## Modelling a chain of HH cell with myelinated axon

When Python is the interpreter, what is a good
design for the interface to the basic NEURON
concepts.

Moderator: hines

subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Modelling a chain of HH cell with myelinated axon

Hi,
I am a Masters student of Computer Science working on a neuroscience project (completely new to neuroscience). My project is to model nodes of Ranvier (following HH cell dynamics) with myelinated axons to observe the effects of length of the myelin sheath.

I am trying to use NEURON+Python for this. Although I understand the chaining and dynamics of HH and Passive cable, myelinated axon seems to be more accurately modelled by extracellular. So the question is, how do I model a simple myelin sheath using extracellular properties?

Code: Select all

``````Params : {
Myelin resistance (R_m): MOhms/mm
Myelin capacitance: pF/mm
longitudinal resistance: MOhms/mm
resting potential
length and diameter of the cable
}
``````
ted
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### Re: Modelling a chain of HH cell with myelinated axon

This is written for general readership, so please excuse explicit discussion of much that you probably already know.

The first and most important step in any modeling project is to develop a clear conceptual model. I will assume you want a model of an unbranched myelinated axon--a series of nodes of Ranvier alternating with myelinated internodes. Decide what you want the axon to start and end with--the first part could be either a node or internode; ditto for the last. I would probably have a node at each end.

At this point you are ready to delve into implementational details. Your model will be easiest to manage if you think of it as two different sets of sections, implemented as Python lists: one set will contain all the nodes, and the other set will contain all the internodes. Then creating sections, connecting them, and assigning anatomical and biophysical properties to them can take advantage of iteration. For the sake of clarity, a good code pattern is to separate the process of creating and connecting sections from the process of assigning their anatomical and biophysical properties.

You haven't said how many nodes or internodes you want, or what their properties are to be. It would be a good idea to specify these via symbolic constants declared near the top of your program where the're easy to discover, before the any statements that create lists or sections etc..

