6: Membrane Potential and the Passive Electrical Properties of the Neuron

• The Resting Membrane Potential Results from the Separation of Charge Across the Cell Membrane

• The Resting Membrane Potential Is Determined by Nongated and Gated Ion Channels

  • Open Channels in Glial Cells Are Permeable to Potassium Only

  • Open Channels in Resting Nerve Cells Are Permeable to Several Ion Species

  • The Electrochemical Gradients of Sodium, Potassium, and Calcium Are Established by Active Transport of the Ions

  • Chloride Ions Are Also Actively Transported

• The Balance of Ion Fluxes That Maintains the Resting Membrane Potential Is Abolished During the Action Potential

• The Contributions of Different Ions to the Resting Membrane Potential Can Be Quantified by the Goldman Equation

• The Functional Properties of the Neuron Can Be Represented as an Electrical Equivalent Circuit

• The Passive Electrical Properties of the Neuron Affect Electrical Signaling

  • Membrane Capacitance Slows the Time Course of Electrical Signals

  • Membrane and Axoplasmic Resistance Affect the Efficiency of Signal Conduction

  • Large Axons Are More Easily Excited Than Small Axons

  • Passive Membrane Properties and Axon Diameter Affect the Velocity of Action Potential Propagation

• An Overall View

Information is carried within neurons and from neurons to their target cells by electrical and chemical signals. Transient electrical signals are particularly important for carrying time-sensitive information rapidly and over long distances. These transient electrical signals—receptor potentials, synaptic potentials, and action potentials—are all produced by temporary changes in the electric current into and out of the cell, changes that drive the electrical potential across the cell membrane away from its resting value. This current represents the flow of negative and positive ions through ion channels in the cell membrane.

Two types of ion channels—resting and gated—have distinctive roles in neuronal signaling. Resting channels are primarily important in maintaining the resting membrane potential, the electrical potential across the membrane in the absence of signaling. Some types of resting channels are constitutively open and are not gated by changes in membrane voltage; other types are gated by voltage but can open at the negative resting potential of neurons. Most voltage-gated channels, in contrast, are closed when the membrane is at rest and require membrane depolarization to open.

In this and the next several chapters we consider how transient electrical signals are generated in the neuron. We begin by discussing how resting ion channels establish and maintain the resting potential and briefly describe the mechanism by which the resting potential can be perturbed, giving rise to transient electrical signals such as the action potential. We then consider how the passive electrical properties of neurons—their resistive and capacitive characteristics—contribute to the integration and local propagation of synaptic and receptor potentials within the neuron. In Chapter 7 we shall examine the detailed mechanisms by which voltage-gated Na⁺, K⁺, and Ca²⁺ channels generate the action potential, the electrical signal conveyed along the axon. Synaptic and receptor potentials are considered in Chapters 8, 9, 10 and 11 in the contexts of synaptic signaling between neurons and sensory reception.

The neuron's cell membrane has thin clouds of positive and negative ions spread over its inner and outer surfaces. At rest the extracellular surface of the membrane has an excess of positive charge and the cytoplasmic surface an excess of negative charge (Figure 6–1). This separation of charge is maintained because the lipid bilayer of the membrane is a barrier to the diffusion of ions, as explained in Chapter 5. The charge separation gives rise to a difference of electrical potential, or voltage, across the membrane called the membrane potential (V_m), defined as

where V_in is the potential on the inside of the cell and V_out the potential on the outside.

The membrane potential of a cell at rest is called the resting membrane potential (V_r) . Since by convention the potential outside the cell is defined as zero, the resting potential is equal to V_in. Its usual range is −60 mV to −70 mV. All electrical signaling involves brief changes in the resting membrane potential that are caused by electrical currents across the cell membrane.

The electric current is carried by ions, both positive (cations) and negative (anions). The direction of current is conventionally defined as the direction of net movement of positive charge. Thus, in an ionic solution cations move in the direction of the electric current and anions move in the opposite direction. In the nerve cell at rest there is no net charge movement across the membrane. When there is a net flow of cations or anions into or out of the cell, the charge separation across the resting membrane is disturbed, altering the electrical potential of the membrane. A reduction or reversal of charge separation, leading to a less negative membrane potential, is called depolarization. An increase in charge separation, leading to a more negative membrane potential, is called hyperpolarization.

Changes in membrane potential that do not lead to the opening of gated ion channels are passive responses of the membrane and are called electrotonic potentials. Hyperpolarizing responses are almost always passive, as are small depolarizations. However, when depolarization approaches a critical level, or threshold, the cell responds actively with the opening of voltage-gated ion channels, which produces an all-or-none action potential (Box 6–1).

The resting membrane potential is the result of the passive flux of individual ion species through several classes of resting channels. Understanding how this passive ionic flux gives rise to the resting potential enables us to understand how the gating of different types of ion channels generates the action potential, as well as the receptor and synaptic potentials.

No single ion species is distributed equally on the two sides of a nerve cell membrane. Of the four most abundant ions found on either side of the cell membrane, Na⁺ and Cl⁻ are concentrated outside the cell and K⁺ and organic anions (A⁻, primarily amino acids and proteins) inside. Table 6–1 shows the distribution of these ions inside and outside of one particularly well-studied nerve cell process, the giant axon of the squid, whose extracellular fluid has a salt concentration similar to that of seawater. Although the absolute values of the ionic concentrations for vertebrate nerve cells are two- to threefold lower than those for the squid giant axon, the concentration gradients (the ratio of the external to internal ion concentration) are similar.

The unequal distribution of ions raises several important questions. How do ionic gradients contribute to the resting membrane potential? What prevents the ionic gradients from dissipating by diffusion of ions across the membrane through the resting channels? These questions are interrelated, and we shall answer them by considering two examples of membrane permeability: the resting membranes of glial cells, which are permeable to only one species of ion, and the resting membranes of nerve cells, which are permeable to three. For the purposes of this discussion, we shall only consider the resting channels that are not gated by voltage and thus are always open.

