Boundary Element Fast Multipole Method for Enhanced Modeling of Neurophysiological Recordings

Abstract

Objective: A new numerical modeling approach is proposed which provides forward-problem solutions for both noninvasive recordings (EEG/MEG) and higher-resolution intracranial recordings (iEEG).

Methods: The algorithm is our recently developed boundary element fast multipole method or BEM-FMM. It is based on the integration of the boundary element formulation in terms of surface charge density and the fast multipole method originating from its inventors. The algorithm still possesses the major advantage of the conventional BEM - high speed - but is simultaneously capable of processing a very large number of surface-based unknowns. As a result, an unprecedented spatial resolution could be achieved, which enables multiscale modeling.

Results: For non-invasive EEG/MEG, we are able to accurately solve the forward problem with approximately 1 mm anatomical resolution in the cortex within 1-2 min given several thousand cortical dipoles. Targeting high-resolution iEEG, we are able to compute, for the first time, an integrated electromagnetic response for an ensemble (2,450) of tightly packed realistic pyramidal neocortical neurons in a full-head model with 0.6 mm anatomical cortical resolution. The neuronal arbor is comprised of 5.9 M elementary 1.2 μm long dipoles. On a standard server, the computations require about 5 min.

Conclusion: Our results indicate that the BEM-FMM approach may be well suited to support numerical multiscale modeling pertinent to modern high-resolution and submillimeter iEEG.

Significance: Based on the speed and ease of implementation, this new algorithm represents a method that will greatly facilitate simulations at multi-scale across a variety of applications.

Fig. 1.

High-level flowchart of boundary element fast multipole method for EEG/MEG/iEEG modeling along with its major features.

Fig. 2.

Electric potential distribution in V over the scalp surface for nearly radial (a) and nearly tangential (b) cortical dipoles (Q = 1.8 nA · m) at the primary motor cortex (coronal plane). c), d) Solutions with all brain compartments included. e), f) Approximate solutions with only three brain compartments (scalp, inner skull, outer skull) included. Note the differences in the maximum amplitudes of the on-skin voltage in every case.

Fig. 3.

Single-dipole EEG and MEG field distributions computed with the BEM-FMM algorithm for a head model with 2.8 M facets. The dipole moment is Q = 1 nA · m. a), b) Problem geometry in two planes. c) The electric field magnitude in the coronal plane using a logarithmic scale vs 1 μV/m. d) The absolute value of dominant tangential magnetic-field component B_x in the sagittal plane using a logarithmic scale vs 1 aT. A magnetic field lacuna is seen in Fig. 3d.

Fig. 4.

EEG/MEG responses of a cortical equivalent dipole layer computed via BEM-FMM. The total area of the cortical layer is 360 mm². Current dipole moment density q₀ (current dipole moment per unit cross sectional area of the active cortex) in the source region is given by the value q₀ = 1 nA · m/mm² (Okada-Murakami constant). a) Problem geometry. b) Close-up view of the dipole layer. c) The electric potential on the skin surface using a linear scale; d) The magnitude of the total magnetic field 18 mm away from the skin surface using a linear scale. The center of the white cross coincides with projection of the geometrical center of the layer onto a transverse plane.

Fig. 5.

Construction of a layer of realistic neocortical pyramidal neurons. a,b) Neocortical pyramidal neuron ID NMO_86955 from the NeuroMorpho.Org inventory. The apical dendrite arbor is shown blue. c) Conformal cloning the PN in layer II/III with the density of 6.2 per mm². Only two cortical surfaces (WM and GM) are shown here.

Fig. 6.

a) Conformal cloning PNs in layer II/III with the density of 155 per mm². The apical dendrite arbor (5.9 M elementary dipoles) is shown blue; b) Computed total induced electric potential in the coronal plane passing through the cluster center. The log-modulus transformation scale given by Eq. (A15) of Supplement A has been used with φ₀ = 0.1 μV; c) Distribution of the magnitude of the total magnetic field in the same coronal plane, also using a logarithmic scale normalized to 10 fT; d) Distribution of the magnitude of the total electric field in the same coronal plane, also using a logarithmic scale normalized to 1 μV/m.

Fig. 7.

a) Ensemble of neurons from Fig. 6a shown in the coronal plane of the entire macroscopic head model with 2.8 M facets. The apical dendrite arbor (5.9 M elementary dipoles) is shown blue. b) Inner-skull voltage changes from 0 to 450 μV and is strongly localized. c) Magnitude of magnetic field at the inner skull surface reaches its maximum of 55 pT and is strongly localized. d) On-skin voltage changes from 0 to approximately 37 μV; the response is spread over a much wider area. e) Magnitude of magnetic field at the skin surface has the maximum of 6 pT; the response is spread over a much wider area.