action potential

1.1 Introduction to the Action Potential

Theories of the encoding and transmission of information in the nervous system go back to the Greek physician Galen (129-210 AD), who suggested a hydraulic mechanism by which muscles contract because fluid flowing into them from hollow nerves. The basic theory held for centuries and was further elaborated by René Descartes (1596 – 1650) who suggested that animal spirits flowed from the brain through nerves and then to muscles to produce movements. A major paradigm shift occurred with the pioneering work of Luigi Galvani who found in 1794 that nerve and muscle could be activated by charged electrodes and suggested that the nervous system functions via electrical signaling. However, there was debate among scholars whether the electricity was within nerves and muscle or whether the nerves and muscles were simply responding to the harmful electric shock via some intrinsic nonelectric mechanism. The issue was not resolved until the 1930s with the development of modern electronic amplifiers and recording devices that allowed the electrical signals to be recorded. One example is the pioneering work of H.K. Hartline 80 years ago on electrical signaling in the horseshoe crab Limulus . Electrodes were placed on the surface of an optic nerve. (By placing electrodes on the surface of a nerve, it is possible to obtain an indication of the changes in membrane potential that are occurring between the outside and inside of the nerve cell.) Then 1-s duration flashes of light of varied intensities were presented to the eye; first dim light, then brighter lights. Very dim lights produced no changes in the activity, but brighter lights produced small repetitive spike-like events. These spike-like events are called action potentials, nerve impulses, or sometimes simply spikes. Action potentials are the basic events the nerve cells use to transmit information from one place to another.

1.2 Features of Action Potentials

The recordings in the figure above illustrate three very important features of nerve action potentials. First, the nerve action potential has a short duration (about 1 msec). Second, nerve action potentials are elicited in an all-or-nothing fashion. Third, nerve cells code the intensity of information by the frequency of action potentials. When the intensity of the stimulus is increased, the size of the action potential does not become larger. Rather, the frequency or the number of action potentials increases. In general, the greater the intensity of a stimulus, (whether it be a light stimulus to a photoreceptor, a mechanical stimulus to the skin, or a stretch to a muscle receptor) the greater the number of action potentials elicited. Similarly, for the motor system, the greater the number of action potentials in a motor neuron, the greater the intensity of the contraction of a muscle that is innervated by that motor neuron.

Action potentials are of great importance to the functioning of the brain since they propagate information in the nervous system to the central nervous system and propagate commands initiated in the central nervous system to the periphery. Consequently, it is necessary to understand thoroughly their properties. To answer the questions of how action potentials are initiated and propagated, we need to record the potential between the inside and outside of nerve cells using intracellular recording techniques.

1.3 Intracellular Recordings from Neurons

The potential difference across a nerve cell membrane can be measured with a microelectrode whose tip is so small (about a micron) that it can penetrate the cell without producing any damage. When the electrode is in the bath (the extracellular medium) there is no potential recorded because the bath is isopotential. If the microelectrode is carefully inserted into the cell, there is a sharp change in potential. The reading of the voltmeter instantaneously changes from 0 mV, to reading a potential difference of -60 mV inside the cell with respect to the outside. The potential that is recorded when a living cell is impaled with a microelectrode is called the resting potential, and varies from cell to cell. Here it is shown to be -60 mV, but can range between -80 mV and -40 mV, depending on the particular type of nerve cell. In the absence of any stimulation, the resting potential is generally constant.

It is also possible to record and study the action potential. The electrode records a resting potential of -60 mV. The cell has also been impaled with a second electrode called the stimulating electrode. This electrode is connected to a battery and a device that can monitor the amount of current (I) that flows through the electrode. Changes in membrane potential are produced by closing the switch and by systematically changing both the size and polarity of the battery. If the negative pole of the battery is connected to the inside of the cell, an instantaneous change in the amount of current will flow through the stimulating electrode, and the membrane potential becomes transiently more negative. This result should not be surprising. The negative pole of the battery makes the inside of the cell more negative than it was before. A change in potential that increases the polarized state of a membrane is called a hyperpolarization. The cell is more polarized than it was normally. Use yet a larger battery and the potential becomes even larger. The resultant hyperpolarizations are graded functions of the magnitude of the stimuli used to produce them.

