Mathematical Neuroscience

186 Pages·2013·3.805 MB·English

by  Stanislaw Brzychczy, Roman R. Poznanski

Part I
Methods of Nonlinear Analysis
CHAPTER 1
Introduction to Part I
PartIisdevotedexclusivelytothetheoryandmethodsofnonlinearanalysisofinfinite
systems.Theseinfinitesystemsplayanimportantroleintheintegrativeaspectsof
neurosciencemodellingwithfunctionalanalyticmethodsofdescribingnonlinearphenomena
throughsettheoryandvectorspaces.Inmodernmathematicsvectorspacesusuallyarecarried
overfromCartesianspaces,andknowledgeofthesespacesisofhelpinanalyzingmore
generalvectorspaces(e.g.,Banachspaces,Hilbertspaces).However,whenitcomestothe
brainanduncoveringitsdynamicalnature,avectororpointinordinarythree-dimensional
spacenolongermakesanysense.Visualizingneuronaldynamicsbasedonordinary
three-dimensionalCartesianspacebecomescounterproductive,especiallyintheprocessof
integratingacrossscale.Thedynamicsofthebrainwithitsenormouscomplexitycannotbe
understoodthroughtheuseofelementaryfunctionsofclassicalmathematics.Firstly,weneed
abroaderdefinitionoffunctionsothatanoncommittalnotationandthenatureoftheruleor
mappingbecomeirrelevant.Secondly,weneedtoconsiderspacesofcontinuousfunctions
suchasinfinite-dimensionalEuclideanvectorspacesthatcarrynonlinearphenomenafrom
whichnewideascanbeharnessedfromdynamicalcontinuity.
Thenoncommittalapproachunderlyingthefoundationaltheoryofinfinitesystemsenables
newconstructsofmorepowerfulmodels,especiallyintheprocessofintegratingacrossscale.
Forinstance,thesetofstatesofbiologicalneuralnetworksisinfinite,topologically
continuous.Thisisbecausethestateofthenetworkisspecifiednotbythefiringofdigital
spikesinneurons,butintheelectricalpatternsbroughtaboutbytheelectricalfield.However,
thesetofstatesofartificialneuralnetworksisfinite,topologicallydiscreteduetospiking
beingthepredominantparadigmwhererefractorypropertiesofneuronsareincludedafterthey
havefired.
Themethodsofnonlinearanalysispresentedinthebookapplyonlytosecond-orderequations
arisinginneuroscience.Giventheimportanceofmorerealisticneuronalmodelsatthecellular
level,itcanbesaidthatneuronalmodelsarefoundationalunitsofthenervoussystemthatcan
beinterconnectedtoformdendritictrees,networksofdendritictrees,systemsofnetworks,
andsoon.Considerataperingcableofadendriticsegmentwithacontinuouslytaperingcable
asshowninFigure1.1.
MathematicalNeuroscience.http://dx.doi.org/10.1016/B978-0-12-411468-5.00001-6
©2014ElsevierInc.Allrightsreserved. 3
4 Chapter1
Figure1.1 Ataperingcablerepresentingasinglebranchofadendritictree.
TheequationdescribingthedepolarizationV = V(x,t)(inmillivolts)atthespacepointx and
timet foracontinuousradiusofthecylinderr(x)denotedbythedashedlinesisgivenby
∂V 1 ∂V(cid:3) ∂V(cid:4) (cid:5)
C = (cid:2) r2(x) − g (x,t;V)(V −V ). (1.1)
m ∂t 2ρr(x) 1+(dr(x)/dx)2 ∂x ∂x i i
Inessence,byassumingC = 1 µF/cm2, z(t,x) = V(x,t),andtheoperators
m
D = ∂V,D = ∂V,D2 = ∂2V,etc.,theaboveequationsturnouttobe
t ∂t x ∂x xx ∂x2
r(x) (dr(x)/dx)
D z(t,x)− (cid:2) D2 z− (cid:2)
t 2ρ 1+(dr(x)/dx)2 xx ρ 1+(dr(x)/dx)2
(cid:5)
D z − g (x,t;z)(z(t,x)−z ).
