Compartmental neuron with`Ri` as state variable

Description of problem

Myelinated axons cannot be defined as uniform cables. It is desirable to construct them as a series of nodal compartments connected to the internodal ones, with each compartment having its own set of equations. For instance, for the following axon composed of two nodes and three internodal myelinated compartments

... =====+=====+====== ...

one needs specify a set of eqs_nodal on the unmyelinated nodes and a different set eqs_internodal on the myelinated parts. My question is: what is the best way to implement such a structure on brian?

The current implementation of SpatialNeuron raises two challenges:

  1. Specifying different equations at each segment (against the shared equations model)
  2. Specifying compartment-dependent axial resistant (against the shared variable Ri)

In the problem I’m dealing with the nodal part are governed by a variation of Hodgkin-Huxley dynamics whereas the internodal parts are passive. I can circumvent 1 by specifying a general active set of equations (resembling the nodal dynamics) and simply turning off the irrelevant conductances on the internodal segments (as they are state variables). But, I’m not sure how to resolve 2. One idea is to utilize the fact the Cm is actually a state variable, and solve Cm for each compartment (with constant Ri) such that effective axial impedances match. But it won’t work if the axial elements are only resistive.

Thanks in advance for ideas/thoughs!
Arash

Hi @arashgmn,
your observations and conclusions are exactly on point, unfortunately you are running into limitations of the current multi-compartmental neuron support. Regarding your point 1: currently, you can only have one set of equations for all compartments of a SpatialNeuron. As you mentioned, having e.g. g_na : siemens (constant) as part of the equations and setting it to 0 for the myelinated parts is the way to get around this restriction. Regarding Ri, you are again right that this is currently a single value for the whole neuron. I don’t see any workaround for this at the moment. I am not 100% sure (@romainbrette is the expert on these topics), but I think myelinated axons are typically modeled with an increased Cm (myelin ≈ thick membrane), without any change in Ri.
All that said, I think we could support a compartment-specific Ri in the code without too much effort. If this is important to you and if you are willing to help with testing (e.g. coming up with an analytical solution we can compare against), I could give it a try.

Best,
Marcel

Hi @mstimberg
Thanks for the clarification.
Your suggestion regarding increasing capacitors to account for thicker myelin lamella layers is worth trying. That said, I still fancy compartment-specific Ri and would be happy to test new functionalities or write new test units.

Arash

Just an update.

Based on the current state of literature, it seems more realistic to partition a myelinated axon alongside its axis into juxtaparanodes, paranodes, and the nodes (look at Fig.2 of this article, for instance). This is in favor of having compartment-specific Ri.