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Auditory information processing in systems with internally coupled ears [Elektronische Ressource] / Christine Voßen

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160 Pages
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Auditory Information Processingin Systems withInternally Coupled EarsChristine Vo enAThis dissertation has been written in LT X us-Eing the memoir class. Typesetting was done withApdfLT X.EFunding has been provided by the Bernstein Cen-ter for Computational Neuroscience (BCCN) { Mu-nich. TECHNISCHE UNIVERSITAT MUNCHENLehrstuhl fur Theoretische PhysikAuditory Information Processingin Systems withInternally Coupled EarsChristine Vo enVollst andiger Abdruck der von der Fakult at fur Physik der Technischen Universit atMunc hen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaftengenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. F. SimmelPrufer der Dissertation: 1. Univ.-Prof. Dr. J. L. van Hemmen2. Univ.-Prof. Dr. F. BornemannDie Dissertation wurde am 12.07.2010 bei der Technischen Universit at Munc hen eingereichtund durch die Fakult at fur Physik am 27.07.2010 angenommen.. . . es sind uns Dinge als au er uns be ndlicheGegenst ande unserer Sinne gegeben, allein von dem, wassie an sich selbst sein m ogen, wissen wir nichts, sondernkennen nur ihre Erscheinungen, d. i. die Vorstellungen,die sie in uns wirken, indem sie unsere Sinne a zieren.Immanuel Kant (1724 { 1804)Prolegomena, 1783ContentsContents vii1 Introduction 11.1 Sound stimuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Mechanical processing of sound stimuli . . . . . . . . . . . . . . . . . . . . . 31.2.

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viitstenCon1ductiontroIn11.1Soundstimuli...................................2
1.2Mechanicalprocessingofsoundstimuli.....................3
1.2.1Evolutionofdifferentauditorysystems.................3
1.2.2Internallycoupledears..........................3
1.3Neuronalprocessingofsoundstimuli......................5
1.3.1Buildingblocksofneuronalcomputation................5
1.3.2Leakyintegrate-and-fireneuron.....................9
1.3.3Poissonneuron..............................10
1.4Neuronalrepresentationofsoundstimuli....................13
1.4.1Neuronalmaps..............................13
1.4.2Pre-wiringamap.............................13
1.4.3Fine-tuningamap............................14
2ModelingInternallyCoupledEars:TheICEmodel17
2.1Introduction....................................17
2.2DerivationoftheICEmodelwithcylindricalmouthcavity.........21
2.2.1Externalsoundinput...........................22
2.2.2Internalcavity..............................23
2.2.3Vibrationofthemembrane.......................25
2.3Numericalsimulationoftheeigenfunctions
ofrealisticmouthcavities............................31
2.4Evaluationandresults..............................33
2.4.1Directionalityofthemembranevibrationpattern...........34
2.4.2Eigenmodesofarealisticmouthcavity.................39
2.4.3GeneralizationoftheICEmodel....................42
2.4.4Spatialvibrationpatternofthemembrane...............49
2.5Conclusion....................................50
3NeuronalprocessingofiTDsandiADs53
3.1Introduction....................................53
3.2SeparatedpathwaysforiTDandiADprocessing...............54
3.3ProcessingofiTDs................................55

vii

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3.4ProcessingofiADs................................62
3.4.1Experimentalandnaturalstimulus...................63
3.4.2Simulationofnucleusangularis(NA)..................64
3.4.3SimulationofEIneurons........................65
3.5Conclusion....................................69
AuditorySensitivityandInternallyCoupledEars73
4.1Introduction....................................73
4.2Theoreticaldescriptionofmembranevibrations................75
4.2.1Membranevibrationdifferencesofindependentears.........77
4.2.2Membranevibrationdifferencesofinternallycoupledears......78
4.2.3Empiricalanalysis............................79
4.3Conclusion....................................82
4.AMethods:Empiricalanalysis...........................82
Optimalityinmono-andmultisensorymapformation87
5.1Introduction....................................87
5.2Fundamentalconceptofneuronalmaps....................89
5.3Mathematicalmodel...............................91
5.3.1Definitionoftheproblem........................92
5.3.2Optimalreconstruction.........................94
5.3.3Matrixnotation.............................97
5.3.4Relationtothemaximum-likelihoodapproach.............98
5.3.5Neuronalrealizationofthemodel....................99
5.4Arecipeofmakingmaps.............................100
5.5Multimodality...................................101
5.5.1Multimodalinteraction.........................101
5.5.2Developmentofmultisensoryspace...................102
5.6Discussion.....................................104
5.ANonlinearitiesininformationprocessing....................105
5.BSelf-averaging...................................105
5.CRemainingderivationstepsleadingto(5.23)..................106
5.DGaussianblurredsignal.............................109
111formationmapdalMultimo6.1Introduction....................................111
6.2Theintegratedmultimodalteacher.......................114
6.2.1HowdounisensorymapsdetermineiMT?...............114
6.2.2HowdoiMTcharacteristicsinfluencemapadaptation?........118
6.2.3HowdoesiMTcalibratedifferentunimodalmaps?..........122
6.3ApplicationsoftheiMTconcept........................122
6.3.1Experimentspro-vision-guidedmapformation.............122
6.3.2Experimentscontra-vision-guidedmapformation...........126
6.4Discussion.....................................128
6.AOptimalcombinationoftwomodalities.....................129