x i i
Definea(x) = √ r(x) , b(x) = √−(dr(x)/dx) ,and
2ρ 1+(cid:6)(dr(x)/dx)2 ρ 1+(dr(x)/dx)2
f(t,x,z(t,x), z) = g (t,x,z)(z(t,x)−z )sothattheaboveequationbecomes
i i
D z(t,x)−a(x)D2 z(t,x)+b(x)D z(x,t) = f(t,x,z(t,x),z). (1.2)
t xx x
Weconsidercoupledinfinitesystemsofequationswherethecouplingarisesfromboththe
connectivitybetweenbranchesofadendritictreeofasingleneuronandthesynapticcoupling
betweenindividualneuronsinnetworks.Wethereforeintroduceasuperscript“i”todenote,
forexample,themembranedepolarizationoftheithneuronwhere“i”rangesfrom1to∞.
Thecoefficientsa(x)andb(x)haveintroducedsubscripts“j”and“k”todenotethe
electrogeometricalinhomogeneityofthebranchingprocessesandweassumethereare“m”
branchesinaparticulardendritictree,sothat(1.2)becomes
(cid:5)m (cid:5)m
D zi(t,x)− a (x)D2 zi(t,x)+ b (x)D zi(x,t) = fi(t,x,z(t,x),z). (1.3)
t jk xjxk j xj
j,k=1 j=1
IntroductiontoPartI 5
Insettheorynotation(1.3)isconsideredaweaklycoupledinfinitesystemofequationsofthe
form
(cid:5)m
Fi[zi](t,x) := D zi(t,x)− ai (t,x)D2 zi(t,x)
t jk xjxk
j,k=1
(cid:5)m
+ bi(t,x)D zi(t,x) = fi(t,x,z(t,x),z) fori ∈ S (1.4)
j xj
j=1
for(t,x) = (t,x ,...,x ) ∈ D,where D isaboundedcylindricaldomain,
1 m
D := (0,T]×G, 0 < T < ∞, G ⊂ Rm isanopenandboundeddomainwhoseboundary
∂G isan(m −1)-dimensionalsufficientlysmoothsurfacem = 1,2,or3, S isanarbitraryset
ofindices,Fi, i ∈ S,arediagonaloperators1 whichareuniformlyparabolicin D,and
fi, i ∈ S,aregivenfunctions
fi: D×B(S)×C (D) → R, (t,x,y,s) (cid:7)→ fi(t,x,y,s), i ∈ S,
S
where S := {(t,x) : t = 0, x ∈ G}, σ := (0,T]×∂G isthelateralsurfaceand(cid:5) := S ∪σ
0 0
istheparabolicboundaryofthedomain D and D := D∪(cid:5) = [0,T]×G.Theright-hand
sides fi ofthesystem(1.4)arefunctionalswithrespecttothelastvariable.Theydescribethe
localnonlinearityduetothemembraneelectricalproperties,andthediffusiveinteractiondue
toelectricalcurrentflowinneurons.
ThefollowingexamplesofVolterrafunctionalsareconsidered:
(cid:7)
t
f (t,x,z) = m(t −τ)K(z(τ,x))dτ, (1.5)
1
(cid:7)0
t
f (t,x,z) = K(t,τ,x,z(τ,x))dτ, (1.6)
2
(cid:7)0 (cid:7)
t
f (t,x,z) = K(t,τ,x,ξ,z(τ,ξ))dτdξ. (1.7)
3
0 G
Equationsincludingsuchfunctionalsneedtobeconsideredinappropriatelychosendomains.
Otherexamplesinclude:
(cid:5)∞ (cid:5)∞
f1(t,x,z)=−z1(t,x) a1zk(t,x)+ b1z1+k(t,x), (1.8)
