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.

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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 **P _{start}** defined in equation (5).
The voltage is then changed suddenly, and

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. **P ^{x}**).
This is because

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 **n ^{4}**.
Thus, to make this concrete, if at a particular voltage the probability of an

(11)

where **gK _{max}** 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
(**E _{m}**) and the reversal
(equilibrium) potential (

(13)

This equation is a
variant of Ohm's law. The factor **E _{m}-E_{eq}**, which is
a measure of how far the membrane potential is from the equilibrium potential
of the ion in question, is called the

(14)

where **I _{K}** is the K current and

(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 **C _{m} (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 (

[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 **I _{stim}** is 0,
and therefore

Now imagine what
happens if a stimulus is applied to the neuron, so **I _{stim}** is not 0. Initially

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.

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.

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.

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 **g _{K}**
replaces the general probability

The next task was to
determine the values of **n** at each
clamp potential. The K conductance when all channels are fully open (**gK _{max}**) was measured as the
maximum conductance achieved with a very depolarised clamp potential. The
stable K conductance (

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,