7
Membrane Potential
The Resting Membrane Potential Results From the tances. These electrical signals-receptor potentials, Separationof ChargesAcross the Cell Membrane synaptic potentials, and action potentials-are all pro- The Resting Membrane Potential Is Determined by Resting duced by temporary changes in the current flow into Ion Channels and out of the cell that drive the electrical potential RestingChannels in Glial Cells Are Selectivefor acrossthe cell membrane away from its resting value. PotassiumOnly This current flow is controlled by ion channels in RestingChannels in Nerve Cells Are Selectivefor Several the cell membrane. We can distinguish two types of ion Ion Species channels-resting and gated-by their distinctive roles PassiveFlux of Sodium and PotassiumIs Balancedby in neuronal signaling. Resting channels normally are Active Pumping of the Ions open and are not influenced significantly by extrinsic Chloride Ions May Be PassivelyDistributed factors, such as the potential acrossthe membrane.They are primarily important in maintaining the resting The Balanceof Ion Fluxes That Gives Rise to the Resting Membrane Potential Is Abolished During the Action membrane potential, the electrical potential across the Potential membrane in the absenceof signaling. Most gated chan- The Contributions of Different Ions to the Resting nels, in contrast, are closed when the membrane is at Membrane Potential Can Be Quantified by the Goldman rest. Their probability of opening is regulated by the Equation three factors we considered in the last chapter: changes The Functional Properties of the Neuron Can Be Represented in membrane potential, ligand binding, or membrane in an Electrical Equivalent Circuit stretch. EachIon ChannelActs as a Conductor and Battery in In this and succeeding chapters we consider how Parallel transient electrical signals are generated in the neuron. We begin by discussing how resting ion channels estab- An Equivalent Circuit Model of the Membrane Includes Batteries,Conductors, a Capacitor, and a Current lish and maintain the resting potential. We also briefly Generator describe the mechanism by which the resting potential An Overall View can be perturbed, giving rise to transient electrical sig- nals such as the action potential. In Chapter 8 we shall consider how the passive properties of neurons-their resistive and capacitive characteristics-contribute to local signaling within the neuron. In Chapter 9 we shall examine how voltage-gated Na +, K+, and Ca2+ chan- I NFORMATION IS CARRIED WITHIN and between neurons nels generate the action potential, the electrical signal by electrical and chemical signals. Transient electri- conveyed along the axon. Synaptic and receptor poten- cal signals are particularly important for carrying tials are considered in Chapters 10-13 in the context of time-sensitive information rapidly and over long dis- synaptic signaling between neurons. 126 Part II / Cell and Molecular Biology of the Neuron
The Resting Membrane Potential Results 1 From the Separation of Charges Across ... (j (j (j.. Equal . (j (I (j . the Cell Membrane +,- IleG) . . . . (it(j.... . Every neuron has a separation of charges across its cell Extr8cellular 8
[embrane Potential g the electrical potential across inside of the membrane becomesmore positive while the out- l in the late 19405.These tech- side of the membrane becomes more negative. This progres- ~ of both the resting and the sive decreasein the normal separation of charge is called dep0- ! of glass micropipettes filled larization. ion that serve as electrodes. :ed on either side of the cell the back ends of the pipettes to an oscilloscope, which dis- nbrane potential in volts. Be- microelectrode is very small l cell with relatively little dam-
:~ .n.ILJL~~ I--t Inwerd TIme- 5Oms
Figure 7-2C Depolarization.
Small depolarizing current pulses evoke purely electro- tonic (passive) potentials in the cell-the size of the change in potential is proportional to the size of the current pulses. How- NeMIcell ever, sufficiently large depolarizing current triggers the open- ing of voltage-gated ion channels. The opening of these chan- ecording setup. nels leads to the action potential, which differs from electrotonic potentials not only in the way in which it is gener- Itside the cell no electrical p0- ated but also in magnitude and duration. ut as soon as one microelec- Reversing the direction of current flow-making the in- ~ oscilloscope shows a steady tracellular electrode negative with respect to the extracellular otential. In most nerve cells at electrode-makes the membrane potential more negative. round -65 mY. This increasein charge separation is called hyperpolilrization.