4 k k
k=1 k=1
1 OperatorsFi,i ∈S,arediagonalifFi dependsonzi onlyforalli ∈S.
6 Chapter1
(cid:5)j−1 (cid:5)∞
f j(t,x,z)= 1 aj−kzj−k(t,x)zk(t,x)−zj(t,x) ajzk(t,x)
5 2 k k
k=1 k=1
(cid:5)∞ (cid:5)j−1
+ bjzj+k(t,x)− 1zj(t,x) bj−k for j = 2,3..., (1.9)
k 2 k
k=1 k=1
and
(cid:7)
(cid:5)∞
f1(t,x,z)=−z1(t,x) a1(x,ξ)zk(t,ξ)dξ
6 k
k=1 G
(cid:7)
(cid:5)∞
+ B1(x,ξ)z1+k(t,ξ)dξ, (1.10)
k
k=1 G
(cid:7)
(cid:5)j−1
f j(t,x,z)= 1 Aj−k(x,ξ,η)zj−k(t,ξ)zk(t,η)dξdη
7 2 k=1 G×G k
(cid:7) (cid:7)
(cid:5)∞ (cid:5)∞
−zj(t,x) aj(x,ξ)zk(t,ξ)dξ + Bj(x,ξ)zj+k(t,ξ)dξ
k k
k=1 G k=1 G
(cid:5)j−1
− 1zj(t,x) bj−k(x) for j = 2,3,..., (1.11)
2 k
k=1
(cid:8) (cid:8)
where Aj(x,ξ,η)dx = aj(ξ,η)and Bj(x,ξ)dx = bj(ξ)arethenonnegative
G k k G k k
coefficientsofratesaj andbj,while f j, f j, f j,and f j areVolterrafunctionals.
k k 4 5 6 7
Systemswithright-handsidesofthistypeareconsideredasthediscrete
coagulation-fragmentationmodelswithdiffusion.Therearetwodifferent(discreteand
continuous)modelswhichdescribechangesintheconcentrationofclusters(proteins)
undergoingcoagulation(composition)orfragmentation(decomposition).Ifavariable
zj = zj(t,x)describestheconcentrationof j clusterscomprising j identicalbasicelements
knownasmonoclusters,thenthevariableisadiscreteone,takingnonnegativeintegervalues.
Themodelthusobtainedisadiscretecoagulation-fragmentationmodelbuiltofinfinite
countablesystemsofequations.
Thesemodelsareexpressedintermsoftheinfinitecountablesystemsofequationsoftheform
∂zj(t,x) (cid:5)m ∂2zj(t,x)
−d = f j (t,x,z), j ∈ N, (1.12)
∂t j ∂x2 5,6
l=1 l
for(t,x) ∈ (0,T]×G := D,whereG ⊂ Rm isaboundeddomainwithsufficientlysmooth
boundary∂G,withtheinitialcondition
z(0,x) =U (x) ≥ 0 for(t,x) ∈ G, (1.13)
0
IntroductiontoPartI 7
andthesealed-endboundarycondition
dzj(t,x)
= 0 for(t,x) ∈ σ, j ∈ N, (1.14)
dν
wherefunctions f j, f j aregivenby(1.8),(1.9),respectively,anddiffusioncoefficientsd are
4 5 j
positive.
Thismodelisexpressedintermsoftheinfinitecountablesystemsofequationsoftheform
∂zj(t,x)
−Aj[zj](t,x) = f j (t,x,z) for(t,x) ∈ D j ∈ N (1.15)
∂t 6,7
withinitialandboundaryconditions(1.13)and(1.14),wherethediffusionoperators,having
thedivergenceform
(cid:9) (cid:10)
(cid:5)m ∂ ∂zj(t,x)
Aj[zj](t,x) := dj(t,x) , j ∈ N
∂x lk ∂x
l,k=1 j k
areuniformlyellipticin D,diffusioncoefficientsdi arepositiveandfunctions fi, fi are
jk 6 7
givenby(1.10),(1.11),respectively,integraloperators.
Unfortunately,themonotoneiterativemethoddoesnotcoverthesystems(1.12)and(1.15).
Therefore,tosolveproblems(1.12)and(1.15),thetruncationmethodisapplied.
Infinitecountablesystemsofordinarydifferentialequationscanbeusedtosolvesome
problemsfornonlinearequationswiththeuseofthefinite-differencemethod.Inthisvery
importantmethodwhichiscalledthenumericalmethodoflines,spatialderivativesonlyare
discretizedbydifferenceexpressions,leadingtoaninfinitecountablesystemofequations.
