Lecture Notes in Mathematics 2058 Editors: J.-M.Morel,Cachan B.Teissier,Paris EditorsMathematicalBiosciencesSubseries: P.K.Maini,Oxford Forfurthervolumes: http://www.springer.com/series/304 • Mostafa Bachar Jerry Batzel Susanne Ditlevsen Editors Stochastic Biomathematical Models with Applications to Neuronal Modeling 123 Editors MostafaBachar SusanneDitlevsen KingSaudUniversity UniversityofCopenhagen CollegeofSciences DepartmentofMathematicalSciences DepartmentofMathematics Copenhagen,Denmark Riyadh,SaudiArabia JerryBatzel UniversityofGraz InstituteforMathematics andScientificComputing andMedicalUniversityofGraz InstituteofPhysiology Graz,Austria ISBN978-3-642-32156-6 ISBN978-3-642-32157-3(eBook) DOI10.1007/978-3-642-32157-3 SpringerHeidelbergNewYorkDordrechtLondon LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2012949578 MathematicsSubjectClassification(2010):60Gxx,92C20,37N25,92Bxx (cid:2)c Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Why use mathematical or stochastic models to study something as complicated andoftenpoorlyunderstoodasdynamicsin physiology?Hopefully,thisbookcan providesomepartialanswersandpointtotheexcitingproblemsthatstillremainto besolved,wheremathematicalandstochastictoolscanbeuseful. In this book we treat basics of stochastic process theory represented by a stochastic differential equation directed towards biological modeling and review the field of neuronalmodels. Theoreticalmodelsmust be relevantphysiologically to be useful and interesting, and their analysis can provide biological insight and help summarize and interpret experimentaldata. Predictions can be extracted from the model, and experiments verifying or invalidating the model can be suggested,therebyenhancingphysiologicalunderstanding.Evenifthemechanisms are well understood, simulations from models can explore the consequences of extremephysiologicalconditionsthatmightbeunethicalorimpossibletoreproduce experimentally.Theprocessofbuildingatheoreticalmodelforcesonetoconsider anddecideontheessentialcharacteristicsofthephysiologicaldynamics,aswellas which variables and mechanisms to include. Analysis and numerical simulations of the model illustrate quantitatively and qualitatively the consequences of the assumptions implied in the model. The unifying aim of theoretical modeling and experimentalinvestigationistheelucidationoftheunderlyingbiologicalprocesses thatresultinaparticularobservedphenomenon. Manybiologicalsystemsarehighlyirregular,andexperimentsundercontrolled conditionsshowalargetrial-to-trialvariability,evenwhenkeepingtheexperimental setup fixed. This calls for a stochastic, as opposed to deterministic, modeling approach, especially because ignoring the stochastic phenomena in the modeling mayhugelyaffecttheconclusionsofthestudiedbiologicalsystem.Inlinearsystems the noise might only blur the underlyingdynamicswithout qualitatively affecting it, but in nonlinear dynamical systems corrupted by noise, the corresponding deterministic dynamics can be drastically changed. In general, stochastic effects influence the dynamics and may enhance, diminish, or even completely change thedynamicbehaviorofthesystem.Incertainbiologicalsystems,e.g.,inauditory v vi Preface neurons,thenoiseisevenbelievedtoenhancethesignal,thusprovidingabiological justificationforthelargeamountofnoisefoundinlivingsystems. Thebooktreatsstochasticityrepresentedbystochasticdifferentialequations,but isnotmeanttobeacomprehensivetextbookonstochasticmethods.Itisprimarily intended for mathematicians and life scientists who are interested in seeing an in-depth and motivated presentation of an important application of stochastic methods. Our goal is to provide an illuminating example of where and how stochastic methods can enter into modeling physiological systems. Life scientists generally, with some background in mathematics, will also be able to benefit by seeing how these two areas are linked, and we point the interested reader to referencesthatfillinmissingbackgroundinformation.Wehopethatthematerialas presentedwillprovideausefulillustrationofinterdisciplinaryresearch. Thereadershouldhavea basicbackgroundinprobability,differentialandinte- gralcalculus,andordinarydifferentialequations.Wefocusonneuronalmodeling, wherestochasticmodelshavesupplementeddeterministicdynamicalmodelssince the sixties. There exist already many excellent books on nonlinear dynamics, biomathematicalmodeling, and computationalneuroscience. The aim of this vol- ume is to provide a focused, up-to-date presentation of neuronal modeling using stochastic methods, while in addition both motivating the linkage and providing insightintopracticalissues.Moreprofitwillofcoursebederivedfromreadingthe book,ifitiscombinedwiththereadingofsomeintroductorytextsincomputational neuroscienceorbiomathematicalmodeling. Thebookisdividedintotwoparts.Thefirstpartintroducessomemethodology, which is useful when modeling biological systems with stochastic dynamics. Chapter 1 is an introduction to stochastic models and a good place to start if the readerhasnobackgroundinstochasticprocesses.Alsoashortoverviewofstatistical