,Iectrode I lOng potential ':l me- oscope display. n be experimentally changed cted to a second pair of elec- )ne extracellular. When the Figure 7-2D Hyperpolarization. c>sitivewith respect to the ex- re current from the generator The responsesof the cell to hyperpolarization are usually Ie neuron from the intracellu- purely electrotonic-as the size of the current pulse increases, , to the extracellular electrode the hyperpolarization increasesproportionately. Hyperpolar- membrane. As a result, the ization does not trigger an active responsein the cell. 128 Part n / Cell and Molecular Biology of the Neuron
Table 7-1 Distributionof the Major Ions Acrossa NeuronalMembrane at Rest:the GiantAxon of the Squid
Concentration Concentration in Equilibrium Species of in cytoplasm extracellular fluid potential1 ion (mM) (mM) (mV)
-75 +55 -60
the passive (resting) channels? These questions are The separation of charge resulting from the diffusion of interrelated, and we shall answer them by considering K+ gives rise to an electrical potential difference: posi- two examples of membrane permeability: the resting tive outside, negative inside. The more K+ continues to membrane of glial cells, which is permeable to only one flow, the more charge will be separated and the greater speciesof ions, and the resting membrane of nerve cells, will be the potential difference. Since K+ is positively which is permeable to three. For the purposes of this charged, this potential difference tends to oppose the discussion we shall consider only the resting ion chan- further efflux of K+. Thus, ions are subject to two forces nels, which are always open. driving them acrossthe membrane: (1) a chemicaldriving force that depends on the concentration gradient across the membrane and (2) an electricaldriving force that de- Resting Channels in Glial Cells Are Selective pends on the electrical potential difference across the for Potassium Only membrane. Once K+ diffusion has proceeded to a cer- A membrane's overall selectivity for individual ion tain point, a potential develops acrossthe membrane at speciesis determined by the relative proportions of the which the electrical force driving K+ into the cell exactly various types of ion channels in the cell that are open. balances the chemical force driving K+ ions out of the The simplest case is that of the glial cell, which has a cell. That is, the outward movement of K+ (driven by its resting potential of about - 75 mV. Here,the vast major- concentration gradient) is equal to the inward move- ity of resting channels in the membrane are permeable ment of K+ (driven by the electrical potential difference only to K+ . As a result, the glial cell membrane at rest is acrossthe membrane). This potential is called the potas- almost exclusively permeable to K+ ions. A glial cell has sium equilibrium potential, EK (Figure 7-3). In a cell per- a high concentration of K+ and negatively charged or- meable only to K+ ions, EK determines the resting mem- ganic anions on the inside and a high concentration of brane potential, which in most glial cells is about Na+ and Cl- on the outside. -75mV. How do these ionic gradients generate the mem- The equilibrium potential for any ion X can be cal- brane potential of the glial cell? BecauseK+ ions are pre- culated from an equation derived in 1888 from basic sent at a high concentration inside the cell and glial cells thermodynamic principles by the German physical are selectively permeable to them, K+ ions tend to dif- chemist Walter Nernst: fuse from inside to outside the cell, down their chemical RT [XJo concentration gradient. As a result, the outside of the Ex = zF In [XJi' Nernst Equation membrane accumulates a positive charge (due to the slight excessof K+) and the inside a negative charge (be- where R is the gas constant, T the temperature (in de- causeof the deficit of K+ and the resulting slight excess grees Kelvin), z the valence of the ion, F the Faraday of anions). Since opposite charges attract each other, the constant, and [XJoand [XJ1are the concentrations of the excesspositive charges on the outside and the excess ion outside and inside of the cell. (To be precise,chemi- negative charges on the inside collect locally on either cal activities should be used rather than concentrations.) surface of the membrane (seeFigure 7-1), Since RT/F is 25 mY at 25°C (room temperature), The diffusion of K+ out of the cell is self-limiting. and the constant for converting from natural logarithms Chapter 7 / MembranePotential 129
A B "'+0 @ NI+O @ NI+o @ @ HI. NIb- 0
ExtnIoeIuI8r aide . + + Selectively permeebII membr8ne
CytopIumic (9 ' K-:- K+ Concentnltion I Iid8 . . Concentretlon A- gradientfa( K+' '.~.OK+ greclient for 1(+ (;;\ movesK+ out 0 moves 1(+out ~ of cell ofceU . K. "--" 8K+ 8K+ .Ie+ ~0 f;\ 8K+OK. OK. ~ \J c;)(9 (9 <9 Figure 7-3 The flux of K+ across the membrane is deter- charge on the outside of the cell and leaves behind on the in- mined by both the K+ concentration gradient and the elec- side an excess of negative charge.This buildup of charge leads trical potential across the membrane. to a potential difference across the membranethat impedes A. In 8 cell permeableonly to K+ the resting potential is gener- the further efflux of K+. so that eventually an equilibrium is ated by the efflux of K+ down its concentration gradient. reached:the electrical and chemical driving forces are equal and opposite, and as many K+ ions move in as move out. B. The continued efflux of K+ builds up an excess of positive
