Action potential

Figure 1. A. Schematic view of an idealized action potential illustrates its various phases as the action potential passes a point on a cell membrane. B. Actual recordings of action potentials are often distorted compared to the schematic view because of variations in electrophysiological techniques used to make the recording.

An action potential is a pulse-like wave of voltage that can travel on certain types of cell membranes. It occurs most commonly on the membrane of the axon of a neuron, but also appears in other types of excitable cells, such as cardiac muscle cells and even plant cells. The resting voltage across the axonal membrane is typically −60 mV to −70 mV, with the inside being more negative than the outside. This voltage results mainly from a difference in concentrations of potassium inside and outside the cell, as described by the Goldman equation. As an action potential passes through a point, the voltage rises to roughly +40 mV in one millisecond, then returns to −60 mV, usually with an undershoot (Figure 1A). The action potential moves rapidly down the axon, with a conduction velocity as high as 100 meters/second (225 miles per hour). Because of this rapid speed, action potentials are useful in conveying information along neurons, which are sometimes longer than a meter; no material object could be transported as quickly through the body.

An action potential is provoked on a patch of membrane when the membrane is depolarized, i.e., when the voltage of the cell's interior relative to the cell's exterior is increased. This depolarization opens voltage-sensitive channels, which allow positive current to flow inwards, further depolarizing the membrane. Weak depolarizations are damped out, restoring the resting potential. A sufficiently strong stimulus, however, will cause the membrane to "fire", initiating a positive feedback loop that suddenly and rapidly increases the voltage. Membrane voltage is restored to its resting value by a combination of effects: the channels responsible for the initial inward current are inactivated, while the raised voltage opens other voltage-sensitive channels that allow a compensating outward current. Because of the positive feedback, an action potential is all-or-none; there are no partial action potentials. An action potential, once fired, runs its course regardless of the stimulus that provoked it, just as a forest fire cannot be stopped by blowing out the match that started it. In neurons, a typical action potential lasts a few thousandths of a second at any given membrane patch; hence, action potentials are sometimes called impulses or spikes.

The passage of an action potential can leave the ionic channels in a non-equilibrium state, making them more difficult to open, thus inhibiting another action potential at the same spot; such an axon is said to be refractory. The principal ions involved in an action potential are sodium and potassium cations; sodium ions enter the cell, and potassium ions leave it, restoring equilibrium. Relatively few ions are required to cross the membrane for the membrane voltage to change drastically. The ions exchanged during an action potential, therefore, make a negligible change in the interior and exterior ionic concentrations. The few ions that do cross are pumped out again by the continual action of the sodium-potassium pump, which, with other ion transporters, maintains the normal ratio of intra- and extracellular ion concentrations.

The action potential "travels" along the axon without diminishing because it is created anew at each patch of membrane; an action potential at one patch raises the voltage at nearby patches, depolarizing them and provoking a new action potential there. In unmyelinated neurons, the patches are adjacent, but in myelinated neurons, the action potential "hops" between distant patches, making the overall speed much faster and the whole process more efficient metabolically. A physiological action potential generally travels in one direction, from the axon hillock towards the synaptic button(s), although travel in the reverse direction can occur. The axon generally forms several branches, and the action potential often travels along both forks of a branch point. The action potential stops at the termini of these branches, but usually provokes the secretion of neurotransmitters at these synapses. These neurotransmitters diffuse and may bind to receptors on an adjacent excitable cell. These receptors are themselves ionic channels, although—in contrast to the axonal channels—they are generally opened by a neurotransmitter's binding, not by changes in voltage. The opening of these receptor channels can help to depolarize the membrane of the new cell (an excitatory channel) or work against its depolarization (an inhibitory channel). If these depolarizations are sufficiently strong, they can provoke another action potential in the new cell, beginning the process anew.

Biophysical and cellular context

Ions and the forces driving their motion

Main articles: Ion, Diffusion, Electrochemical gradient, and Electrophoretic mobility

Ions are charged atoms or molecules. An ion with a positive or negative charge is called a cation or anion, respectively. Certain types of ions are found throughout the human body and, indeed, throughout most living organisms. The most important cations for the action potential are sodium (Na⁺) and potassium (K⁺) , which carry a single positive charge (monovalent), and also calcium (Ca²⁺), which carries a double positive charge (divalent). The most important anion for the action potential is the chloride (Cl⁻), which carries a single negative charge. These atomic elements are fully ionized in physiological solutions and are usually covered in a single layer of water molecules (the hydration shell), due to the strong electrostatic interactions.

The action potential is primarily an electrical phenomenon, in which the voltage across a patch of cell membrane changes drastically in a few milliseconds, due to small currents across the membrane. (A current is the motion of an electric charge, such as that carried by an ion.) The motion of each type of ion contributes to the total current, and the cell has sophisticated mechanisms for controlling each type of ionic current independently of the others. For comparison, electronic circuits use only one type of charge carrier, the electron, and thus have only one type of current that can be controlled. A patch of membrane is defined as a segment of the membrane small enough that the transmembrane voltage does not vary significantly over its surface. Such a membrane patch can be modeled by an equivalent electronic circuit, as discussed below in the sections on cable theory and the Hodgkin-Huxley model.

Adjacent regions will naturalize try to equalize their ion concentrations; ions (pink circles) will flow from the high concentration (left) to the low concentration (right), causing a current. However, as the ions flow, a voltage across the membrane will build up that opposes the ions' motion. When that voltage reaches the equilibrium voltage, the net flow of ions will cease and the concentrations will remain as they are.

Ionic currents can result from two causes: diffusion and an electric field. In diffusion, ions move randomly in solution, with a typical energy given by the equipartition theorem. If there is a high concentration of ions next to a low concentration, then random motion will tend to even out this concentration gradient; on the average, there will be a net motion of ions from the high concentration to the low, and hence, a current. For the electric field, ions are charged and will move under the influence of an electric field at a terminal velocity that is determined by the drag imposed on the ion by its surroundings. This idea is embodied in Ohm's law, V=IR or equivalently I = GV, where I and V are the current and voltage, respectively, with R being the resistance and its reciprocal, G = 1/R, being the conductance. When the system is at equilibrium, there is no net current; the electric field and diffusion work in opposite directions, so that the current due to the electric field exactly cancels the current due to the concentration gradient. When the two concentrations are separated by a membrane, this equilibrium is described by the Nernst equation, where the equilibrium voltage equals to the equilibrium electric field integrated over the thickness of the membrane. At room temperature, roughly 59 millivolts (mV) are required for every ten-fold ratio of concentrations; for illustration, a hundred-fold ratio of concentrations can be held in equilibrium by 118 mV (2 powers of ten = 2×59 mV), whereas a thousand-fold concentration ratio would require 177 mV (3 powers of ten = 3×59 mV). The equilibrium voltages for different ions need not be the same, if their concentration gradients differ; this fact is exploited by the neuron to make action potentials.

Cell membrane

Main articles: Cell membrane, Active transport, and Passive transport

The hydrophobic cell membrane prevents charged molecules from easily diffusing through it, permitting a potential difference to exist across the membrane.

All cells are enclosed by a cell membrane composed largely of two layers of phospholipids. Similar to soap molecules, phospholipids are amphipathi, meaning that they have both water-soluble and water-insoluble parts: a hydrophilic "head" and two long, hydrophobic "tails". Under physiological conditions, these phospholipids assemble spontaneously to form a bilayer (two stacked layers), with the polar heads pointing outwards into solution, and the non-polar tails clustering together away from the solution, due to hydrophobic interactions.

The non-polar middle of the cell membrane has a low dielectric constant relative to water (roughly 4 vs. 80) and is 50 Å thick, which is much longer than the mean free path of a molecule or ion in solution. Since electric charges are repelled from such low-dielectric regions, ions cross the cell membrane very rarely. Therefore, cells have evolved special systems for transporting ions across the membrane. These systems can be divided into two classed: pores ("channels") that allow passive transport of ions, and ion pumps that use ATP for active transport of ions. Both systems can be quite sophisticated, and can turn off and on depending on their envirnoment. For example, ion channels can open and close in response to a particular voltage across the membrane or to the binding of a specific ligand molecule to its receptor.

Ion channels

Main articles: Ion channel and Passive transport

All-atom figure of the open potassium channel, with the potassium ion shown in purple in the middle. When the channel is closed, the passage is blocked.

Ion channels are integral membrane proteins that provide a pore through which ions can cross the membrane. Most channels are specific for their ion; for example, even though the potassium and sodium differ only slightly in their radius, potassium channels allow very few sodium ions through, and vice versa. The pore through which the ion passes is typically very narrow, so small that the ion must pass through it alone; no water molecules can accompany it. Channels can be open or closed; when closed, ions cannot pass through the channel. When the channel is open, ions flow through the channel by passive transport, i.e., at a rate determined by the voltage and concentration difference across the membrane. The action potential is basically a manifestation of different ion channels opening and closing at different times.

The open and closed states of a channel correspond to different conformations of the protein; both the open and closed states may correspond to more than one conformation. In general, the closed states correspond either to a small contraction of the pore, making it impassable, or to a separate part of the protein stoppering the pore. For example, the voltage-dependent sodium channel undergoes inactivation, in which a portion of the protein swings into the pore to seal it off. This inactivation of sodium current plays a critical role in the action potential, by allowing the membrane voltage to return to its resting potential.

Ion channels can respond to their environment. For example, the ion channels involved in the action potential are voltage-sensitive channels; they open and close in response to the voltage across the membrane. Ligand-gated channels form another important class; these ion channels open and close in response to the binding of a ligand molecule, such as a neurotransmitter. Still other ion channels—such as those of sensory neurons—open and close in response to other stimuli, such as light, temperature or pressure.

Ion pumps

Main articles: Ion transporter and Active transport

The sodium-potassium pump operates in two phases. In the first phase, three sodium ions (magenta) from the intracellular medium bind within the protein, which undergoes a conformational transition and releases them to the extracellular medium. In the second phase, two potassium ions from the extracellular medium bind within the protein, which undergoes the reverse transition and releases them into the intracellular medium.

The ionic currents of the action potential flow in response to concentration differences of the ions across the cell membrane. These concentration differences are established by ion transporters, which are integral membrane proteins that carry out active transport, i.e., use cellular energy (ATP) to "pump" the ions against their concentration gradient. Such ion pumps take in ions from one side of the membrane (decreasing its concentration there) and release them on the other side (increasing its concentration there. The main ion pump relevant to the action potential is the sodium-potassium pump, which transports three sodium ions out out of the cell and two potassium ions in. Consequently, the concentration of potassium ions K⁺ inside the neuron is roughly 20-fold larger than outside the cell, whereas the sodium concentration outside is roughly 9-fold larger than inside. Similarly, other ions have different concentrations inside and outside the neuron, such as calcium, chloride and magnesium.

It's worth emphasizing that ion pumps play no role in the action potential except to establish the relative ratio of intracellular and extracellular ion concentrations. The action potential is primarily a phenomenon involving opening and closing ion channels. If the ion pumps are turned off by an inhibitor such as dinitrophenol (DNP), the neuron can still fire hundreds of thousands of action potentials before the amplitudes begin to decay significantly. In particular, ion pumps play no significant role in the repolarization of the membrane after an action potential; that results from changing the membrane's permeability to ions such as sodium and potassium.

Resting potential

Main articles: Resting potential, Membrane potential, and Reversal potential

The flow of ions across a membrane is driven by two factors: an electric field and diffusion. The electric field results from a difference in voltages across the membrane, whereas the diffusion results from the difference in concentration across the membrane. For a given ratio of concentrations across the membrane, each type of ion has an reversal potential—also called its equilibrium voltage or equilibrium potential—at which the net current of that ion across the membrane is zero. For example, at a typical physiological ratio of concentrations, the potassium equilibrium voltage V_{eq, K} is −70 mV, whereas that for the sodium ion V_{eq, Na} is +45mV. Since these disagree, there is no voltage at which the currents of potassium and sodium ions are both zero. (As an aside, membrane voltages are defined relative to the exterior of the cell; thus, a potential of −70 mV implies that the interior of the cell is negative relative to the exterior.)

However, there is a voltage at which the net current of all ions—that is, the sum of the currents of the individual ions across the membrane—is zero; this voltage is known as the resting potential V_rest. The resting potential depends on the ionic concentrations inside and outside the cell, and also on the permeability P of each ion through the membrane; it is given by the Goldman equation

$V_{\mathrm {rest} }={\frac {RT}{F}}\ln {\left({\frac {\sum _{i=1}^{M}P_{\mathrm {C} _{i}^{+}}[\mathrm {C} _{i}^{+}]_{\mathrm {out} }+\sum _{j=1}^{N}P_{\mathrm {A} _{j}^{-}}[\mathrm {A} _{j}^{-}]_{\mathrm {in} }}{\sum _{i=1}^{M}P_{\mathrm {C} _{i}^{+}}[\mathrm {C} _{i}^{+}]_{\mathrm {in} }+\sum _{j=1}^{N}P_{\mathrm {A} _{j}^{-}}[\mathrm {A} _{j}^{-}]_{\mathrm {out} }}}\right)}$

Here, the symbols C⁺ and A^– represent different types of cations and anions, respectively; the summations are over M types of cations and N types of anions. The constants in this equation are T, the temperature in Kelvins, R the molar gas constant, and F the Faraday, which is the total charge of a mole of electrons.

Thus, the resting potential can be varied by changing the relative permeabilities of the ions, which is done by opening and closing the corresponding ion channels. Under normal, unstimulated conditions, the permeability of potassium dominates, so that the membrane resting potential of −70 mV is close to the potassium equilibrium potential, −75 mV. In the middle of the action potential, however, the sodium permeability dominates, and the voltage quickly converges to a new resting potential of +45 mV, close to the sodium equilibrium voltage of roughly +55 mV. This convergence is rapid, because very few ions need to cross the membrane in order to change its voltage drastically. For illustration, a typical action potential changes the ionic concentrations by roughly 1 part in 10 million. Thus, the process of equilibrating to the resting voltage does not significantly change the ionic concentrations that define the resting voltage.

Action potentials arriving at the synapses of the upper right neuron stimulate currents in its dendrites; these currents depolarize the membrane at its axon hillock, provoking an action potential that propagates down the axon to its synaptic knobs, releasing neurotransmitter and stimulating the post-synaptic neuron (lower left).

Anatomy of a neuron

Main articles: Neuron and Axon

Several types of cells support an action potential, such as plant cells, muscle cells (muscle fibers), and the specialized cells of the heart (cardiac action potential). However, the main excitable cell is the neuron, which also has the simplest mechanism. Therefore, we focus on this cell type, and discuss the differences with other cell types below.

