Custom extracellular plane, Poisson equation and dft..

[mgiugliano]

I need to implement and simulate a network of multi-compartmental neurons, "sitting" on a conductive film (e.g. a giant substrate metallic/non-metallic microelectrode).

Mathematically I know I need to solve a 2D Poisson equation, specifying ad-hoc current sources and sinks (not uniformly distributed in space but assigned from the user). In other words each neuron is not adhering completely to the substrate but "here and there".

I know that one effective way to solve the 2D Poisson equation is by using the spectral methods (i.e. employing the DFT), which is supposed to speed-up enormously the computation.. but I assume it will be extremely tough coding all together with the DFT transform and anti-transform in a *.mod file.

Is it feasible in NEURON ?

Is there any suggestion ? I would appreciate any contribution.

Thanks in advance,

yours sincerely

Michele

[csh]

Hi,

I know that one effective way to solve the 2D Poisson equation is by using the spectral methods (i.e. employing the DFT), which is supposed to speed-up enormously the computation.. but I assume it will be extremely tough coding all together with the DFT transform and anti-transform in a *.mod file.

Is it feasible in NEURON ?

As far as the DFT is concerned, you can compute a one-dimensional FFT by calling fft(), a member function of Vector. This is described here:
http://www.neuron.yale.edu/neuron/stati ... 2.html#fft
Mind that if your Vector's size is not a power of 2, it will be zero-padded to the next higher power of 2. This may introduce spurious signals to the spectrum, so you might want to interpolate your data to the next power of 2 before the fft (for details, see Bach & Meigen, Doc Ophthalmol. 1999;99(1):69-82).
I'm very sorry I can't help you a lot with the other part of your question concerning the Poisson equation. My knowledge about that is restricted to what I've read in the "Numerical Recipes", chapter 19.4, available online from http://www.library.cornell.edu/nr/cbookcpdf.html .
A solution using the (pure) Fourier Transform method might be done entirely from hoc, since you will only need the Fourier transform and its inverse. However, as far as I've understood, the most efficient way to solve your equation is the FACR(l) method. I guess it would be a lot of work to integrate this into NEURON. I've not been able to find a C/C++ implementation that might be easily compiled into NEURON. However, there are a lot of Fortran implementations (one of which is given in this article: Christiansen & Hockney, Comp Phys Comm. 1971,2(3):139-156). You might integrate them into NEURON using a Fortran-to-C translator (f2c), as it has been done with praxis().
Hope I could help,
Christoph

[ted]

I need to implement and simulate a network of multi-compartmental neurons, "sitting" on a conductive film (e.g. a giant substrate metallic/non-metallic microelectrode).

Can be done in the time domain. Easiest f you can assume that the neurons affect the
field, but the field does not affect the neurons. In broad outline:
1. Use NEURON to run a simulation of the model net, recording transmembrane current
at each internal node (i.e. the locatons along each section that are associated with area).
This is the time-consuming step.
2. Afterwards, use the recorded currents to compute the field. Take the following into
account:
--location of each node and the surface area associated with it
--geometry and conductivity of the extracellular medium (does not need to be
anisotropic)
This step is fast if the extracellular medium is purely ohmic; if not, numeric integration
is necessary, or convolution of the individual (recorded current*area) with the impulse
response of the extracellular medium at the xyz coords of each internal node.

This discussion thread
Other questions / extracellular potentials
https://www.neuron.yale.edu/phpBB2/viewtopic.php?t=230
has some more information on this topic, plus a link to a downloadable working
implementation of this approach.

Gold et al. used an approach but treats the nodal currents as line sources rather than
point sources.
Gold, C., Henze, D.A., Koch, C., and Buzsaki, G.
On the origin of the extracellular action potential waveform: A modeling study.
Journal of Neurophysiology 95:3113-3128, 2006.
Whether line source gives more accurate results than the point source approach is yet
to be determined. For one thing, the line source approximation is associated with its own
geometrical artifacts because of the irregular shape of neurites.

If the field can affect the neurons, things are much tougher. Some have used NEURON
in tandem with a general purpose field simulator, such as Gauss--both tools advancing
their solutions in time, one step at a time, passing results back and forth.