Spine scaling explanation

[CCohen]

Hello,

It is said that scaling the specific capacitance UP and specific membrane resistivity DOWN in spine-bearing compartments of a model NEURON (layer 5 pyramidal) cell will correct for the absence of the surface area contribution of spines to these compartments (as spines are not seen by Neurolucida). To recapitulate, spines are absent from compartments of a model NEURON cell, their surface area contribution to the model cell is absent, and the scaling strategy described above will correct for this absent surface area.

Why will this scaling strategy correct for the absent surface area?

How does this correction affect non-spine-bearing compartments?

How does one determine a "spine scale" that is a function of spine density for one's particular cell?


Thank you for taking the time to read or answer my questions.

[ted]

Good questions; they show you're thinking. Here are a couple for you:
1. Who says?
2. What have you found in the scientific literature?

[CCohen]

Hello,

Sorry for not mentioning sources explicitly.

Here's one; Stuart and Spruston (1998):

"Spines were not modeled explicitly, but their effects on membrane area were modeled by decreasing Rm and increasing Cm by a factor of 2 in spiny compartments (Shelton, 1985; Holmes, 1989; Larkman, 1991)."

- in Materials and Methods


Here's another; Schmidt-Hieber et al. (2007):

"'Collapsed' spines were implemented into the model by scaling Rm and Cm of spine-bearing compartments to account for the additional membrane surface area that was assumed to be 1.2 um^2 per single spine (Hama et al., 1989)."

- in Materials and Methods


I have found in the literature (above and elsewhere) that the assertion is made of scaling Cm and Rm in spine-bearing compartments to account for their missing contribution to neuronal surface area, but the explanation of why, I think, is missing.


Thank you.

[ted]

The citation chain is rather diffuse, but one point of convergence is
Holmes, W.R., Cable Theory Modeling of the Effectiveness
of Synaptic Inputs in Cortical Pyramidal Cells, Ph.D. thesis,
University of California, Los Angeles, 1986
which you might be able to find online. I haven't read it myself, so can't testify to the detail or rigor of whatever explanation Bill provided.

If I were explaining it, I'd start with the discretized cable equation, which is just the current balance equation
transmembrane current leaving the jth compartment = sum of axial currents entering the jth compartment from all of its immediate neighbors
or, if you prefer,

Code: Select all

c_j dv_j/dt + i_ion_j = SUMMA (v_k - v_j) / r_jk
                                    k

where the two terms on the left are the capacitive and ionic transmembrane currents leaving the jth compartment, i.e.
c_j is the membrane capacitance of the jth compartment
v_j is the membrane potential of " " "
i_ion_j is the sum of ionic transmembrane currents leaving " " "
and
r_jk is the axial resistance between compartments j and k.

Spines do nothing to r_jk. However, they add extra membrane, so total surface area of compartment j is the sum of the surface area of the neurite per se plus the area of its spines. Suppose that the total surface area is some multiple f of the surface area of the neurite itself, and assume that spine membrane has the same properties as the membrane of the neurite. Then, since c_j and i_ion_j are both proportional to the surface area of compartment j, . . . "The remainder of the derivation is left as an exercise to the reader." (see footnote)

Footnote:
This sentence, or something very similar, occured frequently in the textbook "Calculus with Analytic Geometry" by Johnson and Kiokemeister, where it usually meant that a couple of pages of heavy labor awaited the reader. This particular case is much easier, however.

[ted]

A much easier qualitative rationale:
Total membrane capacitance and conductance in a compartment are proportional to surface area in that compartment. Using the "apparent" surface area Aa, which is based on neurite diameter and length, gives underestimates of true membrane capacitance and conductance. If the ratio of spine surface area to "apparent" surface area is f, then total surface area is (1+f)*Aa, and true membrane capacitance and conductance are also 1+f times larger than would be calculated from Aa. So instead of computing total capacitance and conductance as
C = cm*Aa and G = gm*Aa, where cm and gm are the conductance densities (uf/cm2 and S/cm2)
calculate
C = (1+f)*cm*Aa and G = (1+f)*gm*Aa

[CCohen]

A much easier qualitative rationale:
Total membrane capacitance and conductance in a compartment are proportional to surface area in that compartment. Using the "apparent" surface area Aa, which is based on neurite diameter and length, gives underestimates of true membrane capacitance and conductance. If the ratio of spine surface area to "apparent" surface area is f, then total surface area is (1+f)*Aa, and true membrane capacitance and conductance are also 1+f times larger than would be calculated from Aa. So instead of computing total capacitance and conductance as
C = cm*Aa and G = gm*Aa, where cm and gm are the conductance densities (uf/cm2 and S/cm2)
calculate
C = (1+f)*cm*Aa and G = (1+f)*gm*Aa

Hi,

I understand that total capacitance C and total conductance G are affected by increases in surface area. However, specific membrane capacitance Cm or specific membrane conductivity Gm (or specific membrane resistivity Rm = 1/Gm) are not affected by increases in membrane surface area, as these two are independent of membrane surface area and dependent only upon membrane properties. But the spine scaling strategy is to scale Cm, the specific membrane capacitance, and Gm, the specific membrane conductivity, UP (or Rm, specific membrane resistivity, DOWN) in spine-bearing compartments in the model NEURON cell. Thus the confusion: how does this scaling strategy take into account, for the NEURON model cell, the missing surface area of spines?

