Modeling branching off of a soma compartment

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Merchant
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Modeling branching off of a soma compartment

Post by Merchant » Sat Jun 25, 2011 12:30 am

I'm a bit new to computational neuroscience and I'm having some trouble understanding how branching (especially off the soma) is modeled in compartmental models.

Is there a standard way (or set of ways) to model multiple (more than two) compartmental connections to a soma?

For example, in the Traub, et al. 1991 model of the CA3 neuron, the apical and basal dendrites are modeled by a simplified single apical cable on one side of the soma and a single basal cable on the other side of the soma. How might I go about attaching an axon (or, for example, splitting the basal dendrites into multiple branches off the soma)? Is the typical method to simply connect the soma to each adjacent compartment through some resistance (if so, is this resistance calculated differently from other axial resistances)?

Also, I would imagine that, in an appropriately complex model, currents through the soma would be affected by the relative positions of each connected compartment and the 3D shape of the soma itself. For example, if the axon and basal dendrites are on the bottom of a pyramidal cell, and the apical dendrite is at the top, then I would imagine that currents would be conducted slightly faster and over slightly less resistance between the axon and the basal dendrites than between the axon and the apical dendrite, or the apical dendrite and the basal dendrites (not sure about the faster part). Is this accurate, and if so is it beyond the scope of compartmental modeling (and would this be due to the assumption of a single dimension, x)?



A related question - is there a definitive paper on modeling compartmental branching explicitly (i.e. without a 3/2 power rule or other such simplification), including equations for calculating axial current, and perhaps some discussion of relations to morphology? It seems like there are multiple methods in use for modeling branches (typically dendritic branches), but I can't seem to find any material actually discussing a method in detail. Right now I'm using a sort of "Y" branch method, which is what I think both NEURON and GENESIS use (basically, the point where Vm is calculated at the center of each compartment is connected to two axial resistors on each side, and in a Y branch, 3 resistors from 3 compartments are connected at a central point [the middle of the Y-shape], and the current entering a compartment from the Y branch is calculated by solving the equivalent electrical circuit).



Thank you for your time.

[Also, is this the best place to ask this kind of question?]

ted
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Re: Modeling branching off of a soma compartment

Post by ted » Sat Jun 25, 2011 12:35 pm

I am unaware of any evidence that the site of neurite attachment to the soma makes any significant difference whatsoever to the simulation of electrical signaling in neurons. It appears that a single compartment is sufficient for electrical signals. For cosmetic purposes it may be nice for proximal ends of neurites to be rendered as being contiguous with the surface of the soma, but in terms of simulation results their electrical attachments might just as well be at the center, or all from the same point on the surface.
Merchant wrote:A related question . . .
In NEURON, each neurite is discretized into a user-specified number of compartments of equal length, using the central difference method. Numerical integration calculates solutions for membrane potential and other state variables that are second order accurate at the centers of these compartments (AKA "internal nodes"). If two or more neurites are connected at a common point, the potential at that point is calculated algebraically as the weighted sum of the potentials of the adjacent internal nodes, where the weights are the reciprocals of the axial resistances between the common node and the adjacent internal nodes.
is this the best place to ask this kind of question?
Yes.

Merchant
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Re: Modeling branching off of a soma compartment

Post by Merchant » Sun Jun 26, 2011 10:34 pm

Thanks for the reply.
ted wrote:For cosmetic purposes it may be nice for proximal ends of neurites to be rendered as being contiguous with the surface of the soma, but in terms of simulation results their electrical attachments might just as well be at the center, or all from the same point on the surface.
Interesting. So then, for 3 neurites, I could then treat all the neurite-soma connections as a single branch point at the edge of the soma (i.e. half the soma's full-length axial resistance away from the soma's central [or "internal node"] voltage, Vm), branching into 1 axonal and 2 dendritic compartments, using the same branching math that would be used for a dendritic branch into 3 subtrees, right?
Also, when you say the attachments "might just as well be at the center", do you mean without the axial resistance of the soma compartment? Wouldn't the math be different then?

And regarding the "related question" about modeling branches:
ted wrote:If two or more neurites are connected at a common point, the potential at that point is calculated algebraically as the weighted sum of the potentials of the adjacent internal nodes, where the weights are the reciprocals of the axial resistances between the common node and the adjacent internal nodes.
I'm not sure I follow; when I try to figure out the voltage at the common point, I get this (using a Y branch into two trees):
Suppose:
  • A, B, and C are compartments that connect at a common point D, through resistances Ra, Rb, and Rc, respectively.
  • Va, Vb, and Vc are the voltages at the centers of the compartments, and Vd is the voltage at point D.
  • C is the compartment of focus (i.e. I am solving for the current going into C), so Ic is the current entering compartment C from the branch (that is, Ic is the current through resistor Rc, from Vd to Vc), while Ia and Ib are the currents leaving compartments A and B into the branch (i.e. from Va to Vd and Vb to Vd).
(Note that the full axial resistance of compartment C is Rc*2, as the current only flows through half the axial resistance to get from Vm, at the center of compartment C, to point D, at the end of compartment C.)

From this partial circuit, I derive the following 4 basic equations (and this could be where I'm mistaken, as I'm not too experienced with circuits):
1) Va-Vd=Ia*Ra
2) Vb-Vd=Ib*Rb
3) Vd-Vc=Ic*Rc
4) Ia+Ib=Ic
The first 3 are from Ohm's law and the last is from Kirchhoff's current law (Ia and Ib are flowing into point D and Ic is flowing out from point D).

When I solve for Ic (the current flowing into C), I get:
Ic = (Ra*(Vb-Vc) + Rb*(Va-Vc)) / (Ra*Rb + Ra*Rc + Rb*Rc)
(The same form could be used to find -Ia [i.e. negative Ia] and -Ib, the currents flowing into A and B.)
Finally, if I solve for Vd directly, I get:
Vd = (Va*Rb*Rc + Vb*Ra*Rc + Vc*Ra*Rb) / (Ra*Rb + Ra*Rc + Rb*Rc)
Is this incorrect? It doesn't seem to be the same as:
Vd = Va/Ra + Vb/Rb + Vc/Rc
Which is what I thought you meant by "the potential at that point is calculated algebraically as the weighted sum of the potentials of the adjacent internal nodes, where the weights are the reciprocals of the axial resistances between the common node and the adjacent internal nodes".

Am I doing something wrong?

ted
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Re: Modeling branching off of a soma compartment

Post by ted » Mon Jun 27, 2011 11:24 am

Merchant wrote:So then, for 3 neurites . . .
True.
when you {yes} and {yes but the quantitative difference will be negligible}
And regarding the "related question" about modeling branches:
ted wrote:If two or more neurites are connected at a common point, the potential at that point is calculated algebraically as the weighted sum of the potentials of the adjacent internal nodes, where the weights are the reciprocals of the axial resistances between the common node and the adjacent internal nodes.
I should have included the phrase "divided by the sum of the reciprocals of the axial resistances." Or, expressed in terms of the conductances
vcommon = (SUMMA gi * vi) / SUMMA gi

Merchant
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Re: Modeling branching off of a soma compartment

Post by Merchant » Mon Jun 27, 2011 11:39 pm

Ah, I see now.

Thanks for the help.

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