Neurons are remarkable for the enormous variety of their shapes, as described by Santiago Ramón y Cajal around the turn of the 20th century. Most neurons have several long, thin, branched tendrils called neurites. Neurites are divided into two main types, dendrites and axons; most neurons have numerous dendrites but only one axon; a few types have no axon (amacrine cells) and others have more than one. With rare exceptions, only the axon has the ion channels needed to initiate and propagate the action potential; the dendrites are electrically passive. The beginning of the axon is called the axon hillock, which for convenience may be taken as the point at which action potentials are initiated. The axon is generally long, uniform in diameter, has a smooth surface, branches infrequently and receives few synaptic inputs, although the axonal termini (called the synaptic knobs or buttons) proide synaptic inputs to the dendrites. In contrast, the dendrites are short and tapering, frequently with a "knobby" surface, branch frequently, and receive many synaptic inputs. The synapse is the small space between the two neurons, with the terminating axon known as the pre-synaptic neuron and the other known as the post-synaptic neuron. Most often, these synapses connect an axon terminal with the dendrite of another neuron; however, axons are known to terminate on the soma and even on other axons. The action potential itself does not cross between neurons; rather, the arrival of an action potential stimulates the release of neurotransmitters from the axonal terminal, the so-called synaptic button. These neurotransmitters bind to ligand-gated ion channels in the postsynaptic (usually dendritic) membrane, stimulating postsynaptic potentials, excitatory and inhibitory, that may depolarize the postsynaptic axon hillock enough to provoke a new action potential. This integration of dendritic signals and its translation into a temporal code of action potentials at the axon hillock is a key mechanism of neural computing.

Dendrite

Soma

Axon

Nucleus

Node of
Ranvier

Axon terminal

Schwann cell

Myelin sheath

Myelin sheath

Axons may be myelinated or unmyelinated, which has a profound effect on the conduction velocity of action potentials. Myelin is composed of Schwann cells that wrap themselves multiple times around the axon, forming a thick fatty layer that sheaths the axon and prevents ions from escaping. Ions can flow in and out of the axon only at the nodes of Ranvier, which are the gaps between the Schwann cells, the "chinks" in the myelin armor. As explained below in the propagation section, action potentials can occur only at the nodes of Ranvier; they cannot enter the axonal regions sheathed in myelin. Thus, the action potential proceeds down the axon not as a continuous wave (as it does in unmyelinated axons) but rather by "hopping" from one node to the next. In neurons of similar diameter, this saltatory conduction is much faster than ordinary propagation and more economical in energy.

Phases

The course of the action potential is determined by two basic principles. The first principle is that the membrane voltage converges quickly (within milliseconds) to the resting potential, which is determined the membrane's permeability to various ions. The second principle is that those ionic permeabilities may depend conversely on the membrane voltage, setting up the possibility for positive feedback. A complicating factor is that a single ion channel may have multiple internal "gates" that respond at different rates to changes in voltage. For example, the voltage-sensitive sodium channel has an "inactivation gate" that shuts the channel off at the same voltages that open the other gates, but more slowly; thus, when the voltage is raised, the sodium channels initially open, but then inactivate.

The course of the action potential can be understood as follows. The initial membrane permeability to potassium is low, but much higher than that of sodium, making the resting potential close to the equilibrium potential for potassium. A sufficiently strong depolarization (increase in the voltage) causes the sodium channels to open, which pushes the resting potential closer to the equilibrium sodium potential V_Na, roughly +55 mV, because the sodium permeability has been increased. The increasing voltage causes even more sodium channels to open, which pushes the resting potential still further towards V_Na. This positive feedback continues until the sodium channels are fully open and the membrane potential is close to its equilibrium value. At this point, the sodium channels inactivate, lowering the permeability and driving the resting potential back down. Meanwhile, the potassium channels open more fully, and their increased permeability likewise helps to drive the membrane voltage back down to its normal resting potential, near the potassium equilibrium potential.

The voltages and currents of the action potential in all of its phases were modeled accurately by Alan Lloyd Hodgkin and Andrew Huxley in 1952, for which they were awarded the Nobel Prize in Physiology or Medicine in 1963. However, their model considers only two types of voltage-sensitive ion channels, and makes several assumptions about them, e.g., that their internal gates open and close indepedently of one another. In reality, there are many types of ion channels, and they may not always open and close independently.

Stimulation and rising phase

A typical action potential begins at the axon hillock with a depolarization that makes the cell interior less negative compared to the extracellular medium, for example increasing it from −70 mV to −45 mV. This depolarization is often caused by the injection of extra sodium cations on the interior of the cell; these cations can come from a wide variety of sources, such as chemical synapses, natural pacemaker leakage currents, sensory neuron responses to a stimulus. The factors leading to initiation of the axon potential are discussed more fully below.

The depolarization opens both the sodium and potassium channels, which allows their ionic currents to flow inwards and outwards, respectively. If the depolarization is small (say, under 10 mV), the outwards potassium current overwhelms the inwarsd sodium current and the membrane re-polarizes back to its normal resting potential, around −70 mV. The "failed initiations" shown in Figure 1 illustrate this response. However, if the depolarization is large enough, the inwards sodium current increases more than the outwards potassium current and a runaway condition (positive feedback) results: the more inward current flows, the more positive the cell interior becomes, which in turn increases the inwards current still further. This produces the rapidly increasing voltage that makes up the rising phase of the action potential.

The threshold voltage for this runaway condition is usually around −45 mV, but it depends on the recent activity of the axon. A membrane that has just fired an action potential cannot fire another one immediately, since the ion channels have not returned to their usual state. This is known as the absolute refractory period. At longer times, after some but not all the ion channels have recovered, the axon can be stimulated to produce another action potential, but only with a much stronger depolarization, e.g., −15 mV. This is called the relative refractory period.

Peak and falling phase

The positive feedback of the rising phase finally slows and comes to a halt as the sodium ion channels become maximally open. At this point in the action potential, the local sodium permeability is maximized and the membrane voltage attains a resting voltage ≈45 mV that is just a little below the equilibrium voltage for sodium, +55 mV. At this point, however, the sodium ions have already begun to inactivate. In the technical language of the Hodgkin-Huxley model, the three m-gates of the sodium channel are opened by the elevated voltage, but the inactivation h-gate is closed by the same voltage, albeit more slowly than the m-gates. Closing the h-gate stoppers the channel and shuts off the inward sodium current. At the same time, the voltage-sensitive potassium channels are opened by the raised voltage, allowing a compensating outwards current that helps to repolarize the membrane. Lowering the sodium permeability and increasing the potassium permeability brings the resting potential back down to a value close to the potassium equilibrium.

Hyperpolarization ("undershoot")

The raised voltage opened many more potassium channels than usual, and these do not close right away when the membrane returns to its normal resting voltage of roughly –70 mV. Thus, the potassium permeability of the membrane is transiently unusually high, making the resting voltage even closer to the potassium equilibrium voltage V_K. Hence, there is an undershoot, a hyperpolarization in technical language, that persists until the membrane potassium permeability falls to its usual value.

Refractory period

As mentioned above, the opening and closing of the sodium and potassium channels during an action potential may leave some of them in an "refractory" state, in which they are unwilling to open again until they have recovered. For example, the inactivation of the sodium channels takes time to recover from. In the absolute refractory period, so many ion channels are in such a state that no new action potential can be fired. Eventually, enough have recovered that an action potential is possible to provoke, but only with a stimulus much stronger than usual; this is known as the relative refractory period. These refractory periods are essential for the propagation of the action potential in one direction; if an action potential further down the axon could stimulate a new action potential at an earlier point, all axons would be firing continuously. Some arrhythmias are caused by a failure to block self-stimulation; an action potential in the heart can, under some conditions, travel in a circle and provoke another copy of itself.

Initiation, propagation and termination

A typical action potential is initiated at the axon hillock when the membrane there becomes sufficiently depolarized, i.e., when the membrane voltage reaches threshold. Thence it propagates along the axon without being diminished, being created anew at every step. The axon may branch along its length, and the action potential may fail to propagate one or both of the branches. Finally, action potentials that reach the termini of the axon, the so-called synaptic buttons (or synaptic knobs), generally provoke the release of a neurotransmitter into the synapse. This section describes the initiation and termination of the action potential, whereas a later section describes its propagation.

