Single Compartment Models of Neurons

[Noron]

Hello,

I've recently been given the task of creating a single compartment model of a neuron using experimental data (from whole cell patch clamp recordings). In developing my model, I've chosen morphological parameters (soma diameter and surface area) from a paper that's performed 3-D morphological construction of cells stained with biocytin. I've inserted sodium, delayed rectifier potassium, a-type potassium, calcium, and leakage ionic channels using a .mod file. Finally, I am using a parameter search of maximum ionic conductances (i.e., gionbar) to attempt to reproduce the current-voltage relationships observed in vitro following voltage-clamp experiments (I am using SEClamp in a 'for' loop). Additionally, I am attempting to reproduce spiking patterns observed in vitro following current clamp (I am using IClamp in a 'for' loop). As it turns out, I do find some values that allow me to reproduce the potassium current vs. membrane voltage plots observed in vitro. However, I am not entirely convinced that my simulations make any sense. Which leads me to my main question:

What are single compartment models useful for? I read on a previous post that single compartment models will not respond to extracellular stimulation--what exactly is defined as extracellular stimulation? Voltage clamp? Current clamp? I suppose I'm a bit confused and I want to be sure that I'm building a model that is both interesting and useful.

I should also add that the particular cell I'm studying has extensive gap junctioning, and a recent paper has measured input resistance to these cells by blocking the gap junctions.

Thank you,
N.

[ted]

I've chosen morphological parameters (soma diameter and surface area) from a paper that's performed 3-D morphological construction of cells stained with biocytin.

Depending on the anatomy of the cells in question, you might be eliminating more than 99% of the total surface area, and certainly a large fraction (if not most) of the surface area that lies electrotonically close to the soma. If a good case can be made for approximating the cell by a single compartment (see below), it might be better to base surface area on experimental estimates of cellular capacitance (e.g. from apparent membrane time constant and input resistance as measured at the soma), assuming specific membrane capacitance of 1 uf/cm2.

I've inserted sodium, delayed rectifier potassium, a-type potassium, calcium, and leakage ionic channels using a .mod file.

Based on experimental data from the particular type of cell you're studying? Are these mechanisms taken from other models of the same cell type (not absolutely essential, but nice if possible)?

As it turns out, I do find some values that allow me to reproduce the potassium current vs. membrane voltage plots observed in vitro.

That's encouraging. But does your model cell have the same input resistance and time constant as observed experimentally?

However, I am not entirely convinced that my simulations make any sense.

Healthy skepticism is good.

What are single compartment models useful for?

Easier to answer "what are they used for." For me, they're mostly useful as the computational analog of xenopus oocytes, e.g. as a simplified test system for developing and evaluating individual mechanisms of ionic channels, pumps or other transport mechanisms, intracellular mechanisms such as cytoplasmic reactions involving second messengers. They are also often used for modeling individual cells or networks (e.g. stomatogastric ganglion) under the (often tacit) assumption that extended cellular anatomy isn't essential to whatever phenomena are being studied. Rational arguments can often be made for that assumption.

what exactly is defined as extracellular stimulation? Voltage clamp? Current clamp?

Apply stimulus current to the extracellular medium.

I should also add that the particular cell I'm studying has extensive gap junctioning, and a recent paper has measured input resistance to these cells by blocking the gap junctions.

That's helpful. Anything known about membrane time constant?

[Noron]

Hi Ted,

Thanks very much for the prompt and helpful response. I'm studying AII amacrine cells, and they've got some pretty interesting morphological characteristics. They have lobular appendages and distal dendrites that reach into different layers of the retina. Thus far, I have been modeling the cell soma as a cylinder with length (L) and diameter (diam) as specified in NEURON. I've used diameter values and measured surface area values from literature to work out the length. Additionally, I've been calculating the surface area of the cylinder minus the end caps--it is my understanding that this is how NEURON calculates surface area as well. A question that arises from this is whether single compartment models can include dendrites and lobular appendages, or are these types of models designed specifically to model membrane mechanisms of the soma? Would there be any difference between the two as long as I've used the correct value for surface area? This is an interesting question for me because literature suggests that a majority of voltage-gated calcium channels reside in the lobular appendages of the AII amacrine cells, so perhaps putting them in the soma would make for an inaccurate model of calcium current flow? I'm not sure.

