therefore, for the

single-salt case, Equation 62 reduces to

u - u R T cZrn

In

J

=--

u + u F c,,,

On the other hand, the phase-boundary potentials

will in general be different for the bi-ionic case, and

all three terms in Equation 62 will contribute to

II/.

General considerations.

The two examples we have

jus t given illustrate the general approach taken with

membrane transport. One assumes that the mem-

brane can conceptually be divided into three regions:

two interfacial regions and an interior region. It is

assumed that interfacial processes are

so

fast that

equilibrium conditions prevail there. These processes

establish new boundary conditions Ijust within the

membrane, and transport then takes place through

the membrane interior according to the flux equa-

tions. (This analysis incidentally is equally applica-

ble to nonelectrolyte transport.) The membrane po-

tential is the algebraic sum of two equilibrium poten-

tia ls and one diffusion potential. Since the membrane

substance is discontinuous at the interface, there

exist discontinuities in concentrations and electrical

potential at each interface, but the assumption of

equilibrium is equivalent to assuming th at the elec-

trochemical potential of each species is nevertheless

continuous across the interface. [For example, in a

fixed-charge membrane there are jumps in concen-

tration and electrical potential at the interface (see

Fig. 261, but the electrochemical potential of each ion

on the two sides of the interface is the same. In fact,

these characteristic discontinuities of the Donnan

equilibrium were derived on the basis of equality of

electrochemical potential.] Of course, one could be

more sophisticated and expand these discontinuities

into the actual space-charge regions present at each

interface.

There are two points worth noting about this anal-

ysis of membrane transport and potentials. Fi rst , it is

possible that situations might be found where the

assumption of very fast interfacial processes relative

I C 1 n I

1 )

m )

( 2 )

FIG.

27. Concentration profile

for

an

oil

membrane separat-

ing a single salt at t w o different concentrations.

to diffusion within the membrane begins to break

down. It would th en no longer be possible to assume

equilibrium a t the boundaries, and a n explicit analy -

sis of the boundary kinetics would have to be included

in the overall transport. The previous type of analy-

sis, however, still gives a good qualitative picture

even in this instance. For example, suppose that

interfacial events were not fast enough to maintain

the Donnan conditions. Nevertheless the

sign

of the

change of ion concentrations across the interface can

be obtained from the Donnan analysis. That is, for a

positively fixed-charge membrane, t he anion concen-

tration will be elevated across the interface and the

cation concentration depressed. The interfacial po-

tentials will then be somewhat reduced from their

equilibrium values.

The second point is that the analysis presupposes

the possibility

of

dividing the membrane up into in-

terfacial regions and interior. Clearly this becomes a

very tenuous assumption when the membrane

is

of

the order of

50-p\

thickness, as with th e lipid bilayers.

In fact, the analysis breaks down.

It

is no longer

possible to break up the membrane potential as in

Equation 62, since the space-charge regions extend

throughout the membrane. Here, a n explicit analysis

combining the flux equations and Poissons equation

with the appropriate boundary conditions must be

used. It is interesting that for nonelectrolyte trans-

port across the bilayers (e.g., isotopic water flux), the

procedure of dividing the membrane into three re-

gions leads to predictions in good agreement with

experimental results

(18).

It appears that partition

equilibrium is attained rapidly with respect to diffu-

sion even through a very thin

(-50

A)

region.

MOSAIC MEMBRANES.e have been considering mem-

branes whose properties do not vary in a plane paral-

lel to the membrane surfaces; tha t

is,

the membranes

are homogeneous. We now wish to comment on mo-

saic membranes consisting of regions with different

permeability characteristics. For example, the mem-

brane may contain some regions permselective for

Na +, other regions permselective for

K + ,

and still

others relatively nonselective among univalent ions.

As we shall see later, such membranes are of direct

relevance to biological membranes, particularly ex-

citable membranes, and can be experimentally real-

ized with modified thin lipid membranes. At this

point we are not concerned with t he mechanisms for

the selectivities, but accept them as given. (If the

membrane consisted of cation and anion selective

regions, we could imagine that each region was a

patch of high fixed-charge density ion-exchange

resin; such membranes can be fabricated with ion-

exchange beads imbedded in an inert matrix .)

