Numerical Integration in Computational Neuroscience

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sschmitt
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Numerical Integration in Computational Neuroscience

Post by sschmitt »

Hi!

"Numerical Integration in Computational Neuroscience" (https://www.neuron.yale.edu/ftp/ted/neu ... ration.pdf) explains in section "Explicit Euler method" why the explicit Euler method is not suited for stiff equations.

I'm trying to derive the number of the electrical equivalent circuit. Everything is fine, but I have a question about the coupling resistance between the compartments:

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 ra = 1.0186 meg Ohm
I get the value calculating:

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160 Ohm cm / (spine cross section area) * spine length / 2
I have two questions. Why the factor 2? And why can we neglect the total axial resistance of the soma?

Thanks in advance,

Sebastian
ted
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Re: Numerical Integration in Computational Neuroscience

Post by ted »

Why the factor 2?
Good-you're thinking for yourself. NEURON's spatial discretization uses the central difference approximation to the second spatial derivative of membrane potential. Consequently numerical integration of the discretized cable equation generates solutions at locations that correspond to segment ("compartment") centers. You might find it helpful to read the pdf or online version of this paper
https://neuron.yale.edu/neuron/publicat ... nvironment
why can we neglect the total axial resistance of the soma?
For any real neuronal morphology the soma electrically extremely compact--it is essentially isopotential for all frequencies at which the cable equation is relevant. Think about it--what is the DC or AC (at 100 Hz, 1 kHz, or 10 kHz) length constant for a structure with the size and shape of any real soma, for current flow along the longest diameter of that soma? How does that length constant compare to the anatomical length of that path?
ted
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Re: Numerical Integration in Computational Neuroscience

Post by ted »

For any real neuronal morphology the soma electrically extremely compact--it is essentially isopotential for all frequencies at which the cable equation is relevant.
You can run a simple test to prove this to yourself. Create a model cell in which the soma's morphology is specified using the pt3d syntax, e.g. a soma that is 20 um long, 15 um wide at the 0 end, 2 um wide at the 1 end. Attach an "apical" dendrite to the soma's 1 end--make it 2 um wide and maybe 1000 um long. Use the d_lambda rule to discretize the dendrite. Set the soma's nseg to 25, make all membrane passive, and set Ra to 100 ohm cm in all sections. Attach an SEClamp to soma(0.5) and apply a 50 mV step command. Plot V vs. t at soma 0.5 and at the apical dendrite's 0.5 location. Next change soma's nseg to 1 and repeat the simulation. What difference do you see?

Then return soma nseg to 25, remove the SEClamp, and attach an AlphaSynapse with tau = 2 ms, e = 0 mV, to the dendrite at about 100 um from the soma. Adjust the synapse's peak conductance to elicit a 1 mV epsp at the soma. Plot V vs. t at soma 0.5 and at the apical dendrite's 0.5 location. Next change soma's nseg to 1 and repeat the simulation. What difference do you see?
sschmitt
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Re: Numerical Integration in Computational Neuroscience

Post by sschmitt »

Hi Ted!

Thanks for you replies! I'll try to make the experiment and report back.

Best,

Sebastian
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