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 IC which charges the capacitor C and further components Ik which pass through the ion channels. Thus
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 gL = 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 gNa or gK, 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 three variables m, n, and h are called gating variables. They evolve according to the differential equations
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
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 m0 and n0 increase with u whereas h0 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.
The Hodgkin-Huxley equations (2.4)-(2.6) may also be studied for constant input I(t) = I0 for t > 0. (The input is zero for t 0). If the value I0 is larger than a critical value I 6A/cm2, 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 I0 defines the gain function plotted in Fig. 2.5B.
In the previous example we have seen that a constant input current I0 > I generates regular firing. In this paragraph we study the response of the Hodgkin-Huxley model to a step current of the form
The answer to this question is given by Fig. 2.6B. A large step I facilitates the spike initiation. Even for a target value I2 = 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 I1 < 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 I2 > 6A/cm2, 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 I2 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/cm2, 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/cm2. 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.
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.
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