Open-source tools for dynamical analysis of Liley's mean-field cortex model

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ContentslistsavailableatScienceDirect

Journal

of

Computational

Science

j o u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o c s

Open-source

tools

for

dynamical

analysis

of

Liley’s

mean-ﬁeld

cortex

model

Kevin

R. Green

∗

,

Lennaert

van

Veen

FacultyofScience,UniversityofOntarioInstituteofTechnology,2000SimcoeStreetNorth,Oshawa,L1H7K4Ontario,Canada

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received15October2012

Receivedinrevisedform16April2013

Accepted22June2013

Availableonline5July2013

Keywords:

Mean-ﬁeldmodelling

Hyperbolicpartialdifferentialequations

Numericalpartialdifferentialequations

35Q92 65Y05

a

b

s

t

r

a

c

t

Mean-ﬁeldmodelsofthemammaliancortextreatthispartofthebrainasatwo-dimensionalexcitable medium.Theelectricalpotentials,generatedbytheexcitatoryandinhibitoryneuronpopulations,are describedbynonlinear,coupled,partialdifferentialequationsthatareknowntogeneratecomplicated spatio-temporalbehaviour.WefocusonthemodelbyLileyetal.(Network:ComputationinNeural Sys-tems13(2002)67–113).Severalreductionsofthismodelhavebeenstudiedindetail,butadirectanalysis ofitsspatio-temporaldynamicshas,tothebestofourknowledge,neverbeenattemptedbefore.Here,we describetheimplementationofimplicittime-steppingofthemodelandthetangentlinearmodel,and solvingforequilibriaandtime-periodicsolutions,usingtheopen-sourcelibraryPETSc.Byusingdomain decompositionforparallelization,anditerativesolvingoflinearproblems,thecodeiscapableofparsing somedynamicsofamacroscopicsliceofcorticaltissuewithasub-millimetreresolution.

1. Introduction

Modelsofcorticaldynamicscomeintwomainfamilies: neu-ronalnetworkmodelsandmean-fieldmodels.Theformerdescribe manyinteractingneurons,eachwiththeirowndynamicalrules, while thelatter describeelectrical potentials, generated collec-tivelybymanyneurons,ascontinuousinspaceandtime.These potentialscanbethoughtofasaveragesoveranumberof macro-columns,groupsofhundredsofthousandsofneuronsincolumnar structuresatthesurface ofthecortex. Bothofthesemodelling approachescanbeclassifiedasforward:theyattempttopredict thefuturestateofthecortex,giventhecurrentstateandasetof physiologicalparametervalues.Acomplementaryapproach,that canbe calledbackward, isto dividethecortex into interacting componentsthatcanberegardedasfunctionalunits,and com-putethestrengthofcouplingbetweentheseunits.Thebackward approachisoftenbasedonfunctionalMagneticResonanceImaging (fMRI)experiments.Apromisingmodellingstrategyistodescribe each functional component witha neuronal network or mean-fieldmodel,andthenhavetheminteractaccordingtoempirically determinedcoupling,thuscombiningtheforwardandbackward approaches[1].

∗ Correspondingauthor.Tel.:+19057218668x5368.

E-mailaddress:[email protected](K.R.Green).

Whenconsideringtheforwardmodellingofamacroscopicpiece ofcorticaltissue,afundamentaldifferencebetweentheneuronal networkandmean-fieldfamiliesisapparent.Amodeloftheformer kindshoulddescribebillionsofneurons,andmanytimesmore con-nectionsbetweenthem.Asdemonstratedbyrecentpublications, suchasbyIzhikevichandEdelman[2]orbytheBlueBrainteam[3], progressinsupercomputingallowsforthesimulationofeverlarger neuronalnetworks,thatreflectactualbraindynamics.However, itishardtoseehowtheoutputofsuchmodelscanbeanalyzed, otherthanbypurelystatistical techniques.Modelsof thelatter kind,incontrast,canbeanalyzedassmooth,infinite-dimensional dynamicalsystems.

An added advantageof the mean-fieldapproach is that the electrical potentials, which appear as dependent variables, are observable,macroscopicquantities.Anindirectmeasurementof thesefieldsisprovidedbytheelectroencephalograph(EEG)[4]. TheEEGisusuallymeasuredwithelectrodesonthescalpor,in exceptionalcircumstances,directlyonthesurfaceofthebrain.In eithercase,themeasuredsignalisnotthatofindividualneurons, butthatofmanyneurons,spreadoutoverafewsquare centime-tresormillimetres.Thus,thewaythesignalsofindividualneurons aresmearedoutbythespatialaveragingofmean-fieldmodelling is similartotheway theyare mixedupin EEGmeasurements. BecauseofthelinkbetweenthelocalmeanpotentialandtheEEG, mean-fieldmodelsaresometimescalledEEGmodels(e.g.[5,6]). Thegeometryofthecorticalsurface,however,is nottakeninto considerationindoingso.Thissurfaceisfolded,and electrocor-ticalactivitywillresultindifferentEEGsignalsdependingonthe locationandorientationofthegeneratingtissue.Amoredirectlink

http://dx.doi.org/10.1016/j.jocs.2013.06.001

Open access under CC BY license.

CORE Metadata, citation and similar papers at core.ac.uk

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betweenthemodelvariablesandmeasurementsmaybegivenby theLocalFieldPotential(LFP),whichistypicallymeasuredinvitro orunderanaesthesia.

Theoriginofmean-fieldmodellingliesinthe1970s,when pio-neerslikeFreeman[7],WilsonandCowan[8]andLopesdaSilva etal.[9]startedtomodelcomponentsofthehumancortexwith continuousfields.Overthepastfourdecades,mean-fieldmodels havebeenusedtostudyarangeofopenquestionsinneuroscience, suchasthegenerationofthealpharhythm,8–13Hzoscillationsin theEEG(see,e.g.,[9,5]),epilepsy(see,e.g.,[10–12])andanaesthesia [6].Inadifferentcontext,theyareusedinmodelsfor sensorim-otorcontrol,patterndiscriminationandtarget tracking[13].As discussedabove,mean-fieldmodelsalsoappear ascomponents ofcombinedforward-backwardmodelsthat aimtocapturethe functioningofthecortexasawhole,suchasinHoneyetal.[14].

