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
Abs ac — Goal: T ansc anial s a ic magne ic s imula ion is
a no el nonin asi e me hod o educ ion o he co ical
exci abili y in ce ain neu ological diseases ha , unlike
o dina y ansc anial magne ic s imula ion, makes use o
s a ic magne ic ields gene a ed by pe manen magne s. The
physical p inciple unde lying ansc anial magne ic
s imula ion is well known, ha is, he Fa aday´s law. By
con as , he physical mechanism ha explains he in e ac ion
be ween neu ons and s a ic magne ic ields in ansc anial
s a ic magne ic s imula ion emains unclea , which makes i
di icul o imp o e and ine une he ea men . In he p esen
wo k i is discussed he possibili y ha his mechanism migh
be he Lo en z o ce exe ed on he ions lowing along he
memb ane channels o neu ons. Me hods: To suppo his
hypo hesis, a dimensional analysis i is ca ied ou o compa e
he La mo adius o he ions in he p esence o a s a ic
magne ic ield wi h he dimensions o he c oss sec ion o
human axons and memb ane channels in neu ons. Resul s:
This analysis shows ha whe eas a mode a e s a ic magne ic
ield is no expec ed o a ec he ion lux h ough axons,
ne e heless i can a ec he ion lux along memb ane
channels. Conclusion: The o e all e ec o he s a ic magne ic
ield would be o in oduce an addi ional ic ion be ween he
ions and he walls o he memb ane channels, hus educing i s
conduc ance. Signi icance: Calcula ions pe o med by using a
Hodgkin-Huxley model demons a e ha e en a sligh
educ ion o he conduc ance o he memb ane channels can
lead o he supp ession o he ac ion po en ial, hus inhibi ing
neu onal ac i i y.
Index Te ms— T ansc anial s a ic magne ic s imula ion,
S a ic magne ic ield, Lo en z o ce, b ain s imula ion
I. INTRODUCTION
RANSCRANIAL magne ic s imula ion (TMS) is a
well-es ablished nonin asi e me hod o b ain
s imula ion o diagnosis and ea men o neu ological
diseases ha is based on he applica ion o s ong and sho
pulses o magne ic ield ( ypically 1T o ampli ude and
300𝜇𝑠 o du a ion) gene a ed by cu en - ed coils [1]. The
Submi ed o e iew o IEEE T ansac ions on Biomedical Enginee ing
on Janua y 23 h, 2020.
Manuel J. F ei e is wi h he Depa men o Elec onics and
Elec omagne ísm, Uni e si y o Se ille, Spain (e-mail: [email p o ec ed]).
Joaquín Be nal-Méndez is wi h he Depa men o Applied Physics,
Uni e si y o Se ille, Spain.
physics unde lying TMS is well known and i is based on
he induc ion o cu en s in neu ons by i ue o he
Fa aday´s law. P o ocols o TMS he apy a e well
es ablished, being he he a-bu s p o ocol he mos
ex ended o induce long-las ing neu al changes [2].
T ansc anial s a ic magne ic s imula ion ( SMS) is a no el
nonin asi e o m o b ain s imula ion, ha makes use o
s a ic magne ic ields (SMFs) c ea ed by pe manen
magne s o educe co ical exci abili y in humans
[3][4][5][6]. Expe imen al e idences show ha SMFs o
mode a e alues ( ens o hund eds o mT) can in e e e wi h
physiological b ain unc ions [3][4][5][6]. The e is also
expe imen al e idence o e ec p oduced by e en g ea e
SMFs in Magne ic Field Resonance (MRI) exams [7].
Mo eo e , he in e ac ion o mode a e SMFs wi h exci able
memb anes o di e en biological sys ems has been
ex ensi ely epo ed [8][9][10][11][12]. Despi e hese
e idences, a physical mechanism p o iding a clea
explana ion o he in e ac ion o mode a e SMFs wi h
neu ons has no been iden i ied ye . A be e unde s anding
o he physic phenomena unde lying his in e ac ion would
help o inc ease he e iciency o he SMS. A a
undamen al le el, wo kinds o physical mechanisms seem
o be easible candida es o p o ide his explana ion: he
magne ic beha io o he cons i uen molecules o exci able
memb anes in he p esence o a SMF, and he in e ac ion
be ween a SMF and mo ing ions in neu ons h ough he
Lo en z o ce. Wi hin he i s pe spec i e, i has been
sugges ed ha he eo ien a ion o diamagne ic aniso opic
molecules in he cell memb ane can be esponsible o he
in luence o mode a e SMF on he cell memb ane [8][9].