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Ajourneyofathousandmilesmustbegin
step.esinglawithLaoTzu

Thischapterintroducesthefundamentalsofsensoryinformationprocessing,focusingon
theprocessingofsoundsignalsastheauditorysystemisoneofthemostwidespreadof
thevarioussensorysystems.Theprominentroleoftheauditorysystemcanbeexplained
bythespecificpropertiesofsoundperception,namely,hearingisomnidirectionalandits
processingisveryfast.Incomparisontolight,soundwaveshaveamuchlongerwavelength
sotheyarenotblockedbysmallobjects.Wecan,forinstance,hearsomethingbehinda
treebutwecannotseeit.Amongstotheradvantagesauditionthereforeoffersthepossibility
toreacttoapproachingdangersthatarenotyetvisible.Toexploitthelatteradvantage,
itisessentialthatananimalcanlocalizeasoundsource,otherwisethepreycouldtryto
escapeinthedirectionofthepredator.Todeterminethedirectionofasoundstimulus,
severalstepsarenecessaryintheanimalsstudiedinthisthesis.
First,anobjectevokesanauditorystimulus.Section1.1describeshowsuchastimulus
propagatesthroughthesurroundingmediumandisamplifiedbeforearrivingatthe
animal.theofdetectorsSecond,thearrivingsoundwavesexcitethetympani,apairofthinmembranesthat
arepartofthemechanicalauditorysystem.Theanatomyoftheauditorysystemishighly
variable.Theprimaryfocusofthisworkisoninternallycoupledears(ICE)inwhichone
membranecaninfluencevibrationoftheotherthroughinternalcavities;seeSec.1.2.
Third,tympanicvibrationsareprocessedneuronally.Thefundamentalelementof
neuronalcomputationistheneuronalnet,consistingofneuronsasbuildingblocksand
thevariableconnectionsbetweenthem,calledsynapses.Section1.3reviewsthegeneral
functionandmathematicalmodelingofneuronsandsynapses.
Fourth,asresultofneuronalcomputation,sensorystimuligiverisetoneuronalrep-
resentationsofthestimuli,i.e.,neuronalmaps;seeSec.1.4.Eachneuronofthemap
representsaspecificproperty,e.g.,thestimulusataspecificpointinspace.Neighboring
neuronsrespondtosimilarsensoryinputs.Neuronalmapsreconstructthestimulusaswell

1

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aspossible,i.e.,optimallywithinthelimitsofprocessing.Theprecisecalibrationofthe
synapsesrequiredforstimulusreconstructionresultsfromanexperience-basedlearning
processthattakesintoaccountinputsfromallavailablesensorysystems.

ulistimSound1.1Soundwavesareoscillationsofpressurethataretransmittedthroughtheair.Asound
stimulusiscreatedwhenamovementcompressesthesurroundingairandcreatestraveling
soundwavesthatarriveatthetympaniofananimal.Dependingonthespatialrelationship
betweenthetwotympaniandthesoundsource,thearrivingsoundwavesdifferinphaseand
amplitudebetweenthetympani.Thesedifferencesaredenotedasinterauraltimedifferences
(sideITDofs)theandinhead,terauralforinstance,amplitudearrivesatdifferencesthe(facing,IADs).i.e.A,soundipsilaterwalavtefromympanaumsourceearlieratandone
withhigheramplitudethanattheaverted,i.e.,contralateraltympanum.Asoundwave
fromasourceinfrontofthehead,incontrast,arrivesatbothearssimultaneouslyandwith
thesameamplitude.ThesetwoexamplesdemonstratethatITDsandIADscanencode
thedirectionofasoundsource.AnevaluationofITDsandIADsasperformedbythe
auditorysystemandtheconsequentneuronalcircuitscouldthereforedecodethesound
direction.source

2

Figure1.1:Evolutionofvertebrateears.Duringthetransitionfromwatertoland,tympanic
middleearscapableofreceivingairbornesoundevolvedseparatelyamongtheancestorsof
moderncross-sectionsfrogs,throughturtles,diffelizards,rentbirds,headscroofcothesedilians,animalsand(middlemammals.ears-Diagramsgrayatfill).theOftopinshoterestw
aretoads),theausquamataditorysystems(lizardswithandinsnakternallyes),andacoupledves(bearsirds),(ICE)whereasthefoundtinympanianuraare(frogscoupledand
indepthroughendendifferentears,tlyi.e.,shapedtheintympternalanicavitiescannot;cf.asinfluencewellFig.each4.1.other.Mammals,FigureinduecontoSctrast,hnhauppve
andCarr[193].

ductiontroIn1.

