Hodgkin and Huxley (222) 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 is injected into the cell, it may add further charge on the capacitor, or leak through the channels in the cell membrane. Each channel type is represented in Fig. 2.2 by a resistor. The unspecific channel has a leak resistance , the sodium channel a resistance and the potassium channel a resistance . The diagonal arrow across the diagram of the resistor indicates that the value of the resistance is not fixed, but changes depending on whether the ion channel is open or closed. 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 in Fig. 2.2. Since the Nernst potential is different for each ion type, there are separate batteries for sodium, potassium, and the unspecific third channel, with battery voltages and , respectively.
Let us now translate the above schema of an electrical circuit into mathematical equations. The conservation of electric charge on a piece of membrane implies that the applied current may be split in a capacitive current which charges the capacitor and further components 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 ; cf. Fig. 2.2. From the definition of a capacity where is a charge and the voltage across the capacitor, we find the charging current . Hence from (2.3)
In biological terms, is the voltage across the membrane and is the sum of the ionic currents which pass through the cell membrane.
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 . Since is the total voltage across the cell membrane and the voltage of the battery, the voltage at the leak resistor in Fig. 2.2 is . Using Ohm’s law, we get a leak current .
The mathematics of the other ion channels is analogous except that their conductance is voltage and time dependent. If all channels are open, they transmit currents with a maximum conductance or , respectively. Normally, however, some of the channels are blocked. The breakthrough of Hodgkin and Huxley was that they succeeded to measure how the effective resistance of a channel changes as a function of time and voltage. Moreover, they proposed a mathematical description of their observations. Specifically, they introduced additional ’gating’ variables and to model the probability that a channel is open at a given moment in time. The combined action of and controls the Na channels while the K gates are controlled by . For example, the effective conductance of sodium channels is modeled as , where describes the activation (opening) of the channel and its inactivation (blocking). The conductance of potassium is , where describes the activation of the channel.
In summary, Hodgkin and Huxley formulated the three ion currents on the right-hand-side of Eq. (2.4) as
The parameters , , and are the reversal potentials.
The three gating variables , , and evolve according to differential equations of the form
with , and where stands for , , or . The interpretation of Eq. 2.6 is simple: For a fixed voltage , the variable approaches the target value with a time constant . The voltage dependence of the time constant and asymptotic value is illustrated in Fig. 2.3. The form of the functions plotted in Fig. 2.3 as well as the maximum conductances and reversal potentials in Eq. (2.5) were deduced by Hodgkin and Huxley from empirical measurements.
The number of ion channels in a patch of membrane is finite and individual ion channels open and close stochastically. Thus, when an experimentalist records the current flowing through a small patch of membrane, he does not find a smooth and reliable evolution of the measured variable over time but rather a highly fluctuating current, which looks different at each repetition of the experiment (Fig. 2.5).
The Hodgkin-Huxley equations which describe the opening and closing of ion channels with deterministic equations for the variables , , and , correspond to the current density through a hypothetical, extremely large patch of membrane containing an infinite number of channels or, alternatively, to the current through a small patch of membrane but averaged over many repetitions of the same experiment (Fig. 2.5). The stochastic aspects can be included by adding appropriate noise to the model.
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.
The most important property of the Hodgkin-Huxley model is its ability to generate action potentials. In Fig. 2.6A an action potential has been initiated by a short current pulse of 1 ms duration applied at ms. The spike has an amplitude of nearly 100mV and a width at half maximum of about 2.5ms. After the spike, the membrane potential falls below the resting potential and returns only slowly back to its resting value of -65mV.
In order to understand the biophysics underlying the generation of an action potential we return to Fig. 2.3A. We find that and increase with whereas decreases. Thus, if some external input causes the membrane voltage to rise, the conductance of sodium channels increases due to increasing . 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. The explosive increase comes to a natural halt when the membrane potential approaches the reversal potential of the sodium current.
At high values of the sodium conductance is slowly shut off due to the factor . As indicated in Fig. 2.3B, the ‘time constant’ is always larger than . Thus the variable which inactivates the channels reacts more slowly to the voltage increase than the variable which opens the channel. On a similar slow time scale, the potassium (K) current sets in Fig. 2.6C. 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. 2.6A. The negative overshoot, called hyperpolarizing spike-after potential, is due to the slow de-inactivation of the sodium channel, caused by the -variable.
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