Lecture Notes on Mathematical Modelling in the Life Sciences Paul C. Bressloff Waves in Neural Media From Single Neurons to Neural Fields Lecture Notes on Mathematical Modelling in the Life Sciences SeriesEditors AngelaStevens MichaelC.Mackey Forfurthervolumes: http://www.springer.com/series/10049 Paul C. Bressloff Waves in Neural Media From Single Neurons to Neural Fields 123 PaulC.Bressloff UniversityofUtah DepartmentofMathematics SaltLakeCity,UT,USA ISSN2193-4789 ISSN2193-4797(electronic) ISBN978-1-4614-8865-1 ISBN978-1-4614-8866-8(eBook) DOI10.1007/978-1-4614-8866-8 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013948234 MathematicsSubjectClassification(2010):35C07,82C31,92C20,92C30,92C37,82C31 ©SpringerScience+BusinessMediaNewYork2014 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) To theShmu Preface This is a book on mathematical neuroscience, in which the unifying theme is wavelikephenomenaatmultiplespatialandtemporalscales.Therearealreadysev- eral excellent books on applications of dynamical systems theory and bifurcation theory to neuroscience, but these are mainly concerned with mathematical mod- els of neuronsand networksthat take the form of systems of ordinarydifferential equations (ODEs). Spatial aspects are usually restricted to standard partial differ- entialequation(PDE) modelsof action potentialpropagationalongaxonsandthe spatial spread of voltage along dendrites (the cable equation). However, there are manyotherinterestingspatiotemporalphenomenathatarenotusuallyaddressedin anydetail,atleastwithinasinglebook.Theseincludeelectricalandchemicalwaves alongspinydendrites,calciumwavesinastrocytes,excitableandoscillatorywaves in cortical and subcorticalstructures, binocular rivalry waves and ambiguousper- ception,oscillatorywavesandswimminglocomotion,cytoskeletalwavesandneu- ritogenesis,spreadingdepressionandmigraineauras,thespreadofprionsandother neurodegenerativediseaseswithinthebrain,andepilepticwavesincomplexneural networks. All of these phenomenaare linked by the notionof the time-dependent propagationorspreadofsomequantityinspace,whetheritischemicalconcentra- tion, a protein aggregate, voltage, or spiking activity of a population of neurons. They should be distinguished from another importantclass of spatial phenomena, namely,pattern-forminginstabilities,whichcouldbethesubjectofafuturebook! Inordertodevelopthethemesofthisbook,Ihaveendeavoredtogiveapedagog- icalintroductiontothemathematicalmodelingofbiologicalneuronsandnetworks, as well as a toolbox of analytical techniques used to study wavelike phenomena, includingtheconstructionoftravelingwavesolutionsofPDESandnonlocalequa- tions, stability analysis, phase-plane analysis, singular perturbationtheory, phase- reduction and averaging methods, and homogenization theory. A common thread throughoutthebookisanemphasisonstochasticity.Inrecentyears,therehasbeen an explosion of interest in the role of noise in biological systems, partly driven bythe presenceof lowcopynumbersin geneandbiochemicalnetworks.Noise is also ubiquitousin the brain, ranging from fluctuationsin gene expressionand the opening of ion-channel proteins to the Poisson-like spiking activity of individual vii viii Preface corticalneurons,thepresenceoffluctuationsduringperceptualdecisionmakingand bistableperception,andthegenerationofspontaneousactivityduringrestingbrain states.Hence,thebookcontainsanintroductiontothetheoryofstochasticprocesses andhowitcanbeusedtostudytheeffectsofnoiseonwavepropagationandwave initiation.Inthelattercase,onetypicallyhastosolveanescapeproblem,inwhich noise drives some quantity such as voltage, calcium concentration, or population spikingactivityaboveathreshold.Thisresults,forexample,inthegenerationofa spontaneous action potential, a calcium spark, or a transition between cortical up anddownstates,respectively. Beforegivingamoredetaileddescriptionofthecontentsofthisbook,Ithought that I would elaborate a little on my own approach to neuroscience. This is not to favor one approach over another—a diversity of methodological and concep- tual frameworks is crucial in trying to understand such a complex system as the brain—butrathertohighlightmyowntastesandbiases.Letmebeginwithastory (perhapsapocryphal).Severalyearsagoa well-knownmathematicalbiologyjour- nal was relaunched with a new editorial board. At the final organizational meet- ingoftheeditorsinchief,aftersomegoodfoodandwine,a satisfied glowspread aroundthe room as the list of associate editors was drawn up, coveringthe major areasofbiology.However,someonethensheepishlypointedoutthatoneareawas missing—neuroscience!Isuspectthatneurosciencehasaslightlycomplicatedrela- tionshipwiththerestoftheoreticalbiologyduetotheambiguousnotionof“com- putation.” One sense of the word, which applies to all areas of biology, refers to thehighlevelofbiologicaldetailandcomputationalcomplexityinamodel,which means that it is usually less amenable to mathematical analysis than simpler re- ducedmodels.Thefocustendstobeonissuessuchasparametersearchesanddata fitting,sensitivityanalysis,numericalconvergence,andcomputationalefficiency.It follows that one way to characterize a modeling approach is where it lies on the mathematical–computationalbiologyaxis.Withinthiscontext,thepresentbookis onmathematicalratherthancomputationalneuroscience,sinceIconsiderreduced, mathematicallytractablemodelsofwavelikephenomena. The othersense ofthe word “computation”refersto a functionalinterpretation ofthebrainasabiologicalcomputerthatprocessessensoryinputsviaasequenceof internalbrainstatestoproduceamotoroutputorachangeincognitivestate.Acom- monassumptionisthatifwehadacompleteunderstandingofthephysiologyofthe heartat the molecular,cellular, and systems levels, then we would understandthe functionoftheheart,whereasthesameisnottrueofthebrain.Thatis,whatmakes neurosciencesochallenging,beyondtheincrediblecomplexityofthebrainasabio- logicalorgan,ishowonelinksphysiologicalbrainstatestobehaviorandcognition. Consequently,amajordriveincomputationalneuroscience(inthesecondsense)is to link physiological descriptions of neurons and networks with computation and informationprocessing. Popular themes are spike trains and neural coding,statis- tical correlations, the role of feedback in predictive coding, learning algorithms, Bayesianstatistics,andmaximumentropy.Thecurrentbookisnotconcernedwith this form of computational neuroscience either, since I do not really discuss the possiblefunctionalroleofwavelikeneuralphenomena,althoughtheroleofwaves Preface ix indevelopmentanddiseaseisdiscussedatlength.Therehavebeensomeattempts tointerpretthecomputationalroleofwaves,basedontheobservationthatcortical waves oftenoccurbetween sensory processingevents.However,such an observa- tioniscorrelativeatbestandprovidesverylittleinsightintothecausalmechanisms ofperception.Unfortunately,thesamecouldbesaidofmanyoftheoverhypedre- sults coming fromfunctionalmagnetic resonanceimaging(fMRI), where partsof thebrainlightupwhensomeoneistellingajokeorplanningtocommitacrime. In the media lately, a number of distinguished physicists have had some harsh wordstosayaboutphilosophyanditsrelevancetoscientificquestions.Ipersonally disagreewithsuchattitudes,partlyduetomyowninterestsinphilosophyandalso becauseIhavefounditusefultostandbackoccasionallyinordertoplacemyown workin abroadercontext.Ithinkthatthisisparticularlyimportantin afieldsuch asneuroscience,with allofits potentialclinical,psychological,ethical, sociologi- cal,andculturalramifications.Of course,readingphilosophydoesnotmakeme a philosopher,soIproceedwithcaution.Nevertheless,Iamanaturalist,inthesense thatIdonotbelievethebrainismadeofdifferent“stuff”orobeysdifferentphysical laws from planets, sushi, and hot tubs. On the other hand, I tend not to view the braininfunctionalistterms,thatis,asthehardwaretothemind’ssoftware.Ifavora moredirectbiologicalinterpretationintermsofanevolvedandadaptivesystemthat enablesanorganismtocarryoutgoal-directedbehaviorinachangingphysicaland socialenvironment(althoughmanypeoplebelievethatthiscanbereformulatedin termsofalgorithmsandcomputation).Arelatedideaisthatmeaningorsemantics cannotbegeneratedfromapurelysyntacticprocess,sinceoneneedstotakeintoac- counttheevolutionaryandbiographicalhistoryofanembodiedmindactingwithin asocietalandculturalcontext.AnotherreasonItendtosteerclearofcomputational approaches is that I fear that using such a paradigm is in danger of imposing or- der,rationality,andengineeringdesignprinciplesonarathermessyandhaphazard system.Itendtoagreewiththeideathatthebrainisa“kludge,”inwhichmodules thatevolvedtosolvesimplerproblemshavebeenthrowntogethersuboptimallyas the complexity of an organism has increased. I also feel that randomness plays a significantroleinhowwe“think,”whichcounterstheideaoffollowingacomputer programoralgorithm.Idonotmeanrandomnessinthesenseofexploitingstochas- ticitytoenhanceperformance(e.g.,simulatedannealingorrandomsearch),which itselfcouldbealgorithmic,butrandomnessasirrationality.Peoplecommitsuicide, skateboard down Everest on a piano, and drink root beer (an alien concept to an Englishman). The“elephantintheroom”istheso-calledhardproblemofconsciousness,that is, of explaining how and why we have qualia or phenomenal experiences. (This is a controversialissue, since someview qualia asan illusion basedon misguided folkpsychologyorpoorintuitionpumps.)Arelatedquestionishowonegoesfrom objectivebrainstatestosubjectivementalstates.Iwoulddescribemyselfasanon- reductionist,inthesensethatIbelievementalpropertiessuperveneon(butaredis- tinct from) physical properties and are an emergent feature of the physical brain. However, a systematic framework for understanding this emergence is currently beyond our grasp. An analogy is how one goes from microscopic descriptions of