The Hodgkin-Huxley Model for the Generation of Action Potentials
Building Equations to Produce an Action Potential
Voltage dependency of gate position
Voltage dependency of channel conductance
Equations for the membrane potential
What Hodgkin and Huxley actually did
Second: obtaining parameters for the model
Voltage dependency of alpha and beta
Third: reconstructing the spike
The Hodgkin-Huxley (HH; Hodgkin & Huxley, 1952) model for the generation of the nerve action potential is one of the most successful mathematical models of a complex biological process that has ever been formulated. The basic concepts expressed in the model have proved a valid approach to the study of bio-electrical activity from the most primitive single-celled organisms such as Paramecium, right through to the neurons within our own brains.
Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerves. J. Physiol. (Lond.) 117, 500-544.
Neurosim: If you would like to see the Hodgkin-Huxley model in action (and a lot more besides) then I cordially invite you to check out the new version (5) of my teaching simulation package Neurosim.
The starting point of the original model is that the nerve membrane (specifically, the membrane of the squid giant axon) contains three types of ion channels. The first, known as the leakage channels, has a relatively low conductance that does not change. Although their overall conductance is low, it is higher to potassium (K) ions than to sodium (Na) ions. The leakage channels are mainly responsible for the resting membrane potential. The remaining two types of ion channels, which are the ones responsible for generating the action potential, are both voltage-dependent, i.e. their conductances depend upon the voltage across the membrane. There is one set of voltage-dependent channels that are specifically permeable to Na ions, and another set specifically permeable to K ions.
Each voltage-dependent channel can be pictured as being like a tunnel with a small number of gates arranged one-after-another within it. In order for the individual channel to be open and allow ions to flow through, all the gates within that channel must be open simultaneously. If even one gate is shut, then the whole channel is shut.
The individual gates open and close randomly and quite rapidly, but the probability of a gate being open (the open probability) is dependent on the voltage across the membrane. In molecular terms, the gates are thought to act like charge-carrying particles, and hence the position they occupy within the membrane, which determines whether they are open or shut, is affected by the electrical potential across the membrane (the voltage).
Channel gates falls into one of two classes; activation gates have an open probability that increases with depolarisation, while inactivation gates have an open probability which decreases with depolarisation. The probability of a gate being open at any point in time is known as the activation variable for that gate. Since the activation variable defines the probability that a single gate of that class will be open, it therefore also defines the proportion of gates in the total population of that class which are open. As well as differing in how their activation variables change with voltage, gate classes also differ in the rate at which their activation variables change when the voltage changes.
The HH model proposes that each Na channel contains a set of 3 identical, rapidly-responding, activation gates (the m-gates), and a single, slower-responding, inactivation gate (the h-gate). By convention, the activation variable for the m-gates is known as m, and the activation variable of the h-gates is known as h. These two classes of gates in combination explain the transient increase in Na conductivity which results from membrane depolarisation. The way this works is as follows.
At resting potential, the h-gate is open, but the m-gates are shut, and therefore the channel itself is shut (at least, this is the most likely state of affairs, - since the gates open and shut probabilistically the exact state of any gate cannot be predicted with absolute certainty). If the membrane is then depolarised, the m-gates rapidly open, and for a while the channel itself is open or activated. Then the h-gate shuts, and therefore the channel shuts, even though the membrane is still depolarised. The channel is now in the inactivated state. If the membrane is now repolarised, the m-gates rapidly shut. At this point, if the membrane is again depolarised, the m-gates open, but the h-gate, which has not yet reopened in response to the earlier repolarisation, remains shut, and so the channel itself does not reopen. This is the basis of the absolute refractory period of the action potential. Finally, if the membrane is repolarised the m-gates shut, and if the membrane is held repolarised for some time, the h-gate eventually reopens (de-inactivation). The channel is now back in its original condition; shut, but ready to open in response to depolarisation.
The K channel is somewhat simpler. It contains a single class of gate consisting of 4 individual activation gates (the n-gates), which respond more slowly than the activation gates of the Na channel. Thus if the membrane is depolarised, the n-gates open (slowly), and the K channel opens. The channel remains open for as long as the membrane remains depolarised. When the membrane is repolarised, the n-gates, and hence the K channel, slowly shut. The relatively slow rate at which the K channels shut means that there is an elevated K conductance for some time following an action potential, and this can cause an after-hyperpolarisation which is partly responsible for the relative refractory period.
Building Equations to Produce an Action Potential
We will first describe how a very simple set of basic assumptions about the gates leads to a series of equations that describe the active properties of the nerve membrane. We will then describe how experimental data is used to provide numerical parameters to plug into these equations, so that the equations can reconstruct an action potential.
