Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

ASYMPTOTIC STABLILITY CONVEXITY AND LIPSCHITZ REGULARITY OF DOMAINS IN THE

29 pages
ASYMPTOTIC STABLILITY, CONVEXITY, AND LIPSCHITZ REGULARITY OF DOMAINS IN THE ANISOTROPIC REGIME JEROME VETOIS Abstract. Anisotropic operators appear in several branches of applied sciences and, in par- ticular, in physics. They involve directional derivatives with distinct weights which create distortions in the ambient space. Anisotropic rescaling comes with the notion of asymptoti- cally stable domains. We prove two results, one of geometric nature, the other one of analytic nature, which both guarantee that a given domain is asymptotically stable. We also discuss specific examples. 1. Introduction Anisotropic operators appear in several places in the literature. Recent references can be found in physics [9–11,17,18], in biology [6,7], and in image processing (see, for instance, the monograph by Weickert [34]). By definition, anisotropic operators involve directional deriva- tives with distinct weights. A model of such operators is the anisotropic Laplace operator. In dimension n ≥ 2, given ??p = (p1, . . . , pn) with pi > 1 for i = 1, . . . , n, the anisotropic Laplace operator ∆??p is defined by ∆??p u = n∑ i=1 ∂ ∂xi ?pixiu , where ?pixiu = |∂u/∂xi| pi?2 ∂u/∂xi. We let p be an exponent greater than pi for i = 1, .

  • ??

  • nonlinear anisotropic

  • b? ?

  • any open

  • asymptoti- cally ??p

  • anisotropic operators

  • converging uniformly

  • asymptotically ??p

  • num- bers converging


Voir plus Voir moins
JournalofMathematicalBiologymanuscriptNo.(willbeinsertedbytheeditor)ADecision-MakingFokker-PlanckModelinComputationalNeuroscienceJose´AntonioCarrilloSte´phaneCordierSimonaManciniReceived:date/Accepted:dateAbstractIncomputationalneuroscience,decision-makingmaybeexplainedanalyzingmodelsbasedontheevolutionoftheaveragefiringratesoftwointeractingneuronpopulations,e.g.inbistablevisualperceptionproblems.Thesemodelstypicallyleadtoamulti-stablescenariofortheconcerneddy-namicalsystems.Nevertheless,noiseisanimportantfeatureofthemodeltakingintoaccountboththefinite-sizeeffectsandthedecision’srobustness.ThesestochasticdynamicalsystemscanbeanalyzedbystudyingcarefullytheirassociatedFokker-Planckpartialdifferentialequation.Inparticular,intheFokker-Plancksetting,weanalyticallydiscusstheasymptoticbehaviorforlargetimestowardsauniqueprobabilitydistribution,andweproposeanumericalschemecapturingthisconvergence.Thesesimulationsareusedtovalidatedeterministicmomentmethodsrecentlyappliedtothestochasticdif-ferentialsystem.Further,provingtheexistence,positivityanduniquenessoftheprobabilitydensitysolutionforthestationaryequation,aswellasforthetimeevolvingproblem,weshowthatthisstabilizationdoeshappen.Finally,wediscusstheconvergenceofthesolutionforlargetimestothestationarystate.Ourapproachleadstoamoredetailedanalyticalandnumericalstudyofdecision-makingmodelsappliedincomputationalneuroscience.KeywordsComputationalNeuroscienceFokker-PlanckEquationGeneralRelativeEntropyJ.A.CarrilloICREA(Institucio´CatalanadeRecercaiEstudisAvanc¸ats)andDepartamentdeMatema`tiquesUniversitatAuto`nomadeBarcelona,E-08193Bellaterra,SpainTel.:+34-93-5814548E-mail:carrillo@mat.uab.esS.Cordier,S.ManciniFe´de´rationDenisPoisson(FR2964)BP.6759,DepartmentofMathematics(MAPMOUMR6628)UniversityofOrle´ansandCNRS,F-45067Orle´ans,France
21IntroductionThederivationofbiologicallyrelevantmodelsforthedecision-makingpro-cessesdonebyanimalsandhumansisanimportantquestioninneurophys-iologyandpsychology.Thechoicebetweenalternativebehavioursbasedonperceptualinformationisatypicalexampleofadecisionprocess.Itisquitecommontoobservebi-stabilityinseveralpsychologicalexperimentswidelyusedbyneuroscientists.Archetypicalexamplesofthesemulti-stabledecision-makingprocessesarebistablevisualperception,thatis,twodistinctpossibleinterpretationsofthesameunchangedphysicalretinalimage:Neckercube,Rubinsface-vase,binocularrivalryandbistableapparentmotion[6,12,20].Inordertoexplainthesephenomena,biologicallyrealisticnoise-drivenneu-ralcircuitshavebeenproposedintheliterature[9]andevenusedtoqualita-tivelyaccountforsomeexperimentaldata[23].Thesimplestmodelconsistsoftwointeractingfamiliesofneurons,eachonecharacterizedbyitsaveragedfiringrate(averagednumberofspikesproducedpertime).Correlationoftheseneuronfamiliesisbiggerwiththeirownbehaviorthanwiththeothers.More-over,thismechanismismediatedbyinhibitionfromtherestoftheneuronsandthesensoryinput.Theexternalstimulimayproduceanincreasingactiv-ityofoneoftheneuronfamiliesleadingtoadecisionstateinwhichwehaveahigh/lowactivityratioofthefiringrates.Decision-makinginthesemodelsisthenunderstoodasthefluctuation-driventransitionfromaspontaneousstate(similarfiringratesofbothfamilies)toadecisionstate(high/lowactivitylevelratiobetweenthetwofamilies).Ashasalreadybeenexplainedanddiscussedindifferentworks[11,12,22],thetheoryofstochasticdynamicalsystemsoffersausefulframeworkfortheinvestigationoftheneuralcomputationinvolvedinthesecognitiveprocesses.Noiseisanimportantingredientinthesemodels.Infact,suchneuralfamiliesarecomprisedofalargenumberofspikingneurons,sothatfluctuationsarisenaturallythroughnoisyinputand/ordisorderinthecollectivebehaviourofthenetwork.Moreover,thisisusedtointroduceafinite-sizeeffectoftheneuronfamiliesasdiscussedin[11,12].Manyotherworkscanbefoundintherecentliteratureconcerningdecision-makingstochasticdifferentialmodelsandtheunderstandingoftheevaluationofadecision,seeforinstance[8]forareviewpaper,or[16]forareviewonthedifferentwaystocomputeadecisionatdifferentbiologicallevels.Moreover,wereferto[10,15,24,27]forresultsconcerningtwochoicestaskparadigm.Ontheonehand,allthesemodelsgivegoodapproximationsofthereactiontimes(RT),butontheotherhand,mostofthemconsiderlinearorlinearizedstochasticdifferentialsystems.Thus,itispossibletogiveexplicitlytheirre-ductiontoaone-dimensionaldrift-diffusion(orFokker-Planck)equation,andhencetheycanbeexplicitlysolvedasinthecaseoftheOrnstein-Uhlenbeckprocess.Thestochasticdifferentialsystemweshallconsiderinthesequelisnon-linear,andtheprogressiveKolmogorovequation(ortheFokker-Planck)de-scribingtheprobabilitydensityfunction,inthetwodimensionalparameter
Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin