Physics of EEG/MEG

We consider a head to be  a volume conductor with a nonuniform anisotropic conductivity tensor tex2html_wrap_inline235 , and a magnetic permeability of free space tex2html_wrap_inline237 .

In general, electromagnetic field in the media is described by the system of Maxwell equations:


where tex2html_wrap_inline239 and tex2html_wrap_inline241 are electric and magnetic fields, J is the current density and tex2html_wrap_inline243 is the permitivity of the media.

Another important equation is the continuity equation:


where tex2html_wrap_inline245 is an electric charge density.

In neuromagnetism, frequencies are usually below 100Hz and electric and magnetic field time derivatives are typically much smaller than ohmic currents, so we can neglect the displacement current tex2html_wrap_inline247 and the non-potentiality of the tex2html_wrap_inline239 field tex2html_wrap_inline251 . Thus, below we will be using a quasi-static approximation of Maxwell's equations:




(this last equation simply states that the sum of all the currents entering the volume is zero, or Kirkoff's second law).

It is convenient to split the total current tex2html_wrap_inline253 into tex2html_wrap_inline255 - primary current, or current from the sources, and tex2html_wrap_inline257 , secondary or return current. The primary current tex2html_wrap_inline255 (sources) creates electric field which causes secondary current tex2html_wrap_inline261 . Then



Substituting this equation into continuity equation (2), we get



Since tex2html_wrap_inline265 , the electric field can be represented as a gradient of some potential


Substituting this expression into Eq.(4) we get


which is a Poisson's equation for an electric potential.

The second Maxwell's equation, together with the continuity equation


provides boundary conditions on the boundary interface between two media


Since we consider a boundary between a conducting media and an insulator (in the air tex2html_wrap_inline271 ), the boundary conditions will read


Assuming no current sources in the surface layer tex2html_wrap_inline273 . This gives us the boundary condition for the Poisson equation


on the outside surface (scalp).

There is no current outside of the media, but there still exists the tangential electric field which decays with distance. Since the surface shape is non-trivial, the solution of Eq. (6) with boundary conditions Eq. (7) can only be found numerically.

Since the divergence of a magnetic field is always zero (no magnetic monopoles exists), tex2html_wrap_inline277 , the magnetic field can be represented as a curl of some vector tex2html_wrap_inline281 , where tex2html_wrap_inline283 is called a vector potential. Substitution of this expression in the last Maxwell's equation leads to


Using vector calculus we get


Vector potential tex2html_wrap_inline283 is defined up to a gauge. Choosing the ``Coulomb gauge'' we set tex2html_wrap_inline287 =0 and simplify the last equation to


This is again a Poisson's equation, but now with a different set of boundary conditions. Since magnetic permeability of the media is the same as of the air, there is no boundary conditions exists for the magnetic field. The only physical condition will be that tex2html_wrap_inline289 , no magnetic field at infinity. In this case the Poisson's equation can be solved analytically and the solution for the vector potential is well known and given by the following integral:


where integration is over the entire volume space (currents are non-zero only inside the head!). After taking the curl we get the expression for the magnetic field tex2html_wrap_inline241


Substituting current from Eq.(3,5) into the above formula we get




Applying differentiation by parts and assuming zero conductivity outside the surface, we can rewrite the above integral as


It is clear from the last expression, that if tex2html_wrap_inline295 were constant, then the magnetic field would only be due to the primary source currents. If tex2html_wrap_inline295 can be considered piece-wise constant within certain areas, then the above integral will be non-zero only on the boundary between those areas and can easily be reduced to a sum of the surface integrals over those surfaces. In our case, though, tex2html_wrap_inline299 is anisotropic and varies continuously in space, so we have to compute the second integral explicitly.

The primary current density tex2html_wrap_inline255 can be approximated by a current dipole which is a unidirectional line element of current pumped from a ``sink'' to a ``source''. The convenient mathematical representation for the the current density tex2html_wrap_inline255 can be given through the Dirac delta function (point source)


where tex2html_wrap_inline305 is a current dipole. It can also be interpreted as tex2html_wrap_inline307 - dipole current I times the distance between the ``source'' and the ``sink'. Then the Poisson equation (6) will become



Finally, we can summarize the algorithm for the solution of EEG/MEG forward problems:

  • EEG forward problem
    • Set up current sources tex2html_wrap_inline313 ;
    • Solve tex2html_wrap_inline315 with tex2html_wrap_inline317 boundary conditions;
    • Find potentials tex2html_wrap_inline319 on the scalp.
  • MEG forward problem
    • Repeat first two steps from the above;
    • Compute conduction current tex2html_wrap_inline321
    • Compute tex2html_wrap_inline323 ;
    • Find components of tex2html_wrap_inline325 field at the desired positions.



Zhukov Leonid
Tue Aug 31 13:36:30 MDT 1999

Revised: March , 2005