Voltage dependency of gate position
In the HH model the individual gates act like a first order chemical reaction with two states. This can be written thus:
(1)
The factors α and β are called the transition rate constants. α is the number of times per second that a gate which is in the shut state opens, while β is the number of times per second that a gate which is in the open state shuts. All the gates within a particular class have the same value of α and the same value of β (which is likely to be different from the value of α) at any instant in time, but gates which belong to different classes may have different values of α and β. This gives the different classes their different properties.
[To jump ahead the KEY FACTOR in the HH model which allows
action potentials to be generated is that α and β are VOLTAGE DEPENDENT.]
So how does the open probability of a gate depend upon α and β? For a whole population of gates, let us say a proportion P are in the open state, where P varies between 0 and 1. This means that a proportion 1-P will be in the closed state. The fraction of the total population which open in a given time is dependent on the proportion of gates which are shut, and the rate at which shut gates open:
(2)
and similarly
(3)
If a system is in equilibrium, where the proportion of gates in the open state is not changing, then the fraction of gates opening must equal the fraction of gates closing in any given period of time
(4)
which rearranges as
(5)
Thus if α is high and β is low, the gate has a high probability of being open, and vice versa. (The infinity subscript is used for P because the system only achieves equilibrium if α and β remain stable for a relatively long period of time.)
The voltage dependency of P arises because the fundamental transition rate constants α and β are themselves voltage dependent. Clearly, if the membrane potential changes, and consequently the values of α and β for a particular class of gate change, then the open probability P for that class of gate must also change. For activation gates the voltage dependency of α and β is such that a depolarising shift in membrane potential causes P to increase, while for inactivation gates the change in α and β causes P to decrease.
The HH model assumes that α and β change instantly with a change in voltage. However, this does not lead to an instantaneous change in the value of P. The rate at which P achieves its new value following a change in α and/or β is equal to the difference in the rate of shutting and the rate of opening:
(6)
(Note that if we substitute the steady-state value of P in terms of α and β from equation (5), into the right hand side of this equation, dP/dt becomes 0, as of course it should in steady state conditions.) Thus, following a change in voltage, the rate of change of P, as well as the direction and size of change, is dependent on the values of α and β. Depending on the values of α and β, some classes of gates will respond more rapidly to changes in voltage than others.
The differential equation (6) has a solution
(7)
where
(8)
These equations can be understood as follows. We start with assuming that the system has been at a fixed constant voltage for a long period of time, and therefore P is at a starting equilibrium value Pstart defined in equation (5). The voltage is then changed suddenly, and α and β immediately switch to new values appropriate to the new voltage. P then starts to change, and approaches its new equilibrium value P∞ (also defined in equation 5, but with the new values for α and β) with an exponential time course with a time constant of τ. If either α or β are large, then the time constant is short and P arrives at its new value rapidly. If both are small, then the time constant is long and it takes longer for P to reach equilibrium.
By combining equations (5) and (8) it is possible to express α and β in terms of P∞ and τ:
(9)
and
(10)
There is thus a simple relationship between α and β, and the equilibrium value of P and the time constant with which P attains this equilibrium value.
Voltage dependency of channel conductance
Let us start the next stage of analysis by considering the situation when the voltage is stable, as it is when the membrane is sitting at the resting potential. For each class of gate in each type of channel, α and β have values appropriate for the voltage, and P (the probability of a gate being open) is at its steady-state equilibrium value given in equation (5). If a channel contains several (say x) gates of that class within it, the probability of the whole channel being open is P raised to the power of the number of gates within the channel (i.e. Px). This is because all the gates have to be open for the channel to be open.
For reasons that will be explained later, HH proposed that each K channel has 4 identical activation gates (x = 4). We can replace the general probability value P with the specific probability of a K-channel n-gate being open, n, so the probability of a whole K channel being open is n4. Thus, to make this concrete, if at a particular voltage the probability of an n-gate being open is one half (n = 0.5), then the probability of an individual K channel being open is 0.5*0.5*0.5*0.5, or 1 in 16. By scaling up, we can say that 1 out of every 16 in the whole population of K channels will be open, and thus the actual K conductance (gK) will be 1/16 of the maximum possible K conductance, i.e.
(11)
where gKmax is the membrane K conductance when all the K channels are open.