to base10 logarithms is 2.3, the Nernst equation can also Resting Channels in Nerve Cells Are Selective be written as: for Several Ion Species 58mV [XJ Measurements of the resting membrane potential with Ex = -;- log [XI]' intracellular electrodes and flux studies using radioac- tive tracers show that, unlike glial cells, nerve cells at Thus, for K+,., since z = + 1 and given the concentrations rest are permeableto Na+ and CI- ions in addition to inside and outside the squid axon in Table 7-1: K+ ions. Of the abundant ion speciesin nerve cells only the large organic anions (A -)-negatively charged pro- 58 mY log [20] = -75 mY. EIC= ~ [400] teins and amino acids-are unable to permeate the cell membrane. How can the concentration gradients for the The Nemst equation can be used to find the equili~ three penneant ions (Na+, K+, and CI-) be maintained rium potential of any ion that is present on both sides of across the membrane of a single cell, and how do these a membrane permeable to that ion (the potential is three gradients interact to determine the cell's resting sometimescalled the Nernst potentitU).The Na +, K+, and membrane potential? CI- equilibrium potentials for the distributions of ions To answer these questions, it will be easiest to ex- acrossthe squid axon are given in Table 7-1. amine first only the diffusion of K+ and Na +. Let us re- In our discussion so far we have treated the genera- turn to the simple example of a cell having only K+ tion of the resting potential by the diffusion of ions channels, with concentration gradients for K+, Na +, down their chemical gradients as a passive mechanism, a-, and A-as shown in Table 7-1. Under these condi- one that does not require the expenditure of energy by tions the resting membrane potential, Vr, is determined the cell, for example through hydrolysis of ATP. How- solely by the K+ concentration gradient and will be ever,as we shall seebelow, energy (and ATP hydrolysis) equal to Ek (-75 mY) (Figure 7-4A). is required to set up the initial concentration gradients Now considerwhat happensif a few resting Na+ and to maintain them during the activity of a neuron. channels are added to the membrane, making it slightly 130 Part n / Cell and Molecular Biology of the Neuron
A K+ driving NetdrMngforces Net currents forces Chern. Elec. K+ Na+ K+ Na+ - .~;
0
Na+ drivir)g B forces ~ ec. c~m. 0 0 0 Na+ . ,'.11 ..
~..- 0 0 0 00 0 K+000 0 00 0 . 0 0° 00 00
c 0 0 Na+o 0 0 0 0 9 0
":"" 0 00 0 0 t 00 0 0 0 K+O 0 0 °0 0 00
D E,.,
0 ~ rce on a particularion (that is, the sum of the electrical and hemical driving forces) and the relative sizes of the different et ion fluxes. Three hypotheticalsituations are illustrated. Vm ~. In a resting cell in which only K+ permeant channels are Vit Ifesent. K+ ions are in equilibrium and Vm= EKo Q.Adding a few Na+ channelsto the resting membraneat a fK given time allows Na+ ions to diffuse into the cell, and this in- flux begins to depolarizethe membrane. ~ The resting potential settles at a new resting potential, 'f'Inw~ hich is the value of Vm where IN. = - k In this example the Figure 7-4 The resting potential of a cell is determined by gregate conductance of the K+ channels is much greater the relative proportion of different types of ion channels t an that of the Na+ channels because the K+ channels are that are open, together with the value of their equilibrium ore numerous. As a result, a relatively small net driving force (Nemst) potentials. In this simplified diagram the channels ; r K+ ions drives a current equal and opposite to the Na+ cur- shown represent the entire complement of K+ or Na + channels r nt driven by the much larger net driving force for Na + ions. in the cell membrane. The lengths of the arrows within the is is a steady-state condition, in which neither Na+ nor K+ is channels represent the relative amplitudes of the electrical i~ equilibrium but the net flux of charge is null. (red) and chemical (blue) driving forces acting on Na+ and K+. Q. Illustration of membranevoltage changes during the hypo- The lengths of the arrows on the right denote the net driving thetical situations considered in A, B, and C. Chapter 7 / MembranePotential 131
permeableto Na +. Two forces act on Na + to drive it into the cell. First, Na + is more concentrated outside than in- maintained constant over time. That is, the influx of p0s- itive charge must be balanced by the efflux of positive side and therefore it tends to flow into the cell down its charge.If thesefluxes were not equal, the chargesepara- chemical concentration gradient. Second,Na + is driven tion across the membrane, and thus the membrane p0- into the cell by the negative electrical potential differ- tential, would vary continually. As we have seen, the ence across the membrane (Figure 7-48). The influx of passive movement of K+ out of the cell through resting positive charge (Na +) depoIarizes the cell, but only channelsbalances the passivemovement of Na+ into slightly from the K+ equilibrium potential (-75 mY). the cell. However, these steady ion leaks cannot be al- The new membrane potential does not come close to the lowed to continue unopposed for any appreciable Na + equilibrium potential of + 55 m V becausethere are lengthof time becausethe Na+ and K+ gradientswould many more resting K+ channels than Na + channels in eventually run down, reducing the resting membrane the membrane. potential. As soon as the membrane potential begins to de- Dissipation of ionic gradients is prevented by the polarize from the value of the K+ equilibrium potential, Na +-K+ pump, which moves Na + and K+ againsttheir K+ flux is no longer in equilibrium across the mem- net electrochemical gradients: it extrudes Na + from the brane. The reduction in the negative electrical force cell while taking in K+. The pump therefore requires en- driving K+ into the cell means that there will be a net ef- ergy to run. The energy comes from the hydrolysis of flux of K+ out of the cell, tending to counteract the Na + ATP. Thus, at the resting membrane potential the