JournalofAppliedMathematicsandPhysics,2020,8,2091-2102
https://www.scirp.org/journal/jamp
ISSNOnline:2327-4379
ISSNPrint:2327-4352
ModeloftheNerveImpulsewithAccountof
MechanosensoryProcesses:Stationary
Solutions
AlexanderMengnjo
1
,AlainM.Dikand´e
2
,GideonA.Ngwa
1
1
DepartmentofMathematics,FacultyofScience,UniversityofBuea,Buea,Cameroon
2
DepartmentofPhysics,FacultyofScience,UniversityofBuea,Buea,Cameroon
Howtocitethispap er:Mengnjo,A.,
Dikand´e,A.M.andNgwa,G.A.(2020)Mod-
eloftheNerveImpulsewithAccountof
MechanosensoryProcesses:StationarySolu-
tions.JournalofAppliedMathematicsand
Physics,8,2091-2102.
https://doi.org/10.4236/jamp.2020.810156
Received:July24,2020
Accepted:October12,2020
Published:October15,2020
Copyright
c
2020byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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Inhachabitasseplateadictumst.Aliquam
egettortor.Quisquedapibuspedeinerat.
Nuncenim.Induinulla,commodoat,con-
sectetuernec,malesuadanec,elit.Aliquam
ornaretelluseuurna.Sednecmetus.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehicula purus.Fusceac nislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibus etmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoin nisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptent tacitisociosquadlitoratorquen-
tperconubia nostra,perinceptos hymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel, pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummy fringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictum tempus.Namut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis,aliquam
id, ornarevitae, porttitor non,wisi.Maecenas
luctusportalorem.Donec vitaeligulaeu ante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclacinia auctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbinullanibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel, metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac, mauris.Maecenas etelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat, wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentumconvallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero, malesuadavitae, ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurus sedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis, quam.Etiam semest,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sed scelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareante nuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisi ultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsum anisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturient montes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisi commodofelis,ut adipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselit magna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada, tortormetus el-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultricies accumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris a dui
et tortormattis pretium.Sednullametus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesque odio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inporta congue,velit leofacilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,velposuere nibhlacus nec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sit ameteuismodlectus feliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pedecursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravida ante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,varius eu,commodosed, lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuer varius nul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi, sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsit amet,consectetueradipiscing elit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreet eget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Maurisenim sem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverranisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tus sitamet purus faucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuada malesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin, ultriciesid, libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristique quis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris porta dui,vitae aliquamarcu turpis ac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismodac, tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna, blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnon arcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbi ipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsum vel sem.Nul-
laenim.Proinnonummy felisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odio lobortis arcu,sed sollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendum purus.Nullaquam mauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, dui ut
dignissim auctor, orcilibero fringilla massa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netus etmalesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusin augue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipit sitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringilla tempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbi tristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnased enim.Doneceget erosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semper at,metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem, nonporttitorpurus odioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusut me-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatisvitae,vulputate a,pharetra sit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid, mi.Nullamnec nibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnulla pulvinarjusto, aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagna risusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursus arcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfelis elit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In egetrisus luctus wisiconva-
llis pulvinar.Vivamus sapienrisus, temporin,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisque dignissim, lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligula suscipiturna,vitae portaodioer-
at quissapien.Proinluctus leo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
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tusnibhconguelorem,ut tempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
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libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit, gravidaac, pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehiculaat, conguesit amet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nulla ettellus nonfelis imperdi-
etmattis.Aliquameratvolutpat.
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ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturient montes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
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pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsit ametnibh necpurusdignissim mat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpis diam feugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Namligula arcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisus atmetuspulvinar malesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
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luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nuncbibendum augueluctus
diam.Etiam nec sem.Sed eros turpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
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susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajusto vehiculadiam,eu condimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverra id,velit.
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eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitamet est.Vivamuspulvinar leonon
sapien.Aliquamerat volutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet, mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibh ullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt,leo vitaetristiquefa-
cilisis,ipsum wisiinterdumsem,dapibus sem-
pernulla velit vel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim, sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
Crasaneque.Utrisusdui, interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibus orciluctus et ultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augue erosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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ametneque.Sedcommodomalesuadalectus.
