86
Y. Zhang
Equations (5.8) and (5.9) are referred to as the Neumann boundary conditions.
Now that we have established some of the necessary parameters ruling tissue conductance in the scalp, we can return to a more in-depth discussion of our cortical
dipole itself.
5.1.2 The Current Dipole
The primary current, J p , can be used to describe the current flow characteristics of
any large group of pyramidal cells in a small cortical region that gives rise to EEG
measurements. The source of this primary current can be further modeled as a single
equivalent current dipole with two monopoles located at r source and r sink that are given
opposite signs but an equal strength of I. The dipole position r di p can therefore be
described as the midpoint between r source and r sink . The dipole moment d of a dipole
with the current I and an inter-pole distance of l is defined by an orientation unit vector
e d with the magnitude d I · l. This dipole moment can be further decomposed
into d d x e x + d y e y + d z e z , where e x , e y , e z are unit vectors along the Cartesian
axes and d x , d y , d z are magnitudes of the respective dipole moment components.
Thus, a single current dipole consists of 6 parameters: r di p —which includes the 3
position parameters—and d—which accounts for the 3 dipole moment parameters.
A potential field at position r generated by the current dipole d at position r di p in an
infinite, homogeneous volume conductor with a conductivity of σ is then calculated
using the following equation:
V
r, r di p , d
d ·
r − r di p
4πσ
r − r di p
3
(5.10)
5.1.3 The Forward Problem—Algebra
Generally speaking, the EEG forward problem aims to formulate the potential at any
arbitrary scalp position that can be generated by any current dipole (r di p , d) in the
brain:
V (r ) g
r, r di p , d
g
r, r di p , e d
· d
(5.11)
The function g
r, r di p , e d
describing the measured scalp voltage at an arbitrary
point r, which is generated by a current dipole with position r di p and moment d, is
formulated by solving the aforementioned Poisson’s equation. Following the principle of superposition, a scalp potential generated by multiple current dipole sources
according to (5.11) becomes:
Y. Zhang
Equations (5.8) and (5.9) are referred to as the Neumann boundary conditions.
Now that we have established some of the necessary parameters ruling tissue conductance in the scalp, we can return to a more in-depth discussion of our cortical
dipole itself.
5.1.2 The Current Dipole
The primary current, J p , can be used to describe the current flow characteristics of
any large group of pyramidal cells in a small cortical region that gives rise to EEG
measurements. The source of this primary current can be further modeled as a single
equivalent current dipole with two monopoles located at r source and r sink that are given
opposite signs but an equal strength of I. The dipole position r di p can therefore be
described as the midpoint between r source and r sink . The dipole moment d of a dipole
with the current I and an inter-pole distance of l is defined by an orientation unit vector
e d with the magnitude d I · l. This dipole moment can be further decomposed
into d d x e x + d y e y + d z e z , where e x , e y , e z are unit vectors along the Cartesian
axes and d x , d y , d z are magnitudes of the respective dipole moment components.
Thus, a single current dipole consists of 6 parameters: r di p —which includes the 3
position parameters—and d—which accounts for the 3 dipole moment parameters.
A potential field at position r generated by the current dipole d at position r di p in an
infinite, homogeneous volume conductor with a conductivity of σ is then calculated
using the following equation:
V
r, r di p , d
d ·
r − r di p
4πσ
r − r di p
3
(5.10)
5.1.3 The Forward Problem—Algebra
Generally speaking, the EEG forward problem aims to formulate the potential at any
arbitrary scalp position that can be generated by any current dipole (r di p , d) in the
brain:
V (r ) g
r, r di p , d
g
r, r di p , e d
· d
(5.11)
The function g
r, r di p , e d
describing the measured scalp voltage at an arbitrary
point r, which is generated by a current dipole with position r di p and moment d, is
formulated by solving the aforementioned Poisson’s equation. Following the principle of superposition, a scalp potential generated by multiple current dipole sources
according to (5.11) becomes: