Membranes composed of a lipid bilayer enclose all cells in the body as well as all the organelles within those cells. Being lipid, these membranes are impermeant to small ions and macromolecules alike. A variety of transport proteins, selected from among hundreds available in the genome (Table 3-1), are inserted into the lipid bilayer to bring specific functions to one cell type or another. Together, these cell types in the nervous system function to harness electrical and chemical forces in their efforts to gather information, to analyze data, and to encode meaning as a string of action potentials (APs). This chapter begins with a discussion of the fundamental electrical and chemical properties of the solutions of small ions and large proteins that comprise the interior of the cell and its environs and shows how a neuron uses those forces to ensure its own integrity—and how bacteria and our own immune system broach these defenses to destroy neurons. Next are the electrical properties of the nerve that allow us to sense our environment and to integrate neural data. This chapter then concludes with the active changes in the nerve cell membrane that generate APs to encode data and to bring about distant activity.

Table 3-1 Families of Ion Channels*

*Presented here is a sample of the important channels found in nerve membranes. Individual channels are grouped into families and then into superfamilies on the basis of structural homologies. Most channels have a barrel-stave structural motif, with characteristic numbers of transmembrane segments (TM) and pore domains (P). These channels may be open constitutively; be regulated by membrane voltage, by cell calcium, or by second messengers; or be signaled to open by synaptic transmitters. Abbreviations have been assigned by IUPHAR, the International Union of Basic and Clinical Pharmacology.

AP, action potential.

COMBINED FORCES OF CHEMICAL GRADIENTS AND ELECTRICAL POTENTIALS

Because cell membranes are composed of a lipid bilayer, large molecules such as proteins can neither enter nor leave the cell by simple diffusion, whereas small molecules like water can diffuse with relative ease. These impermeant molecules exert an osmotic force that draws water into the cell, causing it to swell and ultimately to burst unless an opposing force intervenes first. As a result, the very first task of the cell is to ensure its own physical integrity, both by minimizing osmotic flow of water and by removing that excess water that does enter the cell. The mechanisms that perform this feat have been further adapted by the nervous system both to generate specialized fluids necessary for its proper function (such as the cerebrospinal fluid and the perilymph and endolymph of the ear) and to detect stimuli, to transport signals, and to integrate information.

Forces Due to Concentration Gradients

Pure water is 55.5 M (mol/kg) in concentration. The logic behind this startling notion is that the molecular weight of water is 18 and the weight of a liter of water is a kilogram: 1000 ÷ 18 = 55.5. Any solute added to water takes up space, displacing water molecules and so reducing their concentration. Table 3-2 provides one example of how common ions might be found in a peripheral nerve cell and its surrounding fluid. Also shown are the protein concentrations, with the cell protein content being 10-fold greater than the protein in the body’s interstitium and 100-fold greater than that in the cerebrospinal fluid, which is the brain’s interstitial fluid. Because of this higher concentration inside the cell, the cell protein displaces, and thus dilutes, the neuron’s water. Since all substances spontaneously tend to move from regions of a higher concentration to regions of a lower concentration, water will tend to move into the cell from the interstitium, causing the cell to swell.

Table 3-2 Distribution of Osmotically Active Particles

Maintenance of proper cell volume is so important that a variety of systems have evolved to counter the presence of cell proteins and to adjust to changing interstitial conditions. Indeed, pathologists generally see abnormal swelling in metabolically compromised cells, when the processes that counter this tendency are no longer adequately functioning. The following sections explain the physical principles used by these systems.

Osmotic forces measure the tendency of water to move down its concentration gradient; but our analytic instruments measure the solutes (sodium, potassium, chloride, sucrose—the dissolved substance), not the water. Hence it is the solute that gets all the attention and not the water. As a result, we dissemble when we say that osmotic forces tend to move water from a more dilute solution (of solute, that is) to a more concentrated one. In truth, the higher concentration of water is in the solution that has the lower concentration of solute, and water does in fact move as required by the laws of entropy, namely, from the solution of higher (water) concentration to the solution of lower (water) concentration.

Aquaporins (AQPs, Fig. 3-1) are the proteins in cell membranes that allow cells to reach osmotic equilibrium rapidly. AQPs contain transmembrane pores so specialized for the transport of water that they allow it to move almost as fast as in bulk solution—3 × 10⁹ molecules per second for each pore—while excluding all other molecules. (Such a membrane is termed semipermeable.) At least 10 distinct aquaporins are present in various cells in the human body, distributed in a tissue-specific manner. Congenital abnormalities result from the absence of specific AQPs: lack of AQP0 leads to cataracts; lack of AQP4 leads to deafness because it is needed for proper cochlear function; and AQP1 is required for adequate intraocular pressure.