Initiation

Before considering the propagation of action potentials along axons and their termination at the synaptic knobs, it is helpful to consider the methods by which action potentials can be provoked, i.e., initiated at the axon hillock. The basic requirement is that the membrane voltage there be raised above the threshold for firing; but there are several ways in which depolarization can occur.

Neurotransmission

Main article: Neurotransmission

The most common cause of initiating an action potential is excitatory postsynaptic potentials from a presynaptic neuron.

A special case of this occurs at electrical synapses.

Pacemaker potential

Main article: Pacemaker potential

Certain cells undergo spontaneous depolarization to threshold, such as the cardiac cells of the sinoatrial node.

Sensory neurons

Main article: Sensory neuron

Signals from the external environment are transduced into action potentials by sensory neurons. There is a wide variety of specialized neurons that convert an external stimulus, such as pressure, temperature, light, etc. into a train of action potentials.

Propagation

Main article: Conduction velocity

The action potential propagates as a wave along the axon. The currents flowing inwards at a point on the axon during an action potential spread out along the axon, and depolarize the adjacent sections of its membrane. If sufficiently strong, this depolarization provokes a similar action potential at the neighboring membrane patches. This basic mechanism was demonstrated by Hodgkin in 1937. After blocking action potentials in a nerve segment by crushing or cooling, he showed that an action potential arriving on one side of the block could provoke another action potential on the other, provided that the blocked segment was sufficiently short.

In the common orthodromic conduction the action potential propagates "downstream" from the axon hillock towards the synaptic buttons (the axonal termini). The more rarely observed antidromic conduction is propagation in the opposite ("upstream") direction. This unidirectionality results from the refractory period that follows a typical action potential; even though the depolarization caused by the inward currents spreads both upstream and downstream, it does not provoke an action potential in a patch of membrane that has recently undergone an action potential. If, however, the axon is stimulated sufficiently in its middle, then two action potentials will be generated, one traveling upstream towards the axon hillock and the other traveling downstream towards the synaptic buttons.

Myelin and saltatory conduction

Main article: Saltatory conduction

In saltatory conduction, an action potential at one node of Ranvier causes inwards currents that move down the axon, depolarizing the membrane and stimulating a new action action potential at the next node of Ranvier.

If a neuron is myelinated, an action potential at one node of Ranvier provokes another action potential at the next node, although no action potential occurs on the intervening segments of membrane. This "hopping" of the action potential from node to node is known as saltatory conduction. Although the mechanism of saltatory conduction was suggested in 1925 by Lillie, the first experimental evidence for saltatory conduction came from Tasaki and Takeuchi and from Hodgkin and Stämpfli. By contrast, in unmyelinated axons, the action potential provokes another in the membrane immediately adjacent, and moves continuously down the axon like a wave. Whether saltatory or not, the mean conduction velocity of an action potential ranges from 1 m/s to over 100 m/s, and generally increases with axonal diameter, for reasons described below.

A close-up illustration of saltatory conduction of an action potential between two nodes of Ranvier.

Myelination confers several important advantages. Saltatory conduction is generally faster than even the fastest axon potentials in unmyelinated axons. For example, the conduction velocity in a myelinated frog nerve is roughly the same (25 m/sec) as that in a squid axon, but the frog neuron has a roughly 30-fold smaller diameter and 100-fold smaller cross-sectional area. Also, since the ionic conduction is confined to the nodes of Ranvier, fewer ions move across the membrane so less energy is needed. This saving is a significant selective advantage, since the human nervous system uses approximately 20% of the body's metabolic energy.

The length of myelinated segments of axon is important to the success of saltatory conduction. They should be as long as possible to maximize the speed of conduction, but not so long that the arriving signal is too weak to provoke an action potential at the next node of Ranvier. In nature, myelinated segments are generally long enough for the passively propagated signal to travel for at least two nodes while retaining enough amplitude to fire an action potential at the second or third node. Thus, the safety factor of saltatory conduction is high, allowing transmission to bypass nodes in case of injury. However, action potentials may end prematurely in certain places where the safety factor is low, even in unmyelinated neurons; a common example is the branch point of an axon, where it divides into two axons.

Some diseases degrade saltatory conduction and reduce the speed of action potential conductance. The most well-known of these is multiple sclerosis, in which the breakdown of myelin impairs coordinated movement.

Cable theory

Main article: Cable theory

Figure.1: Cable theory's simplified view of a neuronal fiber. The connected RC circuits correspond to adjacent segments of a passive neurite. The extracellular resistances r_e (the counterparts of the intracellular resistances r_i) are not shown, since they are usually negligibly small; the extracellular medium may be assumed to have the same voltage everywhere.

The flow of currents within an axon can be described quantitatively by cable theory and its elaborations, such as the compartmental model. Cable theory was developed in 1855 by Lord Kelvin to model the transatlantic telegraph cable and was shown to be relevant to neurons by Hodgkin and Rushton in 1946. In simple cable theory, the neuron is treated as an electrically passive, perfectly cylindrical transmission cable, which can be described by a partial differential equation

$\tau {\frac {\partial V}{\partial t}}=\lambda ^{2}{\frac {\partial ^{2}V}{\partial x^{2}}}-V$

where V(x, t) is the voltage across the membrane at a time t and a position x along the length of the neuron, and where λ and τ are the characteristic length and time scales on which those voltages decay in response to a stimulus. Referring to the circuit diagram above, these scales can be determined from the resistances and capacitances per unit length

$\tau =r_{m}c_{m}$

$\lambda ={\sqrt {\frac {r_{m}}{r_{i}+r_{e}}}}$

These time- and length-scales can be used to understand the dependence of the conduction velocity on the diameter of the neuron in myelinated and unmyelinated fibers. For example, the time-scale τ increases with both the membrane resistance r_m and capacitance c_m; as the capacitance increases, more charge must be transferred to produce a given transmembrane voltage (by the equation Q=CV), and as the resistance increases, less charge is transferred per unit time, making the equilibration slower. Similarly, if the internal resistance per unit length r_i is lower in one axon than in another (e.g., because the radius of the former is larger), then the spatial decay length λ becomes longer and the conduction velocity of an action potential should increase. If the transmembrane resistance r_m is increased, that lowers the average "leakage" current across the membrane, likewise causing λ to become longer, increasing the conduction velocity.

Comparison of the conduction velocities of myelinated and unmyelinated axons in the cat. The conduction velocity v of myelinated neurons varies roughly linearly with axon diameter d (that is, v ∝ d), whereas the speed of unmyelinated neurons varies roughly as the square root (v ∝√ d).. The red and blue curves are fits of experimental data, whereas the dotted lines are their theoretical extrapolations.

The cable equation is helpful in understanding why myelin increases the conduction velocity, the saltatory conduction described above. Myelin is made up of fatty (lipid-rich) Schwann cells wrapped around the axon, which has two electrical effects: it increases the r_m between the nodes of Ranvier (thereby increasing the decay-length λ) while also lowering the c_m by increasing the spatial gap of low-dielectric material between the inside and outside of the axon (thereby decreasing the time-constant τ). These effects allow for much faster axonal conduction between the well-separated gaps between the Schwann cells and the nodes of Ranvier. Myelination also makes nervous conduction less expensive metabolically as well, since no action potential is generated between the nodes of Ranvier, and no current "leaks" across the membrane there; hence, the ion pumps need to work less to restore the equilibrium ratio of ionic concentrations.

Termination

Action potentials on an axon almost always move in the same direction: "downstream" from the axon hillock to the axonal termini, which are called the synaptic knobs or buttons. This is known as orthodromic conduction. The absolute refractory period of the action potential allows this one-way propagation, since it prevents an action potential that has propagated downstream from re-initiating another action potential in an upstream patch of membrane that has just fired. However, certain arrythmias of the heart result from an action potential stimulating an 'echo" of itself.