Now consider the case in which the positive pole of the battery is connected to the electrode. When the positive pole of the battery is connected to the electrode, the potential of the cell becomes more positive when the switch is closed. Such potentials are called depolarizations. The polarized state of the membrane is decreased. Larger batteries produce even larger depolarizations. Again, the magnitude of the responses are proportional to the magnitude of the stimuli. However, an unusual event occurs when the magnitude of the depolarization reaches a level of membrane potential called the threshold. A totally new type of signal is initiated; the action potential. Note that if the size of the battery is increased even more, the amplitude of the action potential is the same as the previous one. The process of eliciting an action potential in a nerve cell is analogous to igniting a fuse with a heat source. A certain minimum temperature (threshold) is necessary. Temperatures less than the threshold fail to ignite the fuse. Temperatures greater than the threshold ignite the fuse just as well as the threshold temperature and the fuse does not burn any brighter or hotter.

If the suprathreshold current stimulus is long enough, however, a train of action potentials will be elicited. In general, the action potentials will continue to fire as long as the stimulus continues, with the frequency of firing being proportional to the magnitude of the stimulus.

Action potentials are not only initiated in an all-or-nothing fashion, but they are also propagated in an all-or-nothing fashion. An action potential initiated in the cell body of a motor neuron in the spinal cord will propagate in an undecremented fashion all the way to the synaptic terminals of that motor neuron. Again, the situation is analogous to a burning fuse. Once the fuse is ignited, the flame will spread to its end.

1.4 Components of the Action Potentials

The action potential consists of several components. The threshold is the value of the membrane potential which, if reached, leads to the all-or-nothing initiation of an action potential. The initial or rising phase of the action potential is called the depolarizing phase or the upstroke. The region of the action potential between the 0 mV level and the peak amplitude is the overshoot. The return of the membrane potential to the resting potential is called the repolarization phase. There is also a phase of the action potential during which time the membrane potential can be more negative than the resting potential. This phase of the action potential is called the undershoot or the hyperpolarizing afterpotential. The undershoots of the action potentials do not become more negative than the resting potential because they are "riding" on the constant depolarizing stimulus.

1.5 Ionic Mechanisms of Resting Potentials

Before examining the ionic mechanisms of action potentials, it is first necessary to understand the ionic mechanisms of the resting potential. The two phenomena are intimately related. The story of the resting potential goes back to the early 1900's when Julius Bernstein suggested that the resting potential (V_m) was equal to the potassium equilibrium potential (E_K). Where

The key to understanding the resting potential is the fact that ions are distributed unequally on the inside and outside of cells, and that cell membranes are selectively permeable to different ions. K⁺ is particularly important for the resting potential. The membrane is highly permeable to K⁺. In addition, the inside of the cell has a high concentration of K⁺ ([K⁺]_i) and the outside of the cell has a low concentration of K⁺ ([K⁺]_o). Thus, K⁺ will naturally move by diffusion from its region of high concentration to its region of low concentration. Consequently, the positive K⁺ ions leaving the inner surface of the membrane leave behind some negatively charged ions. That negative charge attracts the positive charge of the K⁺ ion that is leaving and tends to "pull it back". Thus, there will be an electrical force directed inward that will tend to counterbalance the diffusional force directed outward. Eventually, an equilibrium will be established; the concentration force moving K⁺ out will balance the electrical force holding it in. The potential at which that balance is achieved is called the Nernst Equilibrium Potential.

An experiment to test Bernstein's hypothesis that the membrane potential is equal to the Nernst Equilibrium Potential (i.e., V_m = E_K) is illustrated to the left.

The K⁺ concentration outside the cell was systematically varied while the membrane potential was measured. Also shown is the line that is predicted by the Nernst Equation. The experimentally measured points are very close to this line. Moreover, because of the logarithmic relationship in the Nernst equation, a change in concentration of K⁺ by a factor of 10 results in a 60 mV change in potential.