The HH model proposes that the Na channel has 3 activation m-gates, and one inactivation h-gate, and so by the same reasoning, the Na conductance is
(12)
Once the conductance of a population of ion channels is known, the ionic current that flows through the channels can be calculated. This is because there is usually a fairly simple relationship between current (I), conductance (g), membrane potential (Em) and the reversal (equilibrium) potential (Eeq) of an ion, when the current is carried by that single ion species:
(13)
This equation is a variant of Ohm's law. The factor Em-Eeq, which is a measure of how far the membrane potential is from the equilibrium potential of the ion in question, is called the driving force on the ion, and is equivalent to straight voltage in Ohm's law. We can make this equation specific for K ions
(14)
where IK is the K current and EK is the K equilibrium potential. A similar equation gives the Na current
(15)
There is a third current we have to consider; - the leakage current. As well as the voltage-dependent channels discussed above, the membrane has a small, non-voltage-dependent conductance to both Na and K. This is known as the leakage conductance, and it is always present and remains constant whatever the voltage. The K leakage conductance is much higher than the Na leakage conductance (although both are small compared to the voltage-dependent conductances when they are activated), and so the leakage conductance acts as if it had an equilibrium potential close to resting potential.
(16)
Equations for the membrane potential
It is intuitively obvious that if there is an imbalance in current across the membrane such that more positive charge enters the cell than leaves it, this will change the membrane potential and cause it to depolarise (and vice versa). The change in membrane potential occurs because the unbalanced current alters the charge on the membrane capacitor. This leads to the following relationship:
(17)
In this equation the
expression Cm (dV/dt) is the
capacity current, and it derives simply from the property of capacitance, which
says that the current into a capacitor is proportional to the size of the
capacitance and the rate of change of voltage (dV/dt) across it. The equation states that the capacity current is equal
to the arithmetic sum of all the currents across the membrane this follows from the fact that if there is an
imbalance between positive and negative membrane current, the “spare” current
has nowhere else to go except into the membrane capacitor. The membrane current consists of the ionic
current Iionic , which is
the sum of the Na, K and leakage currents calculated from the modified Ohm's
law given above in equations (14)
(16), plus any stimulating current Istim that is injected.
[Note that equation (17) strictly applies only to a space-clamped neuron, or a single-compartment model. In other words, it assumes that there is no lateral flow of current within the neuron. If there were such flow, it would have to be added in to the right-hand side of the equation.]
We can re-write equation (17) thus
(18)
In the resting neuron by definition the membrane potential is not changing, i.e. dV/dt (the rate of change of voltage) is 0. There is no stimulus applied, so Istim is 0, and therefore Iionic must also be 0. If Iionic is 0, this means that the inward and outward currents flowing through the ionic channels exactly balance to cancel each other, which, of course, is what you would expect for a resting neuron.
Now imagine what happens if a stimulus is applied to the neuron, so Istim is not 0. Initially Iionic does not change (because none of the right hand sides of equations 14 - 16 change), and therefore the stimulus current flows into the membrane capacitor and dV/dt becomes non-zero. Thus at the next instant in time, the membrane potential V has a new value. This will instantly change the values of α and β for the channel gates, which will start to change the value of P for each of the gate classes (equation 6). If P (i.e. m, n and h) changes, then the channel conductance g will change (equations 11 and 12). A change in both conductance and voltage is likely to result in a change in ionic current (equations 14-16), and this in turn is likely to lead to a further change in voltage (equation 18). In this way an iterative feedback process is initiated. The triumph of the HH model is that when you put all these equations together with the appropriate parameters, the voltage changes have the waveform of an action potential!
What Hodgkin and Huxley Actually Did
The original work of Hodgkin and Huxley (and some others) consisted of a three-stage process.
First: the model assumptions
They proposed the basic model, consisting of independent channels containing gates following first order kinetics, and with currents carried entirely by ions moving down electrochemical gradients. This is simple to state, but since there are very many alternative models that could have been proposed (and indeed had been proposed earlier), this was a very insightful step. This theoretical framework led them to develop the equations described above.
Second: obtaining parameters for the model
In order to make use of the equations described above, appropriate numerical values had to be found to fill in the unknown parameters. There were 3 levels of detail required. First, the macro characteristics of the channel types (ionic specificity, maximum conductances, equilibrium potentials), had to be determined. Second, the number of activation and inactivation gates in each channel type had to be determined. Third, equations had to be found to describe the quantitative voltage dependency of α and β for each gate type in each channel type.
Ionic properties
The fact that Na and K are the major ions involved in generating the squid action potential had been established in earlier work, as had the equilibrium potentials for those ions. HH used ion substitutions to treat Na and K currents separately, since TTX and TEA were not available in those days. They then used the voltage clamp technique to measure the steady-state current at various voltages, and the rate of change of current following a change in voltage. Since the equilibrium potentials were known, the Na and K conductances could be determined from the current records using equations (14) and (15). These conductance data provided the information needed to determine the remaining parameters, as will be described next.