cell is influx. The more the membrane potential is depolarized not in equilibrium but rather in a steadystate: there is a and moves away from the K+ equilibrium potential, the continuous passiveinflux of Na+ and efflux of K+ greater is the electrochemicalforce driving K+ out of the through resting channels that is exactly counterbal- cell and consequently the greater is the K+ efflux. Even- anced by the Na +-K+ pump. tually, the membrane potential reachesa new resting p0- The Na +-K+ pump is a large membrane-spanning tential at which the outward movement of K+ just bal- protein with catalytic binding sites for Na +, K+, and ancesthe inward movement of Na + (Figure 7-4C). This ATP. The sites for Na + and ATP are located on its intra- balancepoint (usually -60 m V) is far from the Na + equi- cellular surface and the sites for K+ on its extracellular librium potential (+55 mY) and is only slightly more surface. With each cycle the pump hydrolyzes one mol- positive than the equilibrium pOtential for K+ ( - 75 m V). ecule of ATP. It then uses this energy to extrude three To understand how this balance point is deter- Na + ions and bring in two K+ ions. The unequal flux of mined, bear in mind that the magnitude of the flux of an Na + and K+ ions causesthe pump to generatea net out- ion across a cell membrane is the product of its electro- ward ionic current. Thus, the pump is said to be electro- chemicaldriving force (the sum of the electrical driving genic.This pump-driven outward flux of positive charge force and the chemical driving force due to the concen- tends to hyperpolarize the membrane to a somewhat tration gradient) and the conductance of the membrane more negative potential than would be achieved by the to the ion: simple passive-diffusion mechanismsdiscussed above. ion flux = (electrical driving force + chemical driving force) Chloride Ions May Be Passively Distributed X membrane conductance. So far we have ignored the contribution of chloride A cell has relatively few resting Na + channelsso at (CI-) to the resting potential, even though many nerve restthe conductanceto Na + is quite low. Thus, despite the cells have Cl- channelsthat are open in the resting large chemical and electrical forces driving Na + into the cell, the influx of Na + is small. In contrast, since there are membrane. This simplification is valid for nerve cells that do not have a mechanism for active transport of Cl- many resting K+ channels,the membrane conductanceof against an electrochemical gradient. In these cells the K+ is relatively large. As a result, the small net outward resting potential is ultimately determined by K+ and force acting on K+ at the resting membrane potential is Na + fluxes because the intracellular concentrations of enough to produce a K+ efflux equal to the Na + influx. K+and Na + are fixed by active transport(the Na+ -K+ pump), whereas the Cl- concentration inside the cell is Passive Flux of Sodium and Potassium Is Balanced affected only by passive forces (electrical potential and by Active Pumping of the Ions concentration gradient). Therefore, the movement of CI- ions tends toward equilibrium across the mem- For a cell to have a steady resting membrane potential brane, so that Ea is equal to the resting potential, V" the charge separation across the membrane must be and there is no net CI- flux at rest. 132 Partn / Cell andMolecular Biology of theNeuron
In many nerve cells the 0 - gradient is controlled nels open during the rising phase of the action potential by an integral membrane protein called a CI- trans- that the cell'spermeability to Na + is muchgreater than porter. Like the Na+ -K+ pump it catalyzesthe move- to either cr or K+. Thus,at the peakof the actionp0- ment of ions across the membrane against an electro- tential the membrane potential approaches the Na + chemical gradient without forming a continuous pore. equilibrium potential, just as at rest (when permeability Unlike the Na + -K+ pump, the transport processdoes tp K+ is predominant) the membrane potential tends to not require the hydrolysis of ATP.Although no chemical CWproachthe K+ equilibrium potential. bond energy is utilized in the transport process,the CI- The membrane potential would remain at this large transporter can move 0 - against its electrochemical positivevalue near the Na+ equilibrium potentialindef- gradient by utilizing the energy stored in a preexisting initely but for two processesthat repolarize the mem- ionic concentration gradient for a different type of ion- brane, thus terminating the action potential. First, as the a processknown as serondaryactive transport.For exam- depolarization continues, the population of voltage- ple, one type of 0- transporter couples the outward gated Na + channelsgradually dosesby the processof movement of one cr ion to the outward movement of inactivation (seeChapters 6 and 9). Second,opening of one K+ ion. Since the electrochemical gradient for K+ is the voltage-gated K+ channels causes the K+ efflux to outward, the energetically favorable outward K+ flux is gradually increase. The increase in K+ permeability is able to drive the energetically unfavorable outward 0- slower than the increasein Na + permeabilitybecause of flux. As a result, the outside-ta-inside ratio of 0- is ~e slower rate of opening of the voltage-gated K+ chan- greater than would result from passive diffusion alone. nels. The delayed increasein K+ efflux combines with a The effect of increasing the CI- gradient is to make the ~se in Na + influx to producea net efflux of posi- equilibrium potential for 0- ions more negative than tive charge from the cell, which continues until the cell the resting membrane potential overall. (Remember,the valence (z) of 0 - is -1.) has repolarized to its resting membrane potential.