Note, however, that there are some deviations in the figure at left from what is predicted by the Nernst equation. Thus, one cannot conclude that V_m = E_K. Such deviations indicate that another ion is also involved in generating the resting potential. That ion is Na⁺. The high concentration of Na⁺ outside the cell and relatively low concentration inside the cell results in a chemical (diffusional) driving force for Na⁺ influx. There is also an electrical driving force because the inside of the cell is negative and this negativity attracts the positive sodium ions. Consequently, if the cell has a small permeability to sodium, Na⁺ will move across the membrane and the membrane potential would be more depolarized than would be expected from the K⁺ equilibrium potential.

1.6 Goldman-Hodgkin and Katz (GHK) Equation

When a membrane is permeable to two different ions, the Nernst equation can no longer be used to precisely determine the membrane potential. It is possible, however, to apply the GHK equation. This equation describes the potential across a membrane that is permeable to both Na⁺ and K⁺.

Note that α is the ratio of Na⁺ permeability (P_Na) to K⁺ permeability (P_K). Note also that if the permeability of the membrane to Na⁺ is 0, then alpha in the GHK is 0, and the Goldman-Hodgkin-Katz equation reduces to the Nernst equilibrium potential for K⁺. If the permeability of the membrane to Na⁺ is very high and the potassium permeability is very low, the [Na⁺] terms become very large, dominating the equation compared to the [K⁺] terms, and the GHK equation reduces to the Nernst equilibrium potential for Na⁺.

If the GHK equation is applied to the same data in Figure 1.5, there is a much better fit. The value of alpha needed to obtain this good fit was 0.01. This means that the potassium K⁺ permeability is 100 times the Na⁺ permeability. In summary, the resting potential is due not only to the fact that there is a high permeability to K⁺. There is also a slight permeability to Na⁺, which tends to make the membrane potential slightly more positive than it would have been if the membrane were permeable to K⁺ alone.

1.7 Membrane Potential Laboratory

Ionic Mechanisms of Action Potentials

Voltage-Dependent Conductances

Na⁺ is critical for the action potential in nerve cells. Action potentials are repeatedly initiated as the extracellular concentration of Na⁺ is modified. As the concentration of sodium in the extracellular solution is reduced, the action potentials become smaller.

The straight line is predicted by the Nernst equation (assuming the membrane was exclusively permeable to Na⁺). There is a good fit between the data and the values predicted by a membrane that is exclusively permeable to Na⁺. The experiment gives experimental support to the notion that at the peak of the action potential, the membrane becomes highly permeable to sodium.

However, there are some deviations between what is measured and what is predicted by the Nernst equation. Why? One reason for the deviation is the continued K⁺ permeability. If there is continued K⁺ permeability, the membrane potential will never reach its ideal value (the sodium equilibrium potential) because the diffusion of K⁺ ions tends to make the cell negative. This point can be understood with the aid of the GHK equation.

An action potential is bounded by a region bordered on one extreme by the K⁺ equilibrium potential (-75 mV) and on the other extreme by the Na⁺ equilibrium potential (+55 mV). The resting potential is -60 mV. Note that the resting potential is not equal to the K⁺ equilibrium potential because, as discussed previously, there is a small resting Na⁺ permeability that makes the cell slightly more positive than EK. In principle, any point along the trajectory of action potential can be obtained simply by varying alpha in the GHK equation. If alpha is very large, the Na⁺ terms dominate, and according to the GHK equation, the membrane potential will move towards the Na⁺ equilibrium potential. The peak of the action potentials approaches but does not quite reach ENa, because the membrane retains its permeability to K⁺.