Althoughmean-ﬁeldmodelshavebeenusedinallthese sett-ings,littleanalysishasbeendoneontheirbehaviourasspatially extendeddynamicalsystems.Inpart,thisisduetotheir stagger-ingcomplexity.TheLileymodel[15]consideredhere,forinstance, consistsoffourteencoupledPartialDifferentialEquations(PDEs) withstrongnonlinearities,imposedbycouplingbetweenthemean membranepotentials andthemeansynapticinputs. Themodel can bereduced to a system of Ordinary Differential Equations (ODEs)byconsideringonlyspatiallyhomogeneoussolutions,and theresultingsystemhasbeenexaminedindetailusingnumerical bifurcationanalysis(see[16]andreferencestherein).Inorderto computeequilibria,periodicorbitsandsuchobjectsforthePDE model,weneedaﬂexible,stablesimulationcodeforthemodel anditslinearizationthatcanruninparalleltoscaleuptoadomain sizeof about2500cm2_, _the_size _of_a _full-grown _human_cortex.

Wealsoneedefﬁcient,iterativesolversforlinearproblemswith large,sparsematrices.Inthispaper,wewillshowthatallthiscan beaccomplishedintheopen-sourcesoftwarepackagePETSc[17]. OurimplementationconsistsofanumberoffunctionsinCthatare availablepublicly[18].

Thegoal of this computational workis to parse the spatio-temporaldynamics ofa full-fledged mean-fieldmodel.We will presentthenumericalimplementationofalgorithmsforthe com-putationofequilibriaandtime-periodicsolutionsandstudytheir stabilityandparameterdependence. Thus,ourgoalissimilarto thatofCoombesetal.,whoanalyzed“spots”:rotationally symmet-ric,localizedsolutionsinamodelofasingleneuronpopulationin twodimensions[19].Thechallengeliesingivinguptherestriction toasinglepopulation,asinglespacedimension,orsolutionswith afixedspatialsymmetry.

1.1. Liley’smodel

ThemodelweusewasﬁrstproposedbyLileyetal.[15].The dependentvariablesarethemeaninhibitoryandexcitatory mem-branepotential,hiandhe,thefourmeansynapticinputs,originating

fromeitherpopulationandconnectingtoeither,Iee,Iei,IieandIii,

andtheexcitatoryaxonalactivityinlong-rangeﬁbres,connecting toeitherpopulation,eeandei.Themodelequationsare:

k

∂

hk

_∂

(x,t) t =hrk−hk(x,t)+ heq_ek−hk(x,t)

_heq ek−hrk

Iek(x,t) +h eq ik−hk(x,t)

_heq ik−hrk

Iik(x,t) (1)

∂

t+ek

2 Iek(x,t)=eekek{NekˇSe[he(x,t)]+pek+ek(x,t)} (2)

∂ ∂t+ik

2 Iik(x,t)=eikik{NikˇSi[hi(x,t)]+pik} (3)

∂

t+

v

2 −3 2

v

2

_∇

2

ek(x,t)=N˛ek

v

22Se[he(x,t)] (4) Sk[hk]=Skmax

1+exp

−√2hk−k k

−1 (5) whereindexk={e,i}denotesexcitatoryorinhibitory.Themeaning oftheparameters,alongwithsomephysiologicalboundsandthe valuesusedinourtests,aregiveninTable1.Adetailed descrip-tionoftheseequationscanbefoundinRefs.[15,16].Here,wewill focusontheaspectsofthemodelmostrelevantforthenumerical implementation.

Therearetwosourcesofnonlinearity,relatedtothecoupling ofthesynapticinputstothemembranepotentialandviceversa. Theformerconnectionisquadraticallynonlinear,whilethelatter isgivenbythesigmoidalfunctionSk,whichdescribestheonsetof

ﬁringasthepotentialexceedsthethresholdvaluek.These

nonlin-earitiestendtoformsharptransitionsofthepotentialsacrossthe domain.Thatisonereasonwhyweoptedforafinite-difference discretizationoverapseudo-spectralapproach.Spectralaccuracy wouldbeoflimitedvalueinthepresenceofsteepgradientsand thefinite-differenceschemecanbeparallelizedmuchmore effi-ciently.Thesecondreasonisthatwewouldliketobeabletochange thegeometryofthedomainandtheboundaryconditionsinfuture work.Thefinite-differenceschemeismoreflexibleinthatrespect. Theonlyspatialderivativesinthemodelarethoseinthe equa-tions for the long-range connections. These are damped wave equations.WewilldiscretizetheLaplacianusingafive-point sten-cilonarectangulargrid.Inpreviouswork,BojakandLileychose asecond-ordercentreddifferenceschemeforthetimederivatives [6].Adisadvantageofthisapproachisthatthestabilitycondition ofthisschemedictatesthatwesetthetimestepinversely pro-portionaltothegridspacing.Inpractice,theyusedatimestepof 0.05ms.Toavoidthisobstacle,wewanttouseimplicit timestepp-ing,andhavecurrentlyimplementedtheunconditionallystable implicitEulermethod,asdescribedinSection3.

Followingearlierworkonthismodel(e.g.[6,20]),weadopt peri-odicboundaryconditionsinbothdimensions.Thisisacommon choiceinthestudyofmean-fieldmodels,andispartiallyjustified bytheobservation,thateachpartofthecortexisconnectedtoeach otherpart.Adiscussionofthisargumentcanbefoundinchapter11 ofNunezandSrinivasan[4].Aresultofthischoiceisthatthemodel PDEswillbeequivariantundertranslationsandreflections.This equivariancehasconsequencesforthebehaviourofthemodel.An in-depthdiscussionoftheseconsequencesisbeyondthescopeof thecurrentpaper,butinSection6wewilldecribehowtocompute periodicsolutionsfortheequivariantsystem.

Otherauthorshaveusedthismodelwithanadditionaldiffusive termintheequationsforthemembranepotentialstomodelgap junctions [21]. Inclusion of these terms can drasticallychange thebifurcation behaviour, as theycan causeTuring transitions to space-dependentequilibria. Without theadditional terms, a Hopfbifurcationfromaspatiallyhomogeneousequilibriumtoa spacedependentperiodicorbitorasaddle-nodebifurcationofthe equilibrium oftenappear tobetheprimaryinstability.The gap junctiontermscanreadilybeincludedinourimplementation,and inSection5wewilldescribehowtosolveforequilibriumstates thatmaydependonspace.