Figure 3-1. The tetrameric aquaporin channel has a fourfold symmetry that centers about rigid protrusions and a central dimple. The four pores are ringed by a highly mobile outer structure (A). A computer-simulated cutaway diagram shows the path for water flux (B). The narrowest part of the channel excludes larger molecules; the inner structure has a high dielectric that effectively substitutes for bulk water, allowing single water molecules to corkscrew through the channel. (B From de Groot BL, Grubmüller H: Water permeation across biological membranes: Mechanism and dynamics of aquaporin-1 and GlpF. Science 294:2353-2357, 2001.)

Cells counter the osmotic force exerted by the high intracellular concentrations of protein by making a predominantly extracellular particle (sodium) impermeant as well. Consequently, sodium is generally more concentrated outside the cell (extracellular) than inside the cell (intracellular) (Table 3-2). The osmotic force (Π for pressure) exerted by this one ionic gradient is huge, being proportional to the difference between the two concentrations (Table 3-3, the van’t Hoff equation):

Table 3-3 Equations Governing Electrical and Chemical Forces at Equilibrium

Thus it is supremely important for the integrity of the cell that the contributions of all osmotically active particles sum to as close to zero as possible. The adjustments available to the cell that correct for small imbalances are the topic of the next section.

Electrical Forces

Cell proteins generally carry negative charges, and this large quantity of impermeant charge has important electrical consequences. In any solution, the number of positive and negative charges must be equal—the principle of microscopic electroneutrality. Because many of the cell’s negative charges are on proteins, the remaining intracellular anions (like chloride) must be reduced relative to their extracellular concentration. In osmotic terms, this concentration gradient acts together with sodium to minimize water flux across the cell membrane. In addition, chloride is also charged, and so electrical forces (measured as voltages) must also come into play as can best be understood by examining the numerical consequences of these interactions.

The best way to visualize the electrical forces involved in this chloride gradient is an approach that is analogous to the van’t Hoff equation (Table 3-3), namely, to calculate the force (in this case a voltage) that is generated by the ionic gradient by use of the Nernst equation (also in Table 3-3). For instance,

This mental image of chemical forces giving rise to electrical forces can be expressed in three equivalent ways: first, this voltage (V_Cl) is the potential at which Cl_o (chloride outside) is in equilibrium with Cl_i (chloride inside); second, the concentration gradient is just offset by electrical forces at V_Cl; and third, at V_Cl, chloride movements into the cell are just equal to chloride movements out of the cell. Hence, this Nernst potential is also called the equilibrium potential for chloride, or simply the chloride potential.

Cations have the opposite valence of anions (the z term in the Nernst equation), and so (by the properties of the logarithm) the concentration gradient is inverted. Thus, for potassium, whose equilibrium potential is similar to that of chloride, the intracellular concentration exceeds the extracellular concentration (Table 3-2).

Returning to the case of sodium, its equilibrium potential is very different from that of potassium or chloride:

This difference is easily tolerated because sodium is the effectively impermeant ion in most cells. Indeed, the steep sodium concentration gradient is used to great advantage in the effective transport of fluid and generation of APs, as we shall soon see.

Membrane Potential

In the long run, individual neurons exist at a steady state, with osmotic forces across the cell membrane balanced and with the concentration gradients of the permeant ions offset by a characteristic voltage (Fig. 3-2). The relationship among these electrochemical parameters was first visualized by Goldman and further developed by Hodgkin and Katz as being governed by the permeabilities of ions across the cell membrane (Table 3-3). This relation is derived from the Nernst-Planck equation, with the algebraic contributions of individual ions being in proportion to P_S, their steady-state membrane permeability. Although the P_S is itself difficult to measure and awkward to express, relative permeabilities are more straightforward concepts. For instance, it is easily demonstrated experimentally that the potassium permeability of a nonmyelinated nerve axon is 100 times that of sodium (and then easy to say that ). By use of these relative permeabilities, and postponing consideration of chloride until later, a more tractable form of the Goldman-Hodgkin-Katz voltage equation would be

Potassium is now obviously the dominant ion in this calculation, with only a modest contribution from extracellular sodium.

