Obviously, this is a complex question, and a little hard to answer as I'm not completely clear on what your actual goal is, however, I would strongly suggest looking into the NEURON package. I know there are even python packages to working with the python abstraction of NEURON that basically do what you are thinking.

If memory serves, the extracellular mechanism wont allow to calculate extracellular voltages arbitrarily far away from the neuron, but only in two excellular "shells".

However, more generally, to deal with completely arbitrary shapes, breaking them up into cubes (or cylinders) sounds completely reasonable, i.e. discretization. Exactly how the resolution of your cubes would need to compare to the shape is a bit hard to say. I would assume that for all reasonable shapes, that with increasingly fine resolution you could get arbitrarily close to the "real" solution. I would guess start with something that you can calculate a closed form solution for (say a cylinder, a sphere and a cone) and then use a discretization approach and see how fine you need to be.

As far as calculating conductances of solid bodies, I would have thought some integral calculus could get you there a lot of the time.

Dear Temenuga,

I am not so familiar with the numerical calculation, but your question reminds me a classical variational problem. 

If we assume the isotropic conductivity, the electric potential inside the conductor is harmonic function which satisfies Laplace equation. 

The variational approach on harmonic function has been intensively studied, and many literature are available. Especially, the principle of least energy dissipation seems to me directly applicable to your question.  

You have only to prepare appropriate trial functions according to the boundary condition imposed at the conductor surfaces, and estimate the overall conductance under the constant current condition.

The obtained value gives the upper limit of the conductance.

Using the reciprocal theorem on variational problems (between maximization problem  and minimization problem), you can also obtain the lower limit of the conductance by another variational problem under the constant voltage condition.

This is my favorite approach, and I hope it is helpful even when you have to try numerical calculation in the next step.

Sincerely,

Ryotaro

I was asked to start this discussion by a colleague of mine, which had problems creating a ResearchGate account. Now, fortunately, this colleague finally has his RG account, so please put all your questions for discussion to him - Gergo Vassilev.  Regards, Temenuga.

Thank you all.

About the precision- Single precision would be sufficient. The current state of the application, that I am developing, uses 32 bit integer, but will switch later to single precision floating point. The 22 bit (4 million) mantissa and the 7 bit exponent (10 powered from -45 to +38) of the single data type is far more than enough. The modern performance GPUs can have 2Teraflops per card, while the low end are above 60 Gigaflops. For double they suffer penalty from factor 3-4 for high end to complete inability to calculate double for the low end.

As my application uses brute force (calculating each node/cell from the 4 direct neighbors) GPU multiprocessor calculation is currently not applicable and CPU will be used instead (almost all modern CPUs are with 64 bit bus thus enabling also the usage of double without penalty).

Anyway single is quite enough and far beyond the accuracy that can be achieved measuring real objects and 1 GB (500*500*500*8byte) is also acceptable for most modern PCs.

About the first question- as the ionic conductivity (while using blocking electrodes as this is usually used for solids) will lead to time dependency, there is also possibility, that metal clusters are also formed or dissolved (silver containing chalcogenide glasses). Complicating the complicated/advanced math formulas with addition of time dependency and probabilities/fluctuations will be too much for me, so I will try to stick with the simplest numerical formulas and passing the results from one formula to another.

In my opinion, it would be easiest to calculate the "real" volume potentials (currents, resistivities, etc.) as dimensionless and then from data for cube to recalculate to real values.