The second hypo hesis has been used o in es iga e, om a
heo e ical poin o iew, he in luence o SMFs on he ion
cu en ha lows along he axon and is associa ed wi h he
p opaga ion o he ac ion po en ial (AP) in ne es [13] [14].
F om he analysis ca ied ou in [13] [14], i ollows ha he
Lo en z o ce exe ed by mode a e SMFs on he ions
lowing along ne es canno app eciably a ec he
p opaga ion o he AP. Ne e heless, he AP is associa ed
no only wi h he ion lux along axons bu also wi h he ion
lux along memb ane channels. Rega ding his, i is
in e es ing o no e ha i has been sugges ed ha ion
channels o neu ons can be modelled as FET ansis o s
[15]. Also, i is well known ha SMFs can a ec he
The Lo en z Fo ce on Ions in Memb ane
Channels o Neu ons as a Mechanism o
T ansc anial S a ic Magne ic S imula ion
Manuel J. F ei e, Senio Membe , IEEE, and Joaquín Be nal-Méndez, Senio Membe , IEEE
T
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pe o mance o FET ansis o s in MRI p eampli ie s due o
he Lo en z o ce in cha ge ca ie s [16]. Thus, in he
p esen wo k, i is discussed he possibili y ha he AP can
be a ec ed by mode a e SMF h ough he Lo en z o ce
exe ed on he ions lowing along he memb ane channels
in neu ons. To suppo his hypo hesis, a dimensional
analysis is ca ied ou o es ima e he a io be ween he
La mo adius o he ions in he p esence o a SMF wi h a
alue ypical o he SMS [3], and he dimensions o he
c oss sec ion o human axons and memb ane channels.
Based on his analysis, i is sugges ed ha , al hough
mode a e SMFs canno a ec he ion lux h ough axons, i
may a ec he ion lux along memb ane channels. I is also
sugges ed ha he e ec o he Lo en z o ce is o in oduce
an addi ional ic ion be ween he ions and he walls o he
memb ane channels. Since he con en ional ic ion
be ween he ions and he walls accoun s o almos 2/3 o
he conduc ance alue o he channels [17], we conclude
ha he ul ima e e ec o he Lo en z o ce is o educe
signi ican ly he conduc ance o channels. Resul s o he
AP ob ained wi h a Hodgkin-Huxley (HH) model [18]
e eal ha a sligh educ ion o he conduc ance o he Na
channel can lead o he supp ession o he AP.
Sec ion II p esen s an analysis ha ules ou he e ec o
Lo en z o ce associa ed wi h mode a e SMFs on ions
lowing along axons as a cause o neu on inhibi ion. Also,
he a io be ween La mo adius and he diame e o he
egion o conduc ion is p esen ed as a sui able benchma k
o de e mine whe he Lo en z o ce can al e he low o
ions. This c i e ion is employed in sec ion III o show ha
memb ane channels migh see i s conduc ance dec eased by
a Lo en z o ce such as ha c ea ed by a mode a e SMF,
and ha he expec ed dec ease can ac ually supp ess he
AP. Finally, conclusions a e p esen ed in sec ion IV.
II. ANALYSIS
As i is well known, he Lo en z o ce is he o ce exe ed
on a cha ged pa icle mo ing in he p esence o a SMF.
Because his o ce is pe pendicula o bo h he eloci y o
he pa icle and he di ec ion o he SMF, i makes he
pa icle o desc ibe a ci cula ajec o y in a plane
pe pendicula o he SMF. The adius o his ajec o y is
e e ed o as he cyclo on adius o La mo adius, 𝑅𝐿, and
i is gi en by 𝑅𝐿=𝑚𝑣/𝑞𝐵, whe e 𝑚, 𝑣 and 𝑞 a e he mass,
eloci y and cha ge o he pa icle, espec i ely, and 𝐵 is
he ampli ude o he SMF.