Figure 3-2. All membranes have pumps and channels. Channel proteins have water-filled pores that selectively allow small molecules to pass through the membrane. Illustrated here are three variations of those channels that are open continuously, one selective for sodium, one for potassium, and one for chloride. According to the concentration gradients diagrammed here, sodium will tend to enter the cell and potassium will tend to leave the cell, as will chloride. Pumps differ from channels because their water-filled cavity is open to only one side of the membrane at a time. Here the sodium pump is shown first accepting three intracellular sodium ions. After being phosphorylated by an ATP, the pump becomes closed to the interior and opens to the exterior, losing its affinity for the sodium ions, which consequently diffuse away. Next, two potassium ions enter the pump, and when the high-energy phosphate group is lost, the pump closes to the outside and opens to the inside, losing its affinity for the potassium ions; consequently, they diffuse away, and the cycle is set to begin again.

This calculation is exactly the reason that we have introduced the concepts of electrical and concentration forces, namely, to demonstrate that the membrane potential is primarily due to the diffusion of potassium from the cell, withdrawing positive charges until the electrical potential across the membrane becomes approximately equal—but opposite—to the force generated by the potassium concentration gradient.

Cell chloride differs from the cations we have discussed not only because it is negatively charged but also because it is often in electrochemical equilibrium with its surroundings, which is to say that V_Cl = V_m. This is true in some nerve and muscle cells in which chloride serves to stabilize the membrane potential. In these cases in which the chloride equilibrium potential equals the membrane potential, V_Cl contributes nothing to the Goldman-Hodgkin-Katz calculation and can be safely omitted, as shown in the previous example. In other cells, chloride ions are pumped out, making V_Cl more negative than V_m. This is true in some postsynaptic nerve terminals, and when inhibitory neurotransmitters open chloride permeant channels, chloride ions diffuse passively into the neuron, transiently making the membrane potential more negative and thus more difficult to trigger an AP. Finally, chloride is actively accumulated in epithelial cells that secrete fluid, which is the subject of the next section.

Fluid Transport by Epithelia

The nervous system requires highly specialized fluids in the extracellular spaces of the brain, cochlea, and eye for the proper function of these organs. The brain is bathed in cerebrospinal fluid (CSF), a solution low in protein that is generated by the choroid plexus and removed through the arachnoid villi. The cochlear endolymph is high in potassium, and the ciliary body of the eye is continuously producing a nutrient solution that flows past the lens and is taken up by specialized veins along the margin of the iris.

In each case, the specialized fluid is generated across an epithelial layer by the judicious placement of membrane pumps and channels (Fig. 3-2). Epithelial cells have two functionally distinct surfaces: the base and the sides (or basolateral surface), which are in contact with the interstitial fluid of the body; and the apical surface, which faces the lumen. Almost all epithelia restrict the sodium pump to the basolateral surface; the two exceptions are the choroid plexus and the retinal pigmented epithelium, in which the sodium pump is exclusively in the apical membrane. Individual epithelia then distinguish themselves by distributing characteristic channels and transporters on their apical and basolateral surfaces.

The sodium pump (a sodium-potassium adenosine triphosphatase, or ATPase) is a molecule present in all cells that moves sodium out of the cell and potassium into the cell (Fig. 3-2). This exchange generates a steep concentration gradient for the two ions, fueled by the chemical energy stored in molecules of ATP (hence, primary active transport). We have already seen that the potassium gradient determines the steady-state membrane potential. The sodium gradient is not only the basis of the AP, but it can also be harnessed to move large quantities of fluid. For instance, Na/2Cl/K cotransport pumps are present on the apical (ventricular) surface of the cells in the choroid plexus. The sodium electrochemical gradient provides the energy for this secondary active transport mechanism to drive potassium and chloride into the cell; specialized chloride channels on the apical surface then allow those ions to diffuse out of the cell. In addition, large quantities of carbonic anhydrase are present in the cells lining the choroid plexus, generating HCO^–₃ ions that accompany chloride into the CSF. These chloride and bicarbonate ions are accompanied by passive movements of sodium and water molecules as dictated by electrical and osmotic forces. (Potassium channels on the basolateral surface recycle that ion back into the interstitium.) Other cotransport mechanisms move nutrients, antibiotics, and a wide variety of other organic molecules into the CSF, generating liters of CSF a week, all the time keeping an effective barrier against erythrocytes, leukocytes, and plasma proteins.