Wewilltestourimplementationbycomparingto,and extend-ing,thecomputationsofoscillationswitha40Hzcomponentby BojakandLiley [20].Oscillationswiththisfrequency arecalled gamma oscillations,and have been hypothesized to aidin the

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Table1

Meaning,rangesandvaluesforthemodelparameters.ThevaluesusedforthetestspresentedinSection7aretakenfromRef.[20].Wemakenotethatpieandpiiarenot

listedhere,astheyaresetto0.

Parameter Deﬁnition Minimum Maximum Value Units

hr

e Restingexcitatorymembranepotential −80 −60 −72.293 mV

hr

i Restinginhibitorymembranepotential −80 −60 −67.261 mV

e Passiveexcitatorymembranedecaytime 5 150 32.209 ms

i Passiveinhibitorymembranedecaytime 5 150 92.260 ms

heq

ee Excitatoryreversalpotential −20 10 7.2583 mV

heq

ei Excitatoryreversalpotential −20 10 9.8357 mV

heq

ie Inhibitoryreversalpotential −90 hrk−5 −80.697 mV

heq

ii Inhibitoryreversalpotential −90 hrk−5 −76.674 mV

ee EPSPpeakamplitude 0.1 2.0 0.29835 mV

ei EPSPpeakamplitude 0.1 2.0 1.1465 mV

ie IPSPpeakamplitude 0.1 2.0 1.2615 mV

ii IPSPpeakamplitude 0.1 2.0 0.20143 mV

ee EPSPcharacteristicrateconstant 100 1000 122.68 s−1

ei EPSPcharacteristicrateconstant 100 1000 982.51 s−1

ie IPSPcharacteristicrateconstant 10 500 293.10 s−1

ii IPSPcharacteristicrateconstant 10 500 111.40 s−1

N˛

ee No.ofcortico-corticalsynapses,targetexcitatory 2000 5000 3228.0 –

N˛

ei No.ofcortico-corticalsynapses,targetinhibitory 1000 3000 2956.9 –

Nˇ

ee No.ofexcitatoryintracorticalsynapses 2000 5000 4202.4 –

Nˇ_ei No.ofexcitatoryintracorticalsynapses 2000 5000 3602.9 –

Nˇ_ie No.ofinhibitoryintracorticalsynapses 100 1000 443.71 –

Nˇ

ii No.ofinhibitoryintracorticalsynapses 100 1000 386.43 –

v Axonalconductionvelocity 100 1000 116.12 cms−1

1/ Decayscaleofcortico-corticalconnectivity 1 10 1.6423 cm

Smax

e Maximumexcitatoryﬁringrate 50 500 66.433 s−1

Smax

i Maximuminhibitoryﬁringrate 50 500 393.29 s−1

e Excitatoryﬁringthreshold −55 −40 −44.522 mV

i Inhibitoryﬁringthreshold −55 −40 −43.086 mV

e Standarddeviationofexcitatoryﬁringthreshold 2 7 4.7068 mV

i Standarddeviationofinhibitoryﬁringthreshold 2 7 2.9644 mV

pee Extracorticalsynapticinputrate 0 10,000 2250.6 s−1

pei Extracorticalsynapticinputrate 0 10,000 4363.4 s−1

communicationbetweengroupsofneurons[22].Bothsimulations andexperimentsindicatethatgammaoscillationsoccurinsubjects performingcognitivetasks(see,e.g.[23]andreferencestherein). GammabandactivitywasfoundintheLileymodeldespitethefact thatitwasinnowaytunedorformulatedtoproducethisbehaviour. TheparametervaluesforthisexperimentarelistedinTable1. The40Hzoscillationsarisespontaneouslyifthenumberoflocal inhibitory-to-inhibitoryconnectionsischangedslightly.We intro-duceascalingparameterrbyreplacingN_iiˇ→rN_iiˇ.Thisistheonly parameterthatwillbevariedinourtests.

1.2. PETScoverview

Ratherthancreatingourcodefromscratch,weoptedtowork withthePortable,Extensible Toolkit forScientiﬁc Computation (PETSc):anopen-source,objectorientedlibrarythatisdesigned forthescalablesolutionandanalysisofPDEs[24,17].PETScis writ-tenintheClanguage,andisusablefromC/C++aswellasFortranand Python.WeusePETScinconjunctionwiththeScalableLibraryfor EigenvalueProblemComputations(SLEPc)[25],forthe computa-tionofeigenspectraofequilibriumandperiodicsolutions.Sinceour implementationusessomefeaturesofPETScthatarerecent addi-tionsandarestillbeingmodiﬁed,weusethedevelopmentversion ofbothprojects.

PETScissplitupintomultiplecomponentstoaddressthe var-iousproblemsassociatedwithsolvingPDEsnumerically.Forour purposes,wetreattheDMcomponent,whichhandlesthe topol-ogyofthediscretization,asthemostfundamental, fromwhich wecaneasilyderivememoryallocationandcommunicationfor distributed vectors(Vec) and matrices (Mat).With vectorsand matrices,wecannowsolvelinearsystems,suchasthosethatarise inNewtoniterationforimplicittime-steppingandthecomputation ofequilibriaandperiodicorbits.PETSc’scomponentforthisiscalled

KSP,andithasnumerousiterativesolversimplemented,aswellas preconditioners,(PC),toincreaseconvergencerates.Forimplicit time-stepping,forexample,weuseGMRES,preconditionedwith incompleteLU(ILU)factorization,combinedwiththeblockJacobi method[26,27].Ontopofthelinearsolverscomethenonlinear solvers,PETSc’sSNEScomponent,whichimplementsafew differ-entmethods,suchasgloballyconvergentNewtoniterationwith linesearch[28]. Finally,PETSc providesa timestepping compo-nent,TS,toobtaintimedependentsolutions.Implementedhere arenumerousexplicitandimplicitschemessuchasadaptive step-sizeRunge–KuttaandimplicitEuler.Theimplicitschemesmake useoftheSNEScomponent.Aschematicofthehierarchydiscussed herecanbefoundinFig.1.

Forourdynamicalsystemscalculationswewillfrequentlyneed tocomputespeciﬁceigenvaluesandeigenvectorsforsystem-sized matrices.Forthisend,weuseSLEPc,whichimplementsiterative eigenvaluesolversusingPETScVecandMatdistributeddata struc-tures.Thecomponent ofSLEPcthat weuseis EPS,whichhasa few algorithmsforiterativelysolvingeigenproblems.Itsdefault algorithmisKrylov–Schuriteration.