Open Channels in Glial Cells Are Permeable to Potassium Only

The permeability of a cell membrane to particular ion species is determined by the relative proportions of the various types of ion channels that are open. The simplest case is that of the glial cell, which has a resting potential of approximately −75 mV. Like most cells, a glial cell has high concentrations of K⁺ and negatively charged organic anions on the inside and high concentrations of Na⁺ and Cl⁻ on the outside. However, most resting channels in the membrane are permeable only to K⁺.

Because K⁺ ions are present at a high concentration inside the cell, they tend to diffuse across the membrane from the inside to the outside of the cell down their chemical concentration gradient. As a result, the outside of the membrane accumulates a net positive charge (caused by the slight excess of K⁺) and the inside a net negative charge (because of the deficit of K⁺ and the resulting slight excess of anions). Because opposite charges attract each other, the excess positive charges on the outside and the excess negative charges on the inside collect locally on either surface of the membrane (see Figure 6–1).

The flux of K⁺ out of the cell is self-limiting. The efflux of K⁺ gives rise to an electrical potential difference—positive outside, negative inside. The greater the flow of K⁺, the more charge is separated and the greater is the potential difference. Because K⁺ is positive, the negative potential inside the cell tends to oppose the further efflux of K⁺. Thus ions are subject to two forces driving them across the membrane: (1) a chemical driving force, a function of the concentration gradient across the membrane, and (2) an electrical driving force, a function of the electrical potential difference across the membrane.

Once K⁺ diffusion has proceeded to a certain point, the electrical driving force on K⁺ exactly balances the chemical driving force. That is, the outward movement of K⁺ (driven by its concentration gradient) is equal to the inward movement of K⁺ (driven by the electrical potential difference across the membrane). This potential is called the K⁺ equilibrium potential, E_K (Figure 6–3). In a cell permeable only to K⁺ ions, E_K determines the resting membrane potential, which in most glial cells is approximately −75 mV.

The equilibrium potential for any ion X can be calculated from an equation derived in 1888 from basic thermodynamic principles by the German physical chemist Walter Nernst:

where R is the gas constant, T the temperature (in degrees Kelvin), z the valence of the ion, F the Faraday constant, and [X]_o and [X]_i the concentrations of the ion outside and inside the cell. (To be precise, chemical activities rather than concentrations should be used.)

Since RT/F is 25 mV at 25°C (77°F, room temperature), and the constant for converting from natural logarithms to base 10 logarithms is 2.3, the Nernst equation can also be written as follows:

Thus, for K⁺, since z = +1 and given the concentrations inside and outside the squid axon in Table 6–1:

The Nernst equation can be used to find the equilibrium potential of any ion that is present on both sides of a membrane permeable to that ion (the potential is sometimes called the Nernst potential). The Na⁺, K⁺, and Cl⁻ equilibrium potentials for the distributions of ions across the squid axon are given in Table 6–1.

In our discussion so far we have treated the generation of the resting potential as a passive mechanism—the diffusion of ions down their chemical gradients—one that does not require the expenditure of energy by the cell, for example through hydrolysis of adenosine triphosphate (ATP). However, energy (from ATP hydrolysis) is required to set up the initial concentration gradients and to maintain them in neurons, as we shall see below.

Open Channels in Resting Nerve Cells Are Permeable to Several Ion Species

Unlike glial cells, nerve cells at rest are permeable to Na⁺ and Cl⁻ ions in addition to K⁺ ions. Of the abundant ion species in nerve cells, only the large organic anions (A⁻) are unable to permeate the cell membrane. How are the concentration gradients for the three permeant ions (Na⁺, K⁺, and Cl⁻) maintained across the membrane of a single cell, and how do these three gradients interact to determine the cell's resting membrane potential?

To answer these questions, it is easiest to examine first only the diffusion of K⁺ and Na⁺. Let us return to the simple example of a cell having only K⁺ channels, with concentration gradients for K⁺, Na⁺, Cl⁻, and A⁻ as shown in Table 6–1. Under these conditions the resting membrane potential, V_r, is determined solely by the K⁺ concentration gradient and is equal to E_K (−75 mV) (Figure 6–4A).

Now consider what happens if a few resting Na⁺ channels are added to the membrane, making it slightly permeable to Na⁺. Two forces drive Na⁺ into the cell: Na⁺ tends to flow into the cell down its chemical concentration gradient, and it is driven into the cell by the negative electrical potential difference across the membrane (Figure 6–4B). The influx of Na⁺ depolarizes the cell, but only slightly from the K⁺ equilibrium potential (−75 mV). The new membrane potential does not come close to the Na⁺ equilibrium potential of +55 mV because there are many more resting K⁺ channels than Na⁺ channels in the membrane.

As soon as the membrane potential begins to depolarize from the value of the K⁺ equilibrium potential, K⁺ flux is no longer in equilibrium across the membrane. The reduction in the negative electrical force driving K⁺ into the cell means that there is now a net flow of K⁺ out of the cell, tending to counteract the Na⁺ influx. The more the membrane potential is depolarized and driven away from the K⁺ equilibrium potential, the greater is the net electrochemical force driving K⁺ out of the cell and consequently the greater the net K⁺ efflux. Eventually, the membrane potential reaches a new resting level at which the increased outward movement of K⁺ just balances the inward movement of Na⁺ (Figure 6–4C). This balance point (usually approximately −65 mV) is far from the Na⁺ equilibrium potential (+55 mV) and is only slightly more positive than the K⁺ equilibrium potential (−75 mV).

