Page 1 of 1

### Vext and the extracellular mechanism

Posted: Tue Oct 24, 2017 8:53 pm
I would first like to ask how Vext is calculated at a given node in a stretch of cable (of defined length and diameter) with one layer of extracellular inserted. Is Vext generated from the Im of that node multiplied by its lateral surface area, xraxial and length? Im here would be subtracted by the leak through xc and xg, I believe.

Second, I would like to ask about a correct strategy for optimizing xraxial, xg and xc based on anatomical and electrophysiological data. Say one were to take the radial dimension component of the extracellular space inserted into account. This should change Vext and xraxial. If the width of this space is delta, would scaling xg and xc up by 1+2*delta correctly scale Vext and xraxial?

Lastly, on a more general note, if xraxial < inf, and some leak is allowed through xc or xg, does some radial membrane current (Im) pass through xraxial, or would Im leak out of xg or xc only (the radial components)? Related to this question, as xraxial approaches infinity, and some leak remains through xg or xc, v should be getting smaller and smaller while vext gets bigger and bigger. Is it fair to say that in terms of v, this cable is not the same as one without xraxial, even if the combined radial resistance and capacitance are the same?

### Re: Vext and the extracellular mechanism

Posted: Wed Oct 25, 2017 12:26 am
vext is calculated from the Jacobian matrix defined by the discretized PDE that describes a multilayer cable. The matrix takes all longitudinal and radial currents into account.
I would like to ask about a correct strategy for optimizing xraxial, xg and xc based on anatomical and electrophysiological data. Say one were to take the radial dimension component of the extracellular space inserted into account. This should change Vext and xraxial. If the width of this space is delta, would scaling xg and xc up by 1+2*delta correctly scale Vext and xraxial?
Suppose for a moment that between the external surface of the axolemma and the internal surface of the myelin sheath there is a gap (i.e. cylindrical shell) with cross sectional area A in um2 that is filled with a conductive medium with resistivity Rex in ohm cm. Then xraxial[0] would equal Rex / A (omitting scale factors to convert between um, cm, ohms, and megohms). xg and xc have units of specific conductance and specific membrane capacitance and can be estimated from the number of layers of myelin and assumptions about the specific conductance and capacitance of a single layer of myelin (for N layers, xg and xc would be 1/N of the values for a single layer).
Lastly, on a more general note, if xraxial < inf, and some leak is allowed through xc or xg, does some radial membrane current (Im) pass through xraxial
to where? What do you think happens to the current that flows through xraxial in paranodal regions? Are you assuming that it just dumps into extracellular space, or do paranodes present a high resistance barrier to extra-axonal longitudinal current flow? If the latter, then current will exit the axon along the "upstream" (more depolarized) half of the internode, and flow back into the axon along the "downstream" half of the internode. In other words, v along an internode will vary linearly along the internode from its upstream end (which will be most depolarized) to its downstream end (which will be most hyperpolarized). The better the myelin is at insulating the internode, the smaller will be the variation of v along the internode.

### Re: Vext and the extracellular mechanism

Posted: Fri Oct 27, 2017 12:22 pm
Hi Ted,

I am not sure I was well understood. I hope you do not mind if I clarify. If periaxonal delta represents its radius, then the first layer of the myelin sheath is at a distance delta from the axolemma. Notwithstanding the radius of the myelin sheath (considering it as collapsed and just outside the distance delta), would xg and xc be required to be scaled by 1+2*delta in order to account for the effective increase in diameter by 1+2*delta?

Regarding my third set of questions, on the flow of current from the axon core to the periaxonal space, the question was whether this current (Im) flowed longitudinally through the periaxonal space to the next periaxonal node (i.e. one with extracellular inserted, as more than one node containing extracellular and adjacent to one another should be assumed), if xraxial < inf ? What I would like to confirm is that even if periaxonal resistance is high, some Im must flow through it longitudinally over its lengthwise definition (from one extracellular-containing node to the next. xg and xc are defined with xg < inf and xc > 0).

Finally, I would like to clarify the question in the latter part of the third paragraph. Could you please confirm that a stretch of cable in which an axon core, a periaxonal space and myelin sheath are defined, is different from one without the periaxonal space (but otherwise identical axon core and myelin sheath) in terms of v, even as xraxial approaches infinity?

### Re: Vext and the extracellular mechanism

Posted: Thu Nov 02, 2017 11:12 am
would xg and xc be required to be scaled by 1+2*delta . . .
xg and xc are supposed to represent the resistance and capacitance of the radial current path. They will be affected by the radius of the myelin sheath.
What I would like to confirm is that even if periaxonal resistance is high, some Im must flow through it longitudinally
If there is a path with finite resistance, and a potential gradient along that path, current will flow along the path.
Could you please confirm that a stretch of cable in which an axon core, a periaxonal space and myelin sheath are defined, is different from one without the periaxonal space (but otherwise identical axon core and myelin sheath) in terms of v, even as xraxial approaches infinity?
As xraxial becomes very large, the difference will become very small and should approach 0 in the limit. Verification by computational experiment "is left as an exercise for the reader", as Johnson and Kiokemeister used to say.