Methods and System for Modeling Electromagnetic Brain Stimulation and Brain Recordings with Boundary Element Approach with Fast Multipole Acceleration

A method for making a new BEM-FMM computational approach applicable to practical electromagnetic brain modeling for clinical applications including both neurophysiological recordings and neuro stimulation, comprising:

• Solving a first integral equation, Eq. (4), and a second integral equation, Eq. (7), for surface charge density ρ(r) residing on tissue conductivity interfaces within a human head model with the new BEM-FMM computational approach,
• wherein a tissue conductivity interface is a boundary between any two distinct tissues from a plurality of cranial and intracranial tissues,
• wherein the first integral equation, Eq. (4), on the tissue conductivity interfaces is discretized into a plurality of pulse bases on a plurality of triangular facets as shown in a first discretizing equation, Eq. (8a),
• wherein the second integral equation, Eq. (7), on an electrode surface is discretized into a plurality of pulse bases on a plurality of triangular facets as shown in a second discretizing equation, Eq. (8b),
• wherein a special diagonal preconditioner for the second discretizing equation, Eq. (8b), on an electrode interface in the form of an equation, Eq. (9), as an inverse integral electric potential of a uniform charge distribution on a surface of a given facet, is applied,
• wherein the resulting system of coupled equations, Eq. (8a) and Eq. (8b), is solved using a standard fast multipole method and a plurality of iterative methods including a generalized residual method and a bi-conjugate gradient method,
• wherein a set of neighbor double surface potential integrals in the form of an Eq. (6) and an Eq. (10) are precomputed analytically,
• Inclusion and modeling at least one non-nested human head topology,
• Inclusion and modeling at least one anisotropic tissue of the head,
• Inclusion and modeling at least one external and at least one internal voltage electrode,
• Inclusion and modeling at least one external and at least one internal current electrode,
• Improving the convergence of an iterative solution using a charge conservation law,
• Restoring at least one near-surface normal electric field from a computed charge distribution without postprocessing.