Coupling the Output of BEM-FMM or FEM Modeling of Brain Stimulation to The Underlying Physics
DOI: Pending DOI
Synopsis
Mathematical modeling of electrical brain stimulation is sufficiently complex as to be a
“black box.” There is little to no direct coupling between modeling output and the underlying
physics. Interpretations of the output therefore tend to be descriptive as opposed to analytical.
The missing coupling is provided by the Sensing Pixel Technique from the physics literature.
Using basic laws of physics, the technique links the electrical charge at any location of interest
to the charges that surround the location. Using four Rules, it provides, without any math,
intuitive interpretations of charge distributions such as that which forms on the CSF/grey matter interface during tES. The Rules apply to bioelectricity in general and facilitate predictions from mathematical simulations.
Background
Mathematical modeling of electrical brain stimulation is a highly developed technology.
Leading software uses either the finite element method (FEM)1,2 or the boundary element
method in combination with the fast multipole method. (BEM-FMM).3,4
Each method uses complex mathematics (different in each case) to solve a matrix equation
with typically millions of entries. The end result is a “black box” with little to no direct coupling between the method’s outputs (e.g., pseudo-color plots of electric field strengths) and the underlying physics. Interpretations of the outputs therefore tend to be descriptive as opposed to analytical, thus reducing the predictive power of the simulations.
Methods
We address this lack of a coupling between modeling output and the underlying physics by
switching our focus from potentials and electric fields (the classic approach) to electrical charges.
Electrical charges accumulate on stimulating electrodes and conductivity interfaces such as the
CSF/grey matter interface during electrical brain stimulation. Importantly, charge distributions
contain the same information as potential/electric field distributions and thus nothing is lost
by changing the focus to charges.
As we will show, when the modeling output is in the form of a charge distribution, direct
coupling between the output and the underlying physics becomes possible. The recently developed “Sensing Pixel Technique” from the physics literature provides the coupling.5,6 Using Coulomb’s law, Ohm’s law, and charge conservation – the basic laws of undergraduate physics – the Sensing Pixel Technique links the charge at any location of interest to the charges
which surround the location. In a nutshell, Sensing Pixels “sense” the charges in their surroundings and adjust their charge to reach steady state, that steady state charge being the charge at the location of interest.
We use the specific problem of tES with a bipolar electrode montage to illustrate how
the technique works. We calculate the charge distribution on the CSF/grey matter interface during tES using the Makarov lab’s BEM-FMM software and then interpret the output of the simulation using Sensing Pixels.
Results
A Sensing Pixel is a modeling construct (not a physical sensor) which one imagines has been
placed at a location of interest on a conductivity interface to understand what is happening
electrically at that location (Fig. 1A).
Figure 1B shows a Sensing Pixel “sensing” a charge qj. The Sensing Pixel has two identical
major faces: a Sσhgh face in a medium where the conductivity is higher (σhgh) and a Sσlow face
in a medium where the conductivity is lower (σlow). The pixel is three-dimensional having side
faces (shown exaggerated in Fig. 1B), which connect the major faces to form a closed volume.
As such, the pixel can contain internal charge. While the side faces have a finite thickness,
they are sufficiently short so that significant amounts of current only enter and leave the pixel
through its major faces.
In accordance with Coulomb’s law, the internal charge (qint in Fig. 1B) produces electric
fields at the major faces, the normal components of which cause current flows across the faces
in accordance with Ohm’s law. External charges such as qj also cause current flows across the
faces, again in accordance with Coulomb’s law and Ohm’s law. The external charges can be
at sources/sinks or they can be charges at other locations on the same conductivity interface
on which the Sensing Pixel is located or on other conductivity interfaces in the system.
Central to the Sensing Pixel Technique is the fact that the Sensing Pixel’s internal charge
qint can have a value which causes the major-face current flows to be in balance. That is, the
internal charge can have a value which puts the pixel into steady state with the charge entering
the pixel equaling the charge leaving, thus satisfying the principle of charge conservation.
The steady state value of qint is the surface charge at the location of the Sensing Pixel.
As detailed in Refs. 5 and 6, the Sensing Pixel Technique can be used quantitatively to
calculate qint values using the solid angles of the pixel as seen from the external charges (Ωj in
Fig. 1B). As further shown in Ref. 6, the sign of qint determines the direction of current flow
across the pixel with +qint corresponding to flow from σhgh to σlow and −qint to flow from σlow
to σhgh.
