2.2 Hodgkin-Huxley Model

Hodgkin and Huxley (Hodgkin and Huxley, 1952) performed experiments on the giant axon of
the squid and found three different types of ion current, viz., sodium,
potassium, and a leak current that consists mainly of Cl^{-} ions. Specific
voltage-dependent ion channels, one for sodium and another one for potassium,
control the flow of those ions through the cell membrane. The leak current
takes care of other channel types which are not described explicitly.

The Hodgkin-Huxley model can be understood with the help of
Fig. 2.2. The semipermeable cell membrane separates the interior of
the cell from the extracellular liquid and acts as a capacitor. If an input
current *I*(*t*) is injected into the cell, it may add further charge on the
capacitor, or leak through the channels in the cell membrane. Because of
active ion transport through the cell membrane, the ion concentration inside
the cell is different from that in the extracellular liquid. The Nernst
potential generated by the difference in ion concentration is represented by a
battery.

Let us now translate the above considerations into mathematical equations.
The conservation of electric charge on a piece of membrane implies that the
applied current *I*(*t*) may be split in a capacitive current *I*_{C} which
charges the capacitor *C* and further components *I*_{k} which pass through
the ion channels. Thus

where the sum runs over all ion channels. In the standard Hodgkin-Huxley model there are only three types of channel: a sodium channel with index Na, a potassium channel with index K and an unspecific leakage channel with resistance

In biological terms,

As mentioned above, the Hodgkin-Huxley model describes three types of channel.
All channels may be characterized by their resistance or, equivalently, by
their conductance.
The leakage channel is described by a
voltage-independent conductance *g*_{L} = 1/*R*; the conductance of the other ion
channels is voltage and time dependent. If all channels are open, they
transmit currents with a maximum conductance
*g*_{Na} or *g*_{K},
respectively. Normally, however, some of the channels are blocked. The
probability that a channel is open is described by additional variables *m*, *n*,
and *h*. The combined action of *m* and *h* controls the Na^{+} channels.
The K^{+} gates are controlled by *n*. Specifically, Hodgkin and Huxley
formulated the three current components as

The parameters

The three variables *m*, *n*, and *h*
are called gating variables.
They
evolve according to the differential
equations

with = d

In order to getter a better understanding of the three equations (2.6), it is convenient to rewrite each of the equations in the form

where

2.2.2 Dynamics

In this subsection we study the dynamics of the Hodgkin-Huxley model for different types of input. Pulse input, constant input, step current input, and time-dependent input are considered in turn. These input scenarios have been chosen so as to provide an intuitive understanding of the dynamics of the Hodgkin-Huxley model.

We see from Fig. 2.3A that *m*_{0} and *n*_{0} increase with *u*
whereas *h*_{0} decreases. Thus, if some external input causes the membrane
voltage to rise, the conductance of sodium channels increases due to
increasing *m*. As a result, positive sodium ions flow into the cell and
raise the membrane potential even further. If this positive feedback is
large enough, an action potential is initiated.

At high values of *u* the sodium conductance is shut off due to the factor
*h*. As indicated in Fig. 2.3B, the `time constant' is
always larger than . Thus the variable *h* which closes the channels
reacts more slowly to the voltage increase than the variable *m* which opens
the channel. On a similar slow time scale, the potassium (K^{+}) current sets
in. Since it is a current in outward direction, it lowers the potential. The
overall effect of the sodium and potassium currents is a short action
potential followed by a negative overshoot; cf. Fig. 2.4A. The
amplitude of the spike is about 100 mV.

In Fig. 2.4A, The spike has been initiated by a short current
pulse of 1 ms duration applied at *t* < 0. If the amplitude of the stimulating
current pulse is reduced below some critical value, the membrane potential
returns to the rest value without a large spike-like excursion; cf. Fig. 2.4B. Thus we have a threshold-type behavior.