To understand how this balance point is determined, bear in mind that the magnitude of the flux of an ion across a cell membrane is the product of its electrochemical driving force (the sum of the electrical and chemical driving forces) and the conductance of the membrane to the ion:

In a resting nerve cell relatively few Na⁺ channels are open, so the membrane conductance of Na⁺ is quite low. Thus, despite the large chemical and electrical forces driving Na⁺ into the cell, the influx of Na⁺ is small. In contrast, many K⁺ channels are open in the membrane of a resting cell so that the membrane conductance of K⁺ is relatively large. Because of the high relative conductance of Na⁺ to K⁺ in the cell at rest, the small net outward force acting on K⁺ is enough to produce a K⁺ efflux equal to the Na⁺ influx.

The Electrochemical Gradients of Sodium, Potassium, and Calcium Are Established by Active Transport of the Ions

For a cell to have a steady resting membrane potential, the charge separation across the membrane must be constant over time. That is, at every instant the influx of positive charge must be balanced by the efflux of positive charge. If these fluxes were not equal, the charge separation across the membrane, and thus the membrane potential, would vary continually.

As we have seen, the passive movement of K⁺ out of the resting cell through open channels balances the passive movement of Na⁺ into the cell. However, this steady leakage of ions cannot be allowed to continue unopposed for any appreciable length of time because the Na⁺ and K⁺ gradients would eventually run down, reducing the resting membrane potential.

Dissipation of ionic gradients is prevented by the sodium-potassium pump (Na⁺-K⁺ pump), which moves Na⁺ and K⁺ against their electrochemical gradients: It extrudes Na⁺ from the cell while taking in K⁺. The pump therefore requires energy, and the energy comes from hydrolysis of ATP (Figure 6–5A). Thus at the resting membrane potential the cell is not in equilibrium but rather in a steady state: There is a continuous passive influx of Na⁺ and efflux of K⁺ through resting channels that is exactly counterbalanced by the Na⁺-K⁺ pump. As we saw in the previous chapter, pumps are similar to ion channels in that they catalyze the movement of ions across cell membranes. However they differ in two important respects. First, whereas ion channels are passive conduits that allow ions to move down their electrochemical gradient, pumps transport ions against their electrochemical gradient by expending chemical energy. Second, ion transport is much faster in channels: Ions typically flow through channels at a rate of 10⁷ to 10⁸ per second, whereas pumps operate at speeds more than 10,000 times slower.

The Na⁺-K⁺ pump is a large membrane-spanning protein with catalytic binding sites for Na⁺ and ATP on its intracellular surface and for K⁺ on its extracellular surface. With each cycle the pump hydrolyzes one molecule of ATP. (Because the Na⁺-K⁺ pump hydrolyzes ATP, it is also referred to as the Na⁺-K⁺ ATPase.) It uses this energy of hydrolysis to extrude three Na⁺ ions from the cell and bring in two K⁺ ions. The un equal flux of Na⁺ and K⁺ ions causes the pump to generate a net outward ionic current. Thus, the pump is said to be electrogenic. This pump-driven efflux of positive charge tends to set the resting potential a few millivolts more negative than would be achieved by the passive diffusion mechanisms discussed above. During periods of intense neuronal activity the increased influx of Na⁺ leads to an increase in Na⁺-K⁺ pump activity that generates a prolonged outward current, leading to a prolonged hyperpolarizing after-potential that can last for several minutes, until the normal Na⁺ concentration is restored. The Na⁺-K⁺ pump is inhibited by ouabain or digitalis plant alkaloids, an action that is important in the treatment of heart failure.

The Na⁺-K⁺ pump is a member of a large family of pumps known as P-type ATPases (because the phosphoryl group of ATP is temporarily transferred to the pump). P-type ATPases include a Ca²⁺ pump that transports Ca²⁺ across cell membranes (see Figure 6–5A). All cells normally maintain a very low cytoplasmic Ca²⁺ concentration, between 50 and 100 nM. This concentration is more than four orders of magnitude lower than the external concentration, which is approximately 2 mM. Calcium pumps in the plasma membrane transport Ca²⁺ out of the cell; other Ca²⁺ pumps located in internal membranes, such as the smooth endoplasmic reticulum, transport Ca²⁺ from the cytoplasm into these intracellular Ca²⁺ stores. Calcium pumps are thought to transport two Ca²⁺ ions for each ATP molecule that is hydrolyzed, with two protons transported in the opposite direction. The Na⁺-K⁺ pump and Ca²⁺ pump have similar structures. They are formed from 110 kD α-subunits, whose large transmembrane domain contains 10 membrane-spanning α-helixes (see Figure 6–5A).

Most neurons have relatively few Ca²⁺ pumps in the plasma membrane. Instead, Ca²⁺ is transported out of the cell by the Na⁺-Ca²⁺ exchanger (Figure 6–5B). This membrane protein is not an ATPase but a different type of molecule called a cotransporter. Cotransporters move one type of ion against its electrochemical gradient by using the energy stored in the electrochemical gradient of a second ion. (The E. coli ClC protein discussed in Chapter 5 is a type of cotransporter.) In the case of the Na⁺-Ca²⁺ exchanger, the electrochemical gradient of Na⁺ drives the efflux of Ca²⁺. The exchanger transports three or four Na⁺ ions into the cell (down the electrochemical gradient for Na⁺) for each Ca²⁺ ion it removes (against the electrochemical gradient of Ca²⁺). Because Na⁺ and Ca²⁺ are transported in opposite directions, the exchanger is termed an antiporter. Ultimately, it is the hydrolysis of ATP by the Na⁺-K⁺ pump that provides the energy (stored in the Na⁺ gradient) to maintain the function of the Na⁺-Ca²⁺ exchanger. For this reason, ion flux driven by cotransporters is often referred to as secondary active transport, to distinguish it from the primary active transport driven directly by ATPases.

Chloride Ions Are Also Actively Transported

So far, for simplicity, we have ignored the contribution of chloride (Cl⁻) to the resting potential. However, in most nerve cells the Cl⁻ gradient is controlled by one or more active transport mechanisms so that E_Cl will differ from V_r. As a result, the opening of Cl⁻ channels will bias the membrane potential toward its Nernst potential. Chloride transporters typically use the energy stored in the gradients of other ions—they are cotransporters.