For a membrane with regions in parallel,

the

equivalent circuit has the form shown in Figure 28.

(For completeness we have included the membrane

capacitance, which we discuss in a later section.)


21/53

CHAPT E R 6:

PHYSICAL PRINCIPLES

AND FORMALIS MS OF ELECTRICAL EXCITABILITY 181

I

- a

'.i

7

t

t 1

1

FIG

28.

Equivalent circuit for

a

mosaic membrane. The indi-

vidual elements are shown to

be

ideally selective

for

a given ion,

but this need not necessarily be the case.

Here we have indicated that in principle the conduct-

ances (g,)can be voltage dependent. (In treating par-

allel circuits, it is more convenient to use conduct-

ances than resistances.) The emfs , however, are in-

variant, being determined by the concentrations of

ions on the two sides of the membrane and the perme-

ability characteristics of the regions. If the regions

are permselective, as in Figure 28, each emf is the

Nernst potential for the permeant ion of tha t region.

This circuit should be contrasted to the one for the

homogeneous membrane given in Figure 16. In that

case there is a single conductance and one emf, both

of which can be voltage dependent. For the mosaic

membrane, we have from elementary circuit theory

z,

= gj 9

Ej)

(63)

for each element of the circuit. Summing over all

elements, we have from Kirchhoff

s

law

where

z = T Z I (65)

Equation 64 bears the same relation to the mosaic

membrane as Equation 38 bears to the homogeneous

membrane. Comparing the two, we see that each

equation has an Ohm's law term and also another

term that may be called the diffusion potential. In a

formal sense the two expressions are quite similar.

Although the flux equations describe the move-

ment of ions across the membrane, the equivalent

circuit of Figure 16 does not indicate that there is ion

flux when Z

=

0. In contrast, the equivalent circuit of

Figure 28 depicts the local currents th at flow through

the elements of the membrane even when no current,

I ,

is being passed across the membrane. Under these

circumstances, the membrane potential is given ex-

clusively by the second term in Equation 64, whereas

Equation 65 becomes

zzj = 0

j

It

should be understood that the dissipative pro-

cesses for the homogeneous and mosaic membranes

are quite different. In the former the ions flow

through a common region, and hence there are no

local current flows. In the mosaic membrane ion

movement is through local currents. Despite these

physical differences,

it

happens tha t the

steady-state

properties of a homogeneous membrane can also be

formally represented by the circuit in Figure

28 (20).

FORMAL CONSEQUENCES OF VOLTAGE-

DEPENDENT CONDUCTANCES

The N a ture

of

Electrical Excitability

In

the

subsection

Ion Transport (the Nern st-

Planck Flux Equations) we developed the properties

of a homogeneous membrane and showed that even

with such a simple membrane as coarse filter paper,

it

is

possible to observe rectification, nonlinearities,

and time transien ts in th e membrane potential. Since

these properties are also characteristic of electrically

excitable biological membranes, the question arises

whether the mechanisms operating in the simple

systems we discussed are also responsible for gener-

ating action potentials and bioelectric phenomena.

We remind the reader th at the nonohmic behavior we

have described is due to the shifting of the ionic

profiles of mobile ions within the membrane in the

face of an applied voltage or current; this change in

the ionic profiles leads to changes in the membrane

conductance and the membrane emf. The question

then is whether such a mechanism could account for

action potentials of nerve and muscle.