2.2.2.2 Example: Mean firing rates and gain function

The Hodgkin-Huxley equations (2.4)-(2.6) may also be studied for
constant input
*I*(*t*) = *I*_{0} for *t* > 0. (The input is zero for *t* 0).
If the value *I*_{0} is larger than a critical value
*I*_{} 6A/cm^{2}, we observe regular spiking; Fig. 2.5A. We may
define a firing rate = 1/*T* where *T* is the inter-spike interval. The
firing rate as a function of the constant input *I*_{0} defines the gain
function plotted in Fig. 2.5B.

In the previous example we have seen that a constant input current
*I*_{0} > *I*_{} generates regular firing. In this paragraph we study the response
of the Hodgkin-Huxley model to a step current of the form

Here (

The answer to this question is given by Fig. 2.6B. A large
step *I* facilitates the spike initiation. Even for a target value
*I*_{2} = 0 (i.e., no stimulation for *t* > 0) a spike is possible, provided that
the step size is large enough. This is an example of inhibitory rebound: A
single spike is fired, if an inhibitory current *I*_{1} < 0 is released. The
letter *S* in Fig. 2.6B denotes the regime where only a single
spike is initiated. Repetitive firing (regime *R*) is possible for
*I*_{2} > 6A/cm^{2}, but must be triggered by sufficiently large current steps.

We may conclude from Fig. 2.6B that there is no unique current
threshold for spike initiation: The trigger mechanism for action potentials
depends not only on *I*_{2} but also on the size of the current step *I*. More generally, it can be shown that the concept of a threshold itself
is questionable from a mathematical point of view (Koch et al., 1995; Rinzel and Ermentrout, 1989). In
a mathematical sense, the transition in Fig. 2.4B, that `looks'
like a threshold is, in fact, smooth. If we carefully tuned the input current
in the regime between 6.9 and 7.0 A/cm^{2}, we would find a family of
response amplitudes in between the curves shown in Fig. 2.4B. For
practical purposes, however, the transition can be treated as a threshold
effect. A mathematical discussion of the threshold phenomenon can be found in
Chapter 3.

In order to explore a more realistic input scenario, we stimulate the
Hodgkin-Huxley model by a time-dependent input current *I*(*t*) that is
generated by the following procedure. Every 2 ms, a random number is drawn
from a Gaussian distribution with zero mean and standard deviation
= 3A/cm^{2}. To get a continuous input current, a linear interpolation was
used between the target values. The resulting time-dependent input current
was then applied to the Hodgkin-Huxley model (2.4). The response to the
current is the voltage trace shown in Fig. 2.7. Note that
action potentials occur at irregular intervals.

2.2.2.5 Example: Refractoriness

In order to study neuronal refractoriness, we stimulate the Hodgkin-Huxley
model by a first current pulse that is sufficiently strong to excite a spike.
A second current pulse of the *same* amplitude as the first one is used to
probe the responsiveness of the neuron during the phase of hyperpolarization
that follows the action potential. If the second stimulus is not sufficient
to trigger another action potential, we have a clear signature of neuronal
refractoriness. In the simulation shown in Fig. 2.7B, a second
spike is possible if we wait at least 15 milliseconds after the first
stimulation. It would, of course, be possible to trigger a second spike after
a shorter interval, if a significantly stronger stimulation pulse was used;
for classical experiments along those lines, see, e.g. (Fuortes and Mantegazzini, 1962).

If we look more closely at the voltage trajectory of Fig. 2.7B, we see that neuronal refractoriness manifests itself in two different forms. First, due to the hyperpolarizing spike afterpotential the voltage is lower. More stimulation is therefore needed to reach the firing threshold. Second, since a large portion of channels is open immediately after a spike, the resistance of the membrane is reduced compared to the situation at rest. The depolarizing effect of a stimulating current pulse decays therefore faster immediately after the spike than ten milliseconds later. An efficient description of refractoriness plays a major role for simplified neuron models discussed in Chapter 4.

Cambridge University Press, 2002

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