In a mix of pseudocode and Python

Code: Select all

``````import necessary libraries like h, numpy, etc.
h.load_file("stdrun.hoc") # defines many useful helper functions
symbolic constants that specify model properties e.g. NNODES (number of nodes), Lnode, Linternode--anything you might want to change in the future)

create nodes and internodes and append them to lists
use connect to assemble the nodes and internodes in alternating sequence
assign anatomical properties
assign biophysical properties``````
Now you're ready to decide how to represent myelination. What is your conceptual model of the internode? A first order conceptual model of an internode would be that it is just a cable in which the core conductor is cytoplasm and the insulation is a lipid bilayer that offers no path for transmembrane ionic flux. An internode with no myelin sheath would have specific membrane conductance of 0 S/cm2, and specific membrane capacitance of 1 uf/cm2. If the internode is wrapped in NLAYERS layers of myelin sheath, and there is no conductive gap between the internode and the sheath or between the membranes of the sheath itself, the specific membrane capacitance would be reduced to 1/(1 + NLAYERS) uf/cm2. That's 1+NLAYERS because you need to take into account the membrane of the internode itself. You could implement this representation of a myelinated internode simply by setting cm to an appropriate value. But don't make the mistake of inserting ion channels into the internode. Why? Read the "Note" at the bottom of this post.

Now, suppose you have been reading about the ultrastructure of the myelin sheath, and the membrane properties of the internodal axolemma (membrane of the axon itself, as distinct from the myelin membrane) and electrical properties of myelin itself. Then you will know that internodes do have some ion channels, and that myelin isn't a perfect insulator. Also, there is a small gap between the external surface of the axon per se and the myelin sheath; this periaxonal space may allow some current to escape from the axon and flow along the outer surface of the internode. And there are other gaps in the sheath (e.g. Schmidt-Lantermann clefts) that allow current to flow "across" the myelin sheath. Finally, in each paranodal region (a few um on either side of each node), the periaxonal space between axolemma and myelin is sealed off by tight junctions, which greatly increases the resistance to longitudinal current flow in the periaxonal space.

If you want your model to take these biological featues into account, you can insert "extracellular" into your internode sections. "extracellular" isn't a "biophysical mechanism" in the same sense that an ion channel or a pump is. It just adds layers to the cable equation. Be sure to read about extracellular in the Programmer's Reference. The default values of its parameters are such that

1. its longitudinal resistances .xraxial[] are effectively infinite, which is equivalent to ignoring the periaxonal gap
2. its transverse capacitances .xc[] and conductivities .xg[] are 0 and infinite, respectively, so there is no interference with radial current flow away from the outer surface of the internodal membrane

which means it won't affect simulation results at all.

If you want your model to explicitly represent the effects of myelin on radial current flow from the axon, but ignore longitudinal current flow in the periaxonal gap, just
1. insert whatever channels you need into the internode sections
2. leave internodal cm unchanged from its default value (1 uf/cm2)
3. insert extracellular into the internode sections
4. set internodal xg[0] and xc[0] (representing the myelin layers just outside the internode) to appropriate values
5. leave internodal xg[1] and xc[1] unchanged from their default values

Example:

inode.insert('extracellular')
inode.xc[0] = CMYEL/NLAYERS # question to readers: why not 1+NLAYERS?
inode.xg[0] = GMYEL/NLAYERS

where CMYEL and GMYEL are the specific capacitance (in uf/cm2) and conductance (in S/cm2) of a single layer of myelin

Note: if you are representing myelination by simply reducing internodal cm and not using extracellular, it does not make sense to insert ion channels into the internodal membrane. Why? Because the v that those channels will "see" is not the actual potential drop across the axonal membrane per se. Instead, it includes the voltage drop across the myelin sheath. This will affect both the gating of the channels, and the driving force for current flow through them, and simulation results will be garbage.
ted
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### Re: Modelling a chain of HH cell with myelinated axon

I said nothing at all about spatial discretization (the value to use for nseg). That's a topic for a different discussion.

Nor did I address parameters. The question that triggered this essay stipulated
Params : {
Myelin resistance (R_m): MOhms/mm
Myelin capacitance: pF/mm
longitudinal resistance: MOhms/mm
resting potential
length and diameter of the cable
}
The meanings of the parameter names "Myelin resistance", "Myelin capacitance", and "longitudinal resistance" are unclear. None of their units are consistent with what NEURON expects. Please provide explicit definitions; operational definitions might be best.
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

Hi Ted,
Thanks a lot, that was of great help in understanding what I actually want. To answer some of your questions:
You haven't said how many nodes or internodes you want, or what their properties are to be