The Balance of Ion Fluxes That Gives Rise The Contributions of Different Ions to the to the Resting Membrane Potential Is Resting Membrane Potential Can Be Quantified Abolished During the Action Potential by the Goldman Equation
+ In the nerve cell at rest the steady Na + influx is balanced J\lthoughNa and K+ fluxes set the value of the resting by a steady K+ efflux, so that the membrane potential is potential, V m is not equal to either EK or ENabut lies be- constant. This balance changes, however, when the tween them. As a general rule, when Vm is determined membrane is depolarized past the threshold for generat- by two or more species of ions, the influence of each ing an action potential. Oncethe membrane potential speciesis determined not only by the concentrations of reaches this threshold, voltage-gated Na + channels tbe ion inside and outside the cell but also by the ease open rapidly. The resultant increase in membrane per- with which the ion crossesthe membrane. In terms of meabilityto Na+ causesthe Na+ influx to exceedthe K+ electrical current flow, the membrane's conductance efflux, creating a net influx of positive charge that (1/resistance) provides a convenient measure of how causesfurther depolarization. The increase in depolar- readily the ion crossesthe membrane. Another conve- ization causesstill more voltage-gated Na + channels to nient measure is the permeability (P) of the membrane open, resulting in a greater influx of Na +, which acceler- to that ion in units of velocity, em/ s. This measure is ates the depolarization even further. similar to that of a diffusion constant, which :p\easures This regenerative, positive feedback cycle develops the rate of solute movement in solution. The depen- explosively, driving the membrane potential toward the d~nce of membrane potential on ionic permeability and Na+ equilibrium potential of +55 mY: concentration is given quantitatively by the Goldman equation: RT [Naja [440J EN. = Fin [Na]i = 58 mY log [SO] = +55 mY. RT PK[K+Jo+ PN.[Na +]0 + Pa[CI-]t Vm = Fin PK[K+]t+ PN.[Na+]t+ Pa[Cl-]o' However, the membrane potential never quite reaches that point becauseK+ efflux continues throughout the Goldman Equation depolarization. A slight influx of CI- into the cell also This equationapplies only when Vm is not chang- counteractsthe depolarizingtendency of the Na+ in- ing. It states that the greater the concentration of a par- flux. Nevertheless, so many voltage-gated Na + chan- tictu1arion speciesand the greater its membrane perme- Chapter7 / MembranePotential 133
bility, the greater its role in determining the membrane A IOtential.In the limit, when permeability to one ion is ~ @o@ "ceptionally high, the Goldman equation reduces to 0 ~o~o ~ @ o~ ONe+ neNernst equationfor that ion. For example,if PK 0 .OK+ 'a or PN8fas in glial cells, the equation becomes
RT [K+]o ExtraceUular Vm 3£ Fin [K+]i' eid$ ,
~ Hodgkin and Bernard Katz used the Goldman quation to analyze changes in membrane potential. :'heyfirst measured the variation in membrane poten- ia1of a squid giant axon while systematically changing he extracellular concentrations of Na+, 0-, and K+. ~~K+.0 0 \:J. t:::;:.. 8K+ :'heyfound that if Vm is measuredshortly after the A- . K+\::V macellular concentration is changed (before the inter- tal ionic concentrations are altered), [K+]o has a strong !ifect on the resting potential, [Cr]o has a moderate ef- B ect, and [Na +]0 has little effect. The data for the mem- .rane at rest could be fit accurately by the Goldman quation using the following permeability ratios: PK:PNa:Pa = 1.0:0.04:0.45. ~t the peak of the action potential, however, the varia- ion of Vm with external ionic concentrations was fit best
Figure 7-6 Chemical and electrical forces contribute to cur- rent flow. A. A concentrationgradient for K+ gives rise to an electromo- A tive force. with a value equal to the K+ Nernst potential. This can be representedby a battery, EK.In this circuit the battery is Extrecellular in series with a conductor,'YK. representing the conductanceof $~ a channel that is selectively permeableto K+ ions. B. The current-voltagerelation for a K+ channel in the presence of both electrical and chemical driving forces. The potential at which the current is zero is equal to the K+ Nernst potential.