The AP p opaga ing h ough he axon o neu ons is
associa ed wi h a longi udinal ion cu en lowing along he
axon. In he p esence o a SMF, due o he Lo en z o ce he
ions lowing along he axon expe ience a de lec ion o hei
ajec o y which p oduces a ans e se cu en . In [13] i is
heo e ically analyzed o he i s ime he o de o
magni ude o he SMF necessa y o p oduce an app eciable
de lec ion in he longi udinal cu en associa ed wi h he
p opaga ion o he AP in he axons o human neu ons. The
calcula ions in [13] show ha a magne ic ield on he o de
o 25T is necessa y o p oduce a de lec ion o educ ion o
10% in he ion cu en along he axon. Such a ield is
se e al o de s o magni ude g ea e han mode a e SMF and
e en an o de o magni ude g ea e han ypical SMF in
MRI sys ems. Mo eo e , in [14] a deepe analysis es ima es
he e ec o his de lec ion in he AP by means o a HH
model whe e a e m ha accoun s o he ans e se cu en
ha appea s as a consequence o he de lec ion is added in
he di e en ial equa ions, his e m being p opo ional o
he alue o he SMF. In [14] i is de ined a a io 𝛼
be ween he ans e se cu en and he longi udinal cu en ,
and i is exp essed as a ela ion be ween he alue o he
SMF, 𝐵, and he ans e se mobili y o he ions, 𝜇, as 𝐵 =
𝛼/𝜇. The calcula ions in [14] show ha , in pa icula , a
mode a e alue o he SMF o 𝐵=11 mT will p oduce a
educ ion o 5% (co esponding o 𝛼=0.05 in [14]) in he
longi udinal cu en in he axon. In [14] i is shown ha
aking his in o accoun in he HH model, his will cause a
supp ession o he AP. This esul en i ely disag ees wi h
he conclusion in [13]. This appa en pa adox can be sol ed
by no ing ha he analysis ca ied ou in [14] assumes an
ion mobili y o 5 m2/Vs, which is h ee o de s o magni ude
la ge han alues expe imen ally epo ed [13]. Fo
example, in [13] he peak axial elec ic ield du ing he
passage o he AP is epo ed o be 𝐸=8 V/m and he ion
eloci y 𝑣𝑑=3.3 × 10−2 m/s. The e o e he ion mobili y is
𝜇 = 𝑣𝑑/𝐸 = 0.004125 m2/Vs. Assuming his much mo e
ealis ic alue o 𝜇, he equi ed SMF o a educ ion o
5% in he longi udinal cu en in [14] will be 14.7 T, which
is close o he o de o magni ude es ima ed in [13] (i.e.,
25 T).
F om he abo e discussion i can be concluded ha
mode a e SMFs canno a ec he p opaga ion o he AP in
human axons. This same conclusion can be also d awn om
a simple al e na i e analysis based on he compa ison o he
La mo adius wi h he diame e o he axons. Conside , o
example, a sodium (Na) ion, whose mass and cha ge a e:
𝑚 = 3.8 × 10−26 kg and 𝑞 = 1.67 ×10−19 C. To es ima e
he La mo adius we can assume an ion eloci y in he
axon o 𝑣𝑑= 3.3 × 10−2 m/s, (i.e., he same alue as in
[13]) and a SMF o alue 𝐵 = 164 mT. This is he alue
measu ed by he au ho s o he same magne used in SMS
in [3], a a dis ance o 2 cm om he su ace o he magne ,
which is he dis ance be ween he scalp and he mo o
co ex. Wi h hose assump ions, he La mo adius is 𝑅𝐿=
𝑚𝑣/𝑞𝐵 =478Å. This is wo o de s o magni ude smalle
han he ypical diame e o he human axon which is 1𝜇𝑚.
The e o e, in he p esence o a mode a e SMF o 164 mT
he ionic cu en is expec ed o low wi hou signi ican
de lec ion h ough he axon. Summing up, i can be
concluded ha due o he di e en o de s o magni ude o
he c oss sec ion o he axon and he La mo adius o
mode a e SMFs, mode a e SMFs canno a ec he anspo
o ions h ough he axon, in acco dance wi h [13].
The discussion p esen ed abo e sugges s ha he
compa ison be ween he size o he c oss sec ion o he
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axon and he La mo adius can be conside ed as a
benchma k o asce ain whe he he Lo en z o ce
associa ed wi h a gi en alue o SMF a ec s he ion ne e
conduc ion. In ac , we ha e jus shown ha his c i e ion
allows o ule ou Lo en z o ce due o a mode a e SMF as
he cause o he AP sup ession in axons. In iew o his, in
his wo k we p opose an al e na i e explana ion o he
e ec on he AP o a mode a e SMF based on he e ec o
Lo en z o ce on he conduc ance o memb ane channels.
