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

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.

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.