B if a quite different set of permeability ratios were as- sumed:! PK:PNa:Pa = 1.0:20:0.45. For these values of permeabilities the Goldman equa- tion approachesthe Nernst equation for Na +: RT [Na +]0 Vma!Fln [Na+]i = +5SmV.
=igure 7-5. Electrical properties of a single K+ channel. Thus at the peak of the action potential, when the mem- brane is much more permeable to Na + than to any other i\. A single K+ channelcan be representedas a conductoror ion, Vm approaches EN., the Nemst potential for Na +. esistor (conductance.'Y. is the inverse of resistance. f). t The current-voltagerelation for a single K+ channel in the Ibsenceof a concentrationgradient. The slope of the relation 1 At the peak of the action potential there is an instant in time when s equal to "ft<- V m is not changing and the Goldman equation is applicable. 134 Partn / Cell andMolecular Biology of theNeuron
Figure 7-7 All of the passive K+ channels in a nervecell membranecan be lumped into a singleequivalent electrical structure com- e:acellular prising a battery(EK) in serieswith a con- SI ductor (91<).The conductanceis gK = NK X 'YK. where ~ is the number of passive K+ channels and 'YKis the conductance of a sin- gle K+ channel. Cytoplasmic side
However, the finite permeability of the membrane to K+ Each Ion Channel Acts as a Conductor and CI- results in K+ efflux and a - influx that oppose and Battery in Parallel Na + influx, thereby preventing V m from quite reaching ~ escribed in Chapter 6, the lipid bilayer of the mem- ENa. b e is a poor conductor of ionic current because it is t permeable to ions. Even a large potential difference t produce practically no current flow across a pure The Functional Propertiesof the Neuron Can Be . id bilayer. Consider the cell body of a typical spinal Representedin an Electrical Equivalent Circuit otor neuron, which has a membrane area of about 10-4 cm2. H the membrane were composed solely of The Goldman equation is limited because it cannot be i4'id bilayer, its electrical conductance would be only used to determine how rapidly the membrane potential af;>out1 pS. In reality, however, the membrane contains changesin responseto a change in permeability. More- U\ousands of resting ion channels through which ions over, it is inconvenient for determining the magnitude !cpnstantly. embrane diffuse, at rest is so about that the 40,000 actualpS conductance or 40 X 10-9of S, the ie, of the individual Na +, K+, and CI- currents. This infor- mation can be obtained with a simple mathematical ,000 times greater than it would be if no ion channels model derived from electrical circuits. Within this ere present. model, called an equivalentcircuit, all of the important In an equivalent circuit each K+ channel can be rep- functional properties of the neuron are represented by nted as a resistor or conductor of ionic current with an electrical circuit consisting only of conductors or re- alsingle-channel conductanceof 'YK(remember, conduc- sistors (representing the ion channels), batteries (repre- tcptce= 1/resistance) (Figure 7-5). H there were no K+ senting the concentration gradients of relevant ions), concentration gradient, the current through the K+ and capacitors (the ability of the membrane to store channel would be given by Ohm's law: iK = 'YKX Vm. charge).Equivalent circuits provide us with an intuitive Since there is normally a K+ concentration gradient, understanding as well as a quantitative description of there will be a chemical force driving K+ across the how current flow due to the movement of ions gener- membrane. In the equivalent circuit this chemical force ates signals in nerve cells. The first step in developing a is representedby a battery, whose electromotive force is circuit is to relate the membrane's discrete physical given by the Nernst potential for K+, EK (Figure 7-6). (A properties to its electrical properties. (A review of ele- source of electrical potential is called an electromotive mentary circuit theory in Appendix A may be helpful force and an electromotive force generated by a differ- before proceeding to the discussion that follows.) encein chemical potentials is called a battery.)
Figure 7-8 Eachpopulation of ion channels selective for Na+, K+, or CI- can be represented by a battery in series with a conductor. Note the directions of poles of batteries. indicatinga negativeelectromotive force for K+ and CI- and a positive one for Na+.
