[Problem Set 3 Spike trains] Problem 4: The Hodgkin-Huxley model

Link of the iPython notebook for the code

PROBLEM 4: The Hodgkin-Huxley model

So far, we have treated the action potential as a simple threshold crossing of the voltage, without further specification of how exactly it comes about. In the Hodgkin-Huxley model, the generation of the action potential itself is explained through the action of active, voltage-dependent ion channels. The membrane voltage is given by

$C\frac{ {\rm d}V}{ {\rm d}t} =g_L (E_L - V) + \bar{g}_K n^4 (E_K - V) + \bar{g}_{N_a} m^3 h (E_{N_a} - V) + I$

where the second term on the right-hand-side describes the current due to the “delayed-rectifier” $K$-channel and the third term the current due to the “fast” $N_a$-channel. The parameters for this model are

• $C = 1 \, {\rm μF/cm^2}$
• $g_L = 0.3 \, {\rm mS/cm^2}$
• $E_L = −54.4 \, {\rm mV}$
• $g_K = 36 \, {\rm mS/cm^2}$
• $E_K = −77 \, {\rm mV}$
• $g_{N_a} = 120 \, {\rm mS/cm^2}$
• $E_{N_a} = 50 \, {\rm mV}$

The channel variables $h, m, n$ all follow first-order kinetics, i.e. rate equations of the form

$\frac{ {\rm d}x}{ {\rm d}t} = α(V)(1 - x) - β(V)x$

and the open and closing rates, $α(V)$, and $β(V)$ are channel-specific and voltage-dependent. You can find the equations for these rates in the textbook by Dayan & Abbott, Eq.(5.22) and Eq. (5.24).

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