Section 13.1
Chapter 13 Electricity external fluid is similar to sea water. Its ionic solutes are mostly positive sodium ions and negative chlorine ions. Inside the axon, the positive ions are mostly potassium ions, and the negative ions are mostly large negatively charged organic molecules.
Because there is a large concentration of sodium ions outside the axon and a large concentration of potassium ions inside the axon, we may ask why the concentrations are not equalized by diffusion. In other words, why don’t the sodium ions leak into the axon and the potassium ions leak out of it?
The answer lies in the properties of the axon membrane.
In the resting condition, when the axon is not conducting an electrical pulse, the axon membrane is highly permeable to potassium and only slightly permeable to sodium ions. The membrane is impermeable to the large organic ions. Thus, while sodium ions cannot easily leak in, potassium ions can certainly leak out of the axon. However, as the potassium ions leak out of the axon, they leave behind the large negative ions, which cannot follow them through the membrane. As a result, a negative potential is produced inside the axon with respect to the outside. This negative potential, which has been measured to be about 70 mV, holds back the outflow of potassium so that in equilibrium the concentration of ions is as we have stated. Some sodium ions do in fact leak into the axon, but they are continuously removed by a metabolic mechanism called the sodium pump. This pumping process, which is not yet fully understood, transports sodium ions out of the cell and brings in an equal number of potassium ions.
13.1.3 Action Potential
The description of the axon that we have so far given applies to other types of cells as well. Most cells contain an excess concentration of potassium ions and are at a negative potential with respect to their surroundings. The special property of the neuron is its ability to conduct electrical impulses.
Physiologists have studied the properties of nerve impulses by inserting a probe into the axon and measuring the changes in the axon voltage with respect to the surrounding fluid. The nerve impulse is elicited by some stimulus on the neuron or the axon itself. The stimulus may be an injected chemical, mechanical pressure, or an applied voltage. In most experiments the stimulus is an externally applied voltage, as shown in Fig. 13.4.
A nerve impulse is produced only if the stimulus exceeds a certain thresh old value. When this value is exceeded, an impulse is generated at the point of stimulation and propagates down the axon. Such a propagating impulse is called an action potential. An action potential as a function of time at one point on the axon is shown in Fig. 13.5. The scales of time and voltage are typical of most neurons. The arrival of the impulse is marked by a sudden
Section 13.1 The Nervous SystemFIGURE 13.4 Measuring the electrical response of the axon.
FIGURE 13.5 The action potential.
rise of the potential inside the axon from its negative resting value to about +30 mV. The potential then rapidly decreases to about −90 mV and returns more slowly back to the initial resting state. The whole pulse passes a given point in a few milliseconds. The speed of pulse propagation depends on the type of axon. Fast-acting axons propagate the pulse at speeds up to 100 m/sec.
The mechanism for the action potential is discussed in a following section.
Impulses produced by a given neuron are always of the same size and propagate down the axon without attenuation. The nerve impulses are produced at a rate proportional to the intensity of the stimulus. There is, however, Chapter 13 Electricity an upper limit to the frequency of impulses because a new impulse cannot begin before the previous one is completed.
13.1.4 Axon as an Electric Cable
In the analysis of the electrical properties of the axon, we will use some of the techniques of electrical engineering. This treatment is more complex than the methods used in the other sections of the text. The added complexity, however, is necessary for the quantitative understanding of the nervous system.
Although the axon is often compared to an electrical cable, there are pro found differences between the two. Still, it is possible to gain some insight into the functioning of the axon by analyzing it as an insulated electric cable submerged in a conducting fluid. In such an analysis, we must take into account the resistance of the fluids both inside and outside the axon and the electrical properties of the axon membrane. Because the membrane is a leaky insulator, it is characterized by both capacitance and resistance. Thus, we need four electrical parameters to specify the cable properties of the axon.
The capacitance and the resistance of the axon are distributed continu ously along the length of the cable. It is, therefore, not possible to represent the whole axon (or any other cable) with only four circuit components. We must consider the axon as a series of very small electrical-circuit sections joined together. When a potential difference is set up between the inside and the outside of the axon, four currents can be identified: the current outside the axon, the current inside the axon, the current through the resistive component of the membrane, and the current through the capacitive component of the membrane (see Fig. 13.6). The electrical circuit representing a small axon section of length x is shown in Fig. 13.6. In this small section, the resistances of the outside and of the inside fluids are Ro and Ri, respectively. The membrane capacitance and resistance are shown as Cm and Rm. The whole axon is just a long series of these subunits joined together. This is shown in Fig. 13.7. Sample values of the circuit parameters for both a myelinated and a nonmyelinated axon of radius 5 × 10−6 m are listed in Table 13.1. (These values were obtained from [13-4].) Note that the values in Table 13.1 are quoted for a 1-m length of the axon. The unit mho for the conductivity of the axon membrane is defined in Appendix B.