My project is to only observe the effect of myelin length on the action potential. In other words, I will have HH node connected by a myelin internode (varying length for different simulations) which is then connected to another HH node. I will use monte carlo simulation to vary different myelin lengths and other properties of the internode to observe the effect on action potential in the second HH node.

General circuit model sketch found here:
Electrical representation found here (2 internode chain representing 2mm myelin):
What is your conceptual model of the internode?
As far as I know, I don't think my internode will have any ion channels, just the resistor and a capacitor. I am trying to convert an already existing model (in SPICE) to Python for easier and more efficient simulations. Hence my lack of knowledge of the domain.

Given the above, how would you suggest I approach the problem?

Again, thank you so much for your help!! I really appreciate it.
ted
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### Re: Modelling a chain of HH cell with myelinated axon

OK, you don't need super-detailed representation of the properties of myelin. No need for extracellular.

There is actually an existing literature on the effects of internode length on spike propagation. The key papers are
Brill et al.
Conduction velocity and spike configuration in myelinated fibres: computed dependence on internode distance.
Journal of Neurology, Neurosurgery, and Psychiatry 40:769-774, 1977.
and
Moore et al.
Simulations of conduction in uniform myelinated fibers. Relative sensitivity to changes in nodal and internodal parameters. Biophysical Journal 21:147-160, 1978.

Both are available online at no charge, and both are well worth reading. The model by Brill et al. assumed the membrane properties used by Fitzhugh (1962) in a model of conduction in frog sciatic nerve. Moore et al. tried two different descriptions for nodal currents: "high density Hodgkin-Huxley" (HH with all conductances--gnabar, gkbar, gleak_hh--multiplied by a factor of 10), and Frankenhaeuser-Huxley. They commented that "conduction velocity was found to be surprisingly insensitive to the specific nodal characteristic assignments" (i.e. FH and "high density HH" produced very similar CV), so it's probabaly OK for you to use "high density HH" for your model's nodes.

ModelDB contains a re-implementation of the Brill et al. model written in hoc for NEURON by Michael Hines. In particular, take a peek at cable.hoc, especially the comments at the start. What's that "end effect" stuff about? Sealed end effect. Read this to find out more:
viewtopic.php?p=17998#p17998
I will have HH node connected by a myelin internode (varying length for different simulations) which is then connected to another HH node. I will use monte carlo simulation to vary different myelin lengths and other properties of the internode to observe the effect on action potential in the second HH node.
You're sure about _two_ nodes? That's won't be enough to show much of anything. Stimulus artifact and sealed end effect will dominate the results. But if that's the assignment, that's the assignment.
As far as I know, I don't think my internode will have any ion channels, just the resistor and a capacitor.
Uh, the resistor in series with the little Vm battery is actually a representation of ion channels with fixed conductance (what neuroscientists might call "leak channels").

The parameters in your previous post remain problematic. I will explain why after a break.
subash774
Posts: 11
Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

There is actually an existing literature on the effects of internode length on spike propagation. The key papers are
Brill et al.
Conduction velocity and spike configuration in myelinated fibres: computed dependence on internode distance.
Journal of Neurology, Neurosurgery, and Psychiatry 40:769-774, 1977.
and
Moore et al.
Simulations of conduction in uniform myelinated fibers. Relative sensitivity to changes in nodal and internodal parameters. Biophysical Journal 21:147-160, 1978.
Thanks for these, I'll give them a read.
You're sure about _two_ nodes? That's won't be enough to show much of anything. Stimulus artifact and sealed end effect will dominate the results.
When I meant 2 nodes I meant 2 HH nodes at the beginning and the end and lots of internodes in the middle to figure out the tipping point of where the AP cuts off.
But if that's the assignment, that's the assignment.
Sadly, this isn't even the assignment, it's just the foundation for me to simulate and generate data from which I can do my actual assignment from.
Uh, the resistor in series with the little Vm battery is actually a representation of ion channels with fixed conductance (what neuroscientists might call "leak channels").
Ahh okay, so I basically model a passive cable with some added properties?
The parameters in your previous post remain problematic. I will explain why after a break.
Again, Thank you so much!! Really appreciate this :)
ted
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### Re: Modelling a chain of HH cell with myelinated axon

I meant 2 HH nodes at the beginning and the end and lots of internodes in the middle
Is that
N-I-N-I-I-I-...-I-N-I-N (i.e. 4 nodes and a bunch of internodes?
or
N-I-N-I-N-I-...-N-I-N (i.e. NNODES nodes and NNODES-1 internodes)
"The slovenliness of our language allows us to think foolish thoughts." from Orwell's "Politics and the English Language" (1946).
Not just in politics, George.

WRT parameters and units--