2. Modelimplementation

2.1. Geometry

Following earlier work by Bojak and Liley (e.g. [6,20]) we considerthePDEsonarectangulardomainwithperiodicboundary conditions. Onthis domain,we usea rectangulargridof Nxby

Ny points. In thetests presented in Section7,the domainand

thegridaresquare.PETSc allowsformorecomplicated domain shapesandgrids,thatcanbeencodedintheDMcomponent, inde-pendentofthehigher-levelcomponents.Thischoiceofperiodic boundaryconditionsisoneofcomputationalconvenience.Aswe

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Fig.1. SchematicrepresentationofthecomponentsofPETScandSLEPcusedinourcode,andtheirrelativehierarchy.

havenoformulationofboundaryconditionsthatcomefromthe physiology,weadoptperiodicboundariesandlookatphenomena thatareonlengthscalesbelowthat ofthesystemsize.Abrief demonstrationofthiscanbeseeninFig.2anditscaption.

WithinDM,PETScprovidesasimplersubcomponent,DMDA,for workingwithﬁnitedifferences onstructuredgridssuchasour rectangle.Ifwespecifyastenciltouseforthespatialderivatives in the DMDA, PETSc will automatically handle numerous things forparallelexecution,suchasmemoryallocationand the com-municationsetupfordistributedvectorsandforthedistributed Jacobianmatrix.

2.2. Fields

TomakeuseofPETSc’ssolvers,themodelmustbewrittenasa systemofequationsthatisﬁrstorderintime.Thisweachieveby introducingnewstatesJjkand ekaccordingto

∂

Ijk

∂

t =Jjk−jkIjk (6)

∂

Jjk

∂

t =ejkjk{N ˇ jkSj[hj]+jk+pjk}−jkJjk (7)

∂

ek

∂

t = ek−

v

22 ek (8) 5 15 25 35 45 104 105 106 107 108 L %o f lo ca lc onne cti on s (1,0) (1,1) (2,0) (2,1) (2,2) (3,3) ≈9.3cm ≈9.2cm ≈9.1cm (0,0)

Fig.2. NeutralstabilitycurveforthespatiallyhomogeneousequilibriumoftheLiley

modelwithparameterssetaccordingtoTable1.Shownisthescalingparameter,

r,versusthelineardomainsize,L,andwavenumbers k=(kx,ky)areshownin

parenthesis.Whenvaryingr,onlyforverysmalldomainstheprimaryinstability

isspatiallyhomogeneous.Fordomainsizesover12.5cm×12.5cm2_the_location

oftheprimaryinstabilityapproachesr=1.045andthelengthscaleoftheleading

instabilityapproachesL/k=9.3cm.

∂

ek

∂

t =

v

22_N˛ ekSe[he]+ 3 2

v

2

_∇

2 ek−

v

22 ek, (9)

withindicesj,k={e,i}.

Weoptedtouseastruct,seeninCode2.1,tostoretheﬁelds, ratherthanatriplyindexedarray.

Code2.1. Structfortheﬁelds.

typedef struct Field{

PetscReal he, hi,

Iee, Jee,

Iie, Jie,

Iei, Jei,

Iii, Jii,

phi ee,psiee,

phiei, psiei;

}Field;

This allows the code tobe more readable in the function and Jacobianevaluation routines.Forexample,one accessestheee

componentatgridpoint(xi,yj)simplyasu[j][i].phiee,

pro-videdthattheelementsofthearray(Field **u;)arestoredon theprocessorinwhichthecallismade.

2.3. Parameters

Allofthemodelparametersarestoredinastructdesignated astheapplicationcontext.TheapplicationcontextishowPETSc getsproblemrelatedparametersintotheuser-deﬁnedfunctions neededbyitssolvers.

Code2.2. Applicationcontextstructwiththemodelparameters.

typedef struct AppCtx{

PassiveReal hre, hri,

taue, taui,

heqee, heqie,

heqei, heqii,

Gammaee, Gammaie,

Gammaei, Gammaii,

gammaee, gammaie,

gammaei, gammaii,

Nalphaee, Nalphaei,

Nbetaee, Nbetaie,

Nbetaei, Nbetaii,

v, Lambda, Smaxe, Smaxi, mue, mui, sigmae, sigmai, pee, pei, pie, pii; ... }AppCtx;

Similar to the ﬁelds, this allows readable code for the parameters. For example, one accesses the ie parameter as

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user->Gammaie,ifuserisdeﬁnedasthepointerAppCtx *user;. Howtheparametersshowupinourstructfortheapplication con-textisshowninCode2.2.

2.4. Usersuppliedfunctions

Inadditiontothestructslistedabove,weneedtoprovidePETSc with(atleast)aCfunctionthatcomputesthevectorfieldforagiven state.WecallthisfunctionFormFunction,andfromthisPETScis capableofapproximatingtheJacobianwithvariousfinite differ-encemethods.However,wealsosupplyaCfunctiontoexplicitly computetheJacobian,namedFormJacobian,becausethisallows formoreefficientcalculations,especiallywhenlookingatstepping thevariationalequationsinSection4.

3. Timestepping

WecurrentlyusetheimplicitEulermethodtotime-stepthe dis-cretizedequations.AsmentionedinSection1.1,thisallowsusto takelargertimestepsthanfeasiblewithexplicitmethods.Since weareaimingtocomputeperiodicorbits,ratherthanto gener-atelongtimeseries,theﬁrstorderaccuracyofthemethodisnot anissue.Onceaperiodicorbitiscomputed,thetime-stepsizecan bereducedtoincreaseaccuracy.Anotheroptionistouse Richard-sonextrapolationtoincreasetheorderofaccuracy,usingthesame nonlinearsolvingasdescribedbelow.

3.1. Mathematicalbasis

Wesymbolicallywritethedynamicalsystemas ˙

u =f(u), f:RN_→_RN ₍₁₀₎

whereNisthetotalnumberofunknownsafterdiscretization,inour case14×Nx×Ny.TheimplicitEulerschemefortimeintegrationis

givenby

un+1=un+dtf(un+1) (11)

wherethesubscriptrepresentsthestepnumber,dtthestepsize, andu0theinitialconditions.Thisnonlinearequationissolvedby

Newtoniteration: uk+1_n+1=uk

n+1+duk, (12)

wherethesuperscriptdenotestheNewtoniterate,andduk_is_the

solutiontothelinearsystem

I−dt

_∂

∂

f u

uk n+1

duk₌_d_t_f₍_uk n+1)−ukn+1+ukn, (13)

where

∂

f/

∂

udenotestheN×NJacobianmatrix.Providedthatthe initialapproximation, u0

n+1,is close enough tothe actual

solu-tionofEq.(11),thisiterationshouldconvergequadratically.This isachievedbymakingtheinitialapproximationtheresultofan explicitEulerstep

u0

n+1=un+dtf(un). (14)

Aswescaleupthesizeofourproblems,itbecomesthelinear solveinEq.(13)thattakesmostofthetime.Thisproblemishandled byusingGMREStosolvethelinearsystem.Forlargetimesteps,the spectrumofthematrixinEq.(13)isspreadout,andweneedto preconditionitforiterativesolving.WemakeuseILU,whichhas showntobereliableforthistypeofproblem[29,30].Ifweusemore thanoneprocessor,PETScusesdistributedstorageforthematrix, andcombinesILUwithblockJacobipreconditioning.

3.2. Implementation

PETScprovidesasimpleinterfacefortimesteppinginitsTS com-ponent.ThebasiccoderequiredtosetupaTSisgiveninCode3.1. WithaTSsetuplikethis,thetimesteppingparametersaresetfrom commandlineargumentsatruntime.Forexample,todoimplicit Eulertimesteppingfor40.67mswitha timestepof0.1ms,one needstoprovidethearguments

-tstype beuler -tsdt 0.1 -tsfinaltime 40.67. Inthisspeciﬁccase,sincetheﬁnaltimeisnotanintegernumber oftimesteps,PETScwillsteppastit,andinterpolateatthedesired time.

Code3.1. PETSccodeforsettingupandrunningthetimestepping. FormFunctionTS andFormJacobianTS areuserprovided func-tionsthatcomputetherhsofEq.(10),anditsJacobianrespectively. JisanappropriatelyallocatedmatrixtoholdtheJacobian,andua vectortoholdthesolutions.

TS ts;

TSCreate(PETSCCOMMWORLD,&ts);

TSSetProblemType(ts,TSNONLINEAR); TSSetExactFinalTime(ts); TSSetRHSFunction(ts,PETSC NULL,FormFunctionTS,&user); TSSetRHSJacobian(ts,J,J,FormJacobianTS,&user); TSSetFromOptions(ts); TSSolve(ts,u);

4. Steppingofthevariationalequations

Thevariationalequationsforthedynamicalsystemarewritten as ˙

v

=

_∂

∂

f u

u

v

,

v

∈RN (15)

andmustbeintegratedsimultaneouslywiththedynamical sys-tem(10).Solvingthevariationalequationsallowustocompute thestabilityofsolutions,andisalsoanessentialingredientforthe treatmentofboundaryvalueproblemssuchasthosethatarisein thecomputationofperiodicorbits.

PerformingimplicitEulertimesteppingonthevariationalEq. (15)requiressolutionsofthelinearproblems

I−dt

_∂

∂

f u

un+1

v

_n+1=

v

n. (16)

Since wealready havetheJacobianof thedynamicalsystemat timestepn+1,steppingthevariationalequationsrequiresonlyone additionalN×Nlinearsolvepertimestep.

4.2. Implementation

In PETSc, we implement thetimestepping ofthe variational equationsasa MATSHELL,effectivelyviewingitasa matrix-free multiplication.WithinaMATSHELL,oneneedstoprovidea con-text for storing the relevant data and write functions for the desiredmatrixoperation(s).Forexample,wepointtheoperation MATOPMULTtoafunctionthattakestheinitialstateofthe varia-tionalsystem

v

(0)asinput,andoutputstheresult

v

(T)attheendof thetimestepping.Thecontextweuseforthetimesteppingofthe variationalequationsisshowninCode4.1.Thefunctionweprovide forMATOPMULTworksbyﬁrsttakingastepoftheTS,thenloading theJacobiancomputedfromthatstepandsolvingEq.(16).Thisis repeateduntiltheTSreachesitsend.

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Code4.1. TheMATSHELLcontextfortimesteppingofthe varia-tionalequations.TheTSholdstherelevantinfoforsteppingthe dynamicalsystem.

typedef struct PeriodIntegrationCtx{

// timestepping of the original eqn

TS ts;

Mat tsJac;

Vec initState,endState,fullSol;

// additional requirements for variational eqn

Mat J,eye;

KSP ksp;

}PeriodIntegrationCtx;

TheMATSHELLthusdeﬁnedcanbeusedbySLEPcfortheiterative computationofeigenpairs.Inparticular,wewillusethisapproach tocomputetheFloquetmultipliersofperiodicorbits.

5. Equilibria

HavingsetupthefunctionFormFunctionfortherighthand side of the dynamical system, and its Jacobian computation FormJacobian,alsousedfortimeintegration,wecansetup equi-libriumcalculationsusingPETSc’sSNEScomponentwithverylittle effort.

Equilibriumsolutionstothedynamicalsystem(10)aresolutions thatsatisfy

f(u)=0. (17)

Tosolvethis,wecansetupaNewtoniterationscheme

uk+1=uk+d_uk (18)

withducomingfromthesolutionofthelinearsystem

∂

f

∂

u

uk duk₌_−f₍_uk₎_. ₍₁₉₎

Aswiththetimestepping,iftheinitialguessisgoodenoughthis willconvergequadraticallyprovidedthat(

∂

_f/

∂

_u)|_ukisnonsingular.

Unlikethecaseoftimestepping,though,wedonotalwayshavea waytoproduceaninitialapproximationthatisgoodenough.For stableequilibriumsolutions,wecanusetimesteppingtogetclose toanequilibrium,butthiswillnotworkforunstableequilibria.One possiblesolutionisusingglobally convergentNewtonmethods. Usingsuchmethodswecanﬁndequilibriafromverycoarseinitial data,atthecostofcomputingmanyiterations.Thelinesearch algo-rithmandthetrustregionapproach(see,e.g.[28])areimplemented intheSNEScomponent.

