The brain is an electric organ. As such, for well over a hundred years, physicists, engineers, biologists, and physicians have used electromagnetic theory to try to understand how the brain works and to diagnose and treat disease. The field of electro-neuroscience is immense with thousands of papers being published each year. In this paper, we provide physics students with an introduction to the field using a conventional model of the head that employs four concentric spheres having different conductivities to represent the scalp, skull, cerebrospinal fluid (CSF), and the brain itself. To ground the discussion, we use the specific case of non-invasive DC electrical stimulation of the brain through electrodes applied to the surface of the scalp, a procedure known as transcranial DC stimulation or tDCS. Currently, tDCS is under clinical investigation for such diseases as depression, anxiety, and chronic pain as well as to enhance the performance of athletes and the training of fighter pilots. We solve the corresponding physics problem from a charge perspective and explain why the charge distributions look the way they do using what we call the “sensing pixel” technique. This paper can introduce students to the ways in which electromagnetic theory is being applied to problems in neuroscience; in this case, the problem of how to non-invasively stimulate the brain to treat disease or improve performance.

1.
B.
Parks
, “
Research-inspired problems for electricity and magnetism
,”
Am. J. Phys.
74
(
4
),
351
354
(
2006
).
2.
M. L.
Meyer
and
M. D.
Lieberman
, “
Why people are always thinking about themselves: Medial prefrontal cortex activity during rest primes self-referential processing
,”
J. Cognitive Neurosci.
30
(
5
),
714
721
(
2018
).
3.
S.
Rush
and
D.
Driscoll
, “
Current distribution in the brain from surface electrodes
,”
Anesth. Analg.
47
(
6
),
717
723
(
1968
).
4.
S.
Rush
and
D.
Driscoll
, “
EEG electrode sensitivity—An application of reciprocity
,”
IEEE Trans. Biomed. Eng.
16
(
1
),
15
22
(
1969
).
5.
M.
Ferdjallah
et al, “
Potential and current density distributions of cranial electrotherapy stimulation (CES) in a four-concentric-spheres model
,”
IEEE Trans. Biomed. Eng.
43
(
9
),
939
943
(
1996
).
6.
P.
Miranda
et al, “
Modeling the current distribution during transcranial direct current stimulation
,”
Clin. Neurophysiol.
117
(
7
),
1623
1629
(
2006
).
7.
A.
Datta
et al, “
Transcranial current stimulation focality using disc and ring electrode configurations: FEM analysis
,”
J. Neural Eng.
5
(
2
),
163
174
(
2008
).
8.
P.
Faria
et al, “
A finite element analysis of the effect of electrode area and inter-electrode distance on the spatial distribution of the current density in tDCS
,”
J. Neural Eng.
8
(
6
),
066017
(
2011
).
9.
J.
Dmochowski
et al, “
The point spread function of the human head and its implications for transcranial current stimulation
,”
Phys. Med. Biol.
57
(
20
),
6459
6477
(
2012
).
10.
S.
Makarov
et al, “
Boundary element fast multipole method for modeling electrical brain stimulation with voltage and current electrodes
,”
J. Neural Eng.
18
(
4
),
0460d4
(
2021
).
11.
S.
Conte
and
J.
Richards
, “
The influence of the head model conductor on the source localization of auditory evoked potentials
,”
Brain Topogr.
34
(
6
),
793
812
(
2021
).
12.
M.
Bikson
et al, “
Rigor and reproducibility in research with transcranial electrical stimulation: An NIMH-sponsored workshop
,”
Brain Stimulation
11
(
3
),
465
480
(
2018
).
13.
S.
Rampersad
et al, “
Simulating transcranial direct current stimulation with a detailed anisotropic human head model
,”
IEEE Trans. Neural Syst. Rehabilitation Eng.
22
(
3
),
441
452
(
2014
).
14.
N.
Carnevale
and
M.
Hines
,
The Neuron Book
(
Cambridge U. P.
,
Cambridge
,
2006
).
15.
P.
Nahirney
and
M.
