Hodgin-Huxley model for a single neuron

[Topologia]

I am viewing (through https://www.edx.org/course ) an introduction course to computational neuroscience. In the second lecture, the Hodgin-Huxley model is considered. I am going over some of the questions and have encountered a problem with one of them (a picture of the exercise is attached below). I have a strong background in mathematics, but my background in biology is yet very poor. I am having a hard time connecting the biology to the math. Can anyone help with this question:

http://www.upf.co.il/preview/930650464/ea89d08e9d2bec2a0ddbefc1c860d757.html
http://www.upf.co.il/preview/730918148/54449c4d3938eb4de89a02a072d8159e.html

Thank you!

 

[Topologia]

http://www.upf.co.il/preview/930650464/ea89d08e9d2bec2a0ddbefc1c860d757.html
http://www.upf.co.il/preview/730918148/54449c4d3938eb4de89a02a072d8159e.html%5b/img

 

[atyy]

Are you asking why the correct answer is that g_ion is the maximal conductance?

 

[Topologia]

My problem is that I don't know how to even start this question..
The formal answer of the course is that all the answers are correct except number 2.
My original answer was that 1,5,7 are correct.
But, basically, I have a problem understanding what considerations I should use..
http://www.upf.co.il/preview/146130318/61dd63a379862f86de55863ed4935eb7.html

 

[atyy]

I wouldn't worry about it too much. I don't think the answer is necessarily correct, but it is conventional. The form of a Hodgkin-Huxley type model is certainly guessed based on having a plausible mechanistic interpretation. However, with respect to the actual channel mechanisms, they should primarily be thought of as phenomenological models or "good curve fits" rather than mechanistic models.

But let me explain the reasoning behind the formal answer of the course. The form of the equation is:

current = conductance * (potential difference from reversal potential).

The reversal potential is the thermodynamic equilibrium potential so it is the potential at which no net current flows. So we know that current flow must be some function of the potential difference from the reversal potential, and we guess that the function is linear for each conformation of the channel. The channel conformation determines the conductance.

The conductance (g * rⁿ¹ * sⁿ²) is determined by the channel conformation. The terms rⁿ¹ and sⁿ² vary between 0 and 1, so they are fractions, and g is the maximal conductance. The terms r_o and s_o determine what r and s will tend to. When the membrane potential is increased, the curves how that r_o increases towards 1 and s_o will decrease towards 0. So r will tend to open the channel, which is why it represents activation, and s will tend to close the channel which is why it represents inactivation. The time constant for r must be shorter than the time constant for s if the channel is to open before it closes (or more technically is inactivated).

(It's really a question of physical chemistry, since the form is motivated by thermodynamics and common mathematical forms for chemical kinetics.)

 

[atyy]

Here is a bit more detail about why you should consider the equations phenomenological rather than mechanistic.

http://www.nature.com/nrn/journal/v8/n6/box/nrn2148_BX3.html
Limitations of the Hodgkin–Huxley model for channel kinetics (from a review by Bruce Bean)

 

[Topologia]

I see.. will think about it a little further.
Thank you for your detailed answer and for the link!