The axon is generally branched, and a traveling action potential may not invade both branches. The inflowing current at the branch point may not suffice to depolarize both branches above threshold. Thus, an action potential traveling along the trunk of the axon may not reach all of its termini. The axonal termini are typically enlarged "knobs" often containing vesicles filled with a neurotransmitter.

Chemical synapses

Main articles: Chemical synapse, Neurotransmitter, Excitatory postsynaptic potential, and Inhibitory postsynaptic potential

When the action potential arrives at the axon terminal (the synaptic knob or button), it causes voltage-sensitive calcium channels to open. The influx of calcium induces vesicles filled with neurotransmitter to fuse with the membrane, releasing its contents into the synaptic cleft. The neurotransmitter molecules diffuse across and bind to receptors; this binding opens chemically gated ion channels, causing either an excitatory or inhibitory postsynaptic potential. The neurotransmitter may then be re-absorbed or degraded, to limits its effects.

The action potentials that do reach the synaptic knobs generally cause a neurotransmitter to be released into synaptic cleft. These neurotransmitters are small molecules that may excite or inhibit action potentials in the postsynaptic cell. The arrival of the pre-synaptic action potential cases voltage-sensitive calcium channels to open. Free calcium ions are very rare inside cells, and therefore they often have a powerful physiological effect. In this case, they provoke the migration of vesicles filled with neurotransmitter to migrate to the cell surface and release their contents (exocytosis), all of which processes require a complex coordination of so-called SNARE proteins such as synaptobrevin and synaptotagmin. The formation of this complex is inhibited by the neurotoxins tetanospasmin and botulinum toxin, which are responsible for tetanus and botulism, respectively.

Molecular machinery reponsible for the release (exocytosis) of neurotransmitter. The action potential allows in calcium ions that bind to synaptotagmin and cause the formation of a SNARE complex. This complex draws together and fuses the membranes of the neuron and its synaptic vesicle, thus releasing the neurotransmitter.

Neuromuscular junction

Main articles: Neuromuscular junction, Acetylcholine receptor, and Cholinesterase enzyme

A special case of a chemical synapse is the neuromuscular junction. In this case, the neurotransmitter is acetylcholine, which binds to the acetylcholine receptor, an integral membrane protein in the membrane (the sarcolemma) of the muscle fiber. However, the acetylcholine does not remain bound forever; rather, it dissociates and is digested by a specialized enzyme, acetylcholinesterase, located in the synapse. This enzyme quickly reduces the stimulus to the muscle, which allows the degree and timing of muscular contraction to be regulated delicately. Some poisons bind to acetylcholinesterase to prevent this delicate control, such as the neurotoxin sarin and tabun, and the insecticides diazinon and malathion.

Cardiac action potential

Main articles: Cardiac action potential, Electrical conduction system of the heart, Cardiac pacemaker, and Arrhythmia

• Action potentials in the human heart
• 
  Phases of a cardiac action potential. The sharp rise in voltage ("0") correspond to the influx of sodium ions, whereas the two decays ("1" and "3", respectively) correspond to the sodium-channel inactivation and the repolarizing eflux of potassium ions. The characteristic plateau ("2") results from the opening of voltage-sensitive calcium channels.
• 
  Cardiac action potentials begin at the sinoatrial node (upper left) and travel to the atrioventricular node in the middle; thence they proceed down the bundle of His to the Purkinje fibers. Since the action potentials of these cells are firing together, their electrical activity can be detected by an electrocardiogram (ECG trace at bottom).
• 
  Schematic representation of the ECG trace of a heartbeat. The depolarization of the atria produces the initial rounded P wave, whereas the ventricular depolarizations are responsible for the QRS complex. The final rounded T wave represents the ventricular repolarization.

The cardiac action potential differs from the neuronal action potential by having an extended plateau, in which the membrane is held at a high voltage for a few hundred milliseconds prior to being repolarized by the potassium current as usual. This plateau is due to the action of slower calcium channels opening and holding the membrane voltage near their equilibrium potential even after the sodium channels have inactivated.

The electrical activity of the heart is highly coordinated. The cardiac cells of the sinoatrial node provide the pacemaker potential that synchronizes the heart. The action potentials of those cells propagate to and through the atrioventricular node (AV node), which is normally the only conduction pathway between the atria and the ventricles. Action potentials from the AV node travel through the bundle of His and thence to the Purkinje fibers.

Several types of arrhythmias are associated with pathologies in the cardiac pacemaker potential in sinoatrial node or the propagation of the cardiac action potential. Some of these are genetic; for example, a mutant form of the potassium channel (gene SCN5A) can produce an abnormally long QT segment. Such arrhythmias may be treated with four basic types of pharmaceuticals, three of which affect the cardiac action potential. Class I consists of drugs that slow the initial upstroke (labeled "0" in the Figure above), such as quinidine (Class IA), lidocaine (Class IB), and flecainide (Class IC). Class III drugs such as amiodarone and bretylium prolong the action potential duration. Finally, Class IV drugs such as verapamil block the slow inward current of calcium that is characteristic of the cardiac action potential.

Other cell types

Muscular action potentials

Main articles: Neuromuscular junction and Muscle contraction

As described above, the arrival of an action potential at the synaptic knob of a neuromuscular junction provokes the release of acetylcholine. This neurotransmitter binds to receptors on the sarcolemma, which is the membrane of the muscle fiber. This binding provokes an action potential in the muscle fiber, which in turn releases calcium ions that free up the tropomyosin and allow the muscle to contract. Given its ubiquity, the acetylcholine receptor is a common target for snake toxins such as cobratoxin and α-bungarotoxin.

Plant action potentials

Many plants also exhibit action potentials that travel via their phloem to coordinate activity. The physiology of these ion movements has been studied most in algae such as charophytes. The main difference between plant and animal action potentials is that plants primarily use potassium and calcium currents while animals typically use currents of potassium and sodium. These signals are used by plants to rapidly transmit information from environmental signals such as temperature, light, touch or wounding.

Taxonomic distribution and evolutionary advantages

Action potentials are found throughout multicellular organisms, ranging from plants and invertebrates such as insects, to vertebrates such as reptiles and mammals. Sponges seem to be the main phylum of multi-cellular eukaryotes that does not transmit action potentials, although some studies have suggested that these organisms have a form of electrical signaling. The resting potential, as wel as the size and duration of the action potential, have not varied much with evolution, although the conduction velocity does vary dramatically with axonal diameter and myelination, as discussed above in the Propagation section.

Comparison of action potentials (APs) from a representative cross-section of animals

Animal	Cell type	Resting potential (mV)	AP increase (mV)	AP duration (ms)	Conduction speed (m/s)
Squid (Loligo)	Giant axon	−60	120	0.75	35
Earthworm (Lumbricus)	Median giant fiber	−70	100	1.0	30
Cockroach (Periplaneta)	Giant fiber	−70	80–104	0.4	10
Frog (Rana)	sciatic nerve axon	−60 to −80	110–130	1.0	7–30
Cat (Felis)	Spinal motor neuron	−55 to −80	80–110	1–1.5	30–120

Given its conservation throughout evolution, the action potential seems to confer evolutionary advantages. One function of action potentials is rapid long-range signaling within the animal; the conduction velocity can exceed 110 m/sec, which is one-third the speed of sound. No material object could convey a signal that rapidly throughout the body; for comparison, a hormone molecule carried in the bloodstream moves at roughly 8 m/sec in large arteries. Part of this function is the tight coordination of mechanical events, such as the contraction of the heart. A second function is the computation associated with its generation. Being an all-or-none signal that does not decay with transmission distance, the action potential has similar advantages to digital electronics. The integration of various dendritic signals at the axon hillock and its thresholding to form a complex train of action potentials is another form of computation, one that has been exploited biologically to form central pattern generators and mimicked in artificial neural networks.