NEURON needs the cytoplasmic resistivity Ra, which is in units of ohm-cm. This is the resistance between the opposite faces of a 1 cm cube of cytoplasm. The "true" value is very hard to determine experimentally; for squid axon it's 35.4 ohm cm, but for neurons from vertebrates people generally use something between 80 and 200 ohm cm. That aside, what you have is not the value of cytoplasmic resistivity, but something called "longitudinal resistance" which is in units of megohms/mm. Presumbably "longitudinal resistance" is the resistance between the two ends of a 1 mm long cylinder of cytoplasm. But you need the diameter of the cylinder to determine the corresponding value of Ra. Given a cylinder of cytoplasm with resistivity Ra ohm cm, the diameter and length of which are diam and L in um, the resistance between the two ends of the cylinder is (Ra * L * 4 / (PI * diam^2)) ohm * cm * um / um^2
which equals
1e4 Ra * L * 4 / (PI * diam^2) ohms = 0.01 Ra * L * 4 / (PI * diam^2) megohms. For convenience let's call "longitudinal resistance" RL. Then
RL = 0.01 Ra * L * 4 / (PI * diam^2)
and
Ra in ohm cm = RL * 25 * PI * diam^2 / L where diam and L are both in microns. You know that L is 1 mm = 1e3 um, so
Ra in ohm cm = RL * 0.025 * PI * diam^2

So if you knew the numerical value of RL, and the value of diam in um, you could calculate Ra. But you don't know diam.

What about membrane capacitance? NEURON needs specific membrane capacitance in uf/cm2. You can assume that nodal membrane's specific membrane capacitance is 1uf/cm2 ("true" value of specific membrane capacitance is hard to measure experimentally; it's probably somewhere between 0.7 and 2 uf/cm2, and most modelers just assume 1 uf/cm2). But you don't have a handle on the effective internodal specific membrane capacitance (I call this "effective specific membrane capacitance" because it's the equivalent capacitance of the axolemmal specific capacitance in series with the much smaller specific capacitance of the myelin sheath). Instead you have something called "myelin capacitance" which is in pf/mm. Presumably this is the net membrane capacitance of a 1 mm length of internode. The total surface area of a cylinder with length 1 mm and diameter diam um is
PI * diam * 1e3
in units of um^2, which is equvalent to
PI * diam * 1e-5
in units of cm^2 (because 1e4 um = 1 cm).
Using CM to signify internodal specific membrane capacitance in uf/cm2, a 1 mm long internode has a total capacitance of CM * PI * diam * 1e-5 uf, which is equivalent to 10 CM * PI * diam pf. In other words,
10 CM * PI * diam = "myelin capacitance"
so
CM = "myelin capacitance" / (10 * PI * diam).

So if you knew the numerical value of "myelin capacitance" and the value of diam in um, you could calculate Ra. But you don't know diam.

Finally we get to the resistor in that equivalent circuit of internodal membrane, which you're calling "myelin resistance" or "R_m", and which has units of megohms/mm. First, the units make no sense at all. In the case of "myelin capacitance," units of capacitance/length make sense because the longer a cylinder is, the more surface area it has and the larger its total capacitance becomes. Given a cylinder with a particular length, calculate the product of length and capacitance/length and you get . . . capacitance. But in the case of membrane resistance, the more surface area you have, the lower the total resistance should be. After all, the area is all in parallel, right? Take two cylinders of a given length, attach one to the other, and now you have twice as much surface area, so total membrane resistance should be half as much, not twice as much.

OK, let's get past that by assuming that whoever wrote MOhms/mm really meant MOhms mm, so that the numerical value of "myelin resistance" is the net resistance of the membrane of an axon that is 1 mm long. But NEURON expects specific membrane conductance gm, which is in units of Siemens/cm2. The total surface area of the internode is
PI * diam * 1e3
in units of um^2, which is equvalent to
PI * diam * 1e-5
in units of cm^2 (because 1e4 um = 1 cm).
so the net conductance of the internodal membrane in Siemens is
gm * PI * diam * 1e-5
and its net resistance is
1e5 / (gm * PI * diam)
in ohms, or
0.1 / (gm * PI * diam)
in megohms. And once again you're stuck, because
gm = 0.1 / (PI * diam * R_m)
but you need to know diam in order to calculate gm.

And you don't know diam.

Two possible outs:
1. If you can discover somewhere the numerical value (and units) of effective specific membrane conductance of myelinated internode, that could be used to calculate diam.
2. You could just arbitrarily assume a particular value of diam. 10 um is a reasonable internal diameter for a myelinated axon in peripheral nerve. A quick literature search on PubMed should probably turn up some freely available references about diameter ranges.
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

Is that
N-I-N-I-I-I-...-I-N-I-N (i.e. 4 nodes and a bunch of internodes?
Something like that: N-I-I-I-I-......-I-I-N

but you need to know diam in order to calculate gm.

And you don't know diam.
Really sorry, my bad. My prof has tested out the parameters on SPICE model and I am meant to follow that. Here is the list of parameters:

Diameter is 14um. I think following what you've said, knowing the diameter, it's doable now. Did you say "extracellular" was overkill? Should I just follow passive cable model then?
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

I tried with the Ra formula and I am not getting any spikes in the ball and stick model, I might be doing something wrong. I just set the cm and Ra as you described, is there more to it?
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

Sorry to bombard you with questions. I've the following code (adapted from cable.hoc implementation of Brill et al).

Code: Select all