Stabilityof equilibrium solutions followsfrom thespectrum oftheJacobian. Duetodiscrete symmetries ofa domain,these canappearingroups.Onasquaredomain,forinstance,asingle eigenvaluewillbeassociatedwithuptoeighteigenvectors,with wavenumbers(±kx,±ky)and(±ky,±kx).

AsdiscussedinSection1.1,themodelisalsoequivariantunder translatationsinbothdimensions.Inthepresenceofthissymmetry, itismorenaturaltosearchforrelativeequilibria,alsocalled travel-lingwaves.Thisleadstotheintroductionoftwoextraunknowns, thatcanbethoughtofasthewavevelocities,intosystem(17),and anextensionbytwoequationsoftheassociatedlinearsystem(19). However,sincewehavesofaronlyobservedspatially homoge-neousequilibriumstates,wewilldiscussthisadjustmentinSection 6onperiodicsolutions.

5.2. Implementation

SettingupandusinganonlinearsolverwithinPETScis straight-forward,asshowninCode5.1.ThedefaultalgorithmusedbySNES isNewton’smethodwithlinesearch.

Code5.1. Codesnippetforsolvingforequilibria.Vectorsrand uarepreallocated,withubeingtheinitialapproximation,andJa preallocatedmatrixfortheJacobian.

SNES snes;

SNESCreate(PETSC COMM WORLD,&snes);

SNESSetFunction(snes,r,FormFunctionSNES,&user); SNESSetJacobian(snes,J,J,FormJacobianSNES,&user); SNESSetFromOptions(snes);

SNESSolve(snes,PETSCNULL,u);

6. Periodicsolutions

TheprimaryinstabilityintheLileymodelisoftenaHopf bifur-cation,andperiodicorbitshavebeenshowntoplayanimportant roleinthedynamicsofODEreductionsofthemodel(e.g.[16,31]). However,spacedependentperiodicorbitshavenotpreviouslybeen computedandstudied.UsingPETScdatastructuresforbordered matrices,inconjunctionwithaMATSHELL,wecansolveforperiodic orbitsbasedonthetimesteppingdescribedinSections3and4. 6.1. Mathematicalbasis

Relativeperiodicorbitssolvetheboundaryvalueproblem

F(u,t)=(t,u)−Tabu=0, (20)

whereistheﬂowofthedynamicalsystem(10),tistheperiod, andTabu(x,y)=u(x−a,y−b)thetranslationoperator.Ourstrategy

forsolvingthisequationisessentiallythatofSanchezetal.[32], namelyNewtoniterationscombinedwithunconditionedGMRES iteration.LinearizingEq.(20)gives

(Du(u,t)−I)du+f((u,t))dt+Tab

_∂

∂

u_xda+Tab

∂

_∂

_yudb

=−F(u,t), (21)

whereDuisamatrixofderivativesoftheﬂowwithrespecttoits

initialcondition.Uponconvergence,thisisthemonodromymatrix oftheperiodicorbit.TheresultisNequationsinN+3unknowns, whichmustbeclosedbyphaseconditions.Forthetemporalphase, weoptedtohandlethiswithaonedimensionalPoincarésection, whichgivesaconstraintontheNewtonupdatestep:

[Du(u,t)]k,.du+fk((u,t))dt=C−k(u,t), (22)

where[Du(u,t)]k,.denotesthekthrowofthematrixDu,andC

thedesiredvalueoftheﬂowonthePoincarésection.Forthespatial phase,werestricttheupdatesteptobeorthogonaltothegenerators ofspatialtranslations:

∂

u

∂

xdu =0,

∂

u

∂

ydu =0. (23)

Thesechoicesgivetheborderedsystem

⎡

⎢

⎣

(Du(u,t)−Tab) f((u,t)) Tab∂u ∂x Tab ∂u ∂y [Du(u,t)]k,. fk((u,t)) 0 0 ∂u ∂x 0 0 0 ∂u ∂y 0 0 0

⎤

⎥

⎦

⎡

⎢

⎣

du dt da db

⎤

⎥

⎦

=

⎡

⎢

⎣

−F(u,t) C−k(u,t) 0 0

⎤

⎥

⎦

, (24)

(7)

thesolutiontowhichcanbeusedtoupdatetheapproximate solu-tion

⎡

⎢

⎣

un+1 tn+1 an+1 bn+1

⎤

⎥

⎦

=

⎡

⎢

⎣

un tn an bn

⎤

⎥

⎦

+

⎡

⎢

⎣

du dt da db

⎤

⎥

⎦

. (25)

ThematrixDu isdense,soweshouldavoidcalculatingand

storingitexplicitly.Iterativesolvingofthelinearproblem,(24), requiresthecomputationofmatrix-vectorproducts,whichare con-structedfromtheintegrationofthevariationalEq.(15)with

v

(0)= duandthevectorﬁeldf((u,t))attheendpointofthe approxi-matelyperiodicorbit.SincethegoverningPDEisdissipative,most oftheeigenvaluesofthemonodromymatrixareclusteredaround zero.ThisaidstheconvergenceofGMRES,withoutany precondi-tioning.Sanchezetal.[32]provideboundsforthenumberofGMRES iterationsfortheNavier–Stokesequation,andtheconvergencewe observefortheLileymodelisqualitativelysimilar.

6.2. Implementation

Theproblemof creatinga bordered matrix systemina dis-tributedenvironmentisnotatrivialone.Thespeciﬁccasethatwe haveisonevector,u,thatissparselyconnectedanddistributed amongprocessors,andthreeparameterst,a,andbthatmustexist andbesynchronizedacrossallprocessors.

PETSc’sDMmodulehassomerecentlyintroducedfunctionality thatallowsustohandlethisinastraightforwardway,lettingus makeuseoftheDMDAalreadyusedintheothertypesofcalculations. DMRedundantcanbe usedfor thea,b,t componentsof our extendedsystem,asithastheprecisebehaviourthatwerequire. Next,weuseaDMCompositetojointogethertheDMDAofthegrid withtheDMRedundantoftheperiodandtranslations.Wecanthen derivevectorsfromthisDMComposite,andusethesevectorsfor PETSc’siterativelinearsolvers.PETSccodethatillustratesthisidea isshowninCode6.1.

Code6.1. AdditionalDMpiecesforextendedvectorsasinEq.(25), assumingthatdaistheDMassociatedwiththegridstructure.The numericalargumentsinDMRedundantCreaterepresentthe pro-cessorwheretheredundantentrieslive(inglobalvectors),and thenumberofredundantentriesrespectively.Notethatthespatial andthetimesymmetriesaretakentobeinseparateDMRedundant entries,thisisjustamatterofpreference.