As far as gating dynamics go, I've been using literature reports from RT-PCR experiments performed to identify specific channel subtypes found in the AII. From there, I use this website to work out some of the values: http://channelpedia.epfl.ch/ionchannels/189
For channels not having models/equations reported by Channelpedia, I adapt values from a model of the retinal ganglion cell (Fohlmeister et al).

Another question concerning surface area is whether I can convert the current density (flux?) in mA/cm^2 value that NEURON outputs into a current in mA by multiplying by either a.) the surface area of the cell (this doesn't really make sense to me, but who knows) or b.) by multiplying by the "patch area" reported by the patch clampers (clampists?). I'm curious as to whether patch area can be calculated by simply using pi*r^2 and the diameter of the patch clamp pipette. I should also mention that a majority of the experimental data I'm using was recorded in the whole cell configuration, which may or may not make the patch area irrelevant for my simulations.

The authors of the gap junction blockage paper report values for gap junctional conductance (gj), membrane resistance (rm), membrane capacitance (Cm), cytoplasmic resistivity (Ri), and input resistance (Rin) using dual voltage clamp recording. They report that Rin is 620 Mohms before gap junctions are blocked, then 4,400 Mohms after gap junctions are blocked. They also report that specific membrane resistance is 41 Kohms*cm^2. It seems they assume a "standard value" for Cm (0.01 pF/um^2) and use their surface area values to work out average capacitance: 9 pF. As far as membrane time constant goes, can I simply do Cm*rm or Cm*Rin? Or is this value something that needs to be measured empirically? There is one other question I have pertaining to these issues: Can I simulate a dual voltage clamp experiment in NEURON? Would it be as simple as placing SEClamps in different locations of the model?

Sorry Ted, lots and lots of questions!

Thank You,
Nick

[ted]

A question that arises from this is whether single compartment models can include dendrites and lobular appendages

What you probably meant to ask is whether the activity of a cell with extended anatomy can be satisfactorily represented by a single compartment model. The answer is maybe, maybe not.

literature suggests that a majority of voltage-gated calcium channels reside in the lobular appendages of the AII amacrine cells, so perhaps putting them in the soma would make for an inaccurate model of calcium current flow?

Again, maybe, maybe not. The only way to find out is by building a model and testing it.

In the 1970s and 80s it was commonly thought that the types of channels expressed by a cell dictated the cell's electrophysiological phenotype, i.e. whether it was silent at rest, spontaneously active, fired in bursts etc.. A reasonable supposition at the time, experimental and theoretical evidence over the past 20 years has shown otherwise. In particular see
Mainen ZF, Sejnowski TJ (1996)
Influence of dendritic structure on firing pattern in model neocortical neurons.
Nature 382:363-6
You can download their model from ModelDB and see for yourself.

So the story is more complex--cell shape and spatial distribution of channels jointly determine how a cell functions.

That said, a strong case can be made for staring with the simplest model possible--a single compartment--and seeing if that is sufficient for the purpose of your investigation. After all, a computational model is just an expression of a hypothesis in computable form. Expect it to fall short of "reality"--all models are simplifications and abstractions of real systems. Add complexity only when it turns out to be absolutely necessary.

As far as gating dynamics go . . .

Good strategy. You're going about this in a sensible way.

whether I can convert the current density (flux?) in mA/cm^2 value that NEURON outputs into a current in mA

What are you interested in: total transmembrane calcium current, or intracellular calcium concentration?

Rin is . . . 4,400 Mohms after gap junctions are blocked. They also report that specific membrane resistance is 41 Kohms*cm^2.

If the cell is electrically compact, surface area would be ~930 um2.

specific membrane resistance is 41 Kohms*cm^2
. . .
they assume a "standard value" for Cm (0.01 pF/um^2) and use their surface area values to work out average capacitance: 9 pF.

so membrane time constant is 41 ms (0.01 pF/um^2 is 1 uf/cm2) and surface area is 900 um2.

Can I simulate a dual voltage clamp experiment in NEURON?

Do you mean an experiment in which two single electrode clamps are attached to a cell, or do you mean an experiment in which one electrode is used to inject current while a second electrode is used to record membrane potential (possibly at a different location)?

