The inverse recovery in EEG is a Calderón-type inverse problem with the goal of recovering source terms and/or conductivity in layers of the human head from electroencephalographic measurements. Fundamentally, this inverse recovery seeks to solve the elliptic partial differential equation given by
∇ ⋅ ( σ ∇ u ) = S {\displaystyle \nabla \cdot (\sigma \nabla u)={\mathcal {S}}} or div ( σ grad u ) = S {\displaystyle {\hbox{div }}(\sigma {\hbox{ grad }}u)={\mathcal {S}}}
where u {\displaystyle u} is the electric potential, σ {\displaystyle \sigma } is the (possibly anisotropic) conductivity, and S {\displaystyle {\mathcal {S}}} represents primary current sources in the brain. Depending on the application, the inverse problem consists of either recovering S {\displaystyle {\mathcal {S}}} from u {\displaystyle u} (the inverse source problem) or recovering σ {\displaystyle \sigma } from u {\displaystyle u} (the inverse conductivity problem). Because the human head is highly inhomogeneous and composed of multiple layers with different conductivities, the inverse EEG problem is severely ill-posed and requires either analytical techniques or numerical approximations to obtain stable solutions (such as the finite element method). Due to the form of the governing equation being a sort of "generalization" of the Laplace-Beltrami operator, this problem has deep connections to generalized analytic function theory, heat conduction, and broader electromagnetics. In fact, the problem can be seen as solving the Poisson equation for an inhomogeneous media, which is indeed how it is derived in the EEG problem.
Problem derivation An overview on the physical foundations of the problem is given by Darbas and Lohrengel. Consider the current density J ( r ) {\displaystyle \mathbf {J} (r)} produced by neural activity,
J ( r ) = J p ( r ) + σ ( r ) E ( r ) {\displaystyle \mathbf {J} (r)=\mathbf {J} ^{p}(r)+\sigma (r)\mathbf {E} (r)}
where J p ( r ) {\displaystyle \mathbf {J} ^{p}(r)} denotes primary current and σ ( r ) E ( r ) {\displaystyle \sigma (r)\mathbf {E} (r)} the return current (composed of macroscopic conductivity and the brain's electric field). Using the quasi-static approximation of Maxwell's equations,
∇ × E = 0 and ∇ ⋅ J = 0 {\displaystyle \nabla \times \mathbf {E} =0\quad {\hbox{and}}\quad \nabla \cdot \mathbf {J} =0}
The first of the above equation implies that the electric field is path-independent and thus we may write E = − ∇ u {\displaystyle \mathbf {E} =-\nabla u} with u {\displaystyle u} the electric potential function. Then,
∇ ⋅ ( J p + σ E ) = − ∇ ⋅ ( σ E ) = ∇ ⋅ J p {\displaystyle \nabla \cdot (\mathbf {J} ^{p}+\sigma \mathbf {E} )=-\nabla \cdot (\sigma \mathbf {E} )=\nabla \cdot \mathbf {J} ^{p}}
which gives
∇ ⋅ ( σ ∇ u ) = ∇ ⋅ J p {\displaystyle \nabla \cdot (\sigma \nabla u)=\nabla \cdot \mathbf {J} ^{p}}
The primary current is modeled by Q {\displaystyle Q} pointwise sources located at some coordinate C q {\displaystyle C_{q}} with dipolar moments p q {\displaystyle p_{q}} (note that p q {\displaystyle p_{q}} is a vector quantity). Using the Dirac delta distribution,
J p = ∑ q = 1 Q p q δ C q {\displaystyle \mathbf {J} ^{p}=\sum _{q=1}^{Q}p_{q}\delta _{C_{q}}}
Thus, the source term is
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