methodstoestimatemodelparametersfromdataisprovided.Chapters2and3need more mathematical preparation and a basic understanding of stochastic processes andprobabilitytheory,aswellassomenotionofmeasuretheory.InChap.2,scale andspeedmeasuresofone-dimensionaldiffusionprocessesarereviewed,whichare used to determineexpressionsfor hitting times or first-escape times, in particular, boundarybehaviors.Thesetoolsareveryusefulwhendealingwithneuronalmodels, as thosepresentedin the secondpartof the book.Chapter3 givesan introduction to the theory of large deviations, providing an asymptotic description of the fluctuationsofastochasticsystem,inparticular,providingexponentialestimatesfor thewaitingtimetorareevents.Chapter4closesthemethodologicalpartindicating howthetheoryfromChap.3canbeimplementedinpractice.Thesetechniquesare naturaltousewhenanalyzingstochasticmodelsofbiologicalsystems. Thesecondpartofthebooktreatsneuronalmodels.Chapter5providesatimely review of existing methods and available analytical results for the most common one-dimensional stochastic integrate-and-fire models. These models of neuronal activity collapse the neuronal anatomy into a single point in space, sacrificing realism for mathematical tractability, although they often succeed in predicting neuronalresponsewithconsiderableaccuracy.Chapter6goesastepfurthertomore realisticmodelsandincludesthespatialdimensionofneuronaldynamics.Stochastic Preface vii partialdifferentialequationmodelsarereviewed,inparticulartheHodgkin–Huxley andtheFitzHugh–Nagumomodelsaretreatedindetail.Chapter7isdedicatedtoa probabilistictreatmentofFitzHugh–Nagumosystems.Finally,Chap.8implements the tools from Chap.6 to a specific application of modeling spreading of cortical depression. This volume was first conceivedas a result of the experienceof designing and holding a combined summer school/workshop event on the subject of stochas- tic modeling in physiology. This event was part of a Marie Curie sponsored series of four training events designed to encourage interdisciplinary research in modeling physiological systems. The events brought together mathematicians, bioengineers, statisticians, medical clinicians, physiologists, and other related life scienceresearchers. Theunderlyingmotivationandinspirationforthisseriesofeventsisthatunrav- eling the complexities of physiological systems and the intricacies of interaction between systems requires development of novel and innovative insights as well as new research approaches and techniques. Furthermore, such new approaches can be strongly stimulated by merging the different perspectives from the mathe- matical/engineeringdisciplines on the one side and the life sciences on the other. These observations motivated the design of the events in which summer schools wouldintroducenewandyoungresearchestoaninterdisciplinarytreatmentofthe modelingofkeyphysiologicalsystemswithemphasisonhowmodelingcanaddress important clinical problems related to these systems. Directly following each summerschoolaninterdisciplinaryscientificworkshopwasheld.Theseworkshops focused on the same themes as the preceding summer school and were designed as stand alone scientific events. Students from the school participated in these workshopsandinthiswaynewandcurrentresearcherscouldinteractanddevelop contactsforcollaboration. ThegeneralwebpagelinkingandreflectingallfourMarieCurietrainingeventscan befoundat:http://www.uni-graz.at/biomedmath/info.html. Theeventrelatedtothisvolumecanbefoundattheeventwebpage: http://www.math.ku.dk/(cid:2)susanne/SummerSchool2008/ Acknowledgements This book would never have existed without the work of the contributors ofthisvolume.Theyallparticipatedinandcontributedtothesuccessofthecombinedsummer school and workshop, and we would like to thank them all for sharing their insights with us. WearegratefultoMichaelSørensenandFranzKappelfortheirhelpandpartoforganizingthe summerschoolandtheworkshop.Wewishtothankallthepeoplewhoattendedthesummerschool and workshop and especially theteachers of thesummer school: Andrea DeGaetano, Susanne Ditlevsen,TereseGraversen,MartinJacobsen,SeemaNanda,BerntØksendal,UmbertoPicchini, Laura Sacerdote, Michael Sørensen, and Gilles Wainrib. Thanks to Flemming H. Jacobsen for typing parts of the book. We also thank Springer Verlag and especially Ute McCrory for their support during the production of the book. Finally, this volume would not have been possible withoutthefunds from different sources, mainlytheEuropean Unionundertheprogram Marie CurieConferencesandTrainingCoursewhichsupportedthesummerschoolandworkshopunder viii Preface theprojectBiomathtech07-10(MSCF-CT-2006-045961).WewerealsosupportedbytheEuropean Society for Mathematical and Theoretical Biology, Forskerskolen i Matematik og Anvendelser, Forskerskolen i Biostatistik, Copenhagen Statistics Network at University of Copenhagen and Biomedical Simulations Resource, University of Southern California. Research by Susanne DitlevsenwassupportedbytheDanishCouncilforIndependentResearchjNaturalSciences,Jerry BatzelwaspartiallyfundedbyFWF(Austria)projectP18778-N13,MostafaBacharwaspartially supported by Deanship of Scientific Research, College of Science Research center, King Saud University,AdelineSamsonwassupportedbytheBonusQualiteRecherchefromUniversiteParis Descartes, Laura Sacerdote and Maria Teresa Giraudo were supported by MIUR PRIN 2008, Miche`leThieullenwasSupportedbyAgenceNationaledeRecherche ANR-09-BLAN-0008-01, andHenryTuckwellthanksProfDrJu¨rgenJostforhisfinehospitalityatMISMPI. Riyadh,SaudiArabia MostafaBachar Graz,Austria JerryBatzel Copenhagen,Denmark SusanneDitlevsen February2012 Contents PartI Methodology 1 IntroductiontoStochasticModelsinBiology............................. 