2 The Hodgkin-Huxley Model

2.4 Summary

The Hodgkin-Huxley model describes the generation of action potentials on the level of ion channels and ion current flow. It is the starting point for detailed biophysical neuron models which in general include more than the three types of currents considered by Hodgkin and Huxley. Electrophysiologists have described an overwhelming richness of different ion channels. The set of ion channels is different from one neuron to the next. The precise channel configuration in each individual neuron determines a good deal of its overall electrical properties.


A nice review of the Hodgkin-Huxley model including some historical remarks can be found in Nelson and Rinzel [1995]. A comprehensive and readable introduction to the biophysics of single neurons is provided by the book of Christof Koch (269). Even more detailed information on ion channels and non-linear effects of the nervous membrane can be found in B. Hille’s book on ‘Ionic channels of excitable membranes’ (221). The rapidly growing knowledge on the genetic description of ion channel families and associated phenotypes is condensed in Channelpedia (419).


  1. 1.

    Nernst Equation . Using the Nernst equation (Eq. 2.2 ) calculate the reversal potential of Ca 2+{}^{2+} at room temperature (21 degree Celsius), given an intracellular concentration of 10 -4{}^{-4}  mM and an extracellular concentration of 1.5 mM.

  2. 2.

    Reversal Potential and Stationary Current-Voltage Relation . An experimentalist studies an unknown ion channel by applying a constant voltage uu while measuring the injected current II needed to balance the membrane current that passes through the ion channel.

    a) Sketch the current-voltage relationship ( II as a function of uu ) assuming that the current follows Iion=gionmh(u-Erev)I_{\rm ion}=g_{\rm ion}m\,h\,\,(u-E_{\rm rev}) with giong_{\rm ion} =1 nS and Erev=0E_{\rm rev}=0  mV where m=0.1m=0.1 and h=1.0h=1.0 are independent of the voltage.

    b) Sketch qualitatively the current-voltage relationship assuming that the current follows Iion=gionmh(u-Erev)I_{\rm ion}=g_{\rm ion}m\,h\,\,(u-E_{\rm rev}) with giong_{\rm ion} =1 nS and Erev=0E_{\rm rev}=0  mV where m0(u)m_{0}(u) and h0(u)h_{0}(u) have the qualitative shape indicated in Fig. 2.16 .

  3. 3.

    Activation Time Constant . An experimentalist holds the channel from Fig. 2.16 A and B at u=-50u=-50  mV for two seconds and then suddenly switches to u=0u=0  mV. Sketch the current passing through the ion channel as a function of time assuming Iion=gionmh(u-Erev)I_{\rm ion}=g_{\rm ion}m\,h\,\,(u-E_{\rm rev}) with giong_{\rm ion} =1 nS and Erev=0E_{\rm rev}=0  mV.

  4. 4.

    The Power of the Exponent . An experimentalist holds an unknown potassium ion channel with activation variable nn with voltage dependence n0(u)n_{0}(u) and time constant τn\tau_{n} at u=-50mVu=-50mV for two seconds and then, at time t=0t=0 , suddenly switches to u=0u=0  mV.

    a) Sketch the activation variable nn , n2n^{2} , n3n^{3} as a function of time for times smaller than τn\tau_{n} .

    b) Show mathematically that for 0<t<τn0<t<\tau_{n} the time course of the activation variable can be approximated n(t)=n0(-50mV)+[n0(0mV)-n0(-50mV)]t/τmn(t)=n_{0}(-50mV)+[n_{0}(0{\rm mV})-n_{0}(-50{\rm mV})]t/\tau_{m}

    c) Do you agree with the statement that “the exponent pp of in the current formula Iion=gionnp(u-Erev)I_{\rm ion}=g_{\rm ion}n^{p}\,(u-E_{\rm rev}) determines the ’delay’ of activation”? Justify your answer.

  5. 5.

    Hodgkin-and-Huxley Parameter Estimation . Design a set of experiments to constrain all the parameters of the two ion channels of the Hodgkin-Huxley model. Assume that the neuron has only the INaI_{\rm Na} and IKI_{\rm K} currents and that you can use tetrodotoxin (TTX) to block the sodium ion channel and tetraethylammonium (TEA) to block the potassium ion channel.

    Hint: Use the results of the previous exercises.

  6. 6.

    Simplified expression of the activation function . Show that with the voltage-dependent parameters αm(u)=1/[1-e-(u+a)/b]\alpha_{m}(u)=1/\,[1-e^{-(u+a)\,/\,b}] and βm(u)=1/[1-e-(u+a)/b]\beta_{m}(u)=1/\,[1-e^{-(u+a)\,/\,b}] (compare Table 2.1 ), the stationary value of the activation variable can be written as m0(u)=0.5[1+tanh[β(u-θact]m_{0}(u)=0.5\,[1+{\rm tanh}[\beta\,(u-\theta_{\rm act}] . Determine the activation threshold θact\theta_{\rm act} and the activation slope β\beta .

    Hint: tanh(x)=[exp(x)-exp(-x)]/[exp(x)+exp(-x)]{\rm tanh}(x)=[\exp(x)-\exp(-x)]/[\exp(x)+\exp(-x)].