Cell membranes contain a number of different types of Cl⁻ cotransporters (Figure 6–5B). Some transporters increase intracellular Cl⁻ to levels greater than those that would be passively reached if the Cl⁻ Nernst potential was equal to the resting potential. In such cells E_Cl is positive to V_r so that the opening of Cl⁻ channels depolarizes the membrane. An example of this type of transporter is the Na⁺-K⁺-Cl⁻ cotransporter. This protein transports two Cl⁻ ions into the cell together with one Na⁺ and one K⁺ ion. As a result, the transporter is electroneutral. The Na⁺-K⁺-Cl⁻ cotransporter differs from the Na⁺-Ca²⁺ exchanger in that the former transports all three ions in the same direction—it is a symporter.

However, in most neurons the Cl⁻ gradient is determined by cotransporters that move Cl⁻ out of the cell. This action lowers the intracellular concentration of Cl⁻ so that E_Cl is typically more negative than the resting potential. As a result, the opening of Cl⁻ channels leads to an influx of Cl⁻ that hyperpolarizes the membrane. The K⁺-Cl⁻ cotransporter is an example of such a transport mechanism; it moves one K⁺ ion out of the cell for each Cl⁻ ion it exports.

Interestingly, in early neuronal development cells tend to express primarily the Na⁺-K⁺-Cl⁻ cotransporter. At this stage the neurotransmitter γ-aminobutyric acid (GABA), which activates Cl⁻ channels, typically has an excitatory (depolarizing) effect. As neurons develop they begin to express the K⁺-Cl⁻ cotransporter, such that in mature neurons GABA typically hyperpolarizes the membrane and thus acts as an inhibitory neurotransmitter. In some pathological conditions in adults, such as certain types of epilepsy or chronic pain syndromes, the expression pattern of the Cl⁻ cotransporters may revert to that of the immature nervous system. This will lead to aberrant depolarizing responses to GABA that can produce abnormally high levels of excitation.

In the nerve cell at rest the steady Na⁺ influx is balanced by a steady K⁺ efflux, so that the membrane potential is constant. However, this balance changes when the membrane is depolarized toward the threshold for an action potential. As the membrane potential approaches this threshold, voltage-gated Na⁺ channels open rapidly. The resultant increase in membrane conductance to Na⁺ causes the Na⁺ influx to exceed the K⁺ efflux once threshold is exceeded, creating a net influx of positive charge that causes further depolarization. The increase in depolarization causes still more voltage-gated Na⁺ channels to open, resulting in a greater influx of Na⁺, which accelerates the depolarization even further.

This regenerative, positive feedback cycle develops explosively, driving the membrane potential toward the Na⁺ equilibrium potential of +55 mV:

However, the membrane potential never quite reaches E_Na because K⁺ efflux continues throughout the depolarization. A slight influx of Cl⁻ into the cell also counteracts the depolarizing effect of the Na⁺ influx. Nevertheless, so many voltage-gated Na⁺ channels open during the rising phase of the action potential that the cell membrane's Na⁺ conductance is much greater than the conductance of either Cl⁻ or K⁺. Thus, at the peak of the action potential the membrane potential approaches the Na⁺ equilibrium potential, just as at rest (when permeability to K⁺ is predominant) the membrane potential tends to approach the K⁺ equilibrium potential.

The membrane potential would remain at this large positive value near the Na⁺ equilibrium potential indefinitely but for two processes that repolarize the membrane, thus terminating the action potential. First, following the peak of the action potential the voltage-gated Na⁺ channels gradually close by the process of inactivation (see Chapters 5 and 7). Second, opening of the voltage-gated K⁺ channels causes the K⁺ efflux to gradually increase. The increase in K⁺ conductance is slower than the increase in Na⁺ conductance because of the slower rate of opening of the voltage-gated K⁺ channels. The slow increase in K⁺ efflux together with the decrease in Na⁺ influx produces a net efflux of positive charge from the cell, which continues until the cell has repolarized to its resting membrane potential.

Although K⁺, Na⁺, and Cl⁻ fluxes set the value of the resting potential, V_m is not equal to E_K, E_Na, or E_Cl but lies at some intermediate value. As a general rule, when V_m is determined by two or more species of ions, the contribution of one species is determined not only by the concentrations of the ion inside and outside the cell but also by the ease with which the ion crosses the membrane. One convenient measure of how readily the ion crosses the membrane is the permeability (P) of the membrane to that ion, which has units of velocity (cm/s). This measure is similar to that of a diffusion constant, which determines the rate of solute movement in solution driven by a local concentration gradient. The dependence of membrane potential on ionic permeability and concentration is given by the Goldman equation:

This equation applies only when V_m is not changing. It states that the greater the concentration of an ion species and the greater its membrane permeability, the greater its contribution to determining the membrane potential. In the limit, when permeability to one ion is exceptionally high, the Goldman equation reduces to the Nernst equation for that ion. For example, if P_K >> P_Cl or P_Na, as in glial cells, the equation becomes as follows:

Alan Hodgkin and Bernard Katz used the Goldman equation to analyze changes in membrane potential in the squid giant axon. They measured the variations in membrane potential in response to systematic changes in the extracellular concentrations of Na⁺, Cl⁻, and K⁺. They found that if V_m is measured shortly after the extracellular concentration is changed (before the internal ionic concentrations are altered), [K⁺]_o has a strong effect on the resting potential, [Cl⁻]_o has a moderate effect, and [Na⁺]_o has little effect. The data for the membrane at rest could be fit accurately by the Goldman equation using the following permeability ratios:

At the peak of the action potential there is an instant in time when V_m is not changing and the Goldman equation is applicable. At that point the variation of V_m with external ionic concentrations is fit best if a quite different set of permeability ratios is assumed:

For these values of permeability the Goldman equation approaches the Nernst equation for Na⁺:

Thus at the peak of the action potential, when the membrane is much more permeable to Na⁺ than to any other ion, V_m approaches E_Na. However, the finite permeability of the membrane to K⁺ and Cl⁻ results in K⁺ efflux and Cl⁻ influx that partially counterbalance Na⁺ influx, thereby preventing V_m from quite reaching E_Na.