Experimental methods

Model neurons

The giant axons of Loligo vulgaris were crucial for scientists to understand the action potential.

Much of the early work on neurophysiology used the giant axons of the squid genus Loligo. These axons are so large in diameter (roughly 1 mm, roughly 100-fold larger than a typical neuron) that they can be seen with the naked eye, making them easy to extract and manipulate. These giant neurons are used by the squid to escape predators by contracting the squid's mantle to make a jet of water through its hyponome; they have evolved to be so large because they provide high conduction speeds, as described above.

Other neurons commonly used as model systems include those from the related cuttlefish, snails and sea slugs such as Aplysia, octopi, toads and frogs (especially Xenopus laevis and Rana pipiens), bristle worms, crayfish, crabs and lobsters, some fish and birds, and mammals such as rabbits, rats and cats. It is important to study neurons from many types of animals, since they may exhibit different properties. For example, the squid axons lack a myelin sheath, and the ion currents generated during their conduction of action potentials are simpler than those of some other types of neurons.

Researchers have often used an oscilloscope to follow the rapidly changing voltage of an action potential.

Electrodes and oscilloscopes

See also: Electrophysiology and Electrode

The study of action potentials requires a method for measuring the voltage across the membrane of a neuron. Since a neuron is typically 1–10 microns in diameter, such measurements require extremely small electrodes. A typical electrode is a fine glass tube filled with a conducting solution, which is then connected to an amplifier. The original glass capillary electrodes of Hodgkin and Huxley were roughly 100 microns in diameter and had to be inserted into the open end of a cut axon. A breakthrough came with the work of Ling and Gerard in 1949, who formed micropipette electrodes by melting the middle of glass tubes under extreme tension; once they melted, the two halves separated quickly, drawing the glass to a 1 micron hollow tip. This method for producing electrodes was quickly adopted by other researchers and was crucial for the success of the work of Hodgkin and Huxley.

The transmembrane voltage of an action potential can be monitored by connecting the electrode to a device such as an oscilloscope. Such devices must be sensitive to microvolts (10⁻⁶ volts) and picoamps (10⁻¹² amperes), able to follow the rapidly changing voltages, and must have a very high input impedance, so that the measurement itself does not affect the voltage being measured. A typical input impedance for electrophysiology research might be as high as 100 Megaohms (MΩ). Oftentimes, the data are collected in a Faraday cage, to prevent ambient electric fields from interfering with the sensitive measurements.

Voltage clamping and patch clamping

Main article: Voltage clamp

The voltage clamp circuit provides just enough current to hold the voltage across the membrane fixed at any given value. This allowed scientists to study the ionic currents as a function of transmembrane voltage, without the complication of the capacitative current I = C dV/dt.

The technique of voltage clamping was also instrumental to the successful study of the action potential. As noted above, electrophysiology experiments are quite demanding; the instrumentation must be made very sensitive and the electrical noise minimized. One important source of noise is the capacitative current C dV/dt that arises from the capacitance of the membrane and the changing voltage. To eliminate this source of noise, and top focus on the currents passing through the membrane itself, Kenneth Cole and others developed the voltage clamp, a negative feedback circuit that supplies whatever current is necessary to keep the transmembrane voltage at an arbitrary set-point. By systematically changing this set-point, and studying the variation in the needed current, Hodgkin and Huxley were able to isolate and monitor the currents passing through the voltage-dependent sodium and potassium channels. From those data, they proposed their model of gated channels, e.g., a potassium channel with four voltage-sensitive gates, and a sodium channel with likewise four voltage-sensitive gates, one of which is responsible for inactivation.

In patch clamping, the conductance of a single ion channel (shown in red) can be monitored over time, revealing the distinct states of the channel and different types of channels. Earlier work looked at the overall current through the membrane, which is the net effect of many channels.

Patch clamping

Main articles: Patch clamp and Patch clamp recording

Conductance of a single ion channel monitored by a patch clamp electrode. The channel clearly oscillated between two discrete states, open (high conductance) and closed (low conductance).

The technique of patch-clamping allows the researcher to measure the conductance of a single ion channel in the membrane, and to study its response to various drugs, inhibitors and electrical stimuli. The basic approach requires extremely fine glass-pipette electrodes that suck up a tiny patch of membrane, together with the ion channel. The glass walls of the pipette and the plasma membrane both form an electrically insulating barrier to the flow of ions; hence, if current is to flow, the ions must pass through the ionic channel. Chemicals can be introduced either in the external medium or into the pipette itself. The patch-clamp method was developed by Erwin Neher and Bert Sakmann, for which they were awarded the Nobel Prize in Physiology or Medicine in 1991. For comparison, previous methods recorded the conductance of much larger patches of membranes, in which many ion channels were present; thus, they were able to observe only the average behavior of the ion channels, rather than the specific behavior of one channel.

Patch-clamping methods verified that ionic channels have discrete states of conductance, such open, closed and inactivated. The channel switches back and forth between these states stochastically, consistent with statistical mechanics. The application of voltages or chemicals can alter the relative probabilities of the channel being in one or another state.

Electronic chips

Main article: Neurochip

Action potentials are measured with the recording techniques of electrophysiology and more recently with neurochips containing EOSFETs. An oscilloscope recording the membrane potential from a single point on an axon shows each stage of the action potential as the wave passes. These phases trace an arc that resembles a distorted sine wave; its amplitude depends on whether the action potential wave has reached that point on the membrane or has passed it and if so, how long ago.

Optical methods

Most neurons are not visible with the naked eye, being typically 10 microns in diameter. Therefore, most neurophysiological experiments on individual neurons are done under a microscope, with the electrodes being maneuvered using micromanipulators. Since the neuron is under a microscope, it made sense to develop optical methods to monitor ionic events within the cell. The first of these was the use of the jellyfish protein aequorin to monitor the release of free calcium ions, such as those released by the action potential when it reaches the synaptic button. Dyes that change their fluorescence or absorbance were developed in the early 1970's to monitor action potentials, and have been improved ever since.

Neurotoxins and inhibitors

The tetrodotoxin of the pufferfish inhibits the voltage-sensitive sodium channel, halting action potentials. Supremely lethal, it causes fugu poisoning and has no antidote.

A wide variety of inhibitors for specific channels have been isolated and, in some cases, synthesized. Such inhibitors serve an important research purpose, by allowing scientists to "turn off" channels at will, thus dissecting their contribution; they may also be useful in purifying ion channels by affinity chromatography or in assaying their concentration. However, such inhibitors also make effective neurotoxins, and have been considered for chemical warfare. For example, tetrodotoxin from pufferfish and saxitoxin from Gonyaulax (the dinoflagellate genus responsible for "red tides") both inhibit the voltage-sensitive sodium channel, preventing action potentials; similarly , dendrotoxin from the black mamba snake inhibits the voltage-sensitive potassium channel. Many other neurotoxins act on the neuro-muscular synapse, with by blocking the acetylcholine receptor, preventing its opening, or by inhibiting the cholinesterase enzyme; examples include curare, bungarotoxin, and sarin. Neurotoxins aimed at insects have been effective insecticides, such as permethrin which prolongs the activation of the sodium channels involved in action potentials; the ion channels of insects are sufficiently different from their human counterparts that there is little chance of side-effects on humans.

History

Image of two Purkinje cells (labeled as A) drawn by Santiago Ramón y Cajal. Large trees of dendrites feed into the soma, from which a single axon emerges and moves generally downwards with a few branch points. The smaller cells labeled B are granule cells.

The role of electricity in the nervous systems of animals was first observed in dissected frogs by Luigi Galvani, who studied it from 1791 to 1797. Galvani's results stimulated Alessandro Volta to develop the Voltaic pile—the earliest known electric battery—with which he studied animal electricity (such as electric eels) and the physiological responses to applied direct-current voltages.

Scientists of the 19th century studied electrical signals in whole nerves (i.e., bundles of neurons) and demonstrated that nervous tissue was made up of cells, instead of an interconnected network of tubes (a reticulum). Carlo Matteucci followed up Galvani's studies and demonstrated that cell membranes had a voltage across them and could produce direct current. Matteucci's work inspired the German physiologist, Emil du Bois-Reymond, who discovered the action potential in 1848. The conduction velocity of action potentials were first measured in 1850 by du Bois-Reymond's friend, Hermann von Helmholtz. Evoked potentials in the brain were discovered in 1875 by Richard Caton. To establish that nervous tissue was made up of discrete cells, the Spanish physician Santiago Ramón y Cajal and his students used a stain developed by Camillo Golgi to reveal the many various shapes of neurons, which they rendered painstakingly. For their discoveries, Golgi and Ramón y Cajal were awarded the 1906 Nobel Prize in Physiology. Their work resolved a long-standing controversy in the neuroanatomy of the 19th century; Golgi himself had argued for the network model of the nervous system.

Ribbon diagram of the sodium-potassium pump in its E2-Pi state. The estimated boundaries of the lipid bilayer are shown as blue (intracellular) and red (extracellular) planes.

The 20th century was a golden era for electrophysiology. In 1902 and again in 1912, Julius Bernstein advanced the hypothesis that the action potential resulted from a change in the permeability of the axonal membrane to ions. Bernstein's hypothesis was confirmed by Cole and Curtis, who showed that membrane conductance increases during an action potential. In 1949, Hodgkin and Katz refined Bernstein's hypothesis by considering that the axonal membrane might have different permeabilities to different ions; in particular, they demonstrated the crucial role of the sodium permeability for the action potential. This line of research culminated in the five 1952 papers of Hodgkin, Huxley and Katz, in which they applied the voltage clamp technique to determine the dependence of the axonal membrane's permeabilities to sodium and potassium ions on voltage and time, from which they were able to reconstruct the action potential quantitatively. Hodgkin and Huxley correlated the properties of their mathematical model with discrete ion channels that could exist in several different states, including "open", "closed", and "inactivated". Their hypotheses were confirmed in the mid-1970's and 1980's by Erwin Neher and Bert Sakmann, who developed the technique of patch clamping to examine the conductance states of individual ion channels. In the 21st century, researchers are beginning to understand the structural basis for these conductance states and for the selectivity of channels for their species of ion, through the atomic-resolution crystal structures, fluorescence distance measurements and cryo-electron microscopy studies.

Julius Bernstein was also the first to introduce the Nernst equation for resting potential across the membrane; this was generalized by David E. Goldman to the eponymous Goldman equation in 1943. The sodium-potassium pump was identified in 1957 and its properties gradually elucidated, culminating in the determination of its atomic-resolution structure by X-ray crystallography. The crystal structures of related ionic pumps have also been solved, giving a broader view of how these molecular machines work.

Mathematical models

See also: Biological neuron models

Several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e., to reproduce the measurements of current and voltage exactly. The renowned Hodgkin-Huxley model of the axon from the Loligo squid exemplifies such models. Although qualitatively correct, the H-H model does not describe every type of excitable membrane accurately, since it considers only two ions (sodium and potassium), each with only one type of voltage-sensitive channel. However, other ions such as calcium may be important and there is a great diversity of channels for all ions. As an example, the cardiac action potential illustrates how differently shaped action potentials can be generated on membranes with voltage-sensitive calcium channels and different types of sodium/potassium channels. The second type of mathematical model is a simplification of the first type; the goal is not to reproduce the experimental data, but to understand qualitatively the role of action potentials in neural circuits. For such a purpose, detailed physiological models may be unnecessarily complicated and may obscure the "forest for the trees". The Fitzhugh-Nagumo model is typical of this class, which is often studied for its entrainment behavior. Entrainment is commonly observed in nature, for example in the synchronized lighting of fireflies, which is coordinated by a burst of action potentials; entrainment can also be observed in individual neurons. Both types of models may be used to understand the behavior of small biological neural networks, such as the central pattern generators responsible for some automatic reflex actions. Such networks can generate a complex temporal pattern of action potentials that is used to coordinate muscular contractions, such as those involved in breathing or fast swimming to escape a predator.

Hodgkin-Huxley model

Main article: Hodgkin-Huxley model

Equivalent electrical circuit for the Hodgkin-Huxley model of the action potential. I_m and V_m represent the current through, and the voltage across, a small patch of membrane, respectively. The C_m represents the capacitance of the membrane patch, whereas the four g's represent the conductances of four types of ions. The two central conductances, for potassium (K) and sodium (Na), are shown with arrows to indicate that they can vary with the applied voltage, corresponding to the voltage-sensitive ion channels.

In 1952 Alan Lloyd Hodgkin and Andrew Huxley developed a set of equations to fit their experimental voltage-clamp data on the axonal membrane. The model assumes that the membrane capacitance C is constant; thus, the transmembrane voltage V changes with the total transmembrane current I_tot according to the equation

$C{\frac {dV}{dt}}=I_{\mathrm {tot} }=I_{\mathrm {ext} }+I_{\mathrm {Na} }+I_{\mathrm {K} }+I_{\mathrm {L} }$

where I_Na, I_K, and I_L are currents conveyed through the local sodium channels, potassium channels, and "leakage" channels (a catch-all), respectively. The initial term I_ext represents the current arriving from external sources, such as excitatory postsynaptic potentials from the dendrites or a scientist's electrode.

The model further assumes that a given ion channel is either fully open or closed; if closed, its conductance is zero, whereas if open, its conductance is some constant value g. Hence, the net current through an ion channel depends on two variables: the probability p_open of the channel being open, and the difference in voltage from that ion's equilibrium voltage, V − V_eq. For example, the current through the potassium channel may be written as

$I_{\mathrm {K} }=g_{\mathrm {K} }\left(V-V_{\mathrm {eq,K} }\right)p_{\mathrm {open,K} }$

which is equivalent to Ohm's law. By definition, no net current flows (I_K = 0) when the transmembrane voltage equals the equilibrium voltage of that ion (when V = V_eq,K).