Modifying the placement and nature of the cell’s pumps and channels alters the composition of the secreted fluid. For instance, cochlear endolymph is high in potassium because its epithelial cells have their potassium channels on the apical side, causing the potassium and chloride pumped into the cell by the basolateral Na/2Cl/K cotransporter to exit the cell together into the scala media. Thus it can be seen that the large and effective concentration gradients and the osmotic and electrical forces present in neurons require energy input by the sodium-potassium pump. However, the body has adapted these forces, each required for the physical integrity of the cell, to a wide variety of other purposes.

Ohm’s Law and the Equivalent Electrical Circuit of the Neuron

The resting membrane potential refers to the neuron at a steady state and is largely the result of the potassium, chloride, and sodium diffusion potentials; in addition, membrane currents—electrical charges carried by ions crossing the cell membrane—modify this voltage as described by Ohm’s law:

Thus current (I, in amperes) flowing through a conductance (G, in siemens) or across a resistance (R, in ohms) will generate a voltage drop (Fig. 3-3). By a convention established by Benjamin Franklin, electrical current is the flow of positive charges: positive charges leaving the cell are defined as a positive current. Equivalently, negative charges entering the cell are also a positive current. Conversely, positive charges entering the cell are a negative current, as is the exit of negative charges. A familiar example is the sodium pump, which is electrogenic because it cycles three sodium ions out of the cell for every two potassium ions in; this net positive current removes positive charges from the cell interior, causing the membrane potential to be more negative than predicted by the Goldman-Hodgkin-Katz voltage equation. In a cell as large as a skeletal muscle fiber, this amounts to ~2 to 5 mV. In small nerve terminals, where the input resistance is much greater, this current can hyperpolarize the membrane by 15 mV or more.

Figure 3-3. In a simple resistive electrical circuit, voltage (V) is imposed by a battery, much like an ionic concentration gradient across a cell membrane. Current (I) will flow through a resistor (R), which has a conductance (G). The conductance of resistors in parallel, such as channels in a membrane, sums algebraically. In a circuit where a voltage is impressed across a capacitor (C), such as the lipid bilayer of a membrane, a charge (Q, in coulombs) can be held by the capacitor and is proportional to V × C. Capacitors in series, such as found in the many-fold wrappings of the myelin sheath, add as their inverse, which makes myelinated nerves very well insulated with very little membrane capacitance to charge during an action potential:

Of even more interest is the flow of current through open membrane channels because the number of open channels varies when the nerve is stimulated in any of a wide variety of ways. From Ohm’s law, the magnitude and direction of the flow of an individual ion S through the cell membrane equals the driving force on that ion times its conductance:

The electrical driving force on S is the difference between the voltage across the membrane (V_m) and that voltage where the ion is at electrochemical equilibrium (V_S, the Nernst potential for substance S). These relationships are summarized diagrammatically, for those familiar with electrical circuits, in Figure 3-4. Thus the magnitude of the ionic flow will increase as the driving force—or the conductance—of the ion increases and decrease as it decreases. In the face of an increasing conductance and a decreasing driving force, a situation that is described in the section on the AP, specific calculations are required to determine the final outcome.

Figure 3-4. The cell membrane has conductive paths for sodium, potassium, and chloride, and so the concentration gradients of these ions exert electrochemical forces across the membrane. Because the conductance paths are in parallel, the driving forces of the ions combine in proportion to their relative permeabilities to generate a voltage across the membrane capacitance.

We can use the circuit theory in Figure 3-3 to calculate the number of charges (Q) on a cell membrane that has a given voltage (V) because Q = V × C_m, and the capacitance of a cell membrane (C_m) has been measured to be 0.9 µF/cm². If a hypothetical neuron were spherical (for simplicity) and 20 µm in diameter, it would have a surface area of almost 1300 µm², a capacitance of 11 pF per cell, and thus a charge of 1 pC (picocoulomb) when the membrane voltage is 90 mV. The 1 pC of charge on the membrane represents 6 million ions. Although this may seem to be a lot, the cell volume of this neuron would be 4 pL (picoliters) and contain ~250 billion potassium ions and ~25 billion chloride ions. Together, these two ions alone are ~40,000 times the number required to charge the membrane.

Pain and a Syndrome of Periodic Paralysis

Pathologic conditions may alter the concentration of ions ordinarily seen in nerve cells (Table 3-2). For instance, tissue injury causes a local increase in the potassium concentration as these cells release their contents. This increased extracellular potassium moves the steady-state membrane potential toward 0 mV, which generates APs when it is done quickly enough. Thus one source of pain is simply the direct stimulation of nerve endings by elevated potassium concentration in the tissue interstitium.

