## Passive Conductance & Scaling

The basics of how to develop, test, and use models.
lrh
Posts: 1
Joined: Tue Nov 22, 2011 4:11 pm

### Passive Conductance & Scaling

The answer, I suspect, will be worthy of a *duh* moment, but here is my conundrum:

It is my understanding that g_pas in passive.mod is the in-built term for simulating membrane resistance. Going by that assumption, I'll posit 100 MΩ resistance for a 10 μm long cylindrical soma, radius of 5 μm, no neurites. The open-ended surface area is then pi*1e-6 cm^2. g_pas is in units of S / cm^2, so 100 MΩ = 1e-8 S. This gives 1e-8 S / pi*1e-6 cm^2, or ~0.0031831 S / cm^2. I code my value of g_pas into the hoc, after "insert pas" for this one compartment model, compile, and run (note: Cm is left at default 1 μF/cm^2, as is Ra). A current pulse of 1nA gives a roughly 100mV change, as would be expected, however the membrane time constant is well below 1 msec. This is not expected.

Now if I scale up the soma to 100 μm in length and a diameter of 100 μm, and adjust the surface area accordingly (0.000314159 cm^2), the value I insert for g_pas is 3.1831e-5 S / cm^2. Now, I get the proper response to a 1 nA current injection, and a time constant of around 30 ms, which is much more reasonable.

Insofar as I am aware, there should be no discrepancy between these two situations that scaling doesn't account for.

My two questions are:
1) Is my approach correct? Also, how best should one go about adjusting the membrane time constant?
2) How would I achieve a reasonable membrane time constant for the first case?
ted
Posts: 5795
Joined: Wed May 18, 2005 4:50 pm
Location: Yale University School of Medicine
Contact:

### Re: Passive Conductance & Scaling

lrh wrote:I'll posit 100 MΩ resistance for a 10 μm long cylindrical soma, radius of 5 μm, no neurites. The open-ended surface area is then pi*1e-6 cm^2. g_pas is in units of S / cm^2, so 100 MΩ = 1e-8 S. This gives 1e-8 S / pi*1e-6 cm^2, or ~0.0031831 S / cm^2. I code my value of g_pas into the hoc, after "insert pas" for this one compartment model, compile, and run (note: Cm is left at default 1 μF/cm^2, as is Ra). A current pulse of 1nA gives a roughly 100mV change, as would be expected, however the membrane time constant is well below 1 msec. This is not expected.
because arithmetic error (easy to make) led to incorrect expectation. Here's how to think clearly about this. For specific membrane capacitance = 1 uf/cm2 and specific membrane conductance = 0.001 S/cm2, membrane time constant is 1 ms. You're proposing specific membrane conductance 10/PI times larger than 0.001 S/cm2, so membrane time constant has to be 10/PI times smaller than 1 ms.

Some handy facts: for a surface area of 100 um2
total conductance in uS is numerically identical to specific conductance (conductance density) in S/cm2
total capacitance in pF is numerically identical to specific capacitance in uf/cm2
net membrane current in nA is numerically identical to current density in mA/cm2

So instead of assuming a cylinder that is 10 um long and 10 um in diameter, why not start thinking in terms of a patch of membrane with surface area 100 um2? i.e. cylinder 10 um long and 10/PI um diameter, or 10/PI um long and 10 um diameter

This is why the single compartment model that is generated by the GUI tool
NEURON Main Menu / Build / single compartment
has L = 3.1831 um, diam 10 um.