It

is very difficult to give a

definitive

answer to this

question. It can probably be proved that

it

is impossi-

ble to reproduce electrical excitability with a filter

paper membrane, even with the most bizarre mixture

of ions on the two sides of the membrane. This does

not preclude the possibility that a membrane with a

fixed-charge density varying as arc coth x3", or one

with some continuously varying function of the die-

lectric constant of

its

''oil" interior, or a combination

of these st ruc tures could not generate action poten-

tials. Although no one has demonstrated, either theo-

retically or experimentally, that one can develop an

excitable membrane simply from electrophoretically

moving ionic profiles within the membrane, neither

has anyone presented proof that it is impossible to do

so. [In fact, if one allows the ionic profiles to be

shifted not only by the electric field (i.e., electropho-

resis) but also by solvent drag produced by electro-

osmosis and hydrostatic pressure gradients, then one

can indeed develop, with a membrane made merely of

sintered glass, oscillatory behavior and complex all-

or-none responses th at ar e in some ways phenomeno-

logically similar to excitable tissue (72, 73).] There

are nevertheless strong indications for believing th at

such

a

mechanism cannot be the basis for biological


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182

HANDBOOK

O F PHYSIOLOGY

-

THE NERVOUS

SYSTEM I

excitability, and indeed there is compelling evidence

that excitable membranes are not homogeneous, but

rather mosaic structures. We shall discuss the bases

for the above assertions and a t the same time present

the current view of the na ture of biological excitable

systems.

REASONS F O R

BELIEVING

THAT ELECTRICAL EXCITAB IG

ITY

DOES NOT RESULT

FROM THE SHIFTING

OF IONIC

PROFILES. Th e magn i tude of rect if ication. In excitable

cells such as the squid giant axon, the ratio of the

conductance a t large depolarizations to the conduct-

ance at large hyperpolarizations can be several

hundred to one. Since the major conducting ions are

Na+ (on the outside) and

K +

(in the axoplasm), with

approximately equal total sal t concentration on each

side of the membrane, the situat ion is approximately

bi-ionic. In such a case, rectification is achieved by

exchanging in the membrane a more mobile ion with

a less mobile ion, depending on the sign of the poten-

tial. Assuming tha t the contribution of all other ions

to the membrane conductance can be neglected (an

extreme case), the limiting rectification ratio is given

by the ratio of the sodium and potassium mobilities in

the membrane (i.e.,

u K / u N a ) ,

hich is approximately

1.5 in free aqueous solution.

If

this phenomenon ac-

counted for the observed rectification ratio of several

hundred, either the potassium mobility within the

membrane must be abnormally large, or the sodium

mobility must be abnormally small, or both.

Th e t ime scale of the a ct ion potent ia l . In our discus-

sion of trans ien t behavior during ionic profile shifts,

we omitted the time scale. Since the relaxation of the

profiles is basically a diffusion process, the times

involved will be roughly those required for the root

mean square displacement

(?)I/*

of an ion t o be equal

to the membrane thickness. This

is

given by the well-

known result from Brownian motion theory

(13,

14)

x2 = 2Dt (66)

In aqueous solution, the diffusion coefficient, D,

for

small ions such as Na+ and K+ is of the order

cm2/s. Taking the membrane thickness a s

100

A

(lo- ';

cm), we find from Equation

66

that the time involved

for the displacement of concentration profiles is of the

order of s, that

is,

0.1 p s . The events associated

with a n action potential, occurring

as

they do on the

millisecond time scale, are four orders of magnitude

slower. We would therefore have to postulate that t he

diffusion constants (i .e. , mobilities) of ions within the

membrane are at least a factor of

lo4

times smaller

th an t he mobilities of ions in free solution.

Dissociation

of

sod ium and po tass ium conduc t -

ances. One of the distinguishing features of the

permeability changes occurring during a n action po-

tential (or under voltrage-clamp conditions) is th e dif-

ferent time course of these changes for different ions.

In th e squid giant axon, for example, sodium permea-

bility rises and begins to fall before potassium perme-

-

ability changes significantly. In fact, Hodgkin and

Huxley's analysis of excitability in the giant axon

was possible only because they could clearly resolve

and separate the time course of the sodium and potas-

sium conductance changes (29). Since then, t he belief

that these components are separate has been rein-

forced with the discovery of pharmacological agents

that inhibit one, but not both, conductances. For

example, tetrodotoxin (TTX), the puffer fish poison,

can completely and reversibly block the sodium con-

ductance changes without affecting the magnitude

and the time course of the potassium conductance

transients (49). Conversely, tetraethylammonium

(TEA) reduces the potassium conductance without

significantly affecting the sodium transients (2) .

