Relationship between membrane current and voltage in neurons

Depolarization of neurons leads currents of different magnitudes flow in or out of the cell, and the Sodium and Potassium currents can be separately plotted (Purves): Caption: Relationship between current amplitude and membrane potential, taken from experiments such as the one shown in Figure 3.2. Whereas the late outward current increases steeply with increasing depolarization, the early inward current first increases in magnitude, but then decreases and reverses to outward current at about +55 mV (the sodium equilibrium potential). (After Hodgkin et al., 1952.)

Here we see the dependency of the currents (early: Sodium, late: Potassium) on membrane potential.

What I was wondering was this: It looks like the dependency of the Sodium current on the membrane potential is very non-linear. However, experiments which assess the amplitude of injected current on excitatory postsynaptic potentials (EPSP) state that the amplitude of EPSPs are proportional to the injected current. How is this consistent with the non-linear current/permeability changes of Sodium? Is the Sodium current linearly dependent on voltage (and vice versa) before the action potential threshold is reached?

• I don't have Purves in front of me, can you add the caption there? And careful with the axes: the x-axis is voltage, not time, so saying "a dependency on time would have the "early" function shifted to the left" does not make sense. Jul 29 '19 at 20:04
• I have added the caption. Yes exactly, here the x-axis is voltage. If it would be time, as the name says, the early function would be left-shifted wrt to the late function. I removed that comment now though.
– Pugl
Jul 29 '19 at 20:19

This is a bit of a strange way to plot out these data, I think Figure 3.2 is easier to understand, but basically these are data plotted from early voltage-clamp experiments trying to walk you through how the Hodgkin-Huxley model was developed.

They step the voltage to a particular point and look at the current. There are two phases to this current: an early transient and a later plateau. The shape of the "early" component ultimately results from the multiplication of two functions: the gating curve of the voltage-gated sodium channels and the reversal potential for sodium around 50 mV. If the voltage gated channels were simply open all the time, you would just have a straight line through +50mV, it would follow the right-side shape of the curve. The left-side of the curve is due to more channels opening as you depolarize more.

For the late component, which comes from voltage-gated potassium channels, all of the dependence is on voltage, because the plateau is at a point where all the potassium channels are open. You will also notice, however, that the slope changes a bit in time: it's steeper at the higher voltages, because the potassium channels open more quickly (and also sodium channels inactivate; the combination of the two would make it hard to separate them here).

Your picture is a bit different from these traces plotted in time: figure 3.3 is showing the peak of each component as a function of the voltage step. I think it would be better to present 3.3 as a series of data points rather than a smooth curve to make this more obvious.

Note that the actual numbers here are a bit different than one would see in a mammalian cell, because these original experiments used the squid giant axon which has some differences in voltage dependence, gating, and ion concentrations.

The reason sodium and potassium can be seen separately in these voltage clamp experiments is because:

A) Sodium channels open really fast early on, and then inactivate

B) Therefore, in the long term the potassium current dominates the sodium current.

If you blocked the potassium current and VG sodium channels didn't inactivate, the "early" part would look like a mirror image of the "late" part of the voltage-clamp traces. As-is, the early component is mostly sodium but also includes some voltage-gated potassium current as well that cancels it out and makes it appear smaller than it would otherwise be.

If you look at the late curve in your original figure, it's mostly straight from the point that all the K channels are open. For the late component, the K permeability is at max by -30mV so after that it's all linear from driving force. The little curve at the bottom up to about -30mV is the range at which not all the K channels are open, otherwise it would be a completely straight line, and it would reverse at the K reversal potential instead of asymptoting around zero. But it's not exactly zero because there is also some leak which is mostly potassium, and that does reverse around the K reversal.

• Many thanks for the answer - I think I understand these points, also that in my plot we have max(current) for each point on the x-axis (mV). But my question is this: (1) If I understand correctly, the non-linearity of Na+ follows from the non-linear permeability changes and the time-dependent inactivation of Na+, correct? (2) Does this then also mean that K+ permeability is constant over mV (hence the somewhat linear function in the plot I have used)? But in particular: (3) If Na+ permeability changes are non-linear, how can EPSPs be a linear function of injected current amplitude?
– Pugl
Jul 29 '19 at 22:41
• @Pugl I don't follow what you are asking about EPSPs and injected current. EPSPs are caused by activating ligand-gated ion channels. In that case, the amplitude corresponds to how many ligand-gated channels are bound and opened (aka, the permeability). If you open enough of them to also open voltage-gated sodium channels, then you get a nonlinear response, including possibly an action potential if the EPSP is large enough to reach threshold. One could also inject current to mimic an EPSP, and you'd have the same result. However these experiments are in voltage clamp, not IC. Jul 29 '19 at 22:47
• @Pugl I'd also add that the difference between sodium and potassium here is largely because sodium channels inactivate and because the potassium current dominates the sodium current at a long time. If you blocked the potassium current and VG sodium channels didn't inactivate, the "early" part would look like a mirror image of the "late" part of the voltage-clamp traces. As-is, the early component is mostly sodium but also includes some voltage-gated potassium current as well that cancels it out. Jul 29 '19 at 22:57
• @Pugl Both inactivation and activation, yes. If you look at the late curve in your original figure, it's mostly straight from the point that all the K channels are open. The little curve at the bottom up to about -30mV is the range at which not all the K channels are open, otherwise it would be a completely straight line, and it would reverse at the K reversal potential instead of asymptoting around zero. But it's not exactly zero because there is also some leak which is mostly potassium, and that does reverse around the K reversal. Jul 29 '19 at 23:10
• Yes permeability x driving force. For the late component, the K permeability is at max by -30mV so after that it's all linear from driving force. Jul 29 '19 at 23:11