A layer potential approach to inverse problems in brain imaging
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Paul Asensio
Abstract
We study the inverse source localisation problem using the electric potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electric potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents; in particular, we give details for primary currents that are assumed to be smooth vector fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used, we also solve the transmission problem of the electric potential associated with a recovered source; hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that, on the meshed surfaces, we can compute the double and single layer potentials exactly and at arbitrary points. Because we explicitly study the transmission of the electric potential in head when using magnetic data, the coupling of electric and magnetic data in the source recovery problem is made explicit and shows the advantage of using simultaneous electric and magnetic data. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-18-CE40-0035
Funding statement: The work of M. Nemaire is supported by the Agence Nationale de la Recherche (ANR) through the project ANR-18-CE40-0035.
Acknowledgements
The authors are grateful to the OPAL infrastructure from Université Côte d’Azur for providing resources and support.
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© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Stability properties for a class of inverse problems
- Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition
- Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation
- Direct numerical algorithm for calculating the heat flux at an inaccessible boundary
- The game model with multi-task for image denoising and edge extraction
- On the X-ray transform of planar symmetric tensors
- Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity
- Acquiring elastic properties of thin composite structure from vibrational testing data
- Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges
- Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces
- A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization
- Correctness and regularization of stochastic problems
- A layer potential approach to inverse problems in brain imaging
- Inverse problem for Dirac operators with two constant delays
Articles in the same Issue
- Frontmatter
- Stability properties for a class of inverse problems
- Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition
- Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation
- Direct numerical algorithm for calculating the heat flux at an inaccessible boundary
- The game model with multi-task for image denoising and edge extraction
- On the X-ray transform of planar symmetric tensors
- Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity
- Acquiring elastic properties of thin composite structure from vibrational testing data
- Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges
- Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces
- A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization
- Correctness and regularization of stochastic problems
- A layer potential approach to inverse problems in brain imaging
- Inverse problem for Dirac operators with two constant delays