In rare individuals who have certain genetic abnormalities, extracellular concentration of potassium can fall dramatically when epinephrine or insulin stimulates its uptake by muscle cells, leading to muscle weakness and even paralysis. This condition is called hypokalemic periodic paralysis. Surprisingly, the muscle membrane potentials are less negative than normal, just the opposite of what the Nernst equation predicts. For reasons not yet fully understood, the cell membrane loses its ability to select potassium over sodium, which is to say declines markedly. The effect is so great that the cell is depolarized because the membrane potential moves away from V_K toward V_Na, as the Goldman-Hodgkin-Katz voltage equation predicts. This membrane potential change is slow, allowing the muscle fiber to undergo accommodation and so become inexcitable. (Accommodation is explained more fully in the section on APs.)

Attack by the Immune System: Membrane Attack Complexes and Porins

The nervous system can be attacked by bacteria and even the body’s own immune system in ways that short-circuit membrane potentials and destroy the cell’s integrity. Attack by the immune system depolarizes the cell membrane by insertion of nonselective channels into cell membranes. This mechanism is one weapon of self-defense used by the human body’s own cellular and humoral immune systems. The pore-forming elements of the body’s immune system are the porins, from killer T lymphocytes; defensins, produced by phagocytes and epithelial cells; and two elements of the complement cascade, C8 and C9. C8 forms individual, <3-nm pores; C9 aggregates to form >10-nm pores, termed membrane attack complexes (MACs, Fig. 3-5). (MACs attack myelin sheaths of motor neurons, causing a paralysis discussed later in this chapter.) At 16 nm, porins are even larger.

Figure 3-5. Membrane attack complexes are stable membrane structures formed by elements of the complement system, a polymerized ring of 13 C9 monomers.

The channels through MACs and porins are wide enough to easily pass sodium, potassium, chloride, and sucrose, discriminating little among them. The channels’ conductances are correspondingly large: 2 nS (nanosiemens) for the MAC and 6 nS for porin. As a consequence, the formation of just a single C9 aggregate or insertion of a single porin molecule results in a large flow of ions. As in all nonselective channels, the currents flowing through MAC and porin channels are carried primarily by sodium ions because the magnitude of the driving force for sodium is the greatest: |V_m − V_Na| >> |V_m − V_Cl| > |V_m − V_K|. If V_m = −90 mV (to take a simple example), the driving force on the sodium will be

and the current flowing through the MAC is easy to calculate:

Because one ampere is the flow of 6.3 × 10¹⁸ charges per second, the number of sodium ions flowing through a single MAC is 2 × 10⁹ per second, a rate so fast that the membrane potential of our previous example (the 20-µm neuron) would be neutralized in [(6 × 10⁶) ÷ (2 × 10⁹)] = 3 × 10⁻³ seconds.

By following the movements of the various ions due to the electrical forces on them, it is possible to see that the cell gains osmotic particles and so swells. For instance, because the net driving force on sodium is negative, it will enter the cell, with its positive charge tending to cancel the negativity of the membrane potential. In fact, the cell membrane potential will rapidly go to zero and remain there as long as the MACs are in the membrane. As a consequence, the driving force on chloride will increase; (V_m − V_Cl) being positive, the current will be positive, meaning that chloride is entering the cell along with the sodium. Thus, when sodium and chloride enter the cell, water follows, and the cell swells. Even more important, the MAC complexes are so large that molecules the size of ATP can diffuse from the cell. Thus attack by complement or killer T lymphocytes leads inexorably to cell swelling and lysis both because important metabolic contents are lost and because the osmotic pressures exerted by the remaining cell proteins cause cell swelling and death.

Microbial Attacks: Antibiotics

Insertion of ion channels into cell membranes is also a weapon deployed by many microorganisms. Antibiotics such as amphotericin and gramicidin, and α-staphylotoxins from Staphylococcus aureus, lyse cells by broaching their membranes with large pores. When it is used clinically, amphotericin preferentially attacks fungal cells in fungal meningitis, but there is a narrow therapeutic range because amphotericin also attacks cell membranes in the nervous system. In overwhelming sepsis, α-staphylotoxins attack all cells of the body, leading to multiple organ failure and cardiovascular collapse, primarily due to the loss of the integrity of cell membranes.

GRADED POTENTIALS

Electrical events underlie much of the nerve activity in our body. Indeed, many of our ordinary feelings and sensations begin with graded potentials

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