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aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamuspulvinar.Aliquam eratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
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sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibh pretiumtellus,at dignissimleo eros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumante ipsumprimisin faucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoa justo.Quisquenec au-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenim sitametrisus.Curabitur vitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeu nulla utnullainterdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Mechanotransduction refers to a physiological process by which me-
chanicalforces,suchaspressuresexertedbyionizedfluidsoncel-
lmembranesandtissues,cantriggerexcitationsofelectricalna-
tures thatplay importantroleinthe control ofvarioussensory(i.e.
stimuli-responsive)organsandhomeostasisoflivingorganisms.In
thiswork,theinfluenceofmechanotransductionprocessesonthe
genericmechanismoftheactionpotentialisinvestigatedanalyt-
ically,byconsideringamathematicalmodelthatconsistsoftwo
couplednonlinearpartialdifferentialequations.Oneofthesetwo
equationsistheKorteweg-deVriesequationgoverningthespatio-
temporalevolutionofthedensitydifferencebetweenintracellular
andextracellularfluidsacrossthenervemembrane,andtheother
isHodgkin-Huxleycableequationforthetransmembranevoltage
with a self-regulatory (i.e.diode-type) membrane capacitance.The
self-regulatory feature here refers to the assumption that membrane
capacitancevarieswiththedifferenceindensityofion-carryingin-
tracellular andextracellular fluids,thus ensuring anelectromechan-
icalfeedbackmechanismandconsequentlyaneffectivecouplingof
thetwononlinearequations.Theexactone-solitonsolutiontothe
density-differenceequationisobtainedintermsofapulseexcitation.
With the help ofthis exact pulse solution theHodgkin-Huxley cable
equationisshowntotransform,insteadystate,toalineareigen-
valueproblemsomeboundstatesofwhichcanbeobtainedexactly.
Fewofsuchbound-statesolutionsarefoundanalytically.
Keywords
NerveImpulse,MechanosensoryResponse,Hodgkin-Huxley
Equation,Korteweg-deVriesEquation,AssociatedLegendre
Polynomials
DOI: 10.4236/jamp.2020.810156Oct.15,20202091JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
1.Introduction
Themechanism bywhichtheactionpotentialisgeneratedandtrans-
mitted alongthe neuronalnetwork, is a complexphenomenon that has
beenactivelyinvestigatedinMathematicalPhysics[1–8].Apioneer
modelofthenerve-impulsegenerationwasproposedbyHodgkinand
Huxley(hereafterreferredtoasHHmodel)[1];themodelrestson
thepictureofthenerveimpulseasanelectricvoltagepropagatingin
formofpulsewavealongthenervefiber.ThusaccordingtotheHH
modelthe nerve impulseis aself-regenerativewaveassociatedwith the
electrochemicalactivityofthenervecells,duetoflowofioncurrents
(NaandK)throughspecificionchannels[1].Thiswaveisassumed
topropagatewithaconstantshapefollowingamechanismwhichcan
besummarizedasfollow:Duringthegenerationandtransmissionof
the nerve impulse,the leadingedge ofthe depolarization regionofthe
nervetriggersadjacentmembranestodepolarize.Thiscausesaself-
propagationoftheexcitationrelatedtothetransmembranepotential
downthenervefiber[1,9,10].