Ontheotherhand,whenacontinuousvariable(e.g.,concentrationperunitvolume)taking
nonnegativerealvaluesisusedinthedescription,wearriveatacontinuousmodelof
coagulation-fragmentation,amodelbuiltofinfiniteuncountablesystemsofequations.
Adoptinganimportantassumptionthatthenumberofcreatedclustersisnotbounded,we
concludethatthenumbersofbothequationsandvariablesmaybeinfinite.
Continuousmodelsmaybeexpressedintermsofinfiniteuncountablesystemsofequationsof
thereaction-diffusiontype.Studyinginfiniteuncountablesystemsofequationsisfarfrom
simple.Inparticular,ifwehaveaninfinitesystemofequationsoftheform(1.4)
(nonstationaryreaction-diffusionequations),thenthestudymaybebroughtdowntoan
evolutionequationoftheform
∂u
− A(t)u = F(t,u) fort > 0, x ∈ Rm
∂t
8 Chapter1
withtheinitialcondition
u(0,x) = f(0,x) for x ∈ G,
where−Aisalinearoperatorand F isanonlinearreactionterm,consideredinan
appropriatelydefinedvectorspace.
Forsystem(1.4),wewillconsidertheso-calledFourierfirstinitial-boundaryvalueproblem:
findtheregular(classical)solutionz ofsystem(1.4)in D fulfillingtheinitial-boundary
condition
z(t,x) = φ(t,x) for(t,x) ∈ (cid:5). (1.16)
Weremarkthatitisnotpossibletosolveinfinitesystemsofequations.Therefore,anatural
approachtothestudyoftheexistenceanduniquenessofsolutionsofinitialandboundary
valueproblemsforinfinitesystemsofsuchequationsistostartwithafinitenumberofsuch
equations(thatis,studyingfinitesystemsofsuchequationsandthenextendingtheobtained
resultstotheinfinitesystems).However,seriousbasicdifficultiesemergehere.Thedifficulty
referredtoabovemaybesurmountedbyconsideringinfinitesystemsintheformofsystem
(1.4)andinthediagonalformoftheoperatorsFi, i ∈ S.Themethodwillconsistinthe
constructionofcertainapproximatingsequencestendingtosolutionsofinfinitesystemsin
suitablychosenvectorspaces.
Themethodofsuccessiveapproximationisoneofthemostbasicandsimplestmethodsfor
provingexistencetheoremsforcertaintypesofequations.Numerousvariantsofthismethod
arewellknown,usingdifferentconstructionsofasequenceofsuccessiveapproximationsand
differentwaysofprovingtheconvergence.
Toprovetheexistenceanduniquenessoftheregular(classical)andglobalinatimesolution
(i.e.,thesolutionisdefinedonthewholeinterval[0,∞])ofthisFourierfirstinitialvalue
problem,weapplyvariousmonotoneiterativemethodsinvectorspaces,partiallyorderedby
positivecones.Themethodsofupperandlowersolutionsarewell-knownmethodsofproving
theexistenceofsolutionsfornumerousclassesofinitial-boundaryvalueproblems.Applying
thesemethodsrequiresassumingthemonotonicityoftheright-handsidesoftheequations,
i.e.,thereactionfunctions fi(t,x,y,s), i ∈ S,withrespecttothefunctionargument y and
thefunctionalarguments.Wealsoassumetheexistenceofanorderedpairofaloweru and
0
anupperv solutionstotheproblemconsidered.Moreover,abasicassumptionon
0
fi(t,x,y,s), i ∈ S,isalsotheleft-handsideLipschitzcondition.Theseassumptionsarenot
typical,butweobtainconstructiveexistencetheoremsandaprioriinformationonasector
(cid:10)u ,v (cid:11)inwhichthosesolutionsmustremain.Weremarkthattheright-handsideLipschitz
0 0
conditionisusedtoensuretheuniquenessofasolution.
Wewillassumethatthereactionfunctions fi(t,x,y,s), i ∈ S,areVolterrafunctionals,i.e.,
theysatisfytheVolterraconditionwithrespecttothelastvariables.Thismeansthatthe
valuesofthesefunctionsdependonlyonthepasthistoryoftheprocess.Anexceptionally
IntroductiontoPartI 9
importantroleinthetheoryofnonlinearequationsplaysthenonlinearsuperpositionoperator,
whosepropertiesarestudiedincertainimportantlemmas.