~ Chapter 7 / MembranePotential 135
an ion is an intrinsic property of the membrane that is a measure of the easewith which the ion passesthrough the membrane (in units of em/ s). Permeability depends only on the types and numbersof ion channelspresent in the membrane. Conductance,on the other hand, mea- sures the ability of the membrane (or channel) to carry electrical current (in units of 1/ ohms). Since current is EN. carried by ions, the conductanceof a membrane will de- pend not only on the properties of the membrane but also on the concentration of ions in solution. A mem- brane can have a very high permeability to K+ ions, but Figure7-9 The passive current flow in a neuroncan be if there is no K+ in solution there can be no K+ current modeled using an electrical equivalent circuit. The circuit in- flow and so the conductance of the membrane will be dudes elements representingthe ion-selective membrane zero. In practice, permeability is used in the Goldman channelsand the short-circuit pathways provided by the cyto- equation whereas conductance is used in electrical mea- plasmand extracellular fluid. surements and equivalent circuits. A cell membrane has many resting K+ channels,all of which can be combined into a single equivalent cir- cuit consisting of a conductor in series with a battery (Figure 7-7). In this equivalent circuit the total conduc- tance of all the K+ channels (gK),ie, the K+ conductance In the absenceof voltage across the membrane the of the cell membrane in its resting state, is equal to the normal K+ concentration gradient will cause an out- number N of resting K+ channels multiplied by the con- ward K+ current flow. According to our conventions for ductance of an individual K+ channel ('YK): electrical current flow an outward movement of posi- 8K = Nk, X 'YK. tive charge corresponds to a positive electric current. From the Nernst equation, we also saw that when the Since the battery in this equivalent circuit depends concentration gradient for a positively charged ion, solely on the concentration gradient for K+ and is inde- such as K+, is directed outward (ie, there is a higher K+ pendent of the number of K+ channels, its value is the concentration inside than outside the cell), the equilib- equilibrium potential for K+, EK (Figure 7-7), rium potential for that ion is negative. Thus, the K+ cur- rent that flows solely becauseof its concentration gradi- ent is given by iK = -"YK X EK (the negative sign is required becausea negative equilibrium potential pro- ducesa positive current). Finally, for a real neuron that has both a membrane voltage and K+ concentration gradient, the net K+ cur- rent is given by the sum of the currents due to the elec- trical and chemical driving forces: iJC= ('YJc x Vm) - ('YJc x E.J= 'YJc X (Vat- E.J. The term Vm - EK is called the electrochemicaldriving force. It determines the direction of ionic current flow and (along with the conductance) the magnitude of cur- rent flow. This equation is a modified form of Ohm's law that takes into account that ionic current flow through a membrane is determined not only by the volt- age across the membrane but also by the ionic concen- Figure 7.10 Under steady state conditions the passive Na+ tration gradients. and K+ currents are balanced by active Na+ and K+ fluxes Sofar we have used two terms to indicate the ability (rNa and rK) driven by the Na+-K+ pump. The lipid bilayer of ions to cross membranes: permeability and conduc- endows the membrane with electrical capacitance(4n). Note "Na is 50% greater than" K (and therefore INais 50% greater tance. Although they are related, we should be careful than IK)since the Na+-K+pump transports three Na+ ions out not to confuse them. The permeabilityof a membrane to for every two K+ ions it transports into the cell. 136 Chapter 7 / MembranePotential 137 138 Partn / Cell andMolecular Biology of the Neuron
An Equivalent Circuit Model of the Membrane Capacitance is measured in units of farads, F, where a Includes Batteries, Conductors, a Capacitor, charge separation of 1 coulomb across a 1 farad capaci- and a Current Generator tor produces a 1 volt potential difference.
. A typical value of membranecapacitance for a Like the population of resting K+ channels,all the resting Na + channels can be representedby a single conductor neJtvecell is about 1 jJ.F/ cm2 of membrane area. The ex- in serieswith a single battery, as can the restingCI- chan- ce~ of positive and negative charges separated by the nels (Figure 7-8).Since the K+, Na +, and CI- channelsac- mE!lffibraneof a spherical cell body with a diameter of count for the bulk of the passive ionic current through SOjJ.mand a resting potential of -60 mV is 29 X 1(f the membrane in the cell at rest, we can calculate the rest- ions. Although this number may seem large, it repre- serttsonly a tiny fraction (1/200,000)of the total number ing potential by incorporating thesethree channelsinto a of positive or negative chargesin solution within the cy- simple equivalent circuit of a neuron. To construct this circuit we need only connect the el- toplasm. The bulk of the cytoplasm and the bulk of the ements representing each type of channel at their two extracellular fluid are electroneutral. ends with elements representing the extracellular fluid The use of the equivalent circuit model of the neu- and