B ainField-Ope a o : A Physics-In o med Neu al
Ope a o F amewo k
o Bioelec omagne ic B ain Field Modeling Unde
Ex e nal S imula ion
Aulia Oc a iani
Email: [email p o ec ed]
Gi Hub: h ps://gi hub.com/aoc a ia
Abs ac —Non-in asi e b ain s imula ion echniques such as
ansc anial di ec cu en s imula ion ( DCS) gene a e weak
elec ical ields in he b ain whose dis ibu ion depends c i i-
cally on issue conduc i i y, geome y, and elec ode placemen .
Accu a ely modeling hese ields equi es sol ing pa ial di e -
en ial equa ions (PDEs) such as he Poisson equa ion, which
becomes compu a ionally expensi e o la ge pa ame e sweeps
o op imiza ion asks. To add ess his, we in oduce B ainField-
Ope a o , a physics-in o med neu al ope a o amewo k o as
app oxima ion o b ain elec ic ields. The sys em in eg a es
(1) a laye ed biophysical head model, (2) a 2D Poisson PDE
sol e , (3) andomized elec ode con igu a ions, and (4) a Fou ie
Neu al Ope a o (FNO)-based su oga e capable o mapping
conduc i i y and elec ode masks o po en ial ields. Resul s
show ha he su oga e achie es a mean squa ed e o below
1.2×10−4and ela i e L2 e o unde 10%. This hyb id PDE–
ML amewo k enables apid ield p edic ion and p o ides a
ounda ion o u u e compu a ional neu o echnology pipelines.
The amewo k holds p omise o accele a ing design and op i-
miza ion o s imula ion p o ocols in clinical and esea ch se ings.
I. INTRODUCTION
Non-in asi e b ain s imula ion (NIBS) modali ies such as
ansc anial di ec cu en s imula ion ( DCS) and ansc anial
al e na ing cu en s imula ion ( ACS) ha e gained conside -
able a en ion o hei abili y o modula e neu al exci abil-
i y and plas ici y h ough weak elec ic cu en s applied ia
scalp elec odes [2], [3]. P ecise knowledge o he spa ial
dis ibu ion and magni ude o induced elec ic ields wi hin
b ain issue is c i ical o unde s anding s imula ion e icacy
and ailo ing in e en ion s a egies.
T adi ional app oaches ely on ini e elemen me hod
(FEM) simula ions o Poisson’s equa ion using high- esolu ion
ana omically ealis ic head models o es ima e ields [4], [5].
Despi e hei accu acy, hese nume ical me hods a e com-
pu a ionally in ensi e, especially when e alua ing nume ous
elec ode mon ages, conduc i i y a ia ions, o pa ien -speci ic
ana omies, limi ing hei p ac ical applica ion in clinical wo k-
lows o adap i e s imula ion pa adigms.
Recen ad ances in machine lea ning, pa icula ly he de-
elopmen o neu al ope a o s such as he Fou ie Neu al
Ope a o (FNO) [1], [9], o e a scalable al e na i e by lea ning
a di ec mapping om inpu physical pa ame e s (conduc i i y,
elec ode po en ials) o he co esponding solu ion o PDEs.
Such su oga es enable o de s-o -magni ude accele a ion in
in e ence ime while main aining solu ion ideli y, opening
oppo uni ies o eal- ime modeling and closed-loop con ol.
This wo k p esen s B ainField-Ope a o , a physics-
in o med neu al ope a o model ailo ed o apid es ima ion o
DCS-induced elec ic po en ials and ields in a simpli ied wo-
dimensional laye ed head model. We de ail he model a chi-
ec u e, da ase gene a ion, aining p ocedu es, and alida ion
p o ocols, ollowed by comp ehensi e quali a i e and quan-
i a i e esul s. The amewo k’s ex ensibili y owa d mo e
complex 3D ana omies and empo al s imula ion pa e ns is
also discussed.
II. RELATED WORK
A. Compu a ional Modeling o DCS
Nume ical simula ion amewo ks such as SimNIBS [4]
and ROAST [5] le e age ini e elemen modeling on MRI-
de i ed head meshes o es ima e elec ic ield dis ibu ions
induced by DCS. These ools accoun o issue he e ogenei y
and ealis ic elec ode geome y bu o en equi e signi ican
compu a ional esou ces and expe ise o p ep ocessing and
pa ame e uning [2].
B. Su oga e Modeling in Neu o echnology
Su oga e app oaches including polynomial chaos expan-
sions [6], Gaussian p ocess eg ession [7], and educed-o de
models ha e been p oposed o mi iga e compu a ional cos .
Howe e , hei scalabili y o high-dimensional and nonlinea
pa ame e spaces emains limi ed [7].
C. Neu al Ope a o s o PDE Su oga e Modeling
Neu al ope a o s, including FNO [1], DeepONe [8], and
o he a chi ec u es, ha e demons a ed supe io gene aliza-
ion in lea ning mappings be ween unc ion spaces. FNO’s
spec al app oach e ec i ely cap u es long- ange dependencies
inhe en in PDE solu ions, making i highly sui able o
bioelec omagne ic modeling [9].
III. BIOPHYSICAL AND MATHEMATICAL MODEL
A. Laye ed Head Geome y
A compu a ionally ac able wo-dimensional, laye ed head
model is conside ed, consis ing o he ollowing issue com-
pa men s: b ain (σ= 0.33 S/m), skull (σ= 0.015 S/m),
and scalp (σ= 0.43 S/m). These conduc i i y alues e lec
a e age epo ed alues om li e a u e [2], [10].
B. Go e ning Pa ial Di e en ial Equa ion
Unde he quasi-s a ic app oxima ion o bioelec omagne ic
ields, he elec ic po en ial V(x, y)sa is ies he ellip ic PDE:
∇ · σ(x, y)∇V(x, y)= 0,(1)
wi h Di ichle bounda y condi ions applied a elec ode loca-
ions o simula e su ace po en ials [2], [4].
C. Compu a ion o Elec ic Field
The elec ic ield componen s a e e alua ed as he nega i e
g adien o he po en ial:
Ex=−∂V
∂x , Ey=−∂V
∂y ,
p o iding ec o ield in o ma ion c i ical o in e p e ing
s imula ion e ec s.
IV. DATASET GENERATION PIPELINE
A da ase o 200 samples was gene a ed by sol ing Eq. (1)
ona64×64 spa ial g id wi h andomized elec ode con igu a-
ions placed along he scalp bounda y. Each sample con ains:
•Conduc i i y map σ(x, y)encoding issue p ope ies
•Elec ode po en ial map ep esen ing bounda y condi ions
•Resul an PDE solu ion V(x, y)
•De i ed elec ic ield componen s (Ex, Ey)
This da ase cap u es a iabili y in elec ode placemen and
conduc i i y pa e ns, acili a ing obus su oga e aining.
V. NEURAL OPERATOR SURROGATE
A. Fou ie Neu al Ope a o (FNO)
The su oga e model employs he Fou ie Neu al Ope a o
[1], which lea ns an ope a o Gmapping inpu unc ions
(conduc i i y and elec ode map) o he PDE solu ion V:
G: (σ, Velec)7→ V.
Each FNO laye upda es ea u e ep esen a ions ia a com-
bina ion o local linea ans o ms and global con olu ion in
he spec al domain:
k+1(x)=W k(x)+F−1R(F( k)),
whe e Fdeno es he Fou ie ans o m and Ris he lea ned
spec al ke nel.
TABLE I
FNO ARCHITECTURE HYPERPARAMETERS
Pa ame e Value
Laye s 4
Modes (x,y) (16,16)
Wid h 64
Inpu Channels 2
Ou pu Channels 1
Ac i a ion GELU
Op imize Adam
Epochs 50
Ba ch Size 8
Lea ning Ra e 10−3
B. A chi ec u e De ails
The a chi ec u e comp ises ou FNO laye s wi h (16,16)
Fou ie modes and ea u e wid h 64. Inpu s ha e wo channels
(conduc i i y and elec ode map), while ou pu is a single-
channel p edic ed po en ial map. GELU ac i a ion and Adam
op imize a e employed. T aining was conduc ed o 50 epochs
wi h ba ch size 8.
C. Loss Func ion
The aining objec i e minimizes he mean squa ed e o :
L=∥Vp ed −V ue∥2
2,
which en o ces accu acy in p edic ed po en ial ields.
VI. EXPERIMENTAL SETUP
Expe imen s we e pe o med on a MacBook P o wi h Apple
M2 chip, u ilizing PyTo ch o model implemen a ion and
aining. An 80/20 ain- alida ion spli was used o moni o
gene aliza ion.
VII. RESULTS
A. Quali a i e Visualiza ion
Figu e 1 illus a es a ep esen a i e example o g ound u h
PDE solu ion, FNO p edic ion, and hei di e ence hea map.
The su oga e accu a ely ep oduces smoo h ield pa e ns and
cap u es key spa ial a ia ions.
Fig. 1. Le : G ound u h PDE solu ion. Middle: FNO p edic ion. Righ :
E o hea map.
B. Quan i a i e Me ics
Table II summa izes pe o mance me ics, showing low
mean squa ed e o and ela i e L2 e o below 10%. In e ence
imes a e on he o de o milliseconds o e ing a speedup o
app oxima ely 1000×compa ed wi h nume ical sol e s.
TABLE II
SURROGATE PERFORMANCE
Me ic Value
MSE (lowe is be e ) 1.2×10−4
Rela i e L2 e o 9.5%
In e ence ime 3.4ms
Speedup s PDE sol e ∼1000×
VIII. DISCUSSION
The B ainField-Ope a o su oga e accu a ely app oxima es
PDE solu ions, pa icula ly in b ain egions away om elec-
odes, whe e he elec ic ield g adien s a e smoo he . Highe
e o nea elec ode con ac s is expec ed due o s eep po en ial
g adien s and disc e iza ion limi a ions.
Key s eng hs o he amewo k include:
•Fas in e ence, acili a ing apid e alua ion sui able o
eal- ime applica ions and op imiza ion.
•Gene alizabili y o a bi a y elec ode placemen s wi hin
he modeled domain.
•Ex ensibili y o inco po a e aniso opic conduc i i ies,
3D ana omies, and iche da ase s om MRI sou ces.
IX. LIMITATIONS
Se e al simpli ying assump ions a e no ed:
•Limi a ion o 2D simula ion domain es ic s ana omical
ideli y.
•Omission o aniso opic whi e ma e conduc i i y p o-
iles.
•Simpli ied homogeneous skull ep esen a ion.
•Simpli ied elec ode geome y wi hou de ailed shape
modeling.
X. FUTURE WORK
Fu he expansions include:
•Ex ending su oga e lea ning o 3D FEM models com-
pa ible wi h SimNIBS.
•Neu al ope a o modeling o ime- a ying ( ACS) s imu-
la ion ields.
•Inclusion o di e en iable PDE sol e s o in e se p ob-
lem solu ion and pa ame e es ima ion.
•Inco po a ion o Bayesian neu al ope a o s o unce -
ain y quan i ica ion.
•In eg a ion wi h MRI-de i ed conduc i i y maps o pe -
sonalized modeling.
XI. CONCLUSION
This wo k in oduces B ainField-Ope a o , a physics-
in o med Fou ie Neu al Ope a o amewo k o apid and
accu a e app oxima ion o DCS-induced b ain elec ic ields.
By combining adi ional PDE modeling wi h neu al ope a o
su oga es, we achie e signi ican compu a ional speedups
while main aining high accu acy. This app oach o e s a scal-
able pa hway owa d eal- ime neu os imula ion planning and
compu a ional neu oscience applica ions.
ACKNOWLEDGMENTS
The au ho hanks he open-sou ce neu al ope a o commu-
ni y and esea che s in bioelec omagne ic modeling o hei
ounda ional con ibu ions.
REFERENCES
[1] Z. Li e al., “Fou ie Neu al Ope a o o Pa ame ic Pa ial Di e en ial
Equa ions,” ICLR, 2021.
[2] M. Rahman e al., “Modeling Elec ic Fields in he Human B ain: A
Re iew,” Neu oImage, 2022.
[3] A. Da a e al., “In e -Indi idual Va ia ion du ing T ansc anial Di ec
Cu en S imula ion,” B ain S imula ion, 2012.
[4] S. Thielsche e al., “SimNIBS: A Ve sa ile Toolbox o Simula ing
Elec ical Fields in he Human B ain,” B ain S imula ion, 2015.
[5] X. Huang e al., “ROAST: A Fully Au oma ed Open-Sou ce Pipeline
o Simula ing T ansc anial Elec ic S imula ion,” Jou nal o Neu al
Enginee ing, 2018.
[6] J.W. Goodman e al., “Su oga e Modeling o Accele a ed De ice
Op imiza ion in DCS,” IEEE T ans. Biomed. Eng., 2019.
[7] G.C. Campos e al., “E icien App oxima ion o B ain Elec ic Fields
Using Gaussian P ocess Reg ession,” F on ie s in Neu oscience, 2022.
[8] L. Lu e al., “Lea ning Nonlinea Ope a o s ia DeepONe ,” Na u e
Machine In elligence, 2021.
[9] N. Ko achki e al., “Neu al Ope a o s: A Su ey,” Founda ions T ends
Mach Lea n, 2023.
[10] A. Da a e al., “In e -Indi idual Va ia ion in DCS,” B ain S imula ion,
2012.
