We consider a head to be a volume conductor with a nonuniform anisotropic
conductivity tensor , and a magnetic
permeability of free space
In general, electromagnetic field in the media is described by the system of Maxwell
where and are electric and magnetic fields,
J is the current density and is the permitivity of the media.
Another important equation is the continuity equation:
where 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
non-potentiality of the field .
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 into
- primary current, or
current from the sources, and , secondary or return current. The
primary current (sources) creates electric field which causes
secondary current . Then
Substituting this equation into continuity equation (2), we get
Since , 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 ), the boundary conditions will read
Assuming no current sources in the surface
layer . 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), , the magnetic field can be
represented as a curl of some vector , where is called
a vector potential. Substitution of this expression in the last Maxwell's equation
Using vector calculus we get
Vector potential is defined up to a gauge. Choosing the
``Coulomb gauge'' we set =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 , 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
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
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 were constant,
then the magnetic field would only be due to the primary source
currents. If 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, is
anisotropic and varies continuously in space, so we have to compute the
second integral explicitly.
The primary current density 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
can be given through the Dirac delta function (point source)
where is a current dipole. It can also be interpreted as
- 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 ;
- Solve with boundary conditions;
- Find potentials on the scalp.
- MEG forward problem
- Repeat first two steps from the above;
- Compute conduction current
- Compute ;
- Find components of field at the desired positions.