[ted]

First, an important point: it works for any compartmental modeling approach or program, not just NEURON.

The explanation is pretty straightforward. I really don't know how to make it clearer, except perhaps to use a concrete example. Suppose you have a short piece of neurite with surface area of 100 um2 based on its length and diameter, i.e. ignoring spines. Based on this surface area, and assuming that membrane specific capacitance is 1 uf/cm2, the total capacitance of the neurite is 1e-6 uf, i.e. 1 pf. Now suppose someone tells you that spines are attached to the neurite, and that the surface area of the spines is 100 um2. Then the total capacitance of the neurite + spines is 2 pf. Finally, suppose you have another piece of neurite with surface area X um2 based on its length and diameter, and someone tells you that this comes from part of the cell where spines add only 0.5 um2 per um2 of "neurite" surface area. Then the total capacitance of this piece of neurite will be 1.5 X pf. You may think of the calculation as
(apparent surface area * correction factor) * 1 uf/cm2
or
apparent surface area * (correction factor * 1 uf/cm2)
Got it?

[CCohen]

First, an important point: it works for any compartmental modeling approach or program, not just NEURON.

The explanation is pretty straightforward. I really don't know how to make it clearer, except perhaps to use a concrete example. Suppose you have a short piece of neurite with surface area of 100 um2 based on its length and diameter, i.e. ignoring spines. Based on this surface area, and assuming that membrane specific capacitance is 1 uf/cm2, the total capacitance of the neurite is 1e-6 uf, i.e. 1 pf. Now suppose someone tells you that spines are attached to the neurite, and that the surface area of the spines is 100 um2. Then the total capacitance of the neurite + spines is 2 pf. Finally, suppose you have another piece of neurite with surface area X um2 based on its length and diameter, and someone tells you that this comes from part of the cell where spines add only 0.5 um2 per um2 of "neurite" surface area. Then the total capacitance of this piece of neurite will be 1.5 X pf. You may think of the calculation as
(apparent surface area * correction factor) * 1 uf/cm2
or
apparent surface area * (correction factor * 1 uf/cm2)
Got it?

Hello,

First, thank you for this explanation.

How does NEURON use total capacitance or total conductance to determine specific capacitance or specific conductance? For example, Cm might be given by the user as 1 uF/um^2 as a beginning constant parameter, but, using NEURON's MRF in modeling situation, this parameter may be optimized. Say the user uses spine scaling as described above. How does this spine scaling affect NEURON's computation/optimization of Cm, the specific capacitance?

[ted]

How does NEURON use total capacitance or total conductance to determine specific capacitance or specific conductance?

It doesn't. The parameters one uses to specify membrane properties are specific capacitance and specific conductance, which mean capacitance/area and conductance/area, respectively.

Say the user uses spine scaling as described above. How does this spine scaling affect NEURON's computation/optimization of Cm, the specific capacitance?

You're close to asking the right question. The MRF will adjust whatever parameters you tell it to. It's up to you to make sure that whatever changes it makes will affect the model's system equations. Follow a strategy similar to that described in the 2nd MRF tutorial, and cm will end up with whatever value was necessary to satisfy the fit criterion. The question you meant to ask is "How should the optimized value of cm be interpreted?" It should be interpreted as the product of the "real" cm and the spine scaling factor. To discover the former, divide by the latter.

[CCohen]

It's up to you to make sure that whatever changes it makes will affect the model's system equations. Follow a strategy similar to that described in the 2nd MRF tutorial, and cm will end up with whatever value was necessary to satisfy the fit criterion.

Say I "spine_scale" g_pas and cm in the relevant compartments via the params.hoc file, a strategy made possible by the 2nd MRF tutorial, and one which would satisfy the spine scaling strategy to account for the missing spine surface area, as described above.

"How should the optimized value of cm be interpreted?" It should be interpreted as the product of the "real" cm and the spine scaling factor. To discover the former, divide by the latter.

Is the final optimized value of cm divided by "spine_scale" (where spine_scale > 1) to obtain the "real" value of cm?

[ted]

Say I "spine_scale" g_pas and cm in the relevant compartments via the params.hoc file, a strategy made possible by the 2nd MRF tutorial, and one which would satisfy the spine scaling strategy to account for the missing spine surface area, as described above.

Say you do. What then?

Is the final optimized value of cm divided by "spine_scale" (where spine_scale > 1) to obtain the "real" value of cm?

It's all in the mind of the implementer. If the L and diam of your sections are the L and diam of cylindrical neurites, and you allow the MRF to adjust cm, the cm values are those that are necessary to satisfy some error criterion. What you make of that is up to you. If you think "real" specific membrane capacitance is 1 uf/cm2 but cm in the optimized model is larger, then either your guess about "real" specific membrane capacitance was incorrect, or there really is more surface area than one would calculate assuming that neurites are cylinders, or both. If there is more surface area, maybe it is attributable to spines, or maybe it is attributable to surface irregularities or maybe the neurites have a noncircular cross-section (which would also increase surface area), or maybe any combination of these three. When was the last time you saw a thin section electron micrograph in which all neurite outlines were circular?