How is it possible for a cell to initially have a resting potential of -60 mV and then, in response to some stimulus (a brief transient depolarization which reaches threshold), change in less than one millisecond to having a potential of approximately +40 mV? In the 1950's, Hodgkin and Huxley, two British neurobiologists, provided a hypothesis for this transition. They suggested that the properties of some Na⁺ channels in nerve cells (and muscle cells) were unique in that these channels were normally closed but could be opened by a depolarization. This simple hypothesis of voltage-dependent Na⁺ channels goes a long way toward explaining the initiation of the action potential. Suppose a small depolarization causes some of the Na⁺ channels to open. The key point is that the increase in Na⁺ permeability would produce a greater depolarization, which will lead to an even greater number of Na⁺ channels opening and the membrane potential becoming even more depolarized. Once some critical level is reached a positive feedback or regenerative cycle will be initiated, causing the membrane potential to depolarize rapidly from -60 mV to a value approaching the Na⁺ equilibrium potential.

In order to test the Na⁺ hypothesis for the initiation of the action potential, it is necessary to stabilize the membrane potential at a number of different levels and measure the permeability at those potentials. An electronic device known as a voltage-clamp amplifier can "clamp" or stabilize the membrane potential to any desired level and measure the resultant current required for that stabilization. The amount of current necessary to stabilize the potential is proportional to the permeability. Hodgkin and Huxley clamped the membrane potential to various levels and measured the changes in Na⁺ conductances (an electrical term for permeability, which for the present discussion can be used interchangeably). The more the cell is depolarized, the greater is the Na⁺ conductance. Thus, the experiment provided support for the existence of voltage-dependent Na⁺ channels.

2.2 Na⁺ Inactivation

observation also indicates an important property of the voltage-dependent Na⁺ channels. Note that the permeability increases rapidly and then, despite the fact that the membrane potential is clamped, the permeability decays back to its initial level. This phenomenon is called inactivation. The Na⁺ channels begin to close, even in the continued presence of the depolarization. Inactivation contributes to the repolarization of the action potential. However, inactivation is not enough by itself to account fully for the repolarization.

2.3 Voltage-Dependent K⁺ Conductance

In addition to voltage-dependent changes in Na⁺ permeability, there are voltage-dependent changes in K⁺ permeability. These changes can be measured with the voltage-clamp technique as well. The figure shown to above indicates the changes in K⁺ conductance as well as the Na⁺ conductance. There are two important points.

First, just as there are channels in the membrane that are permeable to Na⁺ that are normally closed but then open in response to a voltage, there are also channels in the membrane that are selectively permeable to K⁺. These K⁺ channels are normally closed, but open in response to depolarization.

Second, a major difference between the changes in the K⁺ channels and the changes in the Na⁺ channels is that the K⁺ channels are slower to activate or open. (Some K⁺ channels also do not inactivate.) Note that the return of the conductance at the end of the pulse is not the process of inactivation. With the removal of the pulse, the activated channels are deactivated.

2.4 Sequence of Conductance Changes Underlying the Nerve Action Potential

Some initial depolarization (e.g., a synaptic potential) will begin to open the Na⁺ channels. The increase in the Na⁺ influx leads to a further depolarization.

A positive feedback cycle rapidly moves the membrane potential toward its peak value, which is close but not equal to the Na⁺ equilibrium potential. Two processes which contribute to repolarization at the peak of the action potential are then engaged. First, the Na⁺ conductance starts to decline due to inactivation. As the Na⁺ conductance decreases, another feedback cycle is initiated, but this one is a downward cycle. Sodium conductance decreases, the membrane potential begins to repolarize, and the Na⁺ channels that are open and not yet inactivated are deactivated and close. Second, the K⁺ conductance increases. Initially, there is very little change in the K⁺ conductance because these channels are slow to open, but by the peak of the action potential, the K⁺ conductance begins to increase significantly and a second force contributes to repolarization. As the result of these two forces, the membrane potential rapidly returns to the resting potential. At the time it reaches -60 mV, the Na⁺ conductance has returned to its initial value. Nevertheless, the membrane potential becomes more negative (the undershoot or the hyperpolarizing afterpotential).