3 SusanneDitlevsenandAdelineSamson 1.1 Introduction ............................................................. 3 1.2 MarkovChainsandDiscrete-TimeProcesses ......................... 4 1.3 TheWienerProcess(orBrownianMotion)............................ 5 1.4 StochasticDifferentialEquations ...................................... 8 1.5 ExistenceandUniqueness.............................................. 13 1.6 Itoˆ’sFormula............................................................ 14 1.7 MonteCarloSimulations............................................... 16 1.7.1 TheEuler–MaruyamaScheme................................ 16 1.7.2 TheMilsteinScheme.......................................... 17 1.8 Inference................................................................. 17 1.8.1 MaximumLikelihood......................................... 18 1.8.2 BayesianApproach............................................ 22 1.8.3 MartingaleEstimatingFunctions............................. 23 1.9 BiologicalApplications................................................. 25 1.9.1 Oncology....................................................... 25 1.9.2 Agronomy ..................................................... 29 References..................................................................... 33 2 One-DimensionalHomogeneousDiffusions............................... 37 MartinJacobsen 2.1 Introduction ............................................................. 37 2.2 DiffusionProcesses..................................................... 39 2.3 ScaleFunctionandSpeedMeasure .................................... 40 2.4 BoundaryBehavior ..................................................... 45 2.5 ExpectedTimetoHitaGivenLevel................................... 54 References..................................................................... 55 ix x Contents 3 ABriefIntroductiontoLargeDeviationsTheory........................ 57 GillesWainrib 3.1 Introduction ............................................................. 57 3.2 SumofIndependentRandomVariables................................ 58 3.3 GeneralTheory.......................................................... 61 3.4 SomeLargeDeviationsPrinciplesforStochasticProcesses.......... 64 3.4.1 SanovTheoremforMarkovChains.......................... 64 3.4.2 SmallNoiseandFreidlin–WentzellTheory.................. 65 3.5 Conclusion .............................................................. 71 References..................................................................... 71 4 SomeNumericalMethodsforRareEventsSimulationand Analysis ....................................................................... 73 GillesWainrib 4.1 Introduction ............................................................. 73 4.2 Monte-CarloSimulationMethods...................................... 75 4.2.1 OverviewoftheDifferentApproaches....................... 75 4.2.2 FocusonImportanceSampling............................... 78 4.3 NumericalMethodsBasedonLargeDeviationsTheory.............. 87 4.3.1 QuasipotentialandOptimalPath ............................. 87 4.3.2 NumericalMethods ........................................... 88 4.4 Conclusion .............................................................. 93 References..................................................................... 95 PartII NeuronalModels 5 Stochastic Integrate and Fire Models: A Review on MathematicalMethodsandTheirApplications.......................... 99 LauraSacerdoteandMariaTeresaGiraudo 5.1 Introduction ............................................................. 99 5.2 BiologicalFeaturesoftheNeuron ..................................... 101 5.3 OneDimensionalStochasticIntegrateandFireModels .............. 102 5.3.1 IntroductionandNotation..................................... 102 5.3.2 WienerProcessModel ........................................ 103 5.3.3 RandomizedRandomWalkModel........................... 105 5.3.4 Stein’sModel.................................................. 106 5.3.5 Ornstein–UhlenbeckDiffusionModel ....................... 107 5.3.6 ReversalPotentialModels .................................... 112 5.3.7 ComparisonBetweenDifferentLIFModels................. 117 5.3.8 JumpDiffusionModels....................................... 119 5.3.9 BoundaryShapes.............................................. 120 5.3.10 FurtherModels ................................................ 121 5.3.11 RefractorinessandReturnProcessModels .................. 122