To unde pin his hypo hesis, we will use he benchma k
index desc ibed abo e o de e mine whe he a mode a e
SMF can a ec he ion lux along memb ane channels. In
his ega d, a key poin o be aken in o accoun is ha he
size o he c oss sec ion o ion channels o exci able
memb anes is se e al o de s o magni ude smalle han he
diame e o he axon.
III. RESULTS AND DISCUSSION
In his sec ion a dimensional analysis is ca ied ou o
compa e he size o he po assium (K+) channel wi h he
La mo adius o K+ ions o mode a e SMFs. To his end,
an es ima ion o he d i eloci y o he ions h ough he
channel is equi ed as a i s s ep. Rega ding his poin , i is
impo an i s o de e mine whe he he low o he ions
h ough he channel can be conside ed an ohmic p ocess (o
ions should be conside ed ballis ic cha ges ins ead).
Scien i ic e idence poin s ou ha ic ion caused by he
po e shape and wall o uosi y play an impo an ole in he
conduc ance [19] [20]. The e o e, i is easonable o
conside he low o ions h ough he channel as an ohmic
p ocess. Unde his assump ion, he ampli ude o he cu en
can be w i en as 𝐼 = 𝐽 ⋅ 𝑆, whe e 𝑆 is he a e age c oss
sec ion o he channel, and he cu en densi y 𝐽 can be
w i en as 𝐽 = 𝑞𝑛𝑣𝑑, whe e 𝑛 is he numbe o ions pe uni
olume and 𝑣𝑑 he d i eloci y o ions. Mo eo e , 𝑛 can
be w i en as 𝑛 = 𝑁/𝑉, whe e 𝑁 is he numbe o ions ha
can occupy simul aneously he channel and 𝑉 is he olume
o he channel, ha can in u n be exp essed as 𝑉 = 𝑆𝐿,
whe e L is he leng h o he channel. The e o e, he d i
eloci y can be exp essed as:
𝑣𝑑=𝐼𝐿
𝑁𝑞.
(1)
The K+ channel ex ends 45Å, wi h a wide segmen o
leng h 23Å and a na owe selec i i y il e o adius 1.5 Å
and leng h 12Å whe e he ions would ha e o shed i s
hyd a ing wa e s o en e [17] [21]. The selec i i y il e
con ains wo K+ ions [19] [21], ha is, he numbe o ions
ha can occupy simul aneously he selec i i y il e is N=2.
Since he ampli ude o he cu en is o he o de o
picoampe es [17], assuming 𝐼 = 1pA and L=12Å, 𝑣𝑑 can be
es ima ed om (1) as 𝑣𝑑= 3.75 ×10−3 m/s. F om his
es ima ion o he d i eloci y, and aking in o accoun ha
he mass o K+ ion is 39.0983 uma = 6.49 ×10−26 kg, he
co esponding La mo adius o a SMF o alue B=164mT
can be calcula ed as: 𝑅𝐿= 𝑚𝑣𝑑/𝑞𝐵 =93Å. This alue is
o he same o de o magni ude as he leng h o he channel,
and wha i is mo e impo an , i is no negligible in
compa ison wi h he wid h o he channel. The e o e, he
componen o he SMF pe pendicula o he axis o he
channel will gi e ise o a Lo en z o ce ac ing on he ions
which will cu e he ajec o y o he ions inside he na ow
channel. This si ua ion is ske ched in Fig. 1.
Fig. 1: Ske ch o memb ane channel and he de lec ed ajec o y o an ion.
The La mo adius is app oxima ely wice he lengh o he channel.
Inside he na ow channels he ions a e o ced o ollow a
na ow and s aigh pa h. The e o e, he Lo en z o ce ac s
pushing he ions agains he walls o he channel, which
imposes a ic ion wi h he walls o he channel. This esul s
in a dec ease o he conduc ance o he ions h ough he
channel.