The Sensing Pixel Technique can also be used qualitatively without any math by applying
the following Rules. We take this qualitative approach next for the yellow and green Sensing
Pixels of Fig. 1C:
Rule qj Sign qj Location Effect on qint
1 pos on σhgh side pos
2 pos on σlow side neg
3 neg on σhgh side neg
4 neg on σlow side pos
Fig. 1C plots surface charge polarities calculated using the Makarov BEM-FMM software
for the CSF/grey matter interface during bipolar tES. Red areas are + (inward current) and
blue areas − (outward current). We assume that the charges on the anode (red) and cathode
(blue) are dominant and thus for our purposes here, the Rules are only applied to their charges.
The yellow Sensing Pixel illustrates cooperation between + charges at the anode and −
charges at the cathode. Specifically, the anode’s + charges are on the Sensing Pixel’s σhgh
side and thus the charges and the pixel have a Rule 1 relationship. As to the cathode, its −
charges are on the Sensing Pixel’s σlow side and thus the charges and the pixel have a Rule 4
relationship. Rules 1 and 4 each seek to make the Sensing Pixel + and thus the yellow Sensing
Pixel is in a red area.
The green Sensing Pixel also illustrates cooperation, but now between Rules 2 and 3, the
− charges of the cathode being on the σhgh side of the Sensing Pixel and the + charges of the
anode being on the σlow side. Rules 2 and 3 both seek to make the Sensing Pixel’s internal
charge − and so the green Sensing Pixel is in a blue area.
Discussion
The cooperation between pairs of Rules where both rules seek to make the Sensing Pixel
have the same sign charge is often the reason that outward currents (blue −) can be found close
to an anode and inward currents (red +) close to a cathode. Feedback effects between charges
on different parts of the CSF/grey matter interface (e.g., feedback between charges on sulci
walls) can also play a role, as can surface charges on other conductivity interfaces, thus leading
to competitive Rule combinations instead of cooperative ones. That said, for a back-of-the-envelope understanding of what happens when the convoluted surface of the CSF/grey matter interface is subjected to electrical stimulation, considering just cooperation between anode and cathode charges is a good place to start.
While it is most convenient to use BEM-FMM software that directly calculates charge
distributions, charge distributions can also be obtained using FEM by transforming its output
from a potential distribution into a charge distribution using approaches like those already in
use to transform potential distributions into electric field distributions.
With its basis in fundamental physics principles, the Sensing Pixel Technique is generally
applicable whenever current flows through a medium having one or more conductivity interfaces.
It can serve as an interpretive tool for bioelectricity simulations in general whether involving
applied currents or currents generated by endogenous biological sources thereby facilitating
extrapolation from existing simulations to new situations.
Disclosures
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The author has been issued patents and has a pending patent application relating to electrophysiology and neural activity. The author has no other conflicts of interest to disclose.
References
[1] G. B. Saturnino, O. Puonti, J. D. Nielsen, D. Antonenko, K. H. Madsen, A. Thielscher,
Simnibs 2.1: a comprehensive pipeline for individualized electric field modelling for transcranial brain stimulation, Brain and human body modeling: computational human modeling
at EMBC 2018 (2019) 3–25.
[2] Y. Huang, A. Datta, M. Bikson, L. C. Parra, Realistic volumetric-approach to simulate
transcranial electric stimulation—roast—a fully automated open-source pipeline, Journal of
neural engineering 16 (2019) 056006.
[3] S. N. Makarov, G. M. Noetscher, T. Raij, A. Nummenmaa, A quasi-static boundary element
approach with fast multipole acceleration for high-resolution bioelectromagnetic models,
IEEE transactions on biomedical engineering 65 (2018) 2675–2683.
[4] S. N. Makarov, L. Golestanirad, W. A. Wartman, B. T. Nguyen, G. M. Noetscher, J. P.
Ahveninen, K. Fujimoto, K. Weise, A. R. Nummenmaa, Boundary element fast multipole
method for modeling electrical brain stimulation with voltage and current electrodes, Journal
of neural engineering 18 (2021) 0460d4
[5] M. M. Klee, Surface charges from a sensing pixel perspective, American Journal of Physics
88 (2020) 649–660
[6] M. M. Klee, A charge analysis of non-invasive electrical brain stimulation, American Journal
of Physics 91 (2023) 264–277.