Figure1.2:Internallycoupledearsinlizards.Left:Alightsourceontheothersideofthe
headilluminatesthetympanicmembranefromtheback.Right:Intersectionthrougha
lizard’shead.LargeEustachiantubes(ET)andthepharynx(P)connectthemiddleear
cavities(MEC)andthetympaniceardrums(TM)andallowthevibrationofonetympanum
toinfluencethevibrationofthetympanumattheotherside.PhotographsduetoJakob
Carr.CatherineandChristensen-Dalsgaard

1.2Mechanicalprocessingofsoundstimuli

1.2.1Evolutionofdifferentauditorysystems
Todetectincomingsoundwaves,ancestorsofmodernfrogs,turtles,lizards,birds,crocodil-
airians,toandtheosmammalssiclesdevinsideeloptheedtmiddleympani,ear;i.e.,cf.thinFig.1.1mem.branesCross-sectionsthattransmitofheadssoundoffromdifferenthet
animalswithtympanichearingshowtwofundamentallydifferentconstructions.Mammals
possesstympanithatarespatiallyseparatedandthereforeacousticallyindependentofeach
other.Incontrast,reptiliasuchaslizards,turtles,crocodiles,andbirdshaveinternally
careoupledeconnectedars(ICE)through(forlargedetails,Eustace.g.,hianreviewstub[e19s,as24,193illustrated])inbwhicyhFigs.tympanic1.2andmem1.3.branesThe
evolutionaryappearanceofindependentandinternallycoupledears(Fig.1.1)suggests
thatthelatterareprobablyearlytympanicears.Ifthisisindeedthecase,themammalian
independentearsmustbederivedfrominternallycoupledears.Thisisreasonablesincethe
enlargingmammalianbraincouldhavegrownintotheinternalcavitiesanddisconnected
thetympani(Manley,personalcommunication).

earscoupledternallyIn1.2.2TatoneympanicofthememtbranesympanicofmemICEarebranescanacousticallypropagatecoupledthroughinthethesenseinternallythatainsignalterconnectedarriving
cavitiesandinfluencethevibrationoftheothertympanicmembrane.Inthisway,ICE
translateincomingsoundwaveswithspecificITDsandIADsintoamplifiedvibrationsof
thetwotympanicmembranes.Inturn,thetympanicvibrationsdifferthroughinternal
thetimeexternaldifferencessignal(iTDsand)andtheininternalternalamplitudecoupling.Indifferencesterestingly(,iADsthe),inbeingternalthecouplingresultofofbtheoth

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ductiontroIn1.

Figure1.3:Schematiccross-sectionthroughalizard’sheadtoillustratethemechanical
processingofauditorystimulithroughinternallycoupledears.Arrivingsoundwavesexcite
thetympanicmembranes(TM)aswellastheairwithinthemiddleearcavities(MEC),
theEustachiantubes(ET),andthepharynx(P).Theresultingvibrationofthetympanum
evokesamovementoftheattachedmiddleear,calledcolumella(C).Theleverconstruction
transmitsthevibrationofthetympanitotheovalwindow(OW),themembraneatthe
entranceofthefluid-filledcochlea.ThroughthevibrationoftheOW,thefluidinthe
cochleaisexcitedandgivesrisetoalocal,frequency-dependentactivationoftheembedded
basilarmembraneandtheunderlyingauditorynervefibers.Theroundwindow(RW)is
amembraneattheendofthecochleathatisneededtocompensatethepressurewithin
thefluid.Altogether,translationoftympanicvibrationsintoneuronalresponsesissimilar
foranimalswithinternallycoupledandindependentearsdespiteseveralmodifications;cf.
Chap.3.Figuretakenfrom[27].