Ineachcase,themonotoneiterativetechniquesasthemethodofprovingtheexistenceof
solutionsconsistofthreesteps:(i)constructinganondecreasingandanonincreasingsequence
ofloweranduppersolutions;(ii)showingtheuniformoralmostuniformconvergenceofthe
constructedsequencesofapproximatesolutions;and(iii)provingthatthelimitfunctionsare
solutions.
Toexaminetheexistenceanduniquenessofasolutionofinfinitesystems,sixmonotone
iterativemethodsareemployedinchapter4.Inthemethodofdirectiteration,Chaplygin
methodanditsmodifications,thesuccessivetermsoftheapproximationsequencesaredefined
assolutionsoflinearequations.Wealsopresentadifferentvariantofamonotoneiterative
method,inwhichweapplytheimportantideaofapseudo-linearizationofnonlinearproblems
(thesuccessivetermsofapproximationsequencesaredefinedassolutionsofequations).
ThemaindifferencebetweentheChaplyginmethodandothermethodsofmonotoneiterations
liesinthedefinitionsoftheapproximatingsequences.Thismethodrequiresastronger
assumptiononthefunctions fi,namelytheconvexityassumption,butyieldssequencesof
successiveapproximationsconvergingtothesolutionoftheproblemunderconsideration
morequicklythantheiterativesequences{u }and{v }constructedothermethods,underthe
n n
obviousassumptionthatallmonotoneiterativemethodsstartfromthesamepairofalower
solutionu andanuppersolutionv ,whoseexistenceisassumed.Thesesequencesare
0 0
convergedquadraticallytothesearchedsolution.
UsingChaplygin’sideawewillusethelinearizationoffunctions fi(t,x,y,s),i ∈ S,with
respecttobotharguments: y ands,simultaneously.Acharacteristicfeatureofthismethodis
constructionofthemonotonesequencesconvergingtothesolutionsoughtwithinthesector
(cid:10)u ,v (cid:11),andwhoseconvergentspeedisquadratic.Wewillusethismethodforinfinite
0 0
systemsofequationswithfunctionals.
Nextwepresentamonotonemethodofdirectiterationsinanunboundeddomain,when
consideredfunctionssatisfysomegrowthcondition.
Twoideasareworthmentioningamongtheproposedapproacheswhichenableustodeletethe
monotonicityconditionfortheright-handsideswithrespecttothefunctionandfunctional
arguments(oratleastthelatterone).Forsimplicity,wewillconsiderthecasewherethe
right-handsidedependsonlyonthefunctionalargument.
Thefirstideaconsistsofthemodificationofeachoftheequationsbyadding,tobothsidesof
theequations,thesamelineartermincludingtheunknownfunction,torenderthenew
right-handsidesoftheequationsmonotone,shouldthisbepossible.Thismeansthatwewill
bereplacingthemonotonicityconditionwithaweakerone,namely,asemi-monotonicity
condition.
10 Chapter1
Theotherideaconsistsofdividingthefunctionargumentsintotwogroups,dependingon
whethertheright-handsideofanequationisincreasingordecreasingwithrespecttoagiven
argument.Wheneverthisisthecase,wespeakaboutmixedmonotonicity.Thesplittingofthe
reactionfunctionleadstofourdifferenttypesoforderedpairsofloweranduppersolutions.
Thesituationisbestreflectedinthechoiceofthepairoffunctionswhichstartthe
approximationprocess.
Itshouldbenotedherethatbothideasweredevelopedtoanswertheneedtostudyspecific
finitesystemsofequationsandtheyadmitdirectextensiontotheinfinitesystemsofequations.
TheproofsoftheoremsarebasedonSzarski’sresultsconcerningthedifferentialinequalities
forweaklycoupledinfinitesystemsofequationsinwhichVolterrafuntionalsarise.The
theoremsonweakandstrongdifferentialinequalities,thecomparisontheoremanda
uniquenesscriterionaregiveninchapter3.Themaximumprinciple,whichplaysa
fundamentalrole(seeChapter3)isreflectedinwhatisknownasapositivitylemma,which
directlyfollowsfromthemaximumprinciple.