HodgkinandHuxleysuggestedthataconvenientwaytodescribe
thepropagationofthistransmembrane potential,istoviewthenerve
fiber asanelectrical transmissionline.Henceinits mostconventional
formulation,theHHelectricalmodel assumescurrents in intracellular
andextracellularfluidstobeohmicsuch thatthenettransmembrane
currentisthesumofionicandcapacitivecurrents.Inthiscontext,
theconservationlawforcurrentflowacrossthemembraneleadsto
thefollowingreaction-diffusionequation[1]:
C
m
∂V
∂t
= D
2
V
∂x
2
F(V),(1)
whereVisthetransmembranevoltage(alsoreferredtoasactionpo-
tential), C
m
isthe membrane capacitance,Disthediffusion coefficient
and F(V) isthe contribution fromioncurrents to thetransmembrane
voltagegeneration.However,besidesthe well-established electricalac-
tivityof thenervemembrane,there areexperimentalevidences [11–16]
ofexistingmechanicalforcesactingonthemembrane[8]alsocon-
tributingtothenerveimpulsegeneration[17].Tothislastpoint,it
iswellknownthatatequilibriumthenervemembraneisinaliquid
phase;howeverundermechanicalforcesthemembraneundergoesa
structuralchangetoagelphaseresultinginanincreaseofitscom-
pressibility,reminiscentofstructuralpropertiesofanharmonicsolid-
s[17,18].Todescribetheformationofdensitypulsesassociatedwith
mechanicalconstraintsactingonthenerve membrane, therehasbeen
proposals toincludenonlinearityinthemembrane compressibilityto-
getherwithdispersiveeffectsrelatedtotheladderstructureofthe
nervefiber[17].Inthisrespectanonlinearmathematicalmodelin-
cludingdispersionswasproposedbyHeimburgandJackson[4],and
later on extended by several authors (see e.g.ref. [7]).The underlying
equation,consideringaone-dimensional(1D)density pulsepropagat-
ingalongabiomembraneofacylindricalshape,isgivenby[4,7]:
2
u
∂t
2
= c
2
0
∂x
(1+pu)
∂u
∂x
h
4
u
∂x
4
,(2)
where uisthe density change across themembrane due tomechanical
contraintson themembrane,c
0
isthe soundspeedin thehomogeneous
regime(i.e.intheabsenceofpressurewaves),handpareconstants
[17](hereafterweshallsetp= 1forsimplicity[7]).
DOI: 10.4236/jamp.2020.8101562092JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
Giventheevident simultaneous contributions ofboth electricaland
mechanicalactivitiesofthenerveinthegeneration oftheactionpoten-
tial, a better description ofthe generic mechanism ofthe action poten-
tialrequiresfullaccountofthesetwoactivities.Inthisworkwepro-
poseamathematicalmodelwhichcombinesthetwononlinearpartial
differentialequations,namelytheonedescribingthespatio-temporal
evolution ofthedensity pulseacrossthe membrane, and theHH cable
equationwithafeedbackfromionflowsacrossthenervemembrane
representedeby adensity-dependentmembranecapacitance.Thefull
modelbeinganalyticallyuntractable,inthepresentcontextwefocus
mainlyonthegenericmechanismoftheactionpotential.Inthisre-
gardweconsiderthesteady-stateregimeofthemodel,whereexact
soliton solutions can be found.In a future work numerical simulations
will be carriedout onthe spatio-temporal model, to checkthe stability
oftheseexactsolitonsolutionsastheypropagatealongthenerve.
2.TheModel
Thenervefibercanbepicturedasacylinderwithwallsmadefrom
the cellmembrane surroundedby intracellular and extracellularfluids
[6].Theintracellularfluidstandsforaconductiveliquidwithahigh
concentrationofpotassiumionsbutlowconcentrationofsodiumand
chlorineions,whilethecellmembraneactslikeabarrierpreventing
ionsfromintracellularliquidfrommixingwithextracellularliquid.
Duetothedifferenceinionconcentrationsbetweenintracellularand
extracellularfluids,arestingpotentialisexpectedtosetupacross
themembrane.Ifthenerveisdepolarized,ormorepreciselyinthe
presenceofastimulus,themembranebecomesselectivelypermeable
toioncurrentsflowingrapidlyintothecell,reversingthepolarityof
theactionpotential.
Inthepictureofelectriccable,theselectiveregulationoftrans-
menbraneionsbythecytoplasmsuggestsa“management”(orself-
regulation)ofthechargesstoredinthenervemembranecapacitance.
Consequentlywecanexpectchangesintheelectrostaticpotentialof
themembrane duringapropagatingdensitypulse,indicatingapossi-
bilityforanelectromechanicalcouplingbetweenthemembraneden-
sityandtheelectrostaticpotential.Suchelectromechanicalcoupling
has been widely reported inneurophysiologicalexperients[19–22], and
canbelinkedwithchangesinthemembranecapacitanceasaresult
ofvariationofthefuiddensityacrossthemembrane.
