# What's the effect of prolonged threshold stimulus currents on action potentials?

I've been confused as to what exactly occurs when a current is injected into a membrane, throughout the duration of an action potential. My main source of confusion has been trying to reconcile 2 things, mainly the invariance of the action potential's magnitude, and the logic that I used during my study of action potentials to explain their origin from equivalent circuits.

To illustrate:

This is a diagram from Cellular Physiology and Neurophysiology, a textbook that I use from time to time. This diagram has been of great usefulness to me, as it provided me with an equivalent circuit description of all phases of the action potential. As you can see during B, a current is injected into the membrane, depositing a net positive charge on the membrane capacitance, causing an outward capacitive current that discharges the membrane capacitance to threshold. Immediately after reaching threshold though, the injected current terminates and the rest of the depolarization is done by the inward Na+ current, which is in a positive feedback cycle.

Now comes my problem: what happens, if I deliberately keep injecting current throughout the whole action potential? Would, say, its magnitude change? An easy answer would be "nothing", as the action potential is always the same regardless of stimulus strength & duration. However, some of my logic is making it seem like the action potential should be changed.

Allow me to explain. My problem comes at the peak of the action potential. This is supposedly a steady state, where the decreasing inward Na+ current is equal and opposite to the increasing outward K+ current. However, for our experiment, you would have to account for the constant inward injected current, which would deposit a net +ve charge on the inside of the membrane. Now comes the inevitable question: wouldn't that increase the peak of the depolarization? For a normal AP, the peak will always be a little less than the equilibrium potential for Na+, however here the presence of the constant inward injected current (and by extension the constant outward capacitive current) makes me feel as if it would nudge the peak a bit higher, before the outward K+ current can eventually eclipse it. Is that right or not? I feel like I'm missing something crucial here.

Another way to put it, in equivalent circuit terms:

. at threshold, Ic = Im + INa

. INa decreases with more depolarization, Ic gradually decreases

. At the peak of the AP: Ic = Im (my proposed nudge beyond ENa)

. After that, Ik > Im + INa, and should be able to repolarize the membrane.

So yeah, I guess it all boils down to me not being able to work out what happens in a neuron when a constant threshold current is injected. I'm very sorry if this is too long, and many thanks for reading! I've tried to explain things as best as I can, please tell me if something is not clear.

Diagram source: Blaustein, M. P., Kao, J. P. Y., & Matteson, D. R. (2019). Cellular physiology and neurophysiology: Mosby physiology series (3rd ed.). Elsevier.

$$\ E_m = \frac{RT}{F} \ln{ \left( \frac{ P_{\text{Na}}[\text{Na}^{+}]_\mathrm{out} + P_{\text{K}}[\text{K}^{+}]_\mathrm{out} + P_{\text{Cl}}[\text{Cl}^{-}]_\mathrm{in} }{ P_{\text{Na}}[\text{Na}^{+}]_\mathrm{in} + P_{\text{K}}[\text{K}^{+}]_{\mathrm{in}} + P_{\text{Cl}}[\text{Cl}^{-}]_\mathrm{out} } \right) }$$
where $$\ p_x$$ indicates the permeability of ion X. If the membrane is depolarized, the Na channels open, which increases Na permeability and draws the membrane potential towards the Na Nernst potential (which is well above positive). When K channels open a little later (together with the closure of Na channels), the potential is pulled back down towards the K Nernst potential, which is below the resting potential. If you continue to inject current during the AP, this is equivalent to introducing a new channel for an ion with a Nernst potential equal to the voltage you apply to the electrode. If you apply a voltage higher than the normal AP potential, the AP amplitude will increase. If the voltage is lower than the normal AP potential, the AP amplitude will be smaller. In essence, you have created a voltage clamp setup, which was the key technology used by Hodkin and Katz when they studied the mechanisms behind the AP, which led to the proposal of the existence of voltage dependent Na and K channels.