The usefulness of the Goldman equation is limited because it cannot be used to determine how rapidly the membrane potential changes in response to a change in permeability. It is also inconvenient for determining the magnitude of the individual Na⁺, K⁺, and Cl⁻ currents. This information can be obtained using a simple mathematical model derived from electrical circuits. The model, called an equivalent circuit, represents all of the important electrical properties of the neuron by a circuit consisting of conductors or resistors, batteries, and capacitors.¹

Equivalent circuits provide us with an intuitive understanding as well as a quantitative description of how current caused by the movement of ions generates signals in nerve cells.

The first step in developing an equivalent circuit is to relate the membrane's discrete physical properties to its electrical properties. We start by considering the lipid bilayer, which endows the membrane with electrical capacitance, the ability of an electrical nonconductor (insulator) to separate electrical charges on either side of it. The nonconducting phospholipid bilayer of the membrane separates the cytoplasm and extracellular fluid, both of which are highly conductive environments. The presence of a thin layer of opposing charges on the inside and outside surfaces of the cell membrane, acting as a capacitor, gives rise to the electrical potential difference across the membrane. The electrical potential difference or voltage across a capacitor is

where Q is the net excess positive or negative charge on each side of the capacitor and C is the capacitance. Capacitance is measured in units of farads (F), and charge is measured in coulombs (where 96,500 coulombs of a univalent ion is equivalent to 1 mole of that ion). A charge separation of 1 coulomb across a capacitor of 1 F produces a potential difference of 1 volt. A typical value of membrane capacitance for a nerve cell is approximately 1 μF per cm² of membrane area. Very few charges are required to produce a large potential difference across such a capacitance. For example, the excess of positive and negative charges separated by the membrane of a spherical cell body with a diameter of 50 μm and a resting potential of −60 mV is 29 × 10⁶ ions. Although this number may seem large, it represents only a tiny fraction (1/200,000) of the total number of positive or negative charges in solution within the cytoplasm. The bulk of the cytoplasm and the bulk of the extracellular fluid are electroneutral.

The membrane is a leaky capacitor because it is studded with ion channels that can conduct charge. Ion channels endow the membrane with conductance and with the ability to generate electromotive force (emf). The lipid bilayer itself has effectively zero conductance or infinite resistance. However, because ion channels are highly conductive, they provide pathways of finite electrical resistance for ions to cross the membrane. Because neurons contain many types of channels selective for different ions, we must consider each class of ion channel separately.

In an equivalent circuit we can represent each K⁺ channel as a resistor or conductor of ionic current with a single-channel conductance γ_K (remember, conductance = 1/resistance) (Figure 6–6). If there were no K⁺ concentration gradient, the current through a single K⁺ channel would be given by Ohm's law: i_K = γ_K × V_m. However, since there is normally a K⁺ concentration gradient, there is also a chemical force driving K⁺ across the membrane, represented in the equivalent circuit by a battery. (A source of electrical potential is called an electromotive force and an electromotive force generated by a difference in chemical potentials is called a battery.) The electromotive force of this battery is given by E_K, the Nernst potential for K⁺ (Figure 6–7).

In the absence of voltage across the membrane, the normal K⁺ concentration gradient causes an outward K⁺ current. According to our convention for current, an outward movement of positive charge across the membrane corresponds to a positive current. According to the Nernst equation, when the concentration gradient for a positively charged ion, such as K⁺, is directed outward (ie, the K⁺ concentration inside the cell is higher than outside), the equilibrium potential for that ion is negative. Thus the K⁺ current that flows solely because of its concentration gradient is given by i_K = −γ_K × E_K (the negative sign is required because a negative equilibrium potential produces a positive current at 0 mV).

Finally, for a real neuron that has both a membrane voltage and a K⁺ concentration gradient, the net K⁺ current is given by the sum of the currents caused by the electrical and chemical driving forces:

The term (V_m − E_K) is called the electrochemical driving force. It determines the direction of ionic current and (along with the conductance) its magnitude. This equation is a modified form of Ohm's law that takes into account the fact that ionic current through a membrane is determined not only by the voltage across the membrane but also by the ionic concentration gradients.

A cell membrane has many resting K⁺ channels, all of which can be combined into a single equivalent circuit consisting of a conductor in series with a battery (Figure 6–8). In this equivalent circuit the total conductance of all the K⁺ channels (g_K), ie, the K⁺ conductance of the cell membrane in its resting state, is equal to the number of resting K⁺ channels (N_K) multiplied by the conductance of an individual K⁺ channel (γ_K):

Because the battery in this equivalent circuit depends solely on the concentration gradient for K⁺ and is independent of the number of K⁺ channels, its value is the equilibrium potential for K⁺, E_K.

Like the population of resting K⁺ channels, all the resting Na⁺ channels can be represented by a single conductor in series with a single battery, as can the resting Cl⁻ channels (Figure 6–9). Because the K⁺, Na⁺, and Cl⁻ channels account for the bulk of the passive ionic current through the membrane in the cell at rest, we can calculate the resting potential by incorporating these three pathways into a simple equivalent circuit of a neuron.

To construct this circuit we need only connect the elements representing each type of channel at their two ends with elements representing the extracellular fluid and cytoplasm. The extracellular fluid and cytoplasm are both good conductors (compared with the membrane) because they have relatively large cross sectional areas and many ions available to carry charge. In a small region of a neuron, the extracellular and cytoplasmic resistances can be approximated by a short circuit—a conductor with zero resistance (Figure 6–10). The membrane capacitance (C_m) is determined by the insulating properties of the lipid bilayer, which separates the extracellular and cytoplasmic conductors.