DM packer, redT,redC;

DMCompositeCreate(PETSCCOMMWORLD,&packer);

DMRedundantCreate(PETSCCOMMWORLD,0,1,&redT);

DMRedundantCreate(PETSCCOMMWORLD,0,2,&redC);

DMCompositeAddDM(packer,da); DMCompositeAddDM(packer,redT); DMCompositeAddDM(packer,redC);

ThematrixmultiplicationisdonethroughaMATSHELL,andthe structthatholdstherelevantdataisfoundinCode6.2.

Code6.2. Forﬁndingperiodicsolution,weneedamethodfor integratingthevariationalequations(theMATSHELLdiscussedin Section4),additionalDMs,andspaceforholdingfevaluatedatthe stateattheendoftheintegration.

typedef struct PeriodFindCtx{

Mat *linTimeIntegration;

DM packer,redT,redC;

Vec endState,fatendState;

}PeriodFindCtx; 0 0.2 0.4 0.6 0.8 1 1.2 −70 −60 −50 −40 −30 −20

%oflocalinhibitory–inhibitoryconnections

ma xi m um of ex ci tat or y m em br ane pot en ti al (m V) stable equilibrium unstable equilibrium unstable periodic H

Fig.3.Partialbifurcationdiagramshowingtheprimarytransitionfromaspatially

homogeneousequilibriumtoaspaceandtimedependentperiodicorbit.Onthe

verticalaxisthescalingparameterrisplotted,andontheverticalaxisthe(maximum

of)theexcitatorymembranepotential.Thebranchofperiodicsolutionsshownwith

adashedlineisaspatiallyhomogeneousbranchthatisunstabletospace-dependent

perturbations.

7. Examplecalculations

In thissection,we presentsomecomputations thatserve to validateourimplementationandtoinvestigateitsefﬁciency.All testsarebasedontheparametersetin Table1,andthescaling ofthenumberoflocalinhibitory-to-inhibitoryconnections,r,is variedaroundtheﬁrstbifurcationfromanequilibrium tomore complicated,spatio-temporalbehaviour.

Fig.2showstheneutralstabilitycurveforthespatially homo-geneousequilibrium,whichistheuniqueattractorofthemodel atsmallvaluesofr.TheprimarytransitionisaHopfbifurcation withspatialwavenumbersthatdependonthesystemsize.For systemssmallerthan2×2cm2_,_the_emerging_periodic_orbit_is

spa-tiallyhomogeneous.For largersystems,spacedependentorbits emerge,andtheirtypicallengthscaleconvergestoabout9.3cmfor largesystemsizes.Thesestabilitycurveswerecomputedbysolving small eigenvalue problems for each combination of wavenum-bers,independentfromthePETScimplementation.Theeigenvalues computedbyKrylov–SchuriterationinSLEPc,presentedinSection 7.2,areingoodagreement.

Apartialbifurcationdiagram,forspatiallyhomogeneous solu-tionsonly,isshowninFig.3.Inthisdiagram,theHopfbifurcation issubcritical,andtimeseriesanalysisindicatesthattheHopf bifur-cationsassociatedwithnonzerowavenumbersare,too.Thetime seriespresentedinSection7.1wasgeneratedbystartingfromthe equilibriumatr=1andaddingaﬁnite-sizeperturbationintheleast stabledirection,withwavenumber|kx|=|ky|=1.

7.1. Timestepping

Forthetimesteppingdemonstration,weusedasystemsizeof 12.8×12.8cm2 _with_0.5_mm_resolution, _resulting_in _a₂₅₆_×₂₅₆

grid, and N=917,504 unknowns intotal. Setting the parameter r=1.0,weinitializewiththestableequilibriumsolutionperturbed byitsleaststableeigenmode,showninFig.7.Sincetheequilibrium solutionisstable,smallperturbationsjustdecay,butsufﬁciently largeperturbationsgrow.ThesnapshotsofFig.4weretakenaftera transienttimeof600ms.Themembranepotentialsshowbehaviour

(8)

Fig.4.Threesnapshotsoftheexcitatorymembranepotential(colourbarsinmV),6msapart,ofasolutionatr=1,neartheprimaryHopfbifurcation.Thedomainsizeis

12.8×12.8cm2_,_the_resolution_is_0.5_mm_and_the_time-step_size₁_ms._The_fourth_panel_shows_the_power_spectrum_of_h

e,averagedovertheregioninsidetheblacksquare.

TheseimagesofthemembranepotentialarereminiscentofobservationsmadeofspontaneouscorticalactivityasinKenetetal.[33].

Fig.5.Walltime forthecomputationof100timestepsof0.1mseachona

25.6×25.6cm2_domain_with_0.5_mm_resolution._The_fully_implicit_Euler_steps_are

computedwithNewtoniterations,eachofwhichissolvedforbyGMRES,

precon-ditionedwithacombinationofblockJacobiandILU.Theinitialguessisgivenby

anexplicitEulerstep.TwoorthreeNewtoniterationsaresufﬁcienttoreducethe

residualbyafactorof108_._About₁₀₀_Krylov_vectors_are_computed_by_GMRES_to

bringtherelativeresidualdownto10−5_._The_number_of_unknowns_is_N₌_3,670,016.

0.96 1 1.04 1.08 1.12 1.16 −0.01 −0.005 0 0.005 0.01 0.015

r

Re

(λ

k

)

(1,1) (0,1) (0,2)

Fig.6.Therealpartsoftheleadingtwoeigenvaluepairsofthespatially

homoge-neousequilibriumtrackedinthescalingparameterr,forsystemsizeL=12.8cm.The

primarytransitionistiedtowavenumbers|kx|=|ky|=1.Theothercurvesshownare

forwavenumberskx=0,ky=±1andkx=±1,ky=0andforkx=0,ky=±2andkx=±2,

(9)

Fig.7.Therealpartoftheleaststableeigenmodeofthestableequilibriumlocatedatr=1.046.Displayedaretheexcitatory(left)andinhibitory(right)membranepotentials.

Theeigenvector,withwavenumbers(1,1),wascomputedbyArnoldiiterationandisscaledtohaveunitL2norm.