The key to understanding the hyperpolarizing afterpotential is in the slowness of the K⁺ channels. Just as the K⁺ channels are slow to open (activate), they are also slow to close (deactivate). Once the membrane potential starts to repolarize, the K⁺ channels begin to close because they sense the voltage. However, even though the membrane potential has returned to -60 mV, some of the voltage-dependent K⁺ channels remain open. Thus, the membrane potential will be more negative than it was initially. Eventually, these K⁺ channels close, and the membrane potential returns to -60 mV.

Why does the cell go through these elaborate mechanisms to generate an action potential with a short duration? Recall how information is coded in the nervous system. If the action potential was about one msec in duration, the frequency of action potentials could change from once a second to a thousand a second. Therefore, short action potentials provide the nerve cell with the potential for a large dynamic range of signaling.

2.5 Pharmacology of the Voltage-Dependent Membrane Channels

Some chemical agents can selectively block voltage-dependent membrane channels. Tetrodotoxin (TTX), which comes from the Japanese puffer fish, blocks the voltage-dependent changes in Na⁺ permeability, but has no effect on the voltage-dependent changes in K⁺ permeability. This observation indicates that the Na⁺ and K⁺ channels are unique; one of these can be selectively blocked and not affect the other. Another agent, tetraethylammonium (TEA), has no effect on the voltage-dependent changes in Na⁺ permeability, but it completely abolishes the voltage-dependent changes in K⁺ permeability.

Use these two agents (TTX and TEA) to test your understanding of the ionic mechanisms of the action potential. What effect would treating an axon with TTX have on an action potential? An action potential would not occur because an action potential in an axon cannot be initiated without voltage-dependent Na⁺ channels. How would TEA affect the action potential? It would be longer and would not have an undershoot.

In the presence of TEA the initial phase of the action potential is identical, but note that it is much longer and does not have an after-hyperpolarization. There is a repolarization phase, but now the repolarization is due to the process of Na⁺ inactivation alone. Note that in the presence of TEA, there is no change in the resting potential. The channels in the membrane that endow the cell with the resting potential are different from the ones that are opened by voltage. They are not blocked by TEA. TEA only affects the voltage-dependent changes in K⁺ permeability.

2.6 Pumps and Leaks

It is easy to receive the impression that there is a "gush" of Na⁺ that comes into the cell with each action potential. Although, there is some influx of Na⁺, it is minute compared to the intracellular concentration of Na⁺. The influx is insufficient to make any noticeable change in the intracellular concentration of Na⁺. Therefore, the Na⁺ equilibrium potential does not change during or after an action potential. For any individual action potential, the amount of Na⁺ that comes into the cell and the amount of K⁺ that leaves are insignificant and have no effect on the bulk concentrations. However, without some compensatory mechanism, over the long-term (many spikes), Na⁺ influx and K⁺ efflux would begin to alter the concentrations and the resultant Na⁺ and K⁺ equilibrium potentials. The Na⁺-K⁺ pumps in nerve cells provide for the long-term maintenance of these concentration gradients. They keep the intracellular concentrations of K⁺ high and the Na⁺ low, and thereby maintain the Na⁺ equilibrium potential and the K⁺ equilibrium potential. The pumps are necessary for the long-term maintenance of the "batteries" so that resting potentials and action potentials can be supported.

2.7 Types of Membrane Channels

So far, two basic classes of channels, voltage-dependent or voltage-gated channels and voltage-independent channels, have been considered. Voltage-dependent channels can be further divided based on their permeation properties into voltage-dependent Na⁺ channels and voltage-dependent K⁺ channels. There are also voltage-dependent Ca²⁺ channels (see chapter on Synaptic Transmission). Indeed, there are multiple types of Ca²⁺ channels and voltage-dependent K⁺ channels. Nevertheless, all these channels are conceptually similar. They are membrane channels that are normally closed and as a result of changes in potential, the channel (pore) is opened. The amino acid sequence of these channels is known in considerable detail and specific amino acid sequences have been related to specific aspects of channel function (e.g., ion selectivity, voltage gating, inactivation). A third major channel class, the transmitter-gated or ligand-gated channels, will be described later.

