Table Of ContentLecture 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
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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
