Why do V_Na and V_K stay unchanged in Hodgkin-Huxley model?

Question

In the Hodgkin-Huxley model, ionic current $i_\mathrm{Na}$ and $i_\mathrm{K}$ are given by $$ i_\mathrm{Na}=g_\mathrm{Na}(V_\mathrm{m}-V_\mathrm{Na})\\ i_\mathrm{K}=g_\mathrm{K}(V_\mathrm{m}-V_\mathrm{K}), $$ where $V_\mathrm{Na}$ and $V_\mathrm{K}$ are the equilibrium potential, and are kept the same during the burst of action potential: $$ V_\mathrm{Na}=\frac{RT}{F}\ln\mathrm{\frac{[Na^+]_o}{[Na^+]_i}}=35~\mathrm{mV},\\ V_\mathrm{K}=\frac{RT}{F}\ln\mathrm{\frac{[K^+]_o}{[K^+]_i}}=-85~\mathrm{mV} $$ This means $\mathrm{[Na^+]_o,~[Na^+]_i,~[K^+]_o,~[K^+]_i}$ are all constants, which is really confusing. In my opinion, the inward/outward current of sodium/potassium ions should definitely change the concentrations of Na+ and K+.

Well, it makes sense that $\mathrm{[Na^+]_o}$ and $\mathrm{[K^+]_i}$ can hardly change because they are large enough compared with the lost ions into the regions with fewer ion concentrations. Then what about $\mathrm{[Na^+]_i}$ and $\mathrm{[K^+]_o}$? They are so small that even very a few ions added to them can change them a lot.

For example, $\mathrm{[K^+]_i}=90~\mathrm{mM}$ and $\mathrm{[K^+]_i}=3~\mathrm{mM}$. If the potassium current leads to a change of $\mathrm{1~mM~[K^+]}$, now $\mathrm{[K^+]_i}=89~\mathrm{mM}$ and $\mathrm{[K^+]_i}=4~\mathrm{mM}$. $\mathrm{[K^+]_i}$ changes by 1.1% which can be neglected, but $\mathrm{[K^+]_o}$ changes by 33.3%, which cannot be neglected. Also, the equilibrium potential of potassium now becomes 25*ln(4/89)=-78 mV! This really changes a lot. How can we ignore it?

Moreover, in the Quantitative Model of Gastric Smooth Muscle Cellular Activation of Corrias and Buist (Annals of Biomedical Engineering, Vol. 35, No. 9, 2007), both $V_\mathrm{Ca}$ and $\mathrm{[Ca^{2+}]_i}$ does change during the burst of action potential as follows: $$ \frac{d[\mathrm{Ca}]_i}{d t}=-\frac{I_{\mathrm{caL}}+I_{\mathrm{caT}}}{2 * F * V_c}-I_{\mathrm{CaExt}}. $$ But they also ignored the change of $\mathrm{[Na^+]_i}$ and $\mathrm{[K^+]_o}$. Hmmm, really confsuing.

Answer 1

As you've figured out, the reversal potentials don't change because they're assuming the concentrations don't change. Even relative to $\mathrm{[Na^+]_o}$ and $\mathrm{[K^+]_i}$ the number of ions that move is small (e.g., see this previous Q&A: Ambiguity about the relation between membrane potential and concentration gradient in neuron cells - the squid giant axon isn't quite as efficient as a mammalian neuron but it's not so bad, either): 1/100,000th of the internal potassium is still on the order of 1/10,000th to 1/1,000th relative to the outside. Changing the concentrations by 0.1% isn't going to appreciably change the reversal potential. The example you give of a 1 mM change is far, far greater than what occurs with typical neuronal activity.

Of course, over a longer time scale there would be changes in concentrations, but those changes aren't observed as long as the sodium-potassium pump is active. The pump is working all the time to maintain the concentration gradients observed in neurons, and if you were to disable it the concentrations would eventually equalize across the membrane.

Calcium is in much lower concentrations than either sodium or potassium. It's also buffered heavily intracellularly so that the "observed" calcium concentration is absolutely miniscule: free calcium ions are important in signaling, but they are very quickly bound strongly by various chelating molecules inside the cell. Inside the cell, just a few hundred or thousand calcium ions make a big difference.

There's a bit of a famous quote/aphorism by statistician George Box and others that goes something like "All models are wrong, but some are useful" - so, that's what you're seeing here. Is the HH model absolutely complete to cover all aspects of neuronal function? No, of course not, but if it were it would be missing way more than just the changes in concentration of the ions involved. For the data that HH were fitting, and for many other circumstances involving neurotransmission, this model does well enough. It certainly does well enough for learning, describing, and teaching the basic mechanisms that are critical to neurophysiology. Including more of the total system would require many more free parameters without much improvement to the model.

In some other circumstances, such as hypoxia where the sodium-potassium pump does not function well, or seizures where the level of activity is extremely high and the extracellular space can get flooded with potassium, it may become more important to account for changing sodium and potassium concentrations. For more typical cases, they can usually be ignored and assumed to be held near enough to constant both inside and out thanks to the sodium-potassium pump.