[Noron]

Hi Ted,

Thanks again for your time and helpful responses. For our model, I have built it having a surface area equivalent to what we've discussed previously. When I'm adding in conductances and membrane parameters, am I essentially constructing a cylindrical "pool" of freely floating ion channels? I'm not sure that I specify where or how many voltage-gated sodium channels, for example, I've put in. Are these issues taken into account with the gating dynamics equations? I need to have a closer look at those. As you've mentioned, perhaps to capture the behavior I'm interested in, this won't be an issue; running simulations will answer this question.

To answer your question, I am interested in total transmembrane ionic currents (sodium, calcium, potassium). However, I am also hoping to reproduce spiking behavior observed in current clamp using the maximum conductance values, morphology, and passive membrane properties I find to satisfy my constraints for transmembrane currents. In a sense, I want to validate my model by reproducing spiking behavior. I have a feeling that this is going to be an ambitious endeavor. Furthermore, I'm very interested in studying how all of the aforementioned phenomena are affected by gap junctions.

With the dual voltage clamp, I would like to inject current at one location and record membrane at a different location. I have an experiment in mind: it involves injecting current into a cell on one side of a gap junction and recording membrane potential from a cell on the other side of the gap junction (would this mean nseg=2, and no longer a single compartment model?)--basically what the authors of the paper I'm referencing have done in vitro. Another experiment, as I've imagined it, may help me to understand a bit more about single compartment modeling. If I'm stimulating from one end of the cell (say, IClamp(0.01)) and recording from another end (V(0.9)), am I performing a nonsensical experiment with a single compartment model? I'm familiar with the circuit analogues of single compartment models, so when I record membrane potential, am I basically placing a voltmeter in parallel with the membrane potential, as well as the ionic driving forces? Therefore, does that make alternative locations of the "voltmeter" meaningless?

Thank You,
Nick

[ted]

am I essentially constructing a cylindrical "pool" of freely floating ion channels? I'm not sure that I specify where or how many voltage-gated sodium channels, for example, I've put in. Are these issues taken into account with the gating dynamics equations?

You would probably find it helpful to read chapters 3-6 of The NEURON Book. At the very least, read
Hines, M.L. and Carnevale, N.T.
The NEURON simulation environment.
Neural Computation 9:1179-1209, 1997
which is available from links at http://www.neuron.yale.edu/neuron/nrnpubs

total transmembrane ionic currents (sodium, calcium, potassium)

As long as your mechanisms are specified by NMODL code that WRITEs ina, ica, or ik, (or by Channel Builders that are configured to generate these currents), total current density attributable to each ionic species will automatically be available with the name ina, ica, or ik. Record these to Vectors, then after a simulation use the Vector class's mul() method to multiply by surface area and the appropriate scale factor, e.g. for a section called soma that has nseg 1 and assuming that the time course of ina has been recorded to a Vector called inavec,
soma inavec.mul(0.01*area(0.5))
converts the recorded values from density units (mA/cm2) to absolute current units (nA).

With the dual voltage clamp, I would like to inject current at one location and record membrane at a different location.

Record membrane what? Membrane potential? That's not a dual voltage clamp experiment. That's just injecting current at one point (use an IClamp) and measuring potential at another. If you really want to voltage clamp the membrane at the location of current injection, use an SEClamp instead of an IClamp. As far as measuring membrane potential is concerned, nothing special is necessary--v is automatically computed at every node of a model.

injecting current into a cell on one side of a gap junction and recording membrane potential from a cell on the other side of the gap junction (would this mean nseg=2

No, it would involve having two separate model cells connected by a representation of a gap junction. Purely resistive gap junctions are easy to implement; a reasonable approach is presented in ModelDB entry 43039
Migliore M, Hines ML, Shepherd GM (2005)
The role of distal dendritic gap junctions in synchronization of mitral cell axonal output.
J Comp Neurosci 18:151-161

If I'm stimulating from one end of the cell (say, IClamp(0.01)) and recording from another end (V(0.9)), am I performing a nonsensical experiment with a single compartment model?

If nseg is 1, the site of current injection and membrane potential will be identical. You need to read about range, range variables, and spatial discretization in NEURON. See the material cited above.