The propagation of electrical signals in the nervous system involves local current loops
The extraordinary functional diversity of ion channel proteins provides a large array of mechanisms by which the membrane potential of a cell can be changed to evoke an electrical signal or biochemical response. However, channels alone do not control the spread of electrical current. Like electricity in a copper wire, the passive spread of current in biological tissue depends on the nature of the conducting and insulating medium. Important factors include geometry (i.e., cell shape and tissue anatomy), electrical resistance of the aqueous solutions and cell membrane, and membrane capacitance. Furthermore, the electrotonic spread of electrical signals is not limited to excitable cells.
Efficient propagation of a change in V_{m} is essential for the local integration of electrical signals at the level of a single cell and for the global transmission of signals across large distances in the body. As we discussed above in this chapter (see Fig. 7-2), action potentials propagate in a regenerative manner without loss of amplitude as long as the depolarization spreads to an adjacent region of excitable membrane and does so with sufficient strength to depolarize the membrane above its threshold. However, many types of nonregenerative, subthreshold potentials also occur and spread for short distances along cell membranes. These graded responses, which we also discussed above, contrast with the all-or-nothing nature of action potentials. Such nonregenerative signals include receptor potentials generated during the transduction of sensory stimuli and synaptic potentials generated by the opening of agonist-activated channels.
With a graded response, the greater the stimulus, the greater the voltage change. For example, the greater the intensity of light that is shined on a mammalian photoreceptor cell in the retina, the greater the hyperpolarization produced by the cell. Similarly, the greater the concentration of acetylcholine applied at a postsynaptic neuromuscular junction, the greater the resulting depolarization (i.e., synaptic potential). Just as subthreshold voltage responses produced by current injections into a cell through a microelectrode decline by passive electrotonic spread, graded signals dissipate over distances of a few millimeters and thus have only local effects. In contrast, action potentials—actively generated by above-threshold depolarization—can propagate over a meter via nerve axons.
Electrotonic spread of voltage changes along the cell occurs by the flow of electrical current that is carried by ions in the intracellular and extracellular medium along pathways of the least electrical resistance. Both depolarizations and hyperpolarizations of a small area of membrane produce local circuit currents. Figure 7-21A illustrates how the transient voltage change that occurs during an action potential at a particular active site results in local current flow. The cytosol of the active region, where the membrane is depolarized, has a slight excess of positive charge compared with the adjacent inactive regions of the cytosol, which have a slight excess of negative charge. N7-18 This charge imbalance within the cytosol causes currents of ions to flow from the electrically excited region to adjacent regions of the cytoplasm. Because current always flows in a complete circuit along pathways of least resistance, the current spreads longitudinally from positive to negative regions along the cytoplasm, moves outward across membrane conductance pathways (“leak channels”), and flows along the extracellular medium back to the site of origin, thereby closing the current loop. Because of this flow of current (i.e., positive charge), the region of membrane immediately adjacent to the active region becomes more depolarized, and V_{m} eventually reaches threshold. Thus, an action potential is generated in this adjacent region as well. Nerve and muscle fibers conduct impulses in both directions if an inactive fiber is excited at a central location, as in this example. However, if an action potential is initiated at one end of a nerve fiber, it will travel only to the opposite end and stop because the refractory period prevents backward movement of the impulse (see p. 302). Likewise, currents generated by subthreshold responses migrate equally in both directions.
FIGURE 7-21 Local current loops during action potential propagation. A, In an unmyelinated axon, the ionic currents flow at one instant in time as a result of the action potential (“active” zone). In the “inactive” zones that are adjacent to the active zone, the outward currents lead to a depolarization. If the membrane is not in an absolute refractory period and if the depolarization is large enough to reach threshold, the immediately adjacent inactive zones will become active and fire their own action potential. In the more distant inactive zones, the outward current is not intense enough to cause V_{m} to reach threshold. Thus, the magnitudes of the outward currents decrease smoothly with increasing distance from the active zone. B, In this example, the “active” zone consists of a single node of Ranvier. As an action potential conducts along a myelinated axon, the ionic current flows only through the nodes, which lack myelin and have a very high density of Na^{+} channels. Ionic current does not flow through the internodal membrane because of the high resistance of myelin. Rather, as an action potential fires at a node, the resistance-capacitance properties of the adjacent internodal membrane (see Fig. 6-11) give rise to charge displacement across this internodal membrane—that is, a capacitative current—that actually is responsible for conducting the action potential through the internodal region. As a result, the current flowing down the axon is conserved, and the current density at the nodes is very high, triggering an action potential at the next node. Thus, the regenerative action potential propagates in a “saltatory” manner by jumping from node to node.
