Effects of reconstructed magnetic field from sparse noisy boundary measurements on localization of active neural source

Abstract

Localization of active neural source (ANS) from measurements on head surface is vital in magnetoencephalography. As neuron-generated magnetic fields are extremely weak, significant uncertainties caused by stochastic measurement interference complicate its localization. This paper presents a novel computational method based on reconstructed magnetic field from sparse noisy measurements for enhanced ANS localization by suppressing effects of unrelated noise. In this approach, the magnetic flux density (MFD) in the nearby current-free space outside the head is reconstructed from measurements through formulating the infinite series solution of the Laplace’s equation, where boundary condition (BC) integrals over the entire measurements provide “smooth” reconstructed MFD with the decrease in unrelated noise. Using a gradient-based method, reconstructed MFDs with good fidelity are selected for enhanced ANS localization. The reconstruction model, spatial interpolation of BC, parametric equivalent current dipole-based inverse estimation algorithm using reconstruction, and gradient-based selection are detailed and validated. The influences of various source depths and measurement signal-to-noise ratio levels on the estimated ANS location are analyzed numerically and compared with a traditional method (where measurements are directly used), and it was demonstrated that gradient-selected high-fidelity reconstructed data can effectively improve the accuracy of ANS localization.

Appendices

Appendix 1: Solutions to magnetic scalar potential in Ω outside the head

Solved by the separation-of-variable method in spherical coordinates, the solutions to Eq. (6c) with BCs (7a–d) are a linear combination of the spherical harmonic functions which are products of trigonometric functions and associated Legendre polynomials $P_{l}^{m} \left( {\cos \theta } \right)$ of degree l and order m:

$$\psi \left( {r,\theta ,\varphi } \right) = \sum\limits_{l = 0}^{\infty } {\sum\limits_{m = 0}^{l} {\left[ {r^{l} (a_{l}^{m} { \cos }m\varphi + b_{l}^{m} { \sin }m\varphi )P_{l}^{m} (\cos \theta ) + r^{ - (l + 1)} (c_{l}^{m} { \cos }m\varphi + d_{l}^{m} { \sin }m\varphi )P_{l}^{m} (\cos \theta )} \right]} }$$

(16)

Noting that $\psi \left| {_{r \to \infty } } \right. = 0$, $a_{l}^{m} = b_{l}^{m} = 0$, and $c_{l}^{m} = d_{l}^{m}$ are determined from the BCs:

$$\, \left[ {\begin{array}{*{20}c} {c_{l}^{m} } \\ {d_{l}^{m} } \\ \end{array} } \right] = \frac{{r_{s}^{{\left( {l + 2} \right)}} }}{{\mu_{0} \left( {l + 1} \right)}}\left( {N_{l}^{m} } \right)^{ - 2} \left[ {\begin{array}{*{20}c} {C_{l}^{m} } \\ {D_{l}^{m} } \\ \end{array} } \right]$$

(17)

where the modulus of spherical harmonics

$$\left( {N_{l}^{m} } \right)^{ - 2} = \left( {\frac{2l + 1}{{2\pi \delta_{m} }}} \right)\frac{(l - m)!}{(l + m)!}\quad {\text{with}}\;\delta_{m} = \left\{ {\begin{array}{*{20}l} {2, \, m = 0} \hfill \\ {1, \, m \ne 0} \hfill \\ \end{array} } \right.$$

(18)

and

$$\left[ {\begin{array}{*{20}c} {C_{l}^{m} } \\ {D_{l}^{m} } \\ \end{array} } \right] = \int_{0}^{2\pi } {\int_{0}^{\pi } {B_{s} \left( {\theta ,\varphi } \right)\left[ {\begin{array}{*{20}c} {\cos m\varphi } \\ {\delta_{m} \sin m\varphi } \\ \end{array} } \right]P_{l}^{m} \left( {\cos \theta } \right)\sin \theta d\theta d\varphi } }$$

(19)

Appendix 2: H-nearest neighborhood interpolation of measured data on BC surface

The second-order fitting polynomial function in Eq. (20) is used to interpolate among the sparse measured data on a three-dimensional surface:

$$B_{b} \left( {\theta ,\varphi } \right) \approx g_{0} + r_{s} {\mathbf{g}}^{T} {\mathbf{x}} + r_{s}^{2} {\mathbf{x}}^{T} {\mathbf{Gx}}$$

(20)

$${\text{where}}\quad {\mathbf{x}} = \left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sin \theta \cos \varphi } \\ {\sin \theta \sin \varphi } \\ {\cos \theta } \\ \end{array} } \right];\quad {\mathbf{g}} = \left[ {\begin{array}{*{20}c} {g_{1} } \\ {g_{2} } \\ {g_{3} } \\ \end{array} } \right]\quad {\text{and}}\;{\mathbf{G}} = \left[ {\begin{array}{*{20}c} {g_{7} } & {g_{4} } & {g_{5} } \\ 0 & {g_{8} } & {g_{6} } \\ 0 & 0 & {g_{9} } \\ \end{array} } \right].$$

The coefficients (g ₀, g, G) in Eq. (20) are determined by LS technique weighted over the H-nearest neighbor measurements, which minimizes the objective function (21) where w _i is determined by the distance δ _i between the ith nearest neighbor measurement B _si and the point being interpolated:

$${\text{Minimize}}\; \, f_{b} \left( {g_{0} ,{\mathbf{g}},{\mathbf{G}}} \right) = \sum\limits_{i = 1}^{H} {w_{i} \left\| {B_{bi} \left( {\theta_{i} ,\varphi_{i} } \right) - B_{si} } \right\|^{2} }$$

(21)

$${\text{where }}w_{i} = 1 - 3t_{i}^{2} + 2t_{i}^{3} {\text{ and }}t_{i} = \frac{{\delta_{i} }}{{{\text{Max}}\left\{ {\delta_{1} ,\delta_{2} , \ldots ,\delta_{H} } \right\}}}$$

(22,23)