An examination of the axon performance shows immediately that the cir cuit in Fig. 13.7 does not explain some of the most striking features of the axon. An electrical signal along such a circuit propagates at nearly the speed of light (3 × 108 m/sec), whereas a pulse along an axon travels at a speed that is at most about 100 m/sec. Furthermore, as we will show, the circuit in Fig. 13.7 dissipates an electrical signal very quickly; yet we know that action potentials propagate along the axon without any attenuation. Therefore,
Section 13.1 The Nervous SystemFIGURE 13.6 (a) Currents flowing through a small section of the axon.
(b) Electrical circuit representing a small section of the axon.
FIGURE 13.7 The axon represented as an electrical cable.
TABLE 13.1 Properties of Sample AxonsPropertyNonmyelinated axon Myelinated axon
Axon radius 5 × 10−6 m 5 × 10−6 m
Resistance per unit length of fluid
6.37 × 109/m 6.37 × 109/m both inside and outside axon (r)
Conductivity per unit length of axon 1.25 × 10−4 mho/m 3 × 10−7 mho/m membrane (gm) Capacitance per unit length of axon (c) 3 × 10−7 F/m 8 × 10−10 F/m
Chapter 13 Electricity we must conclude that an electrical signal along the axon does not propagate by a simple passive process.
13.1.5 Propagation of the Action Potential
After many years of research the propagation of an impulse along the axon is now reasonably well understood. (See Fig. 13.8.) When the magnitude of the FIGURE 13.8 The action potential. (a) The action potential begins with the axon membrane becoming highly permeable to sodium ions (closed circles) which enter the axon making it positive. (b) The sodium gates then close and potassium ions (open circles) leave the axon making the interior negative again.
Section 13.1 The Nervous System voltage across a portion of the membrane is reduced below a threshold value, the permeability of the axon membrane to sodium ions increases rapidly. As a result, sodium ions rush into the axon, cancel out the local negative charges, and, in fact, drive the potential inside the axon to a positive value. This process produces the initial sharp rise of the action potential pulse. The sharp positive spike in one portion of the axon increases the permeability to the sodium immediately ahead of it which in turn produces a spike in that region. In this way the disturbance is sequentially propagated down the axon, much as a flame is propagated down a fuse.
The axon, unlike a fuse, renews itself. At the peak of the action poten tial, the axon membrane closes its gates to sodium and opens them wide to potassium ions. The potassium ions now rush out, and, as a result, the axon potential drops to a negative value slightly below the resting potential. After a few milliseconds, the axon potential returns to its resting state and that portion of the axon is ready to receive another pulse.
The number of ions that flow in and out of the axon during the pulse is so small that the ion densities in the axon are not changed appreciably. The cumulative effect of many pulses is balanced by metabolic pumps that keep the ion concentrations at the appropriate levels. Using Eq. B.5 in Appendix B, we can estimate the number of sodium ions that enter the axon during the rising phase of the action potential. The initial inrush of sodium ions changes the amount of electrical charge inside the axon. We can express this change in charge Q in terms of the change in the voltage V across the membrane capacitor C, that is, Q CV
(13.1)
In the resting state, the axon voltage is −70 mV. During the pulse, the voltage changes to about +30 mV, resulting in a net voltage change across the membrane of 100 mV. Therefore, V to be used in Eq. 13.1 is 100 mV.
The calculations outlined in Exercise 13-1 show that, in the case of the nonmyelinated axon described in Table 13.1, during each pulse 1.87 × 1011 sodium ions enter per meter of axon length. The same number of potassium ions leaves during the following part of the action potential. (Measurements show that actually the ion flow is about three times higher than our simple estimate.) The exercise also shows that, in the resting state, the number of sodium ions inside a meter length of the axon is about 7 × 1014 and the number of potassium ions is 7 × 1015. Thus, the inflow and the outflow of ions during the action potential is negligibly small compared to the equilibrium density.
Another simple calculation using Eq. B.6 yields an estimate of the minimum energy required to propagate the impulse along the axon. During the propagation of one pulse, the whole axon capacitance is successively discharged and Chapter 13 Electricity then must be recharged again. The energy required to recharge a meter length of the nonmyelinated axon is E 1 C(V )2 1 × 3 × 10−7 × (0.1)2 1.5 × 10−9J/m
(13.2)
2
2
where C is the capacitance per meter of the axon. Because the duration of each pulse is about 10−2 sec, and an axon can propagate at most 100 pulses/sec, even at peak operation the axon requires only 1.5 × 10−7 W/m to recharge its capacitance.
13.1.6 An Analysis of the Axon Circuit The circuit in Fig. 13.7 does not contain the pulse-conducting mechanism of the axon. It is possible to incorporate this mechanism into the circuit by connecting small signal generators along the circuit. However, the analysis of such a complex circuit is outside the scope of this text. Even the circuit in Fig. 13.7 cannot be fully analyzed without calculus. We will simplify this circuit by neglecting the capacitance of the axon membrane. The circuit is then as shown in Fig. 13.9a. This representation is valid when the capacitors are fully charged so that the capacitive current is zero. With this model, we will be able to calculate the voltage attenuation along the cable when a steady voltage is applied at one end. The simplified model, however, cannot make predictions about the time-dependent behavior of the axon.
The problem then is to calculate the voltage V(x) at point x when a voltage Va is applied at point x0 (see Fig. 13.9a). The approach is to calculate first the voltage drop across a small incremental cable section of length x cut by lines a and b (see [13-5, 13-6]). We assume that the cable is infinite in length and that the total cable resistance to the right of line b is RT. Thus, the whole cable to the right of line b is replaced by RT as shown in Fig. 13.9b. Because the cable is infinite, the resistance to the right of any vertical cut equivalent to line b is also RT. Specifically, the resistance to the right of line a is RT. We can, therefore, calculate RT by equating the resistance to the right of line a in Fig. 13.9b to RT, that is, RT Ro + Ri + RTRm
(13.3)
RT + Rm
Measurements show that the resistivities inside and outside the axon are about the same. Therefore, Ri Ro R and Eq. 13.3 simplifies to RT 2R + RTRm
(13.4)
RT + Rm
Section 13.1 The Nervous SystemFIGURE 13.9 (a) Approximation to the circuit in Fig. 13.7 with the capacitances neglected. (b) The resistances to the right of line b replaced by the equivalent resistor RT.
The solution of Eq. 13.4 (see Exercise 13-2) yields
1/2
RT R + R2 + 2R Rm
(13.5)
A simple circuit analysis (see Exercise 13-3) shows that
Vb
Va
+
Va
(13.6)
1 + (2R)(RTRm)
1 + β
RTRm where β is the quantity in the square brackets.
We can calculate β from the measured parameters shown in Table 13.1.
The resistances R and Rm are the values for a small axon section of length x. Therefore,
1
R rx and gmx or Rm
1
Rm
gmx
Chapter 13 Electricity From Eq. 13.5, it can be shown (see Exercise 13-4) that if x is very small then
2
1/2
r
RT
(13.7)
gm
and β (2rgm)1/2x x
(13.8)
λ
where
1
1/2
λ
(13.9)
2rgm
Now returning to Eq. 13.6, since x is vanishingly small, β is also very small. Therefore, the term 1/(1 + β) is approximately equal to 1 − β (see Exercise 13-5). Consequently, the voltage Vb at b, a distance x away from a, is
Vb Va 1 − x
(13.10)
λ
To obtain the voltage at a distance x away from line a, we divide this distance into increments of x such that nx x. We can then apply Eq. 13.10 successively down the cable and obtain the voltage at x (see Exercise 13-6) as
n
V(x) Va 1 − x
(13.11)
λ
It can be shown that, for small x and large n, Eq. 13.11 can be written as (see Exercise 13-7) V(x) Vae−x/λ
(13.12)
Equation 13.12 states that if a steady voltage Va is applied across one point in the axon membrane, the voltage decreases exponentially down the axon.
From Table 13.1, for a nonmyelinated axon λ is about 0.8 mm. Therefore, at a distance 0.8 mm from the point of application, the voltage decreases to 37% of its value at the point of application.
Myelinated axons, because of their outer sheath, have a much smaller membrane conductance than axons without myelin. As a result, the value of λ is larger. Using the values given in Table 13.1, we can show that λ is 16 mm for a myelinated axon. This result helps to explain the faster pulse conduction along myelinated axons. As we mentioned earlier, the myelin sheath is in 2-mm-long segments. The action potential is generated only at the nodes between the segments. The pulse propagates through the myelinated segments as a fast conventional electrical signal. Because λ is 16 mm, the pulse decreases by only 13% as it traverses one segment, and it is still sufficiently intense to generate an action potential at the next node.
13.1.7 Synaptic Transmission
So far we have considered the propagation of an electrical impulse down the axon. Now we shall briefly describe how the pulse is transmitted from the axon to other neurons or muscle cells.
At the far end, the axon branches into nerve endings which extend to the cells that are to be activated. Through these nerve endings the axon transmits signals, usually to a number of cells. In some cases the action potential is transmitted from the nerve endings to the cells by electrical conduction. In the vertebrate nervous system, however, the signal is usually transmitted by a chemical substance. The nerve endings are actually not in contact with the cells.
