5 EEG Source Imaging and Multimodal Neuroimaging
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neuronal membrane, while the volume current is the Ohmic return current caused
from the primary current that completes the circuit.
J J p + J v
(5.1)
The electric field resulting from the volume current is given by Ohm’s law as
follows:
J v σ E
(5.2)
And the relationship between the electric field and potential field is given, under
quasi-static conditions (∇ × E 0), is given as:
E −∇(V )
(5.3)
Thus, from (5.1), (5.2), and (5.3), the current density generated by neuronal activity is:
J J p − σ ∇(V )
( 5 . 4 )
Taking the divergence of (5.4), we obtain:
∇ · J ∇ · J p − ∇ · (σ ∇(V ))
(5.5)
Considering that ∇ · J 0, (5.5) becomes the Poisson’s equation:
∇ · J p ∇ · (σ ∇(V ))
(5.6)
With the solution for this equation derived as [54]:
V
1
4πσ
v
J p · ∇
1
r
dv
(5.7)
As the current propagates throughout different brain tissue compartments, two
characteristic behaviors of electrical charge at tissue boundaries are established: (i)
all charge leaving one compartment (with a conductivity of σ 1 ) must enter the second
compartment (assigned a conductivity value of σ 2 ):
σ 1 ∇(V 1 ) n σ 2 ∇(V 2 ) n
(5.8)
where
n is the normal vector at the boundary of the interface; and (ii) no current
leaves the outer surface interfacing with air:
σ outer ∇(V outer ) n 0
(5.9)
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neuronal membrane, while the volume current is the Ohmic return current caused
from the primary current that completes the circuit.
J J p + J v
(5.1)
The electric field resulting from the volume current is given by Ohm’s law as
follows:
J v σ E
(5.2)
And the relationship between the electric field and potential field is given, under
quasi-static conditions (∇ × E 0), is given as:
E −∇(V )
(5.3)
Thus, from (5.1), (5.2), and (5.3), the current density generated by neuronal activity is:
J J p − σ ∇(V )
( 5 . 4 )
Taking the divergence of (5.4), we obtain:
∇ · J ∇ · J p − ∇ · (σ ∇(V ))
(5.5)
Considering that ∇ · J 0, (5.5) becomes the Poisson’s equation:
∇ · J p ∇ · (σ ∇(V ))
(5.6)
With the solution for this equation derived as [54]:
V
1
4πσ
v
J p · ∇
1
r
dv
(5.7)
As the current propagates throughout different brain tissue compartments, two
characteristic behaviors of electrical charge at tissue boundaries are established: (i)
all charge leaving one compartment (with a conductivity of σ 1 ) must enter the second
compartment (assigned a conductivity value of σ 2 ):
σ 1 ∇(V 1 ) n σ 2 ∇(V 2 ) n
(5.8)
where
n is the normal vector at the boundary of the interface; and (ii) no current
leaves the outer surface interfacing with air:
σ outer ∇(V outer ) n 0
(5.9)