2.8 Channelopathies

Ion channel mutations have been identified as a possible cause of a wide variety of inherited disorders. Several disorders involving muscle membrane excitability have been associated with mutations in calcium, sodium and chloride channels as well as acetylcholine receptors and have been labeled ‘channelopathies’. It is possible that movement disorders, epilepsy and headache, as well as other rare inherited diseases, might be linked to ion channels. The manifestations and mechanisms of channelopathies affecting neurons are reviewed in Kullman, 2002. The existence of channelopathies may provide insights into the variety of cellular mechanisms associated with the misfunctioning of neuronal circuits.

2.9 Absolute and Relative Refractory Periods

The absolute refractory period is a period of time after the initiation of one action potential when it is impossible to initiate a second action potential no matter how much the cell is depolarized. The relative refractory period is a period after one action potential is initiated when it is possible to initiate a second action potential, but only with a greater depolarization than was necessary to initiate the first. The relative refractory period can be understood at least in part by the hyperpolarizing afterpotential. Assume that an initial stimulus depolarized a cell from -60 mV to -45 mV in order to reach threshold and then consider delivering the same 15-mV stimulus sometime during the after-hyperpolarization. The stimulus would again depolarize the cell but the depolarization would be below threshold and insufficient to trigger an action potential. If the stimulus was made larger, however, such that it again was capable of depolarizing the cell to threshold (-45 mV), an action potential could be initiated.

The absolute refractory period can be explained by the dynamics of the process of Na⁺-inactivation. Here, two voltage clamp pulses are delivered. The first pulse produces a voltage-dependent increase in the Na⁺ permeability which then undergoes the process of inactivation. If the two pulses are separated sufficiently in time, the second pulse produces a change in the Na⁺ conductance, which is identical to the first pulse. However, if the second pulse comes soon after the first pulse, then the change in Na⁺ conductance produced by the second pulse is less than that produced by the first. Indeed, if the second pulse occurs immediately after the first pulse, the second pulse produces no change in the Na⁺ conductance. Therefore, when the Na⁺ channels open and spontaneously inactivate, it takes time (several msec) for them to recover from that inactivation. This process of recovery from inactivation underlies the absolute refractory period. During an action potential the Na⁺ channels open and then they become inactivated. Therefore, if a second stimulus is delivered soon after the one that initiated the first spike, there will be few Na⁺ channels available to be opened by the second stimulus because they have been inactivated by the first action potential.

3.1 Changes in the Spatial Distribution of Charge

Once an action potential is initiated at one point in the nerve cell, how does it propagate to the synaptic terminal region in an all-or-nothing fashion? Positive charges exist on the outside of the axon and negative charges on the inside. Now consider the consequences of delivering some stimulus to a point in the middle of the axon. If the depolarization is sufficiently large, voltage-dependent sodium channels will be opened, and an action potential will be initiated.

Consider for the moment "freezing" the action potential at its peak value. Its peak value now will be about +40 mV inside with respect to the outside. Unlike charges attract, so the positive charge will move to the adjacent region of the membrane. As the charge moves to the adjacent region of the membrane, the adjacent region of the membrane will depolarize. If it depolarizes sufficiently, as it will, voltage-dependent sodium channels in the adjacent region of the membrane will be opened and a "new" action potential will be initiated. This charge distribution will then spread to the next region and initiate other "new" action potentials. One way of viewing this process is with a thermal analogue. You can think of an axon as a piece of wire coated with gunpowder (the gunpowder is analogous to the sodium channels). If a sufficient stimulus (heat) is delivered to the wire, the gunpowder will ignite, generate heat, and the heat will spread along the wire to adjacent regions and cause the gunpowder in the adjacent regions to ignite.

3.2 Determinants of Propagation Velocity

A great variability is found in the velocity of the propagation of action potentials. In fact, the propagation velocity of the action potentials in nerves can vary from 100 meters per second (223 miles per hour) to less than a tenth of a meter per second (0.22 miles per hour). Why do some axons propagate information very rapidly and others slowly? In order to understand how this process works, it is necessary to consider two so-called passive properties of membranes, the time constant and the space or length constant. Why are these called passive properties? They have nothing to do with any of the voltage-dependent conductances discussed earlier. They have nothing to do with any pumps or exchangers. They are intrinsic properties of all biological membranes.

Time Constant. First, consider a thermal analogue. Place a block of metal at 10^oC on a hotplate at 100^oC. How would the temperature change? It will increase from its initial value of 10^oC to a final value of 100^oC. But the temperature will not change instantly. In fact, it would change as an exponential function of time. An analogous situation occurs in nerve cells, when they receive an instantaneous stimulus. The figure at right represents an idealized nerve cell. The recording electrode initially measures a potential of -60 mV (the resting potential). At some point in time (time 0), the switch is closed. The switch closure occurs instantaneously and as a result of the instantaneous closure, instantaneous current flows through the circuit. (This is equivalent to slamming the block of metal on the hotplate.) Note that despite the fact that this stimulus changes instantly, the change in potential does not occur instantaneously. It takes time for the potential to change from its initial value of -60 mV to its final value of -50 mV. There is a total of 10 mV depolarization, but the change occurs as an exponential function of time.

There is a convenient index of how rapidly exponential functions change with time. The index is denoted by the symbol τ and called the time constant. It is defined as the amount of time it takes for the change in potential to reach 63% of its final value. (Why 63%?) In this example, the potential changes from -60 to -50 and the 63% value is -53.7 mV. Thus, the time constant is 10 msec. The smaller the time constant, the more rapid will be the change in response to a stimulus. Therefore, if this neuron had a time constant of 5 msec, then in 5 msec the membrane potential would reach -53.7 mV. The time constant is analogous to the 0 to 60 rating of a high performance car; the lower the 0 to 60 rating, the faster the car. The lower the time constant, the faster or more rapidly a membrane will respond to a stimulus. The effects of the time constant on propagation velocity will become clear below.

The time constant is a function of two properties of membranes, the membrane resistence (R_m ) and the membrane capacitance (C_m ). R_m is the inverse of the permeability; the higher the permeability, the lower the resistance, and vice versa. Membranes, like the physical devices known as capacitors, can store charge. When a stimulus is delivered, it takes time to charge up the membrane to its new value.

[Please consult a college-level textbook on physics for a review of the basic properties of resistors and capacitors. For an additional review of resistors, capacitors and time constants and the use of hydraulic analogues to understand them see: Byrne, J. H., Understanding Electricity with Water, epub, Lulu.com, 2011.]

Space Constant. Consider another thermal analogue. Take a long, metal rod that is again initially at 10^oC and consider the consequences of touching one end of the rod to a hotplate which is at 100^oC. (Assume that it is placed there for a certain amount of time to allow the temperature changes to stabilize.) How would the temperature be distributed along the length of the rod? There would be a temperature gradient along the rod because of the increasing loss of heat with greater distances from the heat source. The temperature gradient can be described by an exponential function of distance because of the physical processes involved.

An analogous situation occurs in nerve cells. The figure at left represents an idealized nerve cell in which recordings are made from different regions along the axon at 1 mm increments. The cell body is impaled with a stimulating electrode connected to a battery, the value of which changes the potential of the cell body to -50 mV (the equivalent of putting a 10^oC rod on a 100^oC hot plate). This axon, even though it initially had a spatially uniform resting potential of -60 mV, now has a potential of -50 mV in the soma because that is the region in which the stimulus is applied. However, the potential is not -50 mV all along the axon; it varies as a function of distance from the soma. One mm away the potential is -56 mV; at 2 mm away it is even closer to -60 mV; and far enough along the axon, the potential of the axon is -60 mV, the resting potential. Just as there is an index for how a change in potential changes with the time (the time constant), there is also an index denoted by the symbol λ (called the space constant or the length constant) which is an indication of how far a potential will spread along an axon in response to a subthreshold stimulus at another point. In Figure 3.3, the space constant or length constant is 1 mm. In 1 mm the potential will change by 63% of its final value. If λ was greater than 1 mm, the potential would spread a greater distance. If λ was 1/2 mm, the potential would spread less along the axon. Thus, whereas the time constant is an index of how rapidly a membrane would respond to a stimulus in time, the space constant is an index of how well a subthreshold potential will spread along an axon as a function of distance. The space constant is a passive property of membranes. Although it influences the rate of propagation of the action potentials, it is an independent process. It is like the surface of a race track and the action potential is like the race car. If the surface is muddy, the car will go slow, if it is firm and paved, the same car will be able to go much faster.