The equivalent circuit can be made more realistic by incorporating the active ion fluxes driven by the Na⁺-K⁺ pump, which extrudes three Na⁺ ions from the cell for every two K⁺ ions it pumps in. This electrogenic ATP-dependent pump, which keeps the ionic batteries charged, can be added to the equivalent circuit in the form of a current generator (Figure 6–11).

The use of the equivalent circuit to analyze neuronal properties quantitatively is illustrated in Box 6–2. There we see how the equivalent circuit can be used to calculate the resting potential, V_m. We also see how the equivalent circuit can be simplified by combining all of the resting Na⁺, K⁺, and Cl⁻ channels into a single resting current pathway, with a conductance g_r and a battery E_r, which is equal to the resting potential.

Once an electrical signal is generated in part of a neuron, for example in response to a synaptic input on a branch of a dendrite, it is integrated with the other inputs to the neuron and then propagated to the axon initial segment, the site of action potential generation. During signaling, when a stimulus generates action potentials or local (synaptic or sensory generator) potentials in a neuron, the membrane potential changes frequently.

What determines the rate of change in potential with time or distance? What determines whether a stimulus will or will not produce an action potential? Here we consider the neuron's passive electrical properties and geometry and how these relatively constant properties affect the cell's electrical signaling. In the next five chapters we shall consider the properties of the gated channels and how the active ionic currents they control change the membrane potential.

Neurons have three passive electrical properties that are important for electrical signaling. We have already considered the resting membrane conductance or resistance (g_r = 1/R_r) and the membrane capacitance, C_m. A third important property that determines signal propagation along dendrites or axons is their intra cellular axial resistance (r_a). Although, as mentioned above, the resistivity of cytoplasm is much lower than that of the membrane, the axial resistance along the entire length of an extended thin neuronal process can be considerable. Because these three elements provide the return pathway to complete the electrical circuit when active ionic currents flow into or out of the cell, they determine the time course of the change in synaptic potential generated by the synaptic current. They also determine whether a synaptic potential generated in a dendrite will depolarize the trigger zone on the axon initial segment enough to fire an action potential. Finally, the passive properties influence the speed at which an action potential is conducted.

Membrane Capacitance Slows the Time Course of Electrical Signals

The steady-state change in a neuron's voltage in response to subthreshold current resembles the behavior of a simple resistor, but the initial time course of the change does not. A true resistor responds to a step change in current with a similar step change in voltage, but the neuron's membrane potential rises and decays more slowly than the step change in current (Figure 6–15). This property of the membrane is caused by its capacitance.

To understand how the capacitance slows down the voltage response, recall that the voltage across a capacitor is proportional to the charge stored on the capacitor. To alter the voltage, charge (Q) must be added to or removed from the capacitor (C):

To change the charge across the capacitor (the neuron's lipid bilayer), current must flow across the capacitor (I_c). Since current is the flow of charge per unit time (I_c = ΔQ/Δt), the change in voltage across a capacitor is a function of the magnitude of the current and the time that the current flows:

Thus the magnitude of the change in voltage across a capacitor in response to a current pulse depends on the duration of the current, because time is required to deposit and remove charge across the capacitor.

If the membrane had only resistive properties, a step pulse of outward current would change the membrane potential instantaneously. Conversely, if the membrane had only capacitive properties, the membrane potential would change linearly with time in response to the same step of current. Because the membrane has both capacitive and resistive properties in parallel, the actual change in membrane potential combines features of the two pure responses. The initial slope of the relation between V_m and time reflects a purely capacitive element, whereas the final slope and amplitude reflect a purely resistive element (Figure 6–15, upper plot).

In the simple case of the spherical cell body of a neuron, the time course of the potential change is described by the following equation:

where e is the base of the system of natural logarithms and has a value of approximately 2.72, and τ is the membrane time constant, given by the product of the membrane resistance and capacitance (R_m C_m). The time constant can be measured experimentally as the time it takes the membrane potential to rise to 1 – 1/e, or approximately 63% of its steady-state value (Figure 6–15, upper plot). Typical values of for neurons range from 20 to 50 ms. We shall return to the time constant in Chapter 10 where we consider the temporal summation of synaptic inputs in a cell.

Membrane and Axoplasmic Resistance Affect the Efficiency of Signal Conduction

So far we have considered the effects of the passive properties of neurons on signaling only within the cell body. Distance is not a factor in the propagation of a signal in the neuron's soma because the cell body can be approximated as a tiny sphere whose membrane voltage is uniform. However, a subthreshold voltage signal traveling along extended structures such as dendrites, axons, and muscle fibers decreases in amplitude with distance from the site of initiation. To understand how this attenuation occurs, we will consider how the geometry of a neuron influences the distribution of current.

If current is injected into a dendrite at one point, how will the membrane potential change along the length of the dendrite? For simplicity, consider how membrane potential varies with distance after a constant-amplitude current pulse has been on for some time (t ≫ τ). Under these conditions the membrane capacitance is fully charged, so membrane potential reaches a steady value. The variation of the potential with distance thus depends solely on the relative values of the membrane resistance in a unit length of dendrite, r_m (units of Ω · cm), and the axial resistance per unit length of dendrite, r_a (units of Ω/cm). The change in membrane potential becomes smaller with distance along the dendrite away from the current electrode (Figure 6–16A). This decay with distance is exponential and expressed by

where λ is the membrane length constant, x is the distance from the site of current injection, and ΔV₀ is the change in membrane potential produced by the current at the site of injection (x = 0). The length constant is the distance along the dendrite to the site where ΔV_m has decayed to 1/e, or 37% of its initial value (Figure 6–16B). It is a measure of the efficiency of electrotonic conduction, the passive spread of voltage changes along the neuron, and is determined by the values of membrane and axial resistance as follows:

The better the insulation of the membrane (that is, the greater r_m) and the better the conducting properties of the inner core (the lower r_a), the greater the length constant of the dendrite. That is because current is able to spread farther along the inner conductive core of the dendrite before leaking across the membrane at some point x to alter the local membrane potential:

The length constant is also a function of the diameter of the neuronal process. Neuronal processes vary greatly in diameter, from as much as 1 mm for the squid giant axon to 1 μm for fine dendritic branches in the mammalian brain. For neuronal processes with similar ion channel densities (number of channels per unit membrane area) and cytoplasmic composition, the larger the diameter, the longer the length constant. Thus thicker axons and dendrites have longer length constants than do narrower processes and hence can transmit passive electrical signals for greater distances. Typical values for neuronal length constants range from 0.1 to 1.0 mm.

To understand how the diameter affects the length constant, we must consider how the diameter (or radius) of a process affects r_m and r_a. Both r_m and r_a are measures of resistance for a unit length of a neuronal process of a given radius. The axial resistance r_a of the process depends inversely on the number of charge carriers (ions) in a cross section of the process. Therefore, given a fixed cytoplasmic ion concentration, r_a depends inversely on the cross sectional area of the process, 1/(π · radius²). The resistance of a unit length of membrane r_m depends inversely on the total number of channels in a unit length of the neuronal process. Channel density, the number of channels per μm² of membrane, is often similar among different-sized processes. As a result, the number of channels per unit length of a neuronal process increases in direct proportion to increases in membrane area, which depends on the circumference of the process times its length; therefore, r_m varies as 1/(2 π radius). Because r_m/r_a varies in direct proportion to the radius of the process, the length constant is proportional to the square root of the radius.

The efficiency of electrotonic conduction has two important effects on neuronal function. First, it influences spatial summation, the process by which synaptic potentials generated in different regions of the neuron are added together at the trigger zone at the axon hillock (see Chapter 10). Second, electrotonic conduction is a factor in the propagation of the action potential. Once the membrane at any point along an axon has been depolarized beyond threshold, an action potential is generated in that region. This local depolarization spreads passively down the axon, causing successive adjacent regions of the membrane to reach the threshold for generating an action potential (Figure 6–17). Thus the depolarization spreads along the length of the axon by local current driven by the difference in potential between the active and resting regions of the axon membrane. In axons with longer length constants local current spreads a greater distance down the axon, and therefore the action potential propagates more rapidly.

Large Axons Are More Easily Excited Than Small Axons

The influence of axonal geometry on action potential conduction plays an important role in a common neurological exam. In the examination of a patient for diseases of peripheral nerves, the nerve often is stimulated by passing current between a pair of external cutaneous electrodes placed over the nerve, and the population of resulting action potentials (the compound action potential) is recorded farther along the nerve by a second pair of cutaneous voltage-recording electrodes. In this situation the total number of axons that generate action potentials varies with the amplitude of the current pulse.

To drive a cell to threshold, a stimulating current from the positive electrode must pass through the cell membrane into the axon. There it travels along the axoplasmic core, eventually exiting the axon into the extracellular fluid through the membrane to reach the second (negative) electrode. However, most of the stimulating current does not even enter the axon, moving instead through neighboring axons or through the low-resistance pathway of the extracellular fluid. Thus, the axons into which current enters most easily are the most excitable.

In general, axons with the largest diameter have the lowest threshold for excitation. The greater the diameter of the axon, the lower the axial resistance to the flow of current down the axon because the number of charge carriers (ions) per unit length of the axon is greater. Because more current enters the larger axon, the axon is depolarized more efficiently than a smaller axon. For these reasons, larger axons are recruited at low values of current; axons with smaller diameter are recruited only at relatively greater current strengths.

The fact that larger axons conduct more rapidly and have a lower current threshold for excitation aids in the interpretation of clinical nerve-stimulation tests. Neurons that convey different types of information (eg, motor versus sensory) often differ in axon diameter and thus conduction velocity (see Table 22–1). In addition, a specific disease process may preferentially affect a subset of the functional classes of axons. Thus, using conduction velocity as a criterion to determine which classes of axons have defective conduction properties can help one infer the neuronal basis for the neurological deficit.

Passive Membrane Properties and Axon Diameter Affect the Velocity of Action Potential Propagation

The passive spread of depolarization during conduction of the action potential is not instantaneous. In fact, electrotonic conduction is a rate-limiting factor in the propagation of the action potential. We can understand this limitation by considering a simplified equivalent circuit of two adjacent segments of axon membrane connected by a segment of axoplasm.

An action potential generated in one segment of membrane supplies depolarizing current to the adjacent membrane, causing it to depolarize gradually toward threshold (see Figure 6–17). According to Ohm's law, the larger the axoplasmic resistance, the smaller the current between adjacent membrane segments (I = V /R) and thus the longer it takes to change the charge on the membrane of the adjacent segment.

Recall that, since ΔV = ΔQ/C, the membrane potential changes slowly if the current is small because ΔQ, equal to current multiplied by time, changes slowly. Similarly, the larger the membrane capacitance, the more charge must be deposited on the membrane to change the potential across the membrane, so the current must flow for a longer time to produce a given depolarization. Therefore, the time it takes for depolarization to spread along the axon is determined by both the axial resistance r_a and the capacitance per unit length of the axon c_m (units F/cm). The rate of passive spread varies inversely with the product r_a c_m. If this product is reduced, the rate of passive spread increases and the action potential propagates faster.

Rapid propagation of the action potential is functionally important, and two adaptive strategies have evolved to increase it. One is an increase in the diameter of the axon core. Because r_a decreases in proportion to the square of axon diameter, whereas c_m increases in direct proportion to diameter, the net effect of an increase in diameter is a decrease in r_a c_m. This adaptation has been carried to an extreme in the giant axon of the squid, which can reach a diameter of 1 mm. No larger axons have evolved, presumably because of the competing need to keep neuronal size small so that many cells can be packed into a limited space.

The second strategy to increase conduction velocity is the wrapping of a myelin sheath around an axon (see Chapter 4). This process is functionally equivalent to increasing the thickness of the axonal membrane by as much as 100 times. Because the capacitance of a parallel-plate capacitor such as the membrane is inversely proportional to the thickness of the insulation material (see Appendix A), myelination decreases c_m and thus r_a c_m. Myelination results in a proportionately much greater decrease in r_a c_m than does the same increase in the diameter of the axon core because the many layers of membrane wrapped in the myelin sheath produce a very large decrease in c_m. For this reason, conduction in myelinated axons is faster than in nonmyelinated axons of the same diameter.

In a neuron with a myelinated axon the action potential is triggered at the nonmyelinated initial segment of membrane just distal to the axon hillock. The inward current that flows through this region of membrane is available to discharge the capacitance of the myelinated axon ahead of it. Even though the capacitance of the axon is quite small (because of the myelin insulation), the amount of current down the core of the axon from the trigger zone is not enough to discharge the capacitance along the entire length of the myelinated axon.

To prevent the action potential from dying out, the myelin sheath is interrupted every 1 to 2 mm by bare patches of axon membrane approximately 1 μm in length, the nodes of Ranvier (see Chapter 4). Although the area of membrane at each node is quite small, the nodal membrane is rich in voltage-gated Na⁺ channels and thus can generate an intense depolarizing inward Na⁺ current in response to the passive spread of depolarization down the axon. These regularly distributed nodes thus boost the amplitude of the action potential periodically, preventing it from decaying with distance.

The action potential, which spreads quite rapidly along the internode because of the low capacitance of the myelin sheath, slows down as it crosses the high-capacitance region of each bare node. Consequently, as the action potential moves down the axon it jumps quickly from node to node (Figure 6–18A). For this reason, the action potential in a myelinated axon is said to move by saltatory conduction (from the Latin saltare, to jump). Because ionic membrane current flows only at the nodes in myelinated fibers, saltatory conduction is also favorable from a metabolic standpoint. Less energy must be expended by the Na⁺-K⁺ pump in restoring the Na⁺ and K⁺ concentration gradients, which tend to run down as the action potential is propagated.

Various diseases of the nervous system are caused by demyelination, such as multiple sclerosis and Guillain-Barré syndrome. As an action potential goes from a myelinated region to a bare stretch of demyelinated axon, it encounters a region of relatively high c_m and low r_m. The inward current generated at the node just before the demyelinated segment may be too small to provide the capacitive current required to depolarize the segment of demyelinated membrane to threshold. In addition, this local-circuit current does not spread as far as it normally would because it is flowing into a segment of axon that has a short length constant because of its low r_m (Figure 6–18B). These two factors can combine to slow, and in some cases actually block, the conduction of action potentials, causing devastating effects on behavior (see Chapter 7).

The lipid bilayer, which is virtually impermeant to ions, is an insulator separating two conducting solutions, the cytoplasm and the extracellular fluid. Ions can cross the lipid bilayer only by passing through ion channels in the cell membrane. When the cell is at rest, passive ionic fluxes into and out of the cell are balanced, so the charge separation across the membrane remains constant and the membrane potential is maintained at its resting value.

The value of the resting membrane potential in nerve cells is determined primarily by resting channels that conduct K⁺, Cl⁻, and Na⁺. In general, the membrane potential is closest to the equilibrium (Nernst) potential of the ion (or ions) with the greatest membrane permeability. The permeability of a cell membrane for an ion species is proportional to the number of open channels that allow passage of that ion.

The resting membrane potential is close to the Nernst potential for K⁺, the ion to which the membrane is most permeable. The membrane is also somewhat permeable to Na⁺, however, and therefore an influx of Na⁺ shifts the membrane potential slightly positive to the K⁺ Nernst potential. At this potential the electrical and chemical driving forces acting on K⁺ are no longer in balance, so K⁺ diffuses out of the cell. The passive fluxes of K⁺ and Na⁺ are each counterbalanced by active fluxes driven by the Na⁺-K⁺ pump.

Chloride is actively transported out of most but not all neurons in the adult nervous system. When it is not, it is passively distributed so as to be at equilibrium inside and outside the cell. Under most physiological conditions the bulk concentrations of Na⁺, K⁺, and Cl⁻ inside and outside the cell are constant. Changes in membrane potential that generate neuronal signals (action potentials, synaptic potentials, and receptor potentials) are caused by substantial changes in the membrane's relative permeabilities to these three ions. These changes in permeability, caused by the opening of gated ion channels, change the net charge separation across the membrane, but produce only negligible changes in the bulk concentrations of ions.

Two competing requirements determine the functional design of neurons. First, to maximize the computing power of the nervous system, neurons must be small so that large numbers of them can fit into the brain and spinal cord. Second, to maximize the ability of the animal to respond to changes in its environment, neurons must conduct signals rapidly. These two adaptive features are constrained by the materials from which neurons are made.

Because the nerve cell membrane is thin and surrounded by two conducting media, it has a high capacitance. In addition, the currents that change the charge on the membrane capacitance along the length of an axon or dendrite flow through a relatively poor conductor—a thin column of cytoplasm. These two factors combine to slow down the conduction of voltage signals. Moreover, the various ion channels that are open at rest and that give rise to the resting potential also degrade the signaling function of the neuron, as they make the cell leaky and limit how far a signal can travel passively.

To compensate for these physical constraints neurons use voltage-gated channels to generate action potentials. The action potential is continually regenerated along the axon and thus conducted without attenuation. For pathways in which rapid signaling is particularly important, conduction of the action potential is enhanced by either myelination of the axon or an increase in axon diameter, or by both.