Computing the extracellular resistivity

[Sherif]

Hello,

I am trying to use Neuron to apply extracellular stimulation pulses to a realistic 3D neuron model. For the assumption that the extracelular field is isotropic (i.e. resistivity is the same in all directions), the relation for the transfer resistance (resistance between the stimulating electrode and each compartment in the model) is:

Transfer Resistance = (rho/4/PI)*(1/r)
where rho is the extracellular resistivity
r is the radial distance between the electrode and the compartment

I am trying to model a non-isotropic medium in which the extracellular resistivity depends on the direction, I failed to find the relation that describes this situation in any book or reference. I did some quick calculations and got the following relationship:

Transfer Resistance = (1/4/PI)
* (1/((r_x/rho_x)^2 + (r_y/rho_y)^2 + (r_z/rho_z)^2)^0.5)

I don't know if it is the correct relation or not.

Thanks

[ted]

According to Warren Grill (personal communication)--
The potential in an anistropic medium was solved by Nicholson:
Nicholson PW (1967) Experimental models for current conduction in an
anisotropic medium. IEEE Transactions on Biomedical Engineering 14:55-56.
The potential is given by
V(x,y,z) = I/(4*pi)*[sy*sz*x^2 + sx*sz*y^2 + sx*sy*z^2]^(-1/2)
where
sx, sy and sz are the conductivities along the three coordinate axes
and I is the current delivered by a source at 0,0,0.

My comments:
Be careful about units when using this formula. No scale factors are necessary with
voltage in mV, current in mA (a reasonable choice for an extracellular stimulus current),
distance in cm, and conductivity in S/cm. However, in NEURON, distance is in um, so
the RHS would have to be multiplied by a factor of 10^4.