The length constant can be described in terms of the physical parameters of the axon, where d is the diameter of the axon, R_m is, as before, the membrane resistance, the inverse of the permeability, and R_i is the internal resistance (resistance of the axoplasm). R_i is an indicator of the ability of charges to move along the inner surface of the axon. A small subthreshold change in the charge distribution at one point along an axon will spread along the axon, but as it does some will diffuse back out of the membrane and some will continue to move along the axon. If the resistance of the membrane (R_m) is high, less will leak out and relatively more will move along the axon. Increasing R_m is like putting insulation on a metal rod and heating the rod at one end. With more insulation (more resistance to heat loss to the outside of the rod), more heat will travel along the inside of the rod.

Propagation Velocity. How are the time constant and the space constant related to propagation velocity of action potentials? The smaller the time constant, the more rapidly a depolarization will affect the adjacent region. If a depolarization more rapidly affects an adjacent region, it will bring the adjacent region to threshold sooner. Therefore, the smaller the time constant, the more rapid will be the propagation velocity. If the space constant is large, a potential change at one point would spread a greater distance along the axon and bring distance regions to threshold sooner. Therefore, the greater the space constant, the more rapidly distant regions will be brought to threshold and the more rapid will be the propagation velocity. Thus, the propagation velocity is directly proportional to the space constant and inversely proportional to the time constant. There are separate equations that describe both the time constant and the space constant. The insight above allows us to make a new equation that combines the two.

The equation provides insights into how it is possible for different axons to have different propagation velocities. One way of endowing an axon with a high propagation velocity is to increase the diameter. However, there is one serious problem in changing the propagation velocity by simply changing the diameter. To double the velocity, it is necessary to quadruple the diameter. Clearly there must be a better way of increasing propagation velocity than by simply increasing the diameter.

Another way to increase the propagation velocity is to decrease the membrane capacitance. This can be achieved by coating axons with a thick insulating sheath known as myelin. One potential problem with this approach is that the process of covering the axon would cover voltage-dependent Na⁺ channels. If Na⁺ channels are occluded, it would be impossible to generate an action potential. Instead of coating the entire axon with the myelin, only sections are coated and some regions called nodes are left bare.

3.3 Propagation in Myelinated Fibers

Propagation of action potentials in myelinated fibers : Start with an action potential at a node on the left. In the absence of myelin, the action potential would propagate actively through the simple mechanisms discussed above. However, now the myelin occludes all the voltage-dependent sodium channels so the action potential can not propagate actively. (In fact, myelinated axons do not even have sodium channels in the internodal region.) Rather, the potential change produced by the action potential at one node spreads in the internodal region along the axon passively just as the temperature would spread along a long metal rod. The potential spreads, but gets smaller (decrements), just as a temperature change induced at one end of a rod would get smaller as it spreads along a rod.

Now consider the point at which the passively spreading potential reaches the next node. A "new" action potential will be initiated. The stimulus for this action potential is the depolarization that emerges from the end of the myelin. Each node acts as a "relay station" that renews the decremented signal. Think of the gunpowder analogue again, but this time coat the rod with some insulation and put gunpowder only at the bare regions. Because of the insulation, a temperature change produced by the ignition of the gunpowder will spread effectively along the metal rod. Some loss of temperature will occur but it will be sufficient to ignite the gunpowder at the next region and the process will repeat itself.