To fit their data accurately, Hodgkin and Huxley assumed that each type of ion channel had multiple "gates", so that the channel was open only if all the gates were open and closed otherwise. They also assumed that the probability of a gate being open was independent of the other gates being open; this assumption was later validated for the inactivation gate. Hodgkin and Huxley modeled the voltage-sensitive potassium channel as having four gates; letting p_n denote the probability of a single such gate being open, the probability of the whole channel being open is the product of four such probabilities, i.e., p_{open, K} = n⁴. Similarly, the probability of the voltage-sensitive sodium channel was modeled to have three similar gates of probability m and a fourth gate, associated with inactivation, of probability h; thus, p_{open, Na} = m³h. The probabilities for each gate are assumed to obey first-order kinetics

${\frac {dm}{dt}}=-{\frac {m-m_{\mathrm {eq} }}{\tau _{m}}}$

where both the equilibrium value m_eq and the relaxation time constant τ_m depend on the instantaneous voltage V across the membrane. If V changes on a time-scale more slowly than τ_m, the m probability will always roughly equal its equilibrium value m_eq; however, if V changes more quickly, then m will lag behind m_eq. By fitting their voltage-clamp data, Hodgkin and Huxley were able to model how these equilibrium values and time constants varied with temperature and transmembrane voltage. The formulae are complex and depend exponentially on the voltage and temperature. For example, the time constant for sodium-channel activation probability h varies as 3^{(θ−6.3)/10} with the Celsius temperature θ, and with voltage V as

${\frac {1}{\tau _{h}}}=0.07e^{-V/20}+{\frac {1}{1+e^{3-V/10}}}$

In summary, the Hodgkin-Huxley equations are complex, non-linear ordinary differential equations in four independent variables: the transmembrane voltage V, and the probabilities m, h and n. No general solution of these equations has been discovered. A less ambitious but generally applicable method for studying such non-linear dynamical systems is to consider their behavior in the vicinity of a fixed point. This analysis shows that the Hodgkin-Huxley system undergoes a transition from stable quiescence to bursting oscillations as the stimulating current I_ext is gradually increased; remarkably, the axon becomes stably quiescent again as the stimulating current is increased further still. A more general study of the types of qualitative behavior of axons predicted by the Hodgkin-Huxley equations has also been carried out.

Fitzhugh-Nagumo model

Main article: Fitzhugh-Nagumo model

Figure FHN: To mimick the action potential, the Fitzhugh-Nagumo model and its relatives use a function g(V) with negative differential resistance (a negative slope on the I vs. V plot). For comparison, a normal resistor would have a positive slope, by Ohm's law I = GV, where the conductance G is the inverse of resistance G=1/R.

Because of the complexity of the Hodgkin-Huxley equations, various simplifications have been developed that exhibit qualitatively similar behavior. The Fitzhugh-Nagumo model is a typical example of such a simplified system. Based on the tunnel diode, the FHN model has only two independent variables, but exhibits a similar stability behavior to the full Hodgkin-Huxley equations. The equations are

$C{\frac {dV}{dt}}=I-g(V),$

$L{\frac {dI}{dt}}=E-V-RI$

where g(V) is a function of the voltage V that has a region of negative slope in the middle, flanked by one maximum and one minimum (Figure FHN). A much-studied simple case of the Fitzhugh-Nagumo model is the Bonhoeffer-van der Pol nerve model, which is described by the equations

$C{\frac {dV}{dt}}=I-\epsilon \left({\frac {V^{3}}{3}}-V\right),$

$L{\frac {dI}{dt}}=-V$

where the coefficient ε is assumed to be small. These equations can be combined into a second-order differential equation

$C{\frac {d^{2}V}{dt^{2}}}+\epsilon \left(V^{2}-1\right){\frac {dV}{dt}}+{\frac {V}{L}}=0$

This van der Pol equation has stimulated much research in the mathematics of nonlinear dynamical systems. Op-amp circuits that realize the FHN and van der Pol models of the action potential have been developed by Keener.

A hybrid of the Hodgkin-Huxley and FitzHugh-Nagumo models was developed by Morris and Lecar in 1981, and applied to the muscle fiber of barnacles. True to the barnacle's physiology, the Morris-Lecar model replaces the voltage-gated sodium current of the Hodgkin-Huxley model with a voltage-dependent calcium current. There is no inactivation (no h variable) and the calcium current equilibrates instantaneously, so that again, there are only two time-dependent variables: the transmembrane voltage V and the potassium gate probability n. The bursting, entrainment and other mathematical properties of this model have been studied in detail.

The simplest models of the action potential are the "flush and fill" models (also called "integrate-and-fire" models), in which the input signal is summed (the "fill" phase) until it reaches a threshold, firing a pulse and resetting the summation to zero (the "flush" phase). All of these models are capable of exhibiting entrainment, which is commonly observed in nervous systems.

Extracellular potentials and currents

Whereas the above models simulate the transmembrane voltage and current at a single patch of membrane, other mathematical models pertain to the voltages and currents in the ionic solution surrounding the neuron. Such models are helpful in interpreting data from extracellular electrodes, which were common prior to the invention of the glass pipette electrode that allowed intracellular recording. The extracellular medium may be modeled as a normal isotropic ionic solution; in such solutions, the current follows the electric field lines, according to the continuum form of Ohm's Law

$\mathbf {j} =\sigma \mathbf {E}$

where j and E are vectors representing the current density and electric field, respectively, and where σ is the conductivity. Thus, j can be found from E, which in turn may be found using Maxwell's equations. Maxwell's equations can be reduced to a relatively simple problem of electrostatics, since the ionic concentrations change too slowly (compared to the speed of light) for magnetic effects to be important. The electric potential φ(x) at any extracellular point x can be solved using Green's identities

$\phi (\mathbf {x} )={\frac {1}{4\pi \sigma _{\mathrm {outside} }}}\oint _{\mathrm {membrane} }{\frac {\partial }{\partial n}}{\frac {1}{\left|\mathbf {x} -{\boldsymbol {\xi }}\right|}}\left[\sigma _{\mathrm {outside} }\phi _{\mathrm {outside} }({\boldsymbol {\xi }})-\sigma _{\mathrm {inside} }\phi _{\mathrm {inside} }({\boldsymbol {\xi }})\right]dS$

where the integration is over the complete surface of the membrane; ${\boldsymbol {\xi }}$ is a position on the membrane, σ_inside and φ_inside are the conductivity and potential just within the membrane, and σ_outside and φ_outside the corresponding values just outside the membrane. Thus, given these σ and φ values on the membrane, the extracellular potential φ(x) can be calculated for any position x; in turn, the electric field E and current density j can be calculated from this potential field.

See also

• Cardiac action potential
• Ventricular action potential
• Membrane potential
• Depolarization
• Hodgkin-Huxley model
• Hyperpolarization
• Signals (biology)
• Time constant
• Length constant
• Bursting

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• Schmidt-Nielsen K (1997). Animal Physiology: Adaptation and Environment (5th ed. ed.). Cambridge: Cambridge University Press. ISBN 978-0521570985. {{cite book}}: |edition= has extra text (help)
• Stevens CF (1966). Neurophysiology: A Primer. New York: John Wiley and Sons. LCCN 66-0 – 0
• Adrian ED, Lucas K (1912). "On the summation of propagated disturbances in nerve and muscle". J. Physiol. (London). 44: 68–124.

External links

Animations

Voltages and ionic flow

• Demonstration of ion flow during action potential at Blackwell Publishing
• Action Potentials Dawn A. Tamarkin, Springfield Technical Community College, Springfield, Massachusetts
• Three-dimensional animation of action potential Teaching Resources Center, UC Davis
• Action potential Teaching Resources Center, UC Davis

Propagation; saltatory conduction

• Action potential propagation in myelinated vs. unmyelinated axons at Blackwell Publishing
• Saltatory conduction Dawn A. Tamarkin, Springfield Technical Community College, Springfield, Massachusetts

Resting potential

• Life: The Science of Biology, by WK Purves, D Sadava, GH Orians, and HC Heller, 8th edition, New York: WH Freeman, ISBN 978-0716776710.
• Nernst/Goldman Equation Simulator at University of Arizona

Lecture notes

• The Action Potential John Kinnamon, University of Denver
• Excitable Cells John Kimball, Tufts University (retired)
• Resting and Action Membrane Potentials Teaching Resources Center, UC Davis. Animated tutorials

Educational resources for schoolchildren

• Modeling neurons and action potentials at Neuroscience for Kids

• Playground games to teach about the action potential and neurotransmission, likewise from Neuroscience for Kids

Plant action potentials

• Electrochemistry of plant life, May, 2004 from Case Western Reserve University

Open-source software for simulating action potentials

• Open-source software to simulate neuronal and cardiac action potentials at SourceForge