Subsections

• 2.2.1 Definition of the model
• 2.2.2 Dynamics
  • 2.2.2.1 Example: Spike generation
  • 2.2.2.2 Example: Mean firing rates and gain function
  • 2.2.2.3 Example: Step current input
  • 2.2.2.4 Example: Stimulation by time-dependent input
  • 2.2.2.5 Example: Refractoriness

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.

2.2.1 Definition of the model

Figure 2.2: Schematic diagram for the Hodgkin-Huxley model.

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

$$\sum_{k}^{}$$ (2.3)

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 R; cf. Fig. 2.2. From the definition of a capacity C = Q/u where Q is a charge and u the voltage across the capacitor, we find the charging current I_C = C du/dt. Hence from (2.3)

$${{\text{d}}u\over {\text{d}}t} \sum_{k}^{}$$ (2.4)

In biological terms, u is the voltage across the membrane and $\sum_{k}^{}$I_k is the sum of the ionic currents which pass through the cell membrane.

Figure 2.3: Equilibrium function (A) and time constant (B) for the three variables m, n, h in the Hodgkin-Huxley model. The resting potential is at u = 0.

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

$$\sum_{k}^{}$$ (2.5)

The parameters E_Na, E_K, and E_L are the reversal potentials. Reversal potentials and conductances are empirical parameters. In Table 2.1 we have summarized the original values reported by Hodgkin and Huxley (Hodgkin and Huxley, 1952). These values are based on a voltage scale where the resting potential is zero. To get the values accepted today, the voltage scale has to be shifted by -65 mV. For example, the corrected value of the sodium reversal potential is E_Na = 50 mV that of the potassium ions is E_K = - 77 mV.

Table 2.1: The parameters of the Hodgkin-Huxley equations. The membrane capacity is C = 1$\mu$F/cm². The voltage scale is shifted so that the resting potential vanishes.

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

$$\dot{{m}} \alpha_{m}^{} \beta_{m}^{}$$

$$\dot{{n}} \alpha_{n}^{} \beta_{n}^{}$$

$$\dot{{h}} \alpha_{h}^{} \beta_{h}^{}$$ (2.6)

with $\dot{{m}}$ = dm/dt, and so on. The various functions $\alpha$ and $\beta$, given in table 2.1, are empirical functions of u that have been adjusted by Hodgkin and Huxley to fit the data of the giant axon of the squid. Eqs. (2.4) - (2.6) with the values given in Table 2.1 define the Hodgkin-Huxley model.

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

$$\dot{{x}} {1\over \tau_x(u)}$$ (2.7)

where x stands for m, n, or h. For fixed voltage u, the variable x approaches the value x₀(u) with a time constant $\tau_{x}^{}$(u). The asymptotic value x₀(u) and the time constant $\tau_{x}^{}$(u) are given by the transformation x₀(u) = $\alpha_{x}^{}$(u)/[$\alpha_{x}^{}$(u) + $\beta_{x}^{}$(u)] and $\tau_{x}^{}$(u) = [$\alpha_{x}^{}$(u) + $\beta_{x}^{}$(u)]^-1. Using the parameters given by Hodgkin and Huxley (Hodgkin and Huxley, 1952), we have plotted in Fig. 2.3 the functions x₀(u) and $\tau_{x}^{}$(u).

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.

2.2.2.1 Example: Spike generation

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.

Figure 2.4: A. Action potential. The Hodgkin-Huxley model has been stimulated by a short, but strong, current pulse before t = 0. The time course of the membrane potential $\Delta$u(t) = u(t) - u_rest for t > 0 shows the action potential (positive peak) followed by a relative refractory period where the potential is below the resting potential. In the spike response framework, the time course u(t) - u_rest of the action potential for t > 0 defines the kernel $\eta$(t). B. Threshold effect in the initiation of an action potential. A current pulse of 1 ms duration has been applied at t=10 ms. For a current amplitude of 7.0 $\mu$A/cm², an action potential with an amplitude of about 100 mV as in a is initiated (solid line, the peak of the action potential is out of bounds). If the stimulating current pulse is slightly weaker (6.9 $\mu$A/cm²) no action potential is emitted (dashed line) and the voltage $\Delta$u(t) = u(t) - u_rest stays always below 10mV. Note that the voltage scale in B is different from the one in A.

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' $\tau_{h}^{}$ is always larger than $\tau_{m}^{}$. 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) = I₀ for t > 0. (The input is zero for t$\le$ 0). If the value I₀ is larger than a critical value I_$\theta$ $\approx$ 6$\mu$A/cm², we observe regular spiking; Fig. 2.5A. We may define a firing rate $\nu$ = 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.

Figure 2.5: A Spike train of the Hodgkin-Huxley model for constant input current I₀. B. Gain function. The mean firing rate $\nu$ is plotted as a function of I₀.

2.2.2.3 Example: Step current input

In the previous example we have seen that a constant input current I₀ > I_$\theta$ generates regular firing. In this paragraph we study the response of the Hodgkin-Huxley model to a step current of the form

$$\Delta \mathcal {H}$$ (2.8)

Here $\mathcal {H}$(t) denotes the Heaviside step function, i.e., $\mathcal {H}$(t) = 0 for t$\le$ 0 and $\mathcal {H}$(t) = 1 for t > 0. At t = 0 the input jumps from a fixed value I₁ to a new value I₂ = I₁ + $\Delta$I; see Fig. 2.6A. We may wonder whether spiking for t > 0 depends only on the final value I₂ or also on the step size $\Delta$I.

The answer to this question is given by Fig. 2.6B. A large step $\Delta$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 > 6$\mu$A/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 $\Delta$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 $\mu$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.

Figure 2.6: Phase diagram for stimulation with a step current. A. The input current I(t) changes at t = 0 from I₁ to I₂. B. Response of the Hodgkin-Huxley model to step current input. Three regimes denoted by S, R, and I may be distinguished. In I no action potential is initiated (inactive regime). In S, a single spike is initiated by the current step (single spike regime). In R, periodic spike trains are triggered by the current step (repetitive firing). Examples of voltage traces in the different regimes are presented in the smaller graphs to the left and right of the phase diagram in the center.

2.2.2.4 Example: Stimulation by time-dependent input

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 $\sigma$ = 3$\mu$A/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.

Figure 2.7: A. Spike train of the Hodgkin-Huxley model driven by a time dependent input current. The action potentials occur irregularly. The figure shows the voltage u as a function of time. B. Refractoriness of the Hodgkin-Huxley model. At t = 20 ms the model is stimulated by a short current pulse so as to trigger an action potential. A second current pulse of the same amplitude applied at t = 25, 27.5, 30, or 32, 5 ms is not sufficient to trigger a second action potential

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.