membrane ionic current = -capacitive current

[Bill Connelly]

A statement I read all the time is the "total membrane ionic current must be equal and opposite to the total membrane capacitive current"

I appreciated this statement if you replace the word current with charge, i.e. any charge that goes into the cell has to EVENTUALLY come out of the cell. or otherwise you don't have a circuit.

But if at any point of time I-membrane = -I-cap then wouldn't there be no change in membrane voltage? i.e. you need the total of transmembrane current to be equal to some non-zero value to change the transmembrane voltage, right?

[ted]

A statement I read all the time is the "total membrane ionic current must be equal and opposite to the total membrane capacitive current"

This is true as long as you aren't injecting current through a microelectrode. The sum Ii + Ic = Im, the total membrane current, and Im must equal the injected current Iinj. If Iinj is 0, Ii = -Ic.

if at any point of time I-membrane = -I-cap then wouldn't there be no change in membrane voltage?

First, step back from the problem and recall that the voltage across a capacitor is proportional to the charge stored in the capacitor. The charge stored in a capacitor is the integral of Ic. So if current flows through membrane capacitance, membrane potential changes.

Now re-examine the question you posed. Do you mean "I-membrane" or I-ionic? The only situation in which Im = -Ic is when Im, Ii, and Ic are all 0.

[Bill Connelly]

I-ion + I-cap = 0
I-mem = I-ion + I-cap
I-m = 0

No transmembrane current, no change in membrane voltage.

What am I doing wrong here? I know when I plot I-ion and I-cap in neuron, (in a single compartment) they are equal and opposite. If I-ion is inward, does the apparent outward I-cap decharge/hyperpolarize the membrane?

I'm imagining a single compartment of 10pF with no ion channels, apart from a couple AMPA channels. Glutamate binds and lets say 1 pC of Na+ moves into the cell, which should create 100mV across the membrane (at the end of charging). i.e. V=Q/C. That 10pC of charge, once in the cell, moves to one side of the membrane, attracting -10pC of from the outside of the cell, which gives us the apparent I-cap; right? But this doesn't decharge the membrane, or does it?

[ted]

I-ion + I-cap = 0
I-mem = I-ion + I-cap
I-m = 0

No transmembrane current, no change in membrane voltage.

On the contrary.
Im = Ii + Ic
If Im = 0,
Ic = -Ii
but Ic = C dV/dt
so if Ii is nonzero, dV/dt is also nonzero.

[Bill Connelly]

Sorry that I'm not getting this. Everything you have said makes sense to me, but none of it seems to explain how, that if everytime charge moves in via Ii it goes out via Ic, how the membrane potential can change.

But maybe this might be the explanation:

Ic = C dV/dt

I had always thought it was
Imem = C dV/dt; but now that I think about it, that doesn't make sense

So if you're saying it is the current that flows across the capacitor that changes the membrane potential; then Ii doesn't change the membrane potential? It just supplies current to charge? But isn't the direction of current flows across the capacitor in the wrong direction?

[ted]

it is the current that flows across the capacitor that changes the membrane potential

True.

then Ii doesn't change the membrane potential? It just supplies current to charge?

True.

isn't the direction of current flows across the capacitor in the wrong direction?

No. Here's how to think about it.

Imagine a spherical cell. The only way current can enter the cell is by passing through membrane capacitance or through ion channels. Then
Ic + Ii = Im = 0
Suppose all of the ion channels have been blocked pharmacologically. Also suppose that the membrane potential of the cell is 0 mV and that Ena is 40 mV.

Now some sodium channels open. As a result:
Sodium ions flow (into / out of) the cell.
This means that positive charge (enters / exits) the cell through ion channels.
The movement of ions through channels constitutes the (ionic / capacitive) component of membrane current.

The interior of the cell has now (gained / lost) positive charge.
The charge that the cell has (gained / lost) is charge is stored in the cell's membrane capacitance.
Movement of charge onto or off of membrane capacitance constitutes the (ionic / capacitive) component of membrane current.
As a consequence of this (ionic / capacitive) current, membrane potential moves in the (positive / negative) direction.

Going back to the equation Ic + Ii = 0,
so
C dV/dt + (V - Ena)*gna = 0
so
C dV/dt = gna*(Ena - V)
If gna is 0, dV/dt is (negative / zero / positive)
If Na channels open, gna becomes (positive / negative)
so
dV/dt becomes (positive / negative)
and V is forced (toward / away from) Ena.

[Bill Connelly]

Okay. That all makes sense. Thanks a lot Ted. It's not quite as cognitively peaceful as Ohms law. But I'll just have to learn to live with it.