Theuniquenessofsolutionstotheproblemsconsideredinourworkisguaranteedbythe
LipschitzconditionandfollowsfromSzarski’suniquenesscriterion.
Themonotoneiterativemethodsarequiteusefulforcomputationofnumericalsolutionsof
finitesystemsofequations.
Chapter4givesmonotoneiterativemethodsforthecaseofinfinitesystemsofequations
whoseright-handsidedependsonz(t,·),i.e., fi(t,x,z(t,x),z(t,·)).
InChapter5,wegivesomeremarksonthemonotoneiterativemethodsandonthe
constructionsofupperandlowersolutions,forfinitesystemsofequations,andsomeremarks
concerningthepossibilityofextendingthesemethodstomoregeneralinfinitesystemsof
equations.
InChapter6,wedescribethetruncationmethodforinfinitecountableanduncountable
systemsofequationswhichplayaveryimportantroleamongapproximationmethods.The
applicationofthesemethodsleadstosequencesofapproximatesolutions{zNN,ψ}N=1,2,...,
which—underappropriateconditions—tendtotheexactsolutionofagivenproblemas
N → ∞.Letusobservethatwedonotneedtoknowthepreviousapproximationsto
determinethe Nthapproximationasasolutionoffinitesystemsofthefirst N equationswith
N unknownfunctionswhicharecalledtruncatedsystems.Thesefinitesystemsofequations
arediscretizedbythefinite-differencemethod.Therearethreebasicmonotoneiteration
schemesforfinite-differencesystems:aPicarditeration,amodifiedversionofaJacobi
iteration,andtheGauss-Seidelmethod.Theseiterationsalsoprovidenumerical
approximationofsolutionswhichisimportantinpracticalapplications.Iftheinitialiteration
isalwaysapairofknowncoupledloweranduppersolutionsoftheconsideredproblem,then
IntroductiontoPartI 11
thesequenceofPicarditerationconvergesfasterthanthesequenceofaGauss-Seideliteration
whichinturnconvergesfasterthanthesequenceoftheJacobiiteration.Thesemonotone
iterativeschemesprovidethesequencesoflineariterationswhichconvergemonotonicallytoa
uniquesolutionoftheconsideredtruncatedproblem.Thereforethismethodisveryusefuland
commonlyusedinpracticalcomputationofapproximatesolutions.
Tostudytheexistenceanduniquenessofasolutionofinfinitesystems,inChapter7weuse
thetopologicalfixedpointmethod.ConsideringpartiallyorderedBanachspaceofcontinuous
mappings,wegivesomenaturalsufficientconditionsfortheexistenceanduniquenessofthe
solution.TheaprioriestimateswhichappearinapplicationsoftheBanachandSchauder
fixedpointtheoremsareparalleltotheabove-mentionedassumptionsinthetheoryof
monotoneiterativemethods.Finally,toprovetheexistenceanduniquenessofglobalsolutions
inatime,i.e.,thesolutiondefinedonthewholeinterval[0,∞),weapplytheLeray-Schauder
fixedpointtheorembutfirstweextendsomeaprioriestimatesoftheFriedmantype.
InChapter8thetheoremonexistenceofasolutionoftheDirichletproblemforinfinite
systemsofequationsisprovedwithuseofacertainvariantoftheChaplyginmethod.The
resultsobtainedareappliedtostudytheasymptoticbehavioroftheglobaltime-dependent
solutionsinrelationtothesolutionsofthetime-independentproblem.Ithasbeenprovedthat
thelimitofthesolutionast → ∞isasolutionofthetime-independentproblem,obtainedby
applyingthemonotonemethodofassociatedloweranduppersolutions.
Thetoolsofmodernmathematicsinthedevelopmentofaframeworkthatunderpinsthe
understandingofinfinitesystemsingeneralisdeveloped(PartI).Infinitesystemsofequations
ariseinavarietyofphysicalcontexts,especiallywhenmodelingintheneurosciences.New
insightscanbeachievedthatwillleadtoabetterunderstandingofthebrainandnervous
systemfromamathematicalperspective(seePartII).