Tremblay
, “
Brain ultrastructure: Putting the pieces together
,”
Front. Cell Dev. Biol.
9
,
629503
(
2021
).
16.
The software packages of Table I are for the calculation of potentials, fields, and current flows associated with electrical brain stimulation. A variety of software packages are also available for analyzing recordings of the electric and magnetic fields produced by brain activity, i.e., electroencephalograms (EEGs) and magnetoencephalograms (MEGs), including EEGLab, MNE, Brainstorm, and FieldTrip, all of which are open source.
17.
Y.
Huang
et al, “
Realistic volumetric-approach to simulate transcranial electric stimulation-ROAST—A fully automated open-source pipeline
,”
J. Neural Eng.
16
(
5
),
056006
(
2019
).
18.
C.
Lee
et al, “
COMETS2: An advanced MATLAB toolbox for the numerical analysis of electric fields generated by transcranial direct current stimulation
,”
J. Neurosci. Methods
277
,
56
62
(
2017
).
19.
G.
Saturnino
et al, “
SimNIBS 2.1: A comprehensive pipeline for individualized electric field modelling for transcranial brain stimulation
,” in
Brain and Human Body Modeling: Computational Human Modeling at EMBC 2018
, edited by
S.
Makarov
et al (
Springer
Nature Switzerland AG, Cham, Switzerland
,
2019
), Chap. 1, pp.
3
25
.
20.
S.
Schrader
et al, “
DUNEuro—A software toolbox for forward modeling in bioelectromagnetism
,”
PloS One
16
(
6
),
e0252431
(
2021
).
21.
M.
Dannhauer
and
S.
Frisby
, “
BrainStimulator: A SCIRun5-based toolkit for modeling of transcranial electromagnetic stimulation
,” <https://www.sci.utah.edu/images/software/SCIRun/BrainStimulatorTutorial.pdf>, accessed on July 1, 2022.
22.
J.
Dmochowski
et al, “
Optimized multi-electrode stimulation increases focality and intensity at target
,”
J. Neural Eng.
8
(
4
),
046011
(
2011
).
23.
P.
Miranda
et al, “
The electric field in the cortex during transcranial current stimulation
,”
Neuroimage
70
,
48
58
(
2013
).
24.
O.
Seibt
et al, “
Computational finite element method (FEM) forward modeling workflow for transcranial direct current stimulation (tDCS) current flow on MRI-derived head: Simpleware and COMSOL Multiphysics tutorial
,” preprint bioRxiv 704940 (
2019
).
25.
Y.
Huang
et al, “
Measurements and models of electric fields in the in vivo human brain during transcranial electric stimulation
,”
eLife
6
,
e18834
(
2017
).
26.
M.
Klee
, “
Surface charges from a sensing pixel perspective
,”
Am. J. Phys.
88
(
8
),
649
660
(
2020
).
27.
M.
Zahn
,
Electromagnetic Field Theory: A Problem Solving Approach
(
Wiley
,
Hoboken, NJ
,
1979
), p.
166
.
29.
D.
van Essen
et al, “
The WU-Minn human connectome project: An overview
,”
NeuroImage
80
,
62
79
(
2013
);
J.
Elam
et al, “
The human connectome project: A retrospective
,”
ibid.
244
,
118543
(
2021
).
30.
A.
Sommerfeld
,
Electrodynamics
(
Academic
,
New York
,
1952
), pp.
125
130
; first published as Elektrodynamik – Vorlesungen über Theoretische Physik, Band 3 (Dietrich'sche Verlagsbuchhandlung, Wiesbaden, 1948).
31.
R. W.
Chabay
and
B. A.
Sherwood
,
Electric and Magnetic Interactions
(
Wiley
,
New York
,
1995
).
32.
George
Green
,
An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism
(
T. Wheelhouse
,
Nottingham, England
,
1828
).
33.
James Clerk
Maxwell
,
A Treatise on Electricity and Magnetism
, 3rd ed. (
Clarendon Press
,
Oxford
,
1891
), Vol.
I
, pp. ix,
123
.
34.
