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

Link of the iPython notebook for the code

AT2 – Neuromodeling: Problem set #3 SPIKE TRAINS

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).

(a) Simulate the Hodgkin-Huxley model and increase the injected current $I$ from $I = 0$ to $I = 10$. At which value does the neuron start to spike repetitively? What is its lowest firing rate?

(b) Use your simulation to figure out how the action potential comes about. What happens at the spiking threshold?