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In Hodgkin and Huxley's articles (1952, J. Physiol.; 1990, Bulletin of Mathematical Biology), the gate variables are formulated as original form

In particular, as $V$ increases, $\alpha_n$ decreases and $\beta_n$ increases, as shown below:

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However, in many other textbooks, e.g., Spiking Neuron Models and Fundamentals of Computational Neuroscience, as well as quite a few websites like HHsectionOnly.doc (illinois.edu) and Hodgkin-Huxley Model | SpringerLink, there is always a negative sign before $V$.

enter image description here

I CAN'T UNDERSTAND why there should be a negative sign. If so, the monotonicity of $\alpha_n$ and $\beta_n$ will be inverted, which is not the real case.

What makes me even more confused is, when I remove the negative sign, I cannot obtain valid results through programming.

In a word, I wonder why there should be a negative sign before the voltage. What's its physical interpretation? Thank u in advance!

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Voltage is a potential difference. Membrane potential is a difference across a membrane. When you calculate a difference A - B or B - A, you get a number that has the same magnitude but opposite sign. So, if A is inside a cell and B is outside a cell, how do you know if the difference should be noted A - B or B - A? The universe won't tell you, you've got to make a decision. It's helpful if everyone makes the same decision and it becomes a convention.

These sorts of conventions are everywhere in physics: there's no real reason that protons shouldn't be thought of as "negative" and electrons as "positive", for example. All the math works out the same way, but it's helpful if the whole field uses the same convention.

Similarly, the field of neurophysiology has settled on thinking about things from the perspective of the inside of cells. That means, by convention, when we say a membrane has a "negative" voltage, we mean that "inside is negative relative to the outside".

Hodgkin and Huxley initially used the opposite convention in their experiments and papers. There's nothing right or wrong with that, but it means if you are using the typical convention that a "negative membrane voltage" means there is more negativity inside the cell versus outside, you'll need to multiply voltage by -1 when you use the original HH paper equations. See also e.g. http://nelson.beckman.illinois.edu/courses/physl317/part1/Lec3_HHsection.pdf - search for "sign conventions" for the relevant discussion.

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    $\begingroup$ Thank you Bryan. You really helped me a lot. $\endgroup$
    – Jasmine
    Commented Jul 18, 2023 at 1:34

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