It is difficult t o imagine a homogeneous regime

of ions where th e permeability changes for one ion are

not intimately coupled with those of the other ions. In

the bi-ionic case tha t we considered, for example, th e

movement of Na+ into or ou t of th e membrane was

accompanied by the obligatory and opposite move-

ment of K + . It

is

even more difficult t o conceive of an

agent, acting in a homogeneous membrane, that

could inhibit transients associated with one ion and

yet not affect those associated with the other ions in

the membrane. For these reasons, virtually all elec-

trophysiologists believe that excitable membranes

are mosaic structures of the form described in the

subsection

MOSAIC

MEMBRANES, and although Hodg-

kin and Huxley did not explicitly stat e this in their

basic papers (29-331, they clearly had this picture in

mind. Certainly the simplest way in which the so-

dium and potassium conductances could be function-

ally independent is for the regions of sodium and

potassium permeability to be physically separated.

HODCKIN-HUXLEY EQUIVALENT CIRCUIT. nalysis of

electrical excitability of the squid giant axon mem-

brane, and any other excitable membrane, starts

with the equivalent circuit for a mosaic membrane of

the type shown in Figure 28. The ions involved in the

circuit can vary depending on the particular biologi-

cal membrane

o r

the ions in the solution that bathes

the membrane.

Because much is known about the

squid axon, we use this system as representative of

the others, with the understanding t hat the pr inc i -

ples operative there are generally believed to apply to

all excitable membranes. For the squid giant axon in

normal seawater , the equivalent circuit of Figure 28

takes the particular form shown in Figure 29 (32) .

The sodium and potassium conductances are voltage

dependent, whereas the leakage conductance (g,)-

probably a lumping together of several ion permea-

bilities- s constant. The sodium and potassium

emf's are,

of

course, given by the Nernst potentials

for these ions, and are assumed to be constant, be-

cause the concentrations of these ions in seawater

and axoplasm are constant (unde r the experimental

conditions). Since [Na+loutside [Na+linside nd


23/53

CHAPTER 6: PHYSICAL PRINCIPLES

AND

FORMALISMS OF ELECTRICAL EXCITABILITY

183

O u t s i d e ( s e a w a t e r )

c

I n s id e ( o x o p l a s m )

FIG

29. The Hodgkin-Huxley equivalent circuit for the squid

giant axon membrane in normal seawater.

[K+loutside < [K+linside>

ENa and EK are of opposite

polarity. Ultimately this system would run down

(Na+coming in and K + going out), unless some coun-

terbalancing process exists. This energy-requiring

process

is

the role assigned to metabolism. The leak-

age emf

(E,)s,

like the leakage conductance, proba-

bly a lumped parameter. Although the leakage ele-

ment modifies the behavior of the system somewhat,

the dominating factors are the voltage-dependent so-

dium and potassium conductance elements; for the

most part, therefore, we confine o u r comments to

these and ignore the leakage element.

The Hodgkin-Huxley description of the action po-

tent ial is briefly summarized as follows: in the rest-

ing state,

g,

>>gNa o tha t the resting potential sits

near EK.At threshold depolarization,

g N a

has in-

creased enough tha t the membrane furthe r depolar-

izes. This leads to a further increase in g N a , which

leads to further depolarization, and so on. The net

result of thi s process is the rising phase of the action

potential, which tends toward EN, a t

its

peak. The

falling phase results from a turning off of g,, (called

sodium inactivation) and a turning on of

g K ;

both

processes act to move the membrane potential back to

EK.

Regenerative behavior develops because the

membrane potential is determined by the relative

conductances (permeabilities) of the membrane to

potassium and sodium [if we neglect the capacitance

current, th e membrane potential is given at all times

by the last ter m in Equation

64,

which in this case is

(gNaEXa

+

g,EK)/(gNa

+

gK)],

while a t the same time

these conductances are functions of the membrane

potential. The functional dependence ofg,, and

g,

on

membrane potential is determined in voltage-clamp

experiments. Tha t is, the membrane potential is held

fixed (via external electrodes) at various levels and

the transient changes in the sodium and potassium

conductances recorded (29-31, 33).

The physics underlying t he voltage-dependent con-

ductances is still not understood; it

is

a major un-

solved problem in electrophysiolo

physical principles and formalisms of electrical excitability

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