There is a gap, about a nanometer wide (1 nm 10−9 m 10−7 cm) between the nerve ending and the cell body. These regions of interaction between the nerve ending and the target cell are called synapses (see Fig. 13.10). When the impulse reaches the synapse, a chemical substance is released at the nerve ending which quickly diffuses across the gap and stimulates the adjacent cell.
The chemical is released in bundles of discrete size.
Usually a neuron is in synaptic contact with many sources. Often a number of synapses must be activated simultaneously to start the action potential in the target cell. The action potential produced by a neuron is always of the same FIGURE 13.10 Synapse.
Chapter 13 Electricity magnitude. The neuron operates in an all-or-none mode: It either produces an action potential of the standard size or does not fire at all. In some cases the chemicals released at the synapse do not stimulate the cell but inhibit its response to impulses arriving along a different channel. Presumably, these types of interactions permit decisions to be made on a cellular level. The details of these processes are not yet fully understood.
13.1.8 Action Potentials in Muscles
Muscle fibers produce and propagate electrical impulses in much the same way as neurons. The action potential in the muscle fiber is initiated by the impulses arriving from motor neurons. This stimulation causes a reduction of the potential across the fiber membrane which initiates the ionic process involved in the pulse propagation. The shape of the action potential is the same as in the neuron except that its duration is usually longer. In skeletal muscles, the action potential lasts about 20 msec, whereas in heart muscles it may last a quarter of a second.
After the action potential passes through the muscle fiber, the fiber con tracts. In Chapter 5, we briefly discussed some aspects of muscle contraction.
The details of this process are not yet fully understood.
Within the skeletal muscle fibers, mechanoreceptor organs called musclespindles continuously transmit information on the state of muscle contraction.
This information is relayed via neurons for processing and further action. In this way, the movement of muscles is continuously under control.
It is possible to stimulate muscle fibers by an external application of an electric current. This effect was first observed in 1780 by Luigi Galvani who noted that a frog’s leg twitched when an electric current passed through it.
(Galvani’s initial interpretation of this effect was wrong.) External muscle stimulation is a useful clinical technique for maintaining muscle tone in cases of temporary muscle paralysis resulting from nerve disorders.
13.1.9 Surface Potentials
The voltages and currents associated with the electrical activities in neurons, muscle fibers, and other cells extend to regions outside the cells. As an example, consider the propagation of the action potential along the axon. As the voltage at one point on the axon drops suddenly, a voltage difference is produced between this point and the adjacent regions. Consequently, current flows both inside and outside the axon (see Fig. 13.11). As a result, a voltage drop develops along the outer surface length of the axon.
Experiments are sometimes performed on a whole nerve consisting of many axons. As shown in Fig. 13.12, two electrodes are placed along the nerve
Section 13.1 The Nervous SystemFIGURE 13.11 The action potential produces currents both inside and outside the axon.
FIGURE 13.12 Surface potential along a nerve bundle.
bundle, and the voltage between them is recorded. This measured voltage is the sum of the surface potentials produced by the individual axons and yields some information about the collective behavior of the axons.
Electric fields associated with the activities in cells extend all the way to the surface of the animal body. Thus, along the surface of the skin, we can measure electric potentials representing the collective cell activities associated with certain processes in the body. Based on this effect, clinical techniques have been developed to obtain, from the skin surface, information about the activities of the heart (electrocardiography) and the brain (electroencephalography). The measurement of these surface signals will be discussed in Chapter 14.
Surface signals are associated with many other activities, such as move ment of the eye, contractions of the gastrointestinal tract, and movement of muscles. Using a technique called electromyography (EMG), measurement of action potentials and their speed of propagation along muscles can provide information about muscle and nerve disorders. (see [13-5]). Surface
Chapter 13 Electricity potentials associated with metabolic activities have also been observed in plants and bones, as discussed in the following sections.
13.2
Electricity in Plants
The type of propagating electrical impulses we have discussed in connection with neurons and muscle fibers have also been found in certain plant cells. The shape of the action potential is the same in both cases, but the duration of the action potential in plant cells is a thousand times longer, lasting about 10 sec.
The speed of propagation of these plant action potentials is also rather slow, only a few centimeters per second. In plant cells, as in neurons, the action potential is elicited by various types of electrical, chemical, or mechanical stimulation. However, the initial rise in the plant cell potential is produced by an inflow of calcium ions rather than sodium ions.
The role of action potentials in plants is not yet known. It is possible that they coordinate the growth and the metabolic processes of the plant and perhaps control the long-term movements exhibited by some plants.
13.3Electricity in the Bone When certain types of crystals are mechanically deformed, the charges in them are displaced; as a result, they develop voltages along the surface. This phenomenon is called the piezoelectric effect (Fig. 13.13). Bone is composed of a crystalline material that exhibits the piezoelectric effect. It has been suggested FIGURE 13.13 The piezoelectric effect.