To es ima e o wha ex en he e ec desc ibed abo e can
ac ually dec ease he conduc i i y o he channel i is
in e es ing o e ise he ela ionship be ween ic ion,
di usion and conduc ance. In he B ownian mo emen , he
Eins ein ela ion ela es he ic ion o ce wi h he di usion
coe icien 𝐷 as 𝐷 = 𝐾𝑇/𝑚𝛾, K and T being he
Bol zmann´s cons an and empe a u e, espec i ely, and
𝑚𝛾𝑣 being he ic ion o ce in he Lange in’s equa ion
[22]. In [17] he di usion coe icien o K+ in he selec i i y
il e o he memb ane channels is calcula ed and i is on
a e age 1/3 o he bulk alue, whe eas in he wide
segmen o he channel is nea ly he same as he bulk alue.
In he same sense, in [20] i is also epo ed ha he ic ion
is esponsible o he di usion coe icien o K+ o be 3 o 5
imes lowe han in bulk wa e (𝐷 = 0.46 ×10−9 m2/s in
he channel and 2.2 × 10−9m2/s in bulk wa e egion).
Mo eo e , in [19] i is poin ed ou ha he di e en
conduc ance o K+ channels migh ha e di e en causes,
he ic ion among hem. Thus, in [17] i is shown ha he
educ ion o he di usion coe icien in he selec i i y il e
( he na owe pa o he channel) in luences he o e all
channel conduc ance. Those e idences sugges ha he
ic ion in oduced by he Lo en z o ce in he dynamics o
ions h ough memb ane channels can esul in he educ ion
o he conduc ance o he channels. The analysis was
ca ied ou o he K+ channel bu he conclusion can be
gene alized o he es o channels.
Al hough he expec ed educ ion o conduc ance caused
by ic ion due o Lo en z Fo ce is only a ac o o 2 o 3, as
men ioned abo e, his educ ion migh be enough o
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comple ely supp ess he AP. This is due he ac ha he
AP gene a ion is qui e sensi i e o small a ia ions o he
conduc ance alues. To illus a e his, Fig. 2 shows changes
unde gone by he ansmemb ane po en ial o a neu onal
cell segmen in esponse o h ee consecu i e equal s imuli
o h ee di e en alues o he conduc ance o he as Na
channel, which is g ea ly in ol ed in he onse o he AP.
These esul s ha e been calcula ed by sol ing he
di e en ial equa ions o he HH model o AP gene a ion by
means o he HHSim so wa e [18], a ee g aphical
simula o ha p o ides access o he pa ame e s o he HH
model. Fig. 2 shows h ee spikes gene a ed unde s imuli
o h ee di e en alues o he conduc ance o he as Na
channel. The i s spike co esponds o a conduc ance o
120 𝜇𝑆, he second spike co esponds o 80 𝜇𝑆 and he las
spike is o a conduc ance o 60 𝜇𝑆. Fo his las alue, i
can be obse ed ha , e en hough he change in
conduc ance is only a 25% wi h espec o he p e ious
alue, he AP is almos en i ely supp essed.
IV. CONCLUSION
In his wo k i is demons a ed ha whe eas Lo en z
o ce p oduced by mode a e SMF is no expec ed o
p oduce app eciable e ec s on he ions lowing along he
axon o neu ons, i migh well a ec he lux o he ions
along he memb ane channels in neu ons. This is due o he
di e en a ios o he c oss sec ions o axons and memb ane
channels wi h espec o he co esponding La mo adius.
I has been shown ha in he memb ane channels he
Lo en z o ce can e ec i ely p oduce a ic ion o he ions
wi h he walls o he channel, and ha his addi ional
ic ion migh educe he conduc ance o he channels.
Calcula ions o neu on esponses by using a Hodgkin-
Huxley (HH) model ha e illus a ed ha educ ions o
conduc ance o he same o de as hose expec ed can
e ec i ely supp ess he AP in neu ons. The e idences
p o ided by his analysis make o he Lo en z o ce a
easible candida e o be he main physical mechanism
explaining he educ ion o he exci abili y o he mo o
co ex achie ed by he SMS echnique.
Fig 2. Response o he ansmemb ane po en ial (con inuous line) o a neu onal cell segmen o h ee consecu i e equal s imuli (dashed line). Pa ame e s in he
HHSIm so wa e: conduc ances o Na, K, and Cl a e se , espec i ely, o 0.0265𝜇𝑆, 0.07𝜇𝑆 and 0.1𝜇𝑆. The conduc ance o he as Na channel is 120 𝜇𝑆 o he
i s s imulus, 80 𝜇𝑆 o he second s imulus and 60 𝜇𝑆 o he las s imulus. No e ha he AP is almos supp essed in he la e case.
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