Toincludetheelectromechanicalcouplinginthemathematicalmod-
elforthenerveimpulsegeneration,wereformulatetheHHequation
for theaction potential whichleads to thefollowingnew mathematical
modelforthenerve-impulsegenerationandpropagation(neglecting
thecontributionofF(V),forsimplicity):
∂Q
m
∂t
=D
2
V
∂x
2
,(3)
2
u
∂t
2
=c
2
0
∂x
(1u)
∂u
∂x
h
4
u
∂x
4
.(4)
Characteristicparametersintheabovesetofcouplede-
quationsaredefinedasfollow:
Q
m
(u)is the total charge stored in the membrane capac-
itanceattimet,
DOI: 10.4236/jamp.2020.8101562093JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
Disthediffusioncoefficient,
c
0
isthesoundspeedalongthemembrane,
histhefourth-orderdispersioncoefficient[4].
Bydefinition, the terminthe leftof Equation(3) representsthe cur-
rentI
c
acrossthemembrane capacitanceC
m
(u) = ∂Q
m
(u)/∂V.This
meansthatI
c
isrelated totheinstantaneous charge inthemembrane
i.e.Q
m
=C
m
(u)V(assumingthatthemembranecapacitanceisnot
explicitely dependent on thetransmembrane voltage), throughthere-
lationI
c
= ∂Q
m
/∂t.Thetermintheright-handsideofEquation(3)
isthediffusioncurrentrelatedtochargeflowacrossthemembrane
(withD= a/2R,whereaistheradiusofthecylindricnerve andRis
themembraneresistance).
Weconsiderthatatsomegiventimetthemembranecapacitance
C
m
is determinedby thedifference uin densitiesof thefluids crossing
themembrane,andproposealawofdependenceofC
m
onuofthe
followingform:
∂C
m
(x,t)
∂t
= κu(x,t),(5)
whereκisrealandconstant.WithEquation(5)wecantransform
Equation(3)intoareaction-diffusionproblemwithalinearsource
termnamely:
C
m
(x,t)
∂V
∂t
= D
2
V
∂x
2
κu(x,t)V.(6)
Since formula (5)suggeststhatthemembrane capacitanceC
m
(x,t)
isfixedbythenetdifferenceu(x,t)indensitiesoftheionicfluids
crossingthemembraneattimetandpositionx,inthenextsection
wesolveanalyticallyEquation(4)inordertoobtaintheanaytical
expressionofu(x,t).
3.Pulse Solution tothe Density-Difference
Equation
Equation(4)describesthemotionofwavesinonespacedimension.
Letusassumethatthesolutionswe seekarelocalizedcoherent struc-
turesobeying:
lim
x→±∞
u(x,t) = 0,(7)
andpropagatinginthedirectionofpositivex.Thus,usingthecoor-
dinatetransformations:
u(x,t) = ψ(ξ,T)= c(
x
c
0
t),T=
h
c
t,(8)
wecanrewriteEquation(4)as:
∂ψ
∂T
= αψ
∂ψ
∂ξ
β
3
ψ
∂ξ
3
+O(h
2
),(9)
whereαandβareconstantsdependingonc
0
,handc.Notethat
parametersαandβcanbesettoanyvaluebycoordinatetransfor-
mation,howeverthevaluesweshalluseinthisworkareα=6and
β=1forwhichEquation(9)reducestothecanonicalKorteweg-de
DOI: 10.4236/jamp.2020.8101562094JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
Vriesequation[23].Asweareinterestedinexactanalyticalsolutions
tothedensity-waveequation,letusfirstexplicitelysettheassociate
mathematicalproblem.Startingwithaninitialdensitypulseu
0
(x),
weareinterestedintheinitial-valueproblem:
∂u
∂t
=6u
∂u
∂x
3
u
∂x
3
,(10)
u(x,0)=u
0
(x).
Welookfortheexistenceofspecialsolutionscalledsolitons,which
aretravellingwaveswithpermanentshapeobtainedbymeansofthe
Inverse-ScatteringTransform[23].
Asausefulrecall,withintheframeworkoftheInverse-Scattering
Transformanyexactlyintegrablenonlinearevolutionequationfora
functionu=u(x,t)canbemappedontoapairofspectralproblems
so-calledLaxpair[24],namely:
L
t
+[L,A] = 0,(11)
whereLandAaretwolinearoperatorsthecoefficientsofwhichdepend
onthefunctionuanditsderivatives.Suppose:
L=
2
∂x
2
+u(x,t),(12)
and
A=4
3
∂x
3
+3(u
∂x
∂x
u),(13)
thentheassociatedevolutionequation;
L
s
+[L,A] = 0,(14)
turnstothefollowingKorteweg-deVriesequation:
∂u
∂t
6u
∂u
∂x
+
3
u
∂x
3
= 0.(15)
ThefirststepintheInverse-ScatteringTransformistosolvethe
forwardscatteringproblemnamely:
LΨ = λΨ,(16)
wherethespectralparameterλRbecausetheoperatorLisself-
adjoint.Ifthepotentialuin(12)decayssufficientlyquicklyas|x|→
,thenthelinearproblem(16)canbewritten:
Ψ
00
∼−λΨ.(17)
Therearetwopossiblekindsofeigenfunctionsforproblem(17):
1.boundstates:Ifλ<0,thenwemaydefinek=
λtohave:
Ψ(x) = de
kx
+ce
kx
asx→∞.(18)
Thevaluesofkforwhichd=0,sothatΨ(x)isboundedas
x+,arethediscreteeigenvaluesofL,denotedk
n
.The
correspondingfunctionsΨ
n
=c
n
e
k
n
x
areboundstates.Itis
convenienttonormalizeΨ
n
sothat:
Z
R
Ψ
2
n
(x)dx= 1,(19)
with thisnormalization conditionthe constant c
n
is thenormal-
izationconstant.
DOI: 10.4236/jamp.2020.8101562095JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
2.unboundstates:If λ>0, then allthesolutions Ψ ofEquation
(17)are sinusoidal asx→±∞andare consequently unbounded.
Thereforethereexistseigenfunctions,forallλ>0,oftheform:
Ψ(x,k)
e
ikx
+b(k,t)e
ikx
asx→∞,
a(k,t)e
ikx
asx→−∞.
Theconstantaiscalledthetransmissioncoefficientandbthe
reflectioncoefficient.
Theeigenvaluesk,k
n
togetherwiththeconstantsc
n
,bandacon-
stitutethescatteringdata.Thetimedependenceofthesescattering
dataensuesfromthesecondauxiliaryequationi.e.:
Ψ
t
= AΨ,(20)
inthiscasewehavethefollowingrelations:
a(k,t)=a(k,0),b(k,t) = b(k,0)e
8ik
3
t
,
c
n
(t)=c
n
(0)e
8k
2
n
t
,k
n
(t) = k
n
(0).
The inverse scatteringproblemistheproblem ofreconstructingthe
wavepotentialuonthebasisofthescatteringdata.Giventhenor-
malizationconstantc,theeigenvaluesk
2
n
fortheboundstates,and
thereflectionandtransmission coefficients forthecontinuous spectra,
thepotentialu(x,t)isrecoveredfrom:
u(x,t) = 2
d
dx
K(x,x;t),(21)
wheretheimplicitlytime-dependentfunctionK(x,y)satisfiesthe
Gelfand-Levitan-Marchenkoequation[25,26]:
K(x,y)+F(x+y)+
Z
x
K(x,z)F(z+y)dz= 0,(22)
inwhich:
F(x)
1
2π
Z
R
b(k,t)e
ikx
dk+
N
X
n=1
c
n
(t)e
k
n
x
.(23)
Wenowapplythisproceduretotheinitialvalueproblem(10), where
theinitialprofileistakentobe:
u(x,0) = 2sech
2
x.(24)
Wenotethefollowing:
1.Sinceu(x,0)L
2
(R),theoperatorLisself-adjointandthus
hasarealspectrum.
2.u(x,0) L
1
(R)and
R
R
u(x)dx<0,hencethediscretespectrum
inthescatteringproblem(17)withthepotential(24)isnon
empty.
Withtheinitialpotential(24),thescatteringproblem(17)at t= 0
becomes:
Ψ
xx
+(λ+2sech
2
x)Ψ = 0,(25)
DOI: 10.4236/jamp.2020.8101562096JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
whichtransformsintotheAssociatedLegendreequation:
d
(1τ
2
)
dΨ
+
2+
λ
1τ
2
Ψ = 0,
whereτ= tanhx.RecallthattheAssociatedLegendreequationisof
thegeneralform[27]:
d
(1τ
2
)
dy
+
n(n+1)+
k
2
1τ
2
y= 0,(26)
wherenandkare integers.Thecorrespondingbounded solutionsare
givenby:
y
j
(τ) P
k
j
(τ),j= 1,2,···,n,(27)
where:
P
k
j
(x) = (1)
k
(1x
2
)
k
2
d
k
dx
k
1
j!2
j
d
j
dx
j
(x
2
1)
j
(28)
areAssociatedLegendrepolynomials.Hencetheboundedsolutionof
Equation(25),correspondingtoλ=k
2
1
,existsfork=k
1
=1and
thediscreteeigenfunctionistherefore:
Ψ
1
(x) P
1
1
(tanhx) = sechx,(29)
where:
P
n
m
(τ)=(1)
n
(1τ
2
)
n
2
d
n
n
P
m
(τ),
P
m
(τ)=
1
m!2
m
d
m
m
(τ
2
1)
m
.(30)
Since
R
R
sech
2
xdx= 2,thenormalizedeigenfunctionbecomes:
Ψ
1
(x) =
1
2
sechx.
Theasymptoticbehaviorofthissolutionyields:
Ψ
1
(x)
2e
x
asx+,
sothat:
c
1
(0) =
2.
Consequentlyc
1
(t)=
2e
4t
.Thisinformationissufficientforthe
reconstructionofu(x,t),giventhatwehavechosenaninitialprofile
forwhichb(k) = 0forallk.ItfollowsthatEquation(23)becomes:
F(x;t) = 2e
8tx
,
andtheintegralEquation(22)istherefore:
K(x,z;t)+2e
8t(x+z)
+2
Z
x
K(x,y;t)e
8t(y+z)
dy= 0.
ThisimpliesthatK(x,z;t) = L(x,t)e
z
,whereLissuchthat:
L+2e
8tx
+2Le
8t
Z
x
e
2y
dy= 0.(31)
DOI: 10.4236/jamp.2020.8101562097JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
Figure1.Spatialprofileofthepulsesolutiontothedensity-difference
equation.
Equation(31)canbesolveddirectlytoyield:
L(x,t) =
2e
8tx
1+e
8t2x
.(32)
Thus,
u(x,t)=2
∂x
2e
8t2x
1+e
8t2x
=2sech
2
(x4t),(33)
whichisapulsesoliton.Thespatialprofileofthesolutionto
thedensity-differenceequationissketchedinFigure1,indi-
mensionlessunits.
4.TheActionPotential
Having obtainedthe analytical expressionfor the density pulseu(x,t),
letusnow lookforpossiblesolutionstoEquation(6).Inthispurpose
weassumethattheaxonisacylinder oflength L,andthatthenerve
impulselasts forTunits oftime.With theseconsiderations, themodel
turnstothefollowingboundary-valuessystem:
C
m
(x,t)
∂V
∂t
=D
2
V
∂x
2
κu(x,t)V,x[0,L],t>0,(34)
V(0,t)=h(t),V(L,t) = 0,V(x,0) = v(x),(35)
∂u
∂t
=6u
∂u
∂x
3
u
∂x
3
,s[0,T](36)
u(x,0)=u
0
(x).(37)
Equation(34)needstobesolvednumerically,togainaconsisten-
tunderstandingofthespatio-temporalevolutionoftheactionpo-
tentialalongtheaxon.Nonethelessthereexistsaregimeinwhich
exactanalyticalsolutionscanbeobtained.Focusingonthisspecif-
icregime,namelythesteady-stateregimewherethedensitypulse
DOI: 10.4236/jamp.2020.8101562098JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
u(x,t)u
0
(x),theactionpotentialcanbereducedtoitsstationary
formv(x)andasasequelEquation(34)turnsto:
[
2
∂x
2
+ϑ(x)]v(x) = 0(x) = µsech
2
x.(38)
Thisisalinear-operatorequationwithzeroeigenvalueinwhich
µ=
2κ
D
.Equation(38)isactuallysimilartoEquation(25), andforthis
reasonwecanapplythesamevariablechangetofindanAssociated
Legendreequationbutnowwithn(n+ 1)=µandk=0.Forthis
case,boundedsolutionsassuming arbitraryvaluesofthe integer nare
theLegendrepolynomials:
v
n
(τ) =
1
2
n
n!