``````class BallAndStick:
def __init__(self, gid, myelin_length):
self._gid = gid
self.ml = myelin_length
self._setup_morphology()
self._setup_biophysics()

def _setup_morphology(self):
self.hh1 = h.Section(name='hh1', cell=self)
self.myelin = h.Section(name='myelin', cell=self)
self.all = [self.hh1, self.myelin]
self.myelin.connect(self.hh1)
self.hh1.L = 14
self.hh1.diam = 14
self.myelin.L = self.ml
self.myelin.diam = 14
#         self.myelin.nseg = 10

def l2a(self, diam):
return 1/(math.pi * diam) * 1e4

def fac(self, diam):
return math.pi*diam**2/4 * 1e-8

def _setup_biophysics(self):
for sec in self.all:
sec.Ra = 1.26e8 * self.fac(sec.diam)   # Axial resistance in Ohm * cm

self.hh1.insert('hh')
for seg in self.hh1:
seg.hh.gnabar = 1.2  # Sodium conductance in S/cm2
seg.hh.gkbar = 0.09  # Potassium conductance in S/cm2
seg.hh.gl = 0.02    # Leak conductance in S/cm2
seg.hh.el = -54.3     # Reversal potential in mV

self.hh1.cm = 3.14e-9*self.l2a(self.hh1.diam)*1e6 # end up as uF/cm2

# Insert passive current in the myelinrite
self.myelin.insert('pas')
for seg in self.myelin:
seg.pas.g = 5.60e-9*self.l2a(self.myelin.diam) # Passive conductance in S/cm2
seg.pas.e = -62.5    # Leak reversal potential mV
self.myelin.cm =  1.87e-11*self.l2a(self.myelin.diam)*1e6 # end up as uF/cm2

print(self.myelin.L)
def __repr__(self):
return 'BallAndStick[{}]'.format(self._gid)
``````
The spikes look right but the myelin length seems to have no effect on it. cable.hoc has a function called l2a:

Code: Select all

``````func l2a() { // convert per length (/cm) to per area (/cm2) based on diameter
return 1/(PI*diam) * 1e4
}``````
Where does the function get the length from?
ted
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### Re: Modelling a chain of HH cell with myelinated axon

Is that
N-I-N-I-I-I-...-I-N-I-N (i.e. 4 nodes and a bunch of internodes?
Something like that: N-I-I-I-I-......-I-I-N
Two nodes separated by a lot of internodes. You'll never get a propagating action potential.
but you need to know diam in order to calculate gm.
And you don't know diam.
. . . Diameter is 14um.
Then you might just be able to calculate the internode parameters that NEURON needs. Except for one thing:
My prof has tested out the parameters on SPICE model and I am meant to follow that. Here is the list of parameters:
The list is incomplete. What is the length of a node, and what are the presumed properties of the nodal membrane (specific membrane capacitance, descriptions of the ion channel kinetics, channel densities)? Is it just garden-variety Hodgkin-Huxley mechanism, with the default ion channel densities?

What are the operational definitions of Rm, Cm, and Ri? Have you verified that Ri the longitudinal resistance of a 1mm long x 14 um diameter cylinder of axoplasm, and that Cm is the net capacitance between the inside and the outside of a 1 mm long x 14 um diameter internode? These definitions would make sense for Ri and Cm, because longitudinal resistance and net capacitance of an internode X mm long would equal X * Ri and X * Cm--but you need to verify before proceeding.