N7-18
Charge Separation Required to Generate the Membrane Potential
Contributed by Ed Moczydlowski
To generate a membrane potential, there must be a tiny separation of charge across the membrane. How large is that charge? Imagine that we have a spherical cell with a diameter of 10 µm. If [K^{+}]_{i} is 100 mM and [K^{+}]_{o} is 10 mM, the V_{m} according to the Nernst equation would be −61.5 mV (or −0.0615 V) for a perfectly K^{+}-selective membrane at 37°C. What is the charge (Q) on 1 cm^{2} of the “plates” of the membrane capacitor? We assume that the specific capacitance is 1 µF/cm^{2}. From Equation 6-13 in the text, we know that
(NE 7-11)
where Q is measured in coulombs (C), C is in farads (F), and V is in volts (V). Thus,
(NE 7-12)
The Faraday is the charge of 1 mole of univalent ions—or 96,480 C N6-12. To determine how many moles of K^{+} we need to separate in order to achieve an electrical charge of 61.5 × 10^{−9} C cm^{−2} (i.e., the Q in the previous equation), we merely divide Q by the faraday. Because V_{m} is negative, the cell needs to lose K^{+}:
(NE 7-13)
The surface area for a spherical cell with a diameter of 10 µm is 3.14 × 10^{−6} cm^{2}. Therefore,
(NE 7-14)
The volume of this cell is 0.52 × 10^{−12} L. Given a [K^{+}]_{i} of 100 mM,
(NE 7-15)
What fraction of the cell's total K^{+} content represents the charge separated by the membrane?
(NE 7-16)
Thus, in the process of generating a V_{m} of −61.5 mV, our hypothetical cell needs to lose only 0.004% of its total K^{+} content to charge the capacitance of the cell membrane.
Myelin improves the efficiency with which axons conduct action potentials
The flow of electrical current along a cylindrical nerve axon has often been compared with electrical flow through an undersea cable. Similar principles apply to both types of conducting fiber. An underwater cable is designed to carry an electrical current for long distances with little current loss; therefore, it is constructed of a highly conductive (low-resistance) metal in its core and a thick plastic insulation wrapped around the core to prevent loss of current to the surrounding seawater. In contrast, the axoplasm of a nerve fiber has much higher resistance than a copper wire, and the nerve membrane is inherently electrically leaky because of background channel conductance. Therefore, in a biological fiber such as a nerve or muscle cell, some current is passively lost into the surrounding medium, and the amplitude of the signal rapidly dissipates over a short distance.
Animal nervous systems use two basic strategies to improve the conduction properties of nerve fibers: (1) increasing the diameter of the axon, thus decreasing the internal resistance of the cable; and (2) myelinating the fibers, which increases the electrical insulation around the cable. As axon diameter increases, the conduction velocity of action potentials increases because the internal resistance of the axoplasm is inversely related to the internal cross-sectional area of the axon. Unmyelinated nerve fibers of the invertebrate squid giant axon (as large as ~1000 µm in diameter) are a good example of this type of size adaptation. These nerve axons mediate the escape response of the squid from its predators and can propagate action potentials at a velocity of ~25 m/s.
In vertebrates, myelination of smaller-diameter (~1- to 5-µm) nerve axons serves to improve the efficiency of impulse propagation, especially over the long distances that nerves traverse between the brain and the extremities. Axons are literally embedded in myelin, which consists of concentrically wound wrappings of the membranes of certain glial cells (see pp. 292–293). The thickness of the myelin sheath may amount to 20% to 40% of the diameter of a nerve fiber, and the sheath may consist of as many as 300 membrane layers. The glial cells that produce myelin are called Schwann cells in the periphery and oligodendrocytes in the brain. Because resistors in series add directly and capacitors in series add as the sum of the reciprocal, the insulating resistance of a myelinated fiber with 300 membrane layers is increased by a factor of 300 and the capacitance is decreased to 1/300 that of a single membrane. This large increase in membrane resistance minimizes loss of current across the leaky axonal membrane and forces the current to flow longitudinally along the inside of the fiber.