Number of gates
HH observed that during the depolarising step of a voltage clamp experiment the conductance change had a sigmoid shape, but during the repolarising step the conductance change had an exponential shape (e.g. Fig 2 in the key HH paper). HH knew that single first-order reactions of the type proposed for the individual channel gates should produce exponential curves, but that sigmoid curves would result from co-operative processes in which several first order reactions had to occur simultaneously. This fitted with the notion that the channels contained several gates, all of which had to be open at once in order for the channel itself to be open, hence the sigmoid shape of the rising curve. On the other hand, only one gate had to shut for the channel to shut, hence the exponential shape of the falling curve. In co-operative processes, the shape of the sigmoid part of the curve depends on the number of events involved; the greater the number of events, the more pronounced the inflexions on the curve. It was the exact shape of the experimentally-measured sigmoid curve that suggested that 4 would be the best estimate of independent gates within the K channel. Similar analysis of conductance curve shapes for Na suggested that 3 activation gates and one inactivation gate would best fit the data.
Voltage dependency of alpha and beta
We have seen earlier (equations 9 and 10) that for any gate type there is a simple relation between the values of the transition rate constants α and β, the fraction of gates in the open state P, and the time constant with which that fraction approaches its equilibrium value τ.This means that if P and τ can be measured at a particular voltage, then α and β can easily be calculated. This was the approach taken by HH. It will be illustrated in detail for the K channel, but a similar approach was taken for the Na channel.
Equation (7) shows how the n-variable (the open probability of a single n gate in a K channel) changes with time upon a change of the transition rate constants α and β. Equation (11) shows how the K conductance varies as the n-variable changes. Combining these equations yields the following
(19)
Note that this equation (given as equation 11 in the HH paper) is very similar to equation (7), except that the K conductance gK replaces the general probability P, and that several factors are either raised to the fourth power, or the fourth root (this takes care of the fact that there are 4 n-gates per K channel). The equation describes a voltage clamp experiment in which gKstart is the stable K conductance at the holding potential before the clamp pulse, gK∞ is the final K conductance attained during a sustained clamp pulse of a particular voltage, gK(t) is the K conductance at time t after the switch from holding potential to clamp potential, and τn is the time constant of the change in the K activation variable n at the clamp potential. All the values except the last (τn) can be read directly from the results of a voltage clamp experiment. HH performed experiments using a wide range of different clamp potentials, and then found which values of τn gave the best fit of this equation (19) to the data at each clamp potential. In this way they determined the voltage dependency of τ.
The next task was to determine the values of n at each clamp potential. The K conductance when all channels are fully open (gKmax) was measured as the maximum conductance achieved with a very depolarised clamp potential. The stable K conductance (gK∞) measured at other clamp potentials could then be expressed as a fraction of this maximum. The activation variable n was then taken as the fourth root of this fraction (equation 11).
Similar experiments gave the voltage dependency of the activation and inactivation variables for the gates in the Na channels.
The α and β values were then calculated from the P and τ values for each gate type (n, m and h) and plotted against voltage. The plots followed a series of smooth curves that could be fitted by the following equations (where V is the membrane potential in mV).
K activation
Na activation
Na inactivation
These equations are essentially empirical, but are based upon equations that describe the movement of a charged particle in an electric field. Since that is the physical model of a gate moving within a channel to open and close, this seems reasonable.
Third: reconstructing the spike
Having derived the model and its equations as described above, and having determined the appropriate numerical parameters by experiment, HH then worked forwards and "reconstructed" the effects of applying a depolarising stimulus to an axon. This was done by numerical integration of the equations, starting with equation (18). When this was done with the appropriate stimulus parameters, they found that there was a truly excellent correspondence between the predicted values of the membrane potential, and the actual shape of an action potential in a space clamped axon.
The HH model has been amazingly successful in both describing and predicting a large number of neuronal properties. Extensions of this model, incorporating a variety of voltage-dependent channel types beyond the original HH pair, have been very widely used in research throughout the world. However, as HH were themselves well aware, the success of the model does not in itself constitute convincing evidence that the "pictorial" interpretation of the HH equations is a true reflection of the real molecular events. It is therefore very gratifying, although perhaps surprising, the extent to which modern investigations into the molecular structure of the various channels have confirmed the physical reality, or approximate reality, of many aspects of the model.
(Dr W. J. Heitler,