Rm is a problem--if it is the net resistance between the inside and outside of a 1 mm long x 14 um diameter internode, then its units should be megohm * mm, not megohm/mm. Why? Imagine that you double the length of an internode. That will double the internode's longitudinal resistance, and it will also double the net capacitance of the internode (twice as much membrane implies twice as much capacitance). But doubling the membrane area should also cut the resistance between the inside and outside of the internode in half. Instead, Rm's units of resistance/length tell you that doubling the length should double the net resistance.

With regard to resting potential, NEURON's implementation of the Hodgkin-Huxley mechanism (which is called hh) has a resting potential of -65 mV, not -62.5. If you're using NEURON's hh, your model axon should have a resting potential of -65 mV.
Did you say "extracellular" was overkill?
Extracellular is unnecessary. Just insert NEURON's pas mechanism into the internodes. Set e_pas to -65 mV, same as hh's resting potential.
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

Two nodes separated by a lot of internodes. You'll never get a propagating action potential.
Would having the structure like N----I--------N and then vary the length of the internode be a better option? I tried with internode lengths from 1mm to 15mm and the signal seems to be dropping off after 10mm.
... garden-variety Hodgkin-Huxley mechanism, with the default ion channel densities?
Nodes of Ravnier are just modelled using vanilla HH model, yes.
What are the operational definitions of Rm, Cm, and Ri? Have you verified ...
Yepp, Ri and Cm are as you described and checking on the Rm.
its units should be megohm * mm, not megohm/mm
Rm is meant to be Mohm*mm. All seems to check out fine.

Thanks so much for your help!!

I've read the Brill et al paper and seen the cable.hoc from ModelDB but I still don't fully understand how functions such as l2a() and fac work and how I'd have to modify them to fit my case.

Code: Select all

``````func l2a() { // convert per length (/cm) to per area (/cm2) based on diameter
return 1/(PI*diam) * 1e4
}

// ohm/cm must be converted to ohm-cm by multiplying by
// cross sectional area
fac = PI*diam^2/4 * 1e-8
``````
ted
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### Re: Modelling a chain of HH cell with myelinated axon

With regard to your class BallAndStick . . .

Nice clean code, great start. Just a couple of things to resolve, and you'll be ready to extend your program so that a parameter can be used to specify how many nodes and internodes the axon has.

If you're going to define an axon class, the name of the class should probably be Axon rather than BallAndStick.

Why not call the node and internode sections node and internode, or node and inode.

hh1 isn't quite the right name for the nodes because hh is so widely understood to be the name of a very specific mechanistic model of spike generation in squid axon. Unlike "node", "hh" doesn't imply anything about structure, nor does node imply hh; indeed, a node could have one of several different spike mechanisms. Future reuse of your code might be reused to explore the properties of myelinated axons whose nodal membrane properties are something other than hh, and that

myelin doesn't seem right for the internodes, because myelin is only a part of what lies between two adjacent nodes of Ranvier--the complete internode consists of an axolemma around which the myelin is wrapped.

So much for the names of things.

The length of the node is much too big. The model's other parameters aren't quite right, because an instance of this model is spontaneously active--spikes repetitively. If the nodes of Ranvier are supposed to be plain vanilla Hodgkin Huxley, the default parameters of NEURON's hh mechanism should be sufficient, i.e. no need to calculate and assign new values to them.

hh rests at -65 mV. The reversal potential e of the internode's pas mechanism should also be set to -65 mV; if it isn't, the model axon will not have a uniform resting potential (or may not have a resting potential at all, if pas.e is too depolarized).

WRT functions, I'll have to review those and get back to you.
subash774
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Joined: Sun Jul 19, 2020 11:20 pm

### Re: Modelling a chain of HH cell with myelinated axon

Yeah, sorry about the variable names, I am just trying to make it work, not the best names for them. I'll try to document them well once the basics work.
the default parameters of NEURON's hh mechanism should be sufficient
Gotcha! I changed it to just hh and it works the same, thanks.
The model's other parameters aren't quite right
I thought so too, especially the cm and Ra. Also, the code I posted was old, the repetetive firing has stopped after I cahnged hh parameters to default.
Also, the voltage at the first node (where I inserted the IClamp) seems to be lower than second node (5mm internode)? Is that normal or just a result of my incorrect parameters?