Fig.8.ResidualsoftheNewtoniteration(left)andthecorrespondingGMRESiterations(right).ThelatterisnormalizedbythenormoftherighthandsideofEq.(24),i.e.the

Newtonresidual.Thetolerancewassetat10−8fortheNewtoniterationandto10−5fortheGMRESiteration.Notethesuperlinearconvergenceoftheformer.

thatisnearlyperiodic,withadominantperiodof40Hz,as demon-stratedbythepowerspectrumshowninthelastpanel.Thepower spectrumiscomputedfromaspatialaverageovertheblackbox intheotherpanels,whichwasdonetomimicthesmearingofthe signalobservedfromthescalp.Averagingoverotherregionsofthe samesizeproducesqualitativelysimilarpowerspectra.

Sincethetime-steppingcodeliesatthecoreoftheperiodicorbit solver,wealsoinvestigateditsscalingwithanincreasingnumberof processors.Doublingthedomainsize,whilekeepingthegrid spac-ingﬁxed,givesadynamicalsystemwithN=3,670,016degreesof freedom.Wetime-steppedthissystemonvaryingnumbersofIntel XeonprocessorswithInﬁnibandinterconnects(saw.sharcnet.ca) andtheresultsaredisplayedinFig.5.Scalingforfewprocessors isslightlyhinderedbythechangeinpreconditionerthatisused, namelyILUonasingleprocessorversusblockJacobiplusILUin par-allel.Exceptforthisirregularity,thescalingisshowntobelinear upto256processors.

At256processors,weareat14,336unknownsperprocess.As perPETSc’srecommendationsofnotgoingbelow10,000unknowns perprocess[24],wedonotexpectefﬁcientscalingtomanymore processors.

7.2. Equilibrium

Wecomputedthewholeequilibrium curveofFig.3through parametercontinuation,whichisatrivialextensionofthe algo-rithmforcomputingequilibria,presentedinSection5.For each computedequilibrium solution, wetookthe Jacobianand used SLEPctocomputetheeigenvalueswiththelargestrealparts.The resultisshowninFig.6.Aspredictedbytheneutralstabilitycurve computation,the(1,1)modeturnsunstableﬁrst,immediately fol-lowedbythe(1,0)mode.Aroundr=1.08,the(0,2)modecrosses the(0,1)modeandproceedstobecomethemostunstablemode

forlargervaluesofr.Theleaststableeigenmodeforr=1.046,just afteritseigenvaluehascrossedzero,isshowninFig.7.

7.3. Periodicsolutions

Wetestedthecomputationofperiodicorbitsonasmallergrid, namely16×16points,stillwith0.5mmresolution,andwithr=1.2. TheprimaryHopfbifurcationis subcritical,sothereisnoeasy waytocomputethebranchofspace-dependentperiodicsolutions. Instead,wecomputedoneofthespatiallyhomogeneousorbits,for whichanapproximatesolutioncanreadilybeobtainedfrom anal-ysisoftheODEreductionofthemodel.Infact,theupperpartof thebranchofperiodicorbitsshowninFig.3isstabletoallspatially homogeneousperturbations.

Startingfromacoarseinitialapproximation,theNewton iter-ationsconvergedfasterthanlinear,and eachNewtonsteptook between 8 and 12 GMRES iterations, out of a maximum of N+3=3587.ThenonlinearandlinearresidualsareshowninFig.8. Subsequently,wecomputedthemostunstablemultipliers,using SLEPCwiththeMATSHELLforsteppingthevariationalsystemas describedinSection4.2.Themostunstablemultiplieris1=1.111

andcorrespondstoawavenumber(1,1)perturbation.

8. Conclusionandfutureimprovements

Inthecurrentpaper,wehavepresentedthebasic implemen-tationofthemodelandexamplecomputationstovalidateitand testitsperformance.Thecodewillbeavailablepublicly[18].Asit isbuiltontopofPETSc,theuserhasaccesstoarangeof nonlin-earandlinearsolversandpreconditioners,whichcanbeusedto solvetheboundaryvalueproblemsthattypicallyarisein dynami-calsystemsanalysis.Theperiodicorbitcomputation,presentedin Section7.3,isasimpleexampleofsuchaboundaryvalueproblem,

(10)

thathasalltheingredients:amodulefortime-steppingthesystem andperturbationsandarepresentationofuser-speciﬁed,bordered matrices.

Thenextstepinthedevelopmentofthecodeisthe implemen-tationofpseudo-arclengthcontinuationofperiodicorbits.Thiswill enableus,forinstance,tocomplementthebifurcationdiagramof thecurrenttestcase,Fig.3,withthebranchesofspace-dependent periodicsolutions thatactuallyregulatetheobserveddynamics, incontrasttothehighlyunstablespatiallyhomogeneousperiodic orbitscomputedfromtheODEreductionofthemodel.

Weﬁnishwithemphasizingthattherearesomepropagating modeswithinthecortexthatcanbedescribedwithoutconsidering itscouplingto,say,thethalamus,asseeninMullerandDestexhe [34].Althoughwedidnotattempttotunethemodelparametersor initialconditionsforthepurpose,acomparisonofthemodeloutput inFig.4tothevoltagesensitivedyeexperimentspresentedin[34] (seetheirFig.3)showsqualitativeagreementwithrespecttothe spatialextentofthepatterns.

Thisimplementationwillbeusefultostudyingsuchdynamics withintheLiley model,andis easilymodiﬁedtosimilar mean-ﬁeldmodelsofthecortex;examplesbeingthosethatincludethe effectsofgapjunctions[21],andmodelsthatincorporatedifferent propertiesforthelongrangeconnections.

Acknowledgements

L.v.V.wassupportedbyNSERCGrantno.355849-2008.Someof thecomputationsweremadepossiblebythefacilitiesoftheShared HierarchicalAcademicResearchComputingNetwork(SHARCNET: www.sharcnet.ca)andCompute/CalculCanada.

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KevinR.GreenisaPhDcandidateintheModellingand

ComputationalScienceprogrammeatUOIT.Hisresearch

interestsspannonlineardynamics,scientiﬁccomputing,

andneuralmean-ﬁeldmodelling.

LennaertvanVeenisaProfessorintheFacultyof

Sci-enceatUOIT.Hismainresearchinterestistheapplication

ofcomputationaldynamicalsystemsanalysistocomplex

phenomena,suchasﬂuidturbulenceandmean-ﬁeld