In myelinated peripheral nerves, the myelin sheath is interrupted at regular intervals, forming short (~1-µm) uncovered regions called nodes of Ranvier. The length of the myelinated axon segments between adjacent unmyelinated nodes ranges from 0.2 to 2 mm. In mammalian axons, the density of voltage-gated Na^{+} channels is very high at the nodal membrane, whereas K^{+} channels are localized in the paranodal regions flanking each node. The unique anatomy of myelinated axons results in a mode of impulse propagation known as saltatory conduction. Current flow that is initiated at an excited node flows directly to adjacent nodes with little loss of transmembrane current through the internode region (see Fig. 7-21B). In other words, the high membrane resistance in the internode region effectively forces the current to travel from node to node.
The high efficiency of impulse conduction in such axons allows several adjacent nodes in the same fiber to fire an action potential virtually simultaneously as it propagates. Thus, saltatory conduction in a myelinated nerve can reach a very high velocity, up to 130 m/s. The action potential velocity in a myelinated nerve fiber can thus be several-fold greater than that in a giant unmyelinated axon, even though the axon diameter in the myelinated fiber may be more than two orders of magnitude smaller. During conduction of an action potential in a myelinated axon, the intracellular regions between nodes also transiently depolarize as a result of capacitative current. However, no transmembrane current flows in these internodal regions, and therefore no dissipation of ion gradients occurs. The nodal localization of Na^{+} channels conserves ionic concentration gradients that must be maintained at the expense of ATP hydrolysis by the Na-K pump.
The cable properties of the membrane and cytoplasm determine the velocity of signal propagation
Using the analogy of a nerve fiber as an underwater cable, cable theory allows one to model the pathways of electrical current flow along biomembranes. The approach is to use circuit diagrams that were first employed to describe the properties of electrical cables. Figure 7-22A illustrates the equivalent circuit diagram of a cylindrical electrical cable or membrane that is filled and bathed in a conductive electrolyte solution. The membrane itself is represented by discrete elements, each with a transverse membrane resistance (r_{m}) and capacitance (c_{m}) connected in parallel (a representation we used above, in Fig. 6-11A). Consecutive membrane elements are connected in series by discrete resistors, each of which represents the electrical resistance of a finite length of the external medium (r_{o}) or internal medium (r_{i}). The parameters r_{m}, c_{m}, r_{o}, and r_{i} refer to a unit length of axon (Table 7-3).
FIGURE 7-22 Passive cable properties of an axon. A, The axon is represented as a hollow, cylindrical “cable” that is filled with an electrolyte solution. All of the electrical properties of the axon are represented by discrete elements that are expressed in terms of the length of the axon. r_{i} is the resistance of the internal medium. Similarly, r_{o} is the resistance of the external medium. r_{m} and c_{m} are the membrane resistance and capacitance per discrete element of axon length. B, When current is injected into the axon, the current flows away from the injection site in both directions. The current density smoothly decays with increasing distance from the site of injection. C, Because the current density decreases with distance from the site of current injection in B, the electrotonic potential (V) also decays exponentially with distance in both directions. V_{o} is the maximum change in V_{m} that is at the site of current injection.
TABLE 7-3
Cable Parameters N7-19
PARAMETER |
UNITS |
DEFINITION OR RELATIONSHIP |
r_{m} |
Ω × cm |
Membrane resistance (per unit length of axon) |
r_{o} |
Ω/cm |
Extracellular resistance (per unit length of axon) |
r_{i} |
Ω/cm |
Intracellular resistance (per unit length of axon) |
c_{m} |
µF/cm |
Membrane capacitance (per unit length of axon) |
R_{m} = r_{m} × 2πa |
Ω × cm^{2} |
Specific membrane resistance (per unit area of membrane) |
R_{i} = r_{i} × πa^{2} |
Ω × cm |
Specific internal resistance (per unit cross-sectional area of axoplasm) |
C_{m} = c_{m}/(2πa) |
µF/cm^{2} |
Specific membrane capacitance (per unit area of membrane) |
a, Radius of the axon.