In Table II, we refer to the external charge “seeking to make” a pixel's internal charge positive or negative, rather than simply saying that the external charge “makes” the internal charge positive or negative, because through the interplay of all of the charges of the system with the pixel, the charge in a pixel is not merely that which an individual external charge would produce if acting alone, but depends on the net effect of all of the external charges. Thus, while an individual external charge seeks to make the internal charge positive or negative, it does not necessarily achieve that result because the effects of other charges in the system may be stronger and cause the internal charge to have the opposite polarity.
35.
A.
van Oosterom
and
J.
Strackee
, “
The solid angle of a plane triangle
,”
IEEE Trans. Biomed. Eng.
2
,
125
126
(
1983
).
36.
In Ref. 26, the conductor was assumed to be homogeneous, isotropic, and non-polarizable with a conductivity that did not vary with time. It was further assumed that the Drude model applied to the conductor and that inductive effects are insignificant. The same simplifying assumptions are made here for each of the head's conductive layers.
37.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/5.0085625 for supplementary materials A–D, where A derives Eqs. (2) and (3), B applies Eq. (3) to the skull–CSF interface, C contains a qualitative comparison with Ref. 10, and D contains a quantitative comparison with Ref. 4. The supplementary material also includes a Matlab® script for supplementary material D and a LTspice® script for Fig. 7.
38.
M.
Klee
, “
Biology's built-in Faraday cages
,”
Am. J. Phys.
82
(
5
),
451
459
(
2014
).
39.
The σ diff / σ sum ratio has also been referred to in the literature as an interface's “conductivity contrast.” See
S.
Makarov
et al,
Low-Frequency Electromagnetic Modeling for Electrical and Biological Systems Using MATLAB
(
Wiley
,
Hoboken, NJ
,
2016
), p.
312
.
40.
A.
Datta
et al, “
Validation of finite element model of transcranial electrical stimulation using scalp potentials: Implications for clinical dose
,”
J. Neural Eng.
10
(
3
),
036018
(
2013
).
41.
C.
Wenger
et al, “
Improving tumor treating fields treatment efficacy in patients with glioblastoma using personalized array layouts
,”
Int. J. Radiat. Oncol. Biol. Phys.
94
(
5
),
1137
1143
(
2016
).
42.
Reference 39, pp. 292–293. See also Ref. 33, p. 455, Eq. (17); Ref. 27, p. 186, Eqs. (15) and (20);
R.
Plonsey
and
R.
Collin
,
Principles and Applications of Electromagnetic Fields
(
McGraw-Hill
,
New York
,
1961
), p. 179, Eqs. (5.54a) and (5.54b); and
I. W.
McAllister
and
G. C.
Crichton
, “
Analysis of the temporal electric fields in lossy dielectric media
,”
IEEE Trans. Electr. Insul.
26
(
3
),
513
528
(
1991
).
43.
In cases where ω ε / σ 1, where ω is the angular frequency, the electrical behavior of biological systems is often modeled parametrically as the product of a time-varying factor f(t) times a steady state solution. See, for example, Ref. 39, Sec. 1.1.4;
R.
Plonsey
and
D.
Heppner
, “
Considerations of quasi-stationarity in electrophysiological systems
,”
Bull. Math. Biophys.
29
(
4
),
657
664
(
1967
);
C.
Bossetti
et al., “
Analysis of the quasi-static approximation for calculating potentials generated by neural stimulation
,”
J. Neural Eng.
5
(
1
),
44
53
(
2008
). Even for media having very high dielectric constants, the ω ε / σ ratio is small for many of the frequencies used for electrical brain stimulation. The low end of those frequencies is on the order of 1–45 Hz, which is used for transcranial AC stimulation (see The Stimulated Brain: Cognitive Enhancement Using Non–Invasive Brain Stimulation, edited by R. Kadosh (Elsevier, Amsterdam, Netherlands, 2014)), while the high end is on the order of 100–300 kHz, which is used to treat brain tumors (see Ref. 41). For these frequencies, a dielectric constant of 5,000 ε 0, and a conductivity of 0.25 S/m, the ω ε / σ ratio ranges from 1 × 10 6 to 3 × 10 1. Thus, the parametric model is justified in some cases, but not in others. Compare
G.
Gaugain
et al., “
Quasi-static approximation error of electric field analysis for transcranial current stimulation
,” preprint arXiv:2204.03555 (
2022
), which questions whether capacitive effects can be ignored in full brain models even for very low frequencies.
44.
P.
Nunez
and
R.
Srinivasan
,
Electric Fields of the Brain: The Neurophysics of EEG
, 2nd ed. (
Oxford U. P
.,
New York
,
2006
), p.
166
.
45.
R.
Plonsey
,
Bioelectric Phenomena
(
McGraw-Hill
,
New York
,
1969
), pp.
203
204
.
46.
J.
Malmivuo
and
R.
Plonsey
,
Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields
(
Oxford U. P
.,
New York
,
1995
), pp.
456
457
.
47.
Reference 44, pp. 166–169, 592–597.
48.
G.
Scharf
and
C.
Scharf
, “
Electrophysiology of living organs from first principles
,” preprint arXiv:1006.3453 (
2010
); see also, for example,
O. D.
Jefimenko
,
Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields
, 2nd ed. (
Electret Scientific Company
,
Star City, WV
,
1989
), p.
297
: “…as far as the calculation of the stationary electric field inside (or outside) a current-carrying conductor is concerned, the current as such may be completely ignored, and only the charge distribution on the surface and in the interior of the conductor needs to be taken into account. Thus the stationary fields of current-carrying conductors, just like the electrostatic fields, have electric charges as their sources.”.
49.
Some students will be familiar with the fact that charges accumulate at the locations where current is introduced into a conductor as a result of having been assigned problem 34 of Chapter 3 of Ref. 27, or having studied Example 7.2 of
D. J.
Griffiths
,
Introduction to Electrodynamics
, 4th ed. (
Pearson
,
Boston
,
2013
) or Example 9-5.1 of Ref. 48.
50.
N.
Grossman
et al, “
Noninvasive deep brain stimulation via temporally interfering electric fields
,”
Cell
169
(
6
),
1029
1041
(
2017
).
51.
M.
Vöröslakos
et al, “
Direct effects of transcranial electric stimulation on brain circuits in rats and humans
,”
Nat. Commun.
9
(
1
),
1
17
(
2018
).
52.
Kadosh, Ref. 43, Chap. 2.
53.
A.
Liu
et al, “
Immediate neurophysiological effects of transcranial electrical stimulation
,”
Nat. Commun.
9
(
1
),
1
12
(
2018
). (Despite more than 4000 publications (PubMed) on TES [Transcranial Electrical Stimulation] in the past decade, we lack a mechanistic understanding of the mechanism (or mechanisms) by which this technique produces beneficial or deleterious effects.);
A.
Fertonani
and
C.
Miniussi
, “
Transcranial electrical stimulation: What we know and do not know about mechanisms
,”
Neuroscientist
23
(
2
),
109
123
(
2017
).
54.
K.
Caulfield
and
M.
George
, “
Optimizing transcranial direct current stimulation (tDCS) electrode position, size, and distance doubles the on-target cortical electric field: Evidence from 3000 Human Connectome Project models
,” preprint bioRxiv https://doi.org/10.1101/2021.11.21.469417 (
2021
);
A.
Molero-Chamizo
et al., “
Standard non-personalized electric field modeling of twenty typical tDCS electrode configurations via the computational finite element method: Contributions and limitations of two different approaches
,”
Biology
10
(
12
),
1230
1252
(
2021
).
55.
M.
Nitsche
and
W.
Paulus
, “
Excitability changes induced in the human motor cortex by weak transcranial direct current stimulation
,”
J. Physiol.
527
(
3
),
633
639
(
2000
).
56.
S.
Grimnes
and
Ø.
Martinsen
,
Bioimpedance & Bioelectricity Basics
, 3rd ed. (
Academic Press
,
Cambridge, MA
,
2015
).