cytoplasm. The extracellular fluid and cytoplasm are rort to analyze neuronal properties quantitatively is illustrated in Box 7-2. both excellent conductors becausethey have relatively large cross-sectionalareas and many ions available to carry charge. Both can be approximated by a short i circuit-a conductor with zero resistance(Figure 7-9). ArI.Overall View The equivalent circuit of the neuron can be made more accurate by adding a current generator. As de- The lipid bilayer, which is virtually impermeant to ions, scribed earlier in this chapter, steady fluxes of Na + and is "n insulator separating two conducting solutions, the K+ ions through the passive membrane channels are ex- cytoplasm and the extracellular fluid. Ions can cross the actly counterbalancedby active ion fluxes driven by the lip~d bilayer only by passing through ion channels in Na+ -K+ pump, which extrudes three Na + ions from the the cell membrane. When the cell is at rest, the passive cell for every two K+ ions it pumps in. This electrogenic iOlUcfluxes into and out of the cell are balanced, so that th~ charge separation across the membrane remains ATP-dependent pump, which keeps the ionic batteries charged, can be added to the equivalent circuit in the cot}stant and the membrane potential remains at its rest- form of a current generator (Figure 7-10). ing value. Finally, we can complete the equivalent circuit of , The value of the resting membrane potential in the neuron by incorporating its capacitance,the third im- neIjve cells is determined primarily by resting channels selective for K+, CI-, and Na +. In general, the mem- portant passive electrical property of the neuron. Capacitanceis the property of an electric nonconductor brane potential will be closest to the equilibrium (insulator) that permits the storage of charge when op- (N,rnst) potential of the ion (or ions) with the greatest posite surfacesof the nonconductor are maintained at a m$1brane permeability. The permeability for an ion difference of potential. For the neuron, the nonconduc- speciesis proportional to the number of open channels tor (or capacitor) is the cell membrane, which separates that allow passageof that ion. the cytoplasm and extracellular fluid, both of which are At rest, the membrane potential is close to the Nt t potential for K+, the ion to which the membrane highly conductive environments. Strictly speaking, the membrane is a leaky capacitor becauseit is penetrated is ost permeable. The membrane is also somewhat by ion channels. However, since the density of the peI!meableto Na +, however, and therefore an influx of Na-+-drives the membrane potential slightly positive to ion channels is low, the insulating portion of the the! K+ Nernst potential. At this potential the electrical membrane-the lipid bilayer-occupies at least 100 times the area of all the ion channels combined. Mem- an4 chemical driving forces acting on K+ are no longer brane capacitanceis included in the equivalent circuit in in ~ce, so K+ diffuses out of the cell. Thesetwo pas- sivt fluxes are each counterbalanced by active fluxes Figure 7-10. driven by the Na+ -K+ pump. The electrical potential difference acrossa capacitor, , Chloride is actively pumped out of some, but not V, is expressedas: all, Icells. When it is not, it is passively distributed so as v - O/C, to 1j>eat equilibrium inside and outside the cell. Under moSt physiological conditions the bulk concentrations where Q is the excessof positive or negative chargeson of Na+, K+ , and CI- inside and outside the cell are con- each side of the capacitor and C is the capacitance. s~t. During signaling the changesin membrane poten- Chapter 7 / MembranePotential 139
t tial (action potentials, synaptic potentials, and receptor Hodgkin AL. 1992.Chance and Design.Cambridge: Cam- potentials) are caused by substantial changes in the bridge Univ. Press. membrane's relative permeabilities to these three ionsf not by changesin the bulk concentrations of ions, which References are negligible. Thesechanges in permeability, caused by BernsteinJ. [1902) 1979.Investigations on the thermodynam- the opening of gated ion channels, causechanges in the ics of bioelectric currents. Pfliigers Arch 92:521-562. net charge separation acrossthe membrane. Translated in: GR Kepner (ed). Cell MembranePermeability and Transport, pp. 184-210. Stroudsburg, fA: Dowden, Hutchinson &t Ross. Goldman DE. 1943.Potential, impedance, and rectification in membranes. J Gen PhysioI27:37~. Hodgkin AL, Katz B. 1949.The effect of sodium ions on the electrical activity of the giant axon of the squid. J Physiol (Lond) 108:37-77. Nernst W. [1888] 1979.On the kinetics of substancesin solu- tion. Z Physik Chem. 2:613-622, 634-637. Thmslated in: Selected Readings GR Kepner (ed). Cell MembranePermeability and Transport, pp. 174-183. Stroudsburg, fA: Dowden, Hutchinson &t Ross. Orkand RK. 1977.Glial cells. In: ER Kandel (ed). Handbookof Physiology:A Critical, ComprehensivePresentation of Physio- logical Knowledgeand Concepts,Sect. 1, TheNervous Svstem. Vol. I, Cellular Biology of Neurons, Part 2, pp. 855-875. Bethesda,MD: American Physiological Society. Siegel GJ, Agranoff BW, Albers RW (eds). 1999.&sit: Neuro- chemistry:Molecular, Cellular, and Medical Aspects,6th ed, Philadelphia: Lippincott-Raven.
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