How do the various electrical components of the cable model influence the electrotonic spread of current along an axon? To answer this question, we inject a steady electrical current into an axon with a microelectrode to produce a constant voltage (V_{0}) at a particular point (x = 0) along the length of the axon (see Fig. 7-22B). This injection of current results in the longitudinal spread of current in both directions from point x = 0. The voltage (V) at various points along the axon decays exponentially with distance (x) from the point of current injection (see Fig. 7-22C), according to the following equation:
(7-5)
The parameter λ has units of distance and is referred to as the length constant or the space constant. One length constant away from the point of current injection, V is 1/e, or ~37% of the maximum value of V_{0}. The decaying currents that spread away from the location of a current-passing electrode are called electrotonic currents. Similarly, the passive spread of subthreshold voltage changes away from a site of origin is referred to as electrotonic spread, unlike the regenerative propagation of action potentials.
The length constant depends on the three resistance elements in Figure 7-22A: N7-19
(7-6)
N7-19
Resistance and Capacitance Units for Cable Properties
Contributed by Emile Boulpaep, Walter Boron
The purpose of this webnote is to justify the units given in the first four rows of Table 7-3. Our approach is to present these four electrical units per unit length of axon and then to show how these units make sense when we calculate the resistance or capacitance of an entire axon.
The longitudinal resistances of the intracellular fluid (r_{o}) and extracellular fluids (r_{i}) are expressed in units of ohms per centimeter. In each case, we can think of the total resistance of either the intra- or extracellular fluid as being the resistance of a stack of resistors in series—each resistor representing 1 cm of fluid length along the axis of the cable. Thus, the total resistance of either the intra- or extracellular fluid (in ohms) increases in proportion to increasing fiber length (in centimeters), and the proportionality factor is the longitudinal resistance of each of the N segments (in ohms per centimeter):
Because each N is in fact 1 cm,
(NE 7-17)
The transverse membrane resistance (r_{m}) is expressed in the units ohms × centimeters. We can think of the total resistance of the membrane as being the resistance of a stack of resistors in parallel—each resistor representing the membrane resistance of a segment of axon that is 1 cm long. Because the N resistors are arranged in parallel,
Rearranging,
(NE 7-18)
Finally, inserting the units for the individual resistance, and realizing that each N is in fact 1 cm, we obtain
(NE 7-19)
One way to think of this is that a longer section of axon membrane has more channels, a greater total conductance, and thus a lower total resistance.
The membrane capacitance (c_{m}) has the units microfarads per centimeter. We can think of the total capacitance of the membrane as being the capacitance of a stack of capacitors in parallel—each capacitor representing the membrane capacitance of a segment of axon that is 1 cm long. Because the N capacitors are arranged in parallel,
Because each N is in fact 1 cm,
(NE 7-20)
Thus, just as longitudinal resistances in series summate, membrane capacitances in parallel summate; both are thus expressed per unit length.
We can simplify this expression by noting that internal resistance is much larger than external resistance, so the contribution of r_{o} to the denominator can be ignored. Thus,
(7-7)
The significance of the length constant is that it determines how far the electrotonic spread of a local change in membrane potential can extend to influence neighboring regions of membrane. The longer the length constant, the farther down the axon a voltage change spreads.
How does the diameter of an axon affect the length constant? To answer this question, we must replace r_{m} and r_{i} (expressed in terms of axon length) in Equation 7-7 with the specific resistances R_{m} and R_{i}(expressed in terms of the area of axon membrane or cross-sectional area of axoplasm). Making the substitutions according to the definitions in Table 7-3, we have N7-20
(7-8)
N7-20
Units of the “Length Constant”
Contributed by Emile Boulpaep, Walter Boron
In our discussion of cable properties in the text, we presented Table 7-3, which summarizes the units of cable parameters in two ways: (1) resistance per unit length (top three rows in Table 7-3) and (2) specific resistance (fifth and sixth rows in Table 7-3).
Resistance per Unit Length
In the text, we presented Equation 7-6 (shown here as Equation NE 7-21):
(NE 7-21)
The above equation uses resistance per unit length. If we substitute the appropriate units from Table 7-3 into the above equation, we obtain
(NE 7-22)
Thus, the length constant has units of distance (centimeters).
Specific Resistance
In the text, we presented Equation 7-8 (shown here as Equation NE 7-23):
(NE 7-23)
The above equation uses specific resistance. If we substitute the appropriate units from Table 7-3 into the preceding equation, we obtain
(NE 7-24)
Thus, the length constant again has units of distance (centimeters).
Thus, the length constant (λ) is directly proportional to the square root of the axon radius (a). Equation 7-8 confirms basic intuitive notions about what makes an efficiently conducting electrical cable:
1. The greater the specific membrane resistance (R_{m}) and cable radius, the greater the length constant and the less the loss of signal.
2. The greater the resistance of the internal conductor (R_{i}), the smaller the length constant and the greater the loss of signal.
These relationships also confirm measurements of length constants in different biological preparations. For example, the length constant of a squid axon with a diameter of ~1 mm is ~13 mm, whereas that of a mammalian nerve fiber with a diameter of ~1 µm is ~0.2 mm.
So far, we have been discussing the spatial spread of voltage changes that are stable in time. In other words, we assumed that the amount of injected current was steady. What happens if the current is not steady? For example, what happens at the beginning of a stimulus when we (or a physiological receptor) first turn the current “on”? To answer these questions, we need to know how rapidly V_{m} changes in time at a particular site, which is described by a second cable parameter called the membrane time constant (τ_{m}). Rather than determining the spread of voltage changes in space, as the length constant does, the time constant influences the spread of voltage changes in time and thus the velocity of signal propagation. We previously discussed the time constant with respect to the time course of the change in V_{m} caused by a stepwise pulse of current (see Fig. 6-12A). Because the membrane behaves like an RC circuit, the voltage response to a square current pulse across a small piece of membrane follows an exponential time course with a time constant that is equal to the product of membrane resistance and capacitance:
(7-9)
We introduced this expression above as Equation 6-17. The shorter the time constant, the more quickly a neighboring region of membrane will be brought to threshold and the sooner the region will fire an action potential. Thus, the shorter the time constant, the faster the speed of impulse propagation and vice versa. In contrast, conduction velocity is directly proportional to the length constant. The greater the length constant, the further a signal can spread before decaying below threshold and the greater the area of membrane that the stimulus can excite. These relationships explain why, in terms of relative conduction velocity, a high-resistance, low-capacitance myelinated axon has a distinct advantage over an unmyelinated axon of the same diameter for all but the smallest axons (<1 µm in diameter; see p. 302).
In summary, the cable parameters of length constant and time constant determine the way in which graded potentials and action potentials propagate over space and time in biological tissue. These parameters are in turn a function of material properties that include resistance, capacitance, and geometric considerations. The dependence of impulse conduction velocity on fiber diameter has been studied experimentally and analyzed theoretically for unmyelinated and myelinated nerve axons. For unmyelinated axons, conduction velocity increases roughly with the square root of the axon's diameter, just as the length constant increases with the square root of the axon's diameter or radius (see Equation 7-8). In contrast, the conduction velocity of myelinated fibers is a linear function of diameter and increases ~6 m/s per 1-µm increase in outer diameter. Thus, a mammalian myelinated axon with an outer diameter of ~4 µm has roughly the same impulse velocity as a squid giant axon with a diameter of 500 µm! However, for myelinated fibers with a very small diameter (<1 µm), the adverse effect of high internal resistance of the axoplasm predominates, and conduction is slower than in unmyelinated axons of the same outer diameter. For outer diameters that are greater than ~1 µm, the increased membrane resistance and reduced capacitance caused by myelination result in much faster conduction velocities.
The physiological importance of myelin in action potential propagation is most dramatically illustrated in the pathology that underlies human demyelinating diseases such as multiple sclerosis. This disease is an autoimmune disorder in which the myelin sheath surrounding CNS axons is progressively lost (see Box 12-1). Gradual demyelination is responsible for an array of neurological symptoms that involve various degrees of paralysis and altered or lost sensation. As myelin is eliminated, the loss of membrane resistance and increased capacitance mean that propagated action potentials may ultimately fail to reach the next node of Ranvier and thus result in nerve blockage.