57.
Practical Guide to Transcranial Direct Current Stimulation: Principles, Procedures and Applications
, edited by
H.
Knotkova
et al (
Springer
,
New York
,
2019
);
H.
Thair
et al, “
Transcranial direct current stimulation (tDCS): A beginner's guide for design and implementation
,”
Front. Neurosci.
11
,
641
(
2017
);
A.
Woods
et al, “
A technical guide to tDCS, and related non-invasive brain stimulation tools
,”
Clin. Neurophysiol.
127
(
2
),
1031
1048
(
2016
).
58.
A.
Datta
et al, “
Gyri-precise head model of transcranial direct current stimulation: Improved spatial focality using a ring electrode versus conventional rectangular pad
,”
Brain Stimul.
2
(
4
),
201
207
(
2009
).
59.
A.
Datta
et al, “
Inter-individual variation during transcranial direct current stimulation and normalization of dose using MRI-derived computational models
,”
Front. Psychiatry
3
(
91
),
91
(
2012
).
60.
M.
Bikson
et al, “
Safety of transcranial direct current stimulation: Evidence based update 2016
,”
Brain Stimul.
9
(
5
),
641
661
(
2016
).
61.
A.
Datta
et al, “
Individualized model predicts brain current flow during transcranial direct-current stimulation treatment in responsive stroke patient
,”
Brain Stimul.
4
(
3
),
169
174
(
2011
).
62.
Z.
Gimbutas
and
L.
Greengard
, “
Simple FMM libraries for electrostatics, slow viscous flow, and frequency-domain wave propagation
,”
Commun. Comput. Phys.
18
(
2
),
516
528
(
2015
).
63.
Jean-Pascal
Lefaucheur
, “
A comprehensive database of published tDCS clinical trials (2005–2016),”
Clin. Neurophysiol.
46
(
6
),
319
398
(
2016
).
64.
For a review of various recent references, see
E.
Gianni
et al., “
tDCS randomized controlled trials in no-structural diseases: A quantitative review
,”
Sci. Rep.
11
(
1
),
1
18
(
2021
).
65.
See, for example,
R.
Fields
, “
Amping up brain function: Transcranial stimulation shows promise in speeding up learning
,”
Sci. Am.
25
(
2011
), available at http://www.scientificamerican.com/article/amping-up-brain-function/;
J.
Choe
et al, “
Transcranial direct current stimulation modulates neuronal activity and learning in pilot training
,”
Front. Human Neurosci.
10
(
34
),
1
25
(
2016
);
E.
Strickland
, “
A new kind of juice
,”
IEEE Spectrum
53
,
34
40
(
2016
);
A.
Kamali
et al, “
Transcranial direct current stimulation to enhance athletic performance outcome in experienced bodybuilders
,”
PloS One
14
(
8
),
1
20
(
2019
).
66.
Efforts to calculate the potentials produced by active neurons typically use, a potential-centric approach. A charge-first approach can be found in
S.
Makarov
et al, “
Boundary element fast multipole method for enhanced modeling of neurophysiological recordings
,”
IEEE Trans. Biomed. Eng.
68
(
1
),
308
318
(
2021
).
67.
G.
Buzsáki
et al, “
The origin of extracellular fields and currents-EEG, ECoG, LFP and spikes
,”
Nat. Rev. Neurosci.
13
(
6
),
407
420
(
2012
).
68.
Alliance for Advancing Bioelectronic Medicine
, “
Building a bioelectronic medicine movement 2019: Insights from leaders in industry, academia, and research
,”
Bioelectronic Med.
6
,
1
11
(
2020
);
see also
L.
Peeples
, “
Core concept: The rise of bioelectric medicine sparks interest among researchers, patients, and industry
,”
Proc. Natl. Acad. Sci.
116
(
49
),
24379
24382
(
2019
);
A.
Park
, “
Why it's time to take electrified medicine seriously
,” TIME, October 24,
2019
, available at https://time.com/5709245/bioelectronic-medicine-treatments/.

Supplementary Material

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