d
n
n
(τ
2
1)
n
,n= 1,2,3,···.(39)
Exactanalyticalexpressionsforthefirstthreeboundstatesare
givenbelow:
v
1
(x)=tanhx,D= 2κ,(40)
v
2
(x)=
1
2
(3tanh
2
x1),D= 6κ,(41)
v
3
(x)=
1
2
(5tanh
2
x3)tanhx,D= 12κ.(42)
Thethreeboundstatesaresketchedin(Figure2).
Notethattheseboundstatesarenottheonlypossiblesolutions
tothesteady-stateequationfortheactionpotential,otherpossible
solutionscanbeobtainedfordifferent valuesoftheparameterµ=
2ε
D
linkingthediffusioncoefficientDwiththecoefficientκcouplingthe
twoequations.
Itisworthwhiletostressherethatthesteady-statesolutionsob-
tainedanalytically,areexactinitialprofilesoftheactionpotential
andhenceareexpectedtoplayasignificantroleinnumericalsimula-
tionsofEquation(34),inordertofindexactspatio-temporalprofiles
forthepropagatingactionpotentialV(x,t).Owingtotheextremely
richvarietyofpossiblesteady-stateprofilesfortheactionpotential,
representedbydistinctLegendrepolynomialsv
n
(τ)fordifferentval-
uesofµ=
2ε
D
= n(n+1),togainaconsistentinsightfromnumerical
simulations itwillbe useful toconsider areasonablenumber of bound
states.Thereforenumericalsimulationsof themodel deserve a specific
frameworkofstudyandwillbeconsideredinafuturework.
Figure2.Sketchesofthefirstthreeboundedmodesfortheactionpotential,inthesteadystate.
DOI: 10.4236/jamp.2020.8101562099JournalofAppliedMathematicsandPhysics
A.Mengnjoetal.
5.Conclusion
Wehaveintroducedamathematicalmodeltodescribetheinfluence
ofmechanosensoryprocesses,onthegenerationandpropagationof
thenerveimpulse.Themodelrestsonapictureoftheaxonhaving
theformofalongcylinder,withthemembraneregardedasacapac-
itivediodeforwhichthecapacitanceisconstantlyadjustedbythe
differenceindensitiesofintracellular andextracellularionicfluids,on
eithersideofthemembrane.Mathematicallythemodelconsistsof
aKorteweg-deVriesequationfortheionicdensityexchangeacross
thenervemembrane,coupledtotheHHcableequationwithavari-
ablecapacitance.Whilethefullmodelturnsouttobeanalytically
untractable,weobtainedthatitsstationaryregimerepresentsavery
richsourceofvariouspossibleexactsolitonsolutions.Namelyweob-
tained asingle-pulsesoliton forthedensity difference,and several dis-
tinctpossiblesolitary-waveprofilesfortheactionpotentialincluding
apulse,akink,apulse-in-kinkstructureandsoon.Given theirlarge
numbers,theissueoftheirpropagationrequiressolvingnumerical-
lythecorrespondingspatio-temporalequation.Thisnumericalstudy
musttakeintoconsiderationseveraloftheseprofiles,whichrequires
aspecificframeworkofstudy.Thislastaspectwillbeconsideredina
separatework.
Acknowledgements
TheauthorsacknowledgethegrantsandsupportoftheCameroon
MinistryofHigherEducation,throughtheinitiativeforthemoderni-
sationofresearchinHigherEducation.
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterest.
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