tthreeympanic[17].mInemaddition,branesinenhancesternallyiTDscoupledinearscomparisonproducetolargeincomingiADsevITDsenbforyupsmalltoainfactorterauralof
distances[27,28].Chap.2and[221]presentageneralmodelofinternallycoupledears,
theICEmodel,thatdescribesvibrationsofthetympaniandtherealisticmouthcavity
dependentonthecharacteristicsofincomingsoundstimuli.Onceasoundwavehasexcited
thetympanicmembranes,thetranslationintoneuronalresponsesissimilartomechanical
soundprocessingasfoundinmammals.Tympanicvibrationsaretransmittedbythe
columella(themiddleearbonethatisattachedtothetympanum)totheovalwindow,a
membraneattheentranceofthecochlea;seeFig.1.3.Vibrationoftheovalwindowthen
resultsinvibrationofthefluidwithinthecochleaandoftheembeddedbasilarmembrane.
Duetoitssystematicallyvaryingstiffness,everypartofthebasilarmembranereactsonly
toaspecificfrequency.Auditorynervefibersenervatedbythemovementofarestricted
regionofthebasilarmembranethereforerespondtoasmallrangeoffrequenciesandin
thismanner,thebasilarmembranedecomposessoundfrequencies.

4

ductiontroIn1.

1.3Neuronalprocessingofsoundstimuli
Assoonassoundwavesareprocessedmechanicallythroughthetympaniandbasilar
membranes,processingcontinuesonaneuronallevel.Thefollowingsectionreviews
importantconceptsofneuronalcomputation.Foradetailedintroductiontocomputational
neurosciencesee,e.g.,Koch[125]orIzhikevich[97].

1.3.1Buildingblocksofneuronalcomputation
InPrize1906,intheMedicineSpanish“inanatomistrecognitionofSanthisiagoworkRamononthyeCajalstructureof(1852-1934)thenervreceousivedsystem”.theNobHisel

Figure1.4:TwoofRamonyCajal’sextraordinarydrawings[236].Left:Drawingofasection
throughtheoptictectumofasparrowshowingindividualneurons.Right:Apyramidalcell
fromthemotorcortexextendedbyvisualizationofthethreestagesofneuronalinformation
processing:collectionofinputsthroughdendrites(lightblue),processingofsignalsthrough
thesoma(blue),andtransmissionofoutputsignalsthroughtheaxon(green).

workconsistedofasystematicanalysisoftheneuraltissue.Basedonhisrevolutionary
observcellularationselemen(cf.ts[Fig.237].1.4The),Cajalexistenceshowofedthesethatthesmallnervousstructuralsystemandconsistsfunctionalofindivunitsidualhad
ThanksalreadybtoeenCajal,suggestedscienintists1891acceptedbyWthataldeytheer-Hartzneuronal[220]whotissuewasdenotednottheconmtinasuousneurbutons.
constructedfromdiscreteelementsthatreceive,process,andtransmitsignals.Accordingly,
aneuronconsistsofthreeparts:dendritestocollectinputsfromotherneurons,thesoma
toprocessinputs,andtheaxontotransmitoutputsignalstootherneurons;cf.Fig.1.4,
t.righ

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a)FigureFollo1.5:wingtheEmergenceconcenoftrationtheion-spgradient,ecificK+Nernstionspdiffuseotentialintoatthetheexampleextra-cellularofK+domain.ions.
b)SinceK+ionsareelectricallycharged,thechangeintheirconcentrationsfromthe
intra-cellulartoextra-cellulardomaincreatesanelectricpotentialleadingtoanincreasing
forceintheoppositedirection.c)Thepotentialinwhichelectricanddiffusiveforcesare
balancedistheion-specificNernstpotentialdenotedasEKforK+ions.Figuredueto
Izhikevich[97,p.27].

StartingwithCajal’sindividualneurons,researchershavesteadilyextendedourknowl-
edgeofneuronalpropertiesandfunction.Today,weknowthattheneuronalcellmembrane
isessentialforprocessingneuronal,i.e.,electricsignals.Themembraneseparatesion
solutionsintheintra-andextra-cellularspaces.Thesesolutionscontaindifferentconcen-
trationsofsodium(Na+),potassium(K+),chloride(Cl-),andcalcium(Ca2+).K+ions
dominatetheintra-cellularsolutionandNa+ionstheextra-cellularsolution.Sincethe
ionsusuallycannotpassthroughthemembranebutinsteadassembleatitssurface,the
membraneactsasacapacitorwithcapacitanceCm.Anelectricpotentialappearsacross
themembrane,calledmembranepotentialV(t).Theionscanonlycrossthemembrane
throughmembrane-embeddedion-selectivechannels.Ioncurrentsthroughthesechannels
aredeterminedbythemembranepotentialandthegradientoftheionconcentrations
betweentheintra-andextra-cellularspaces.Asaconsequence,ionsmovealonganion-
specificreversalpotential,theNernstpotential;seeFig.1.5fordetails.Intherestingstate,
primarilyK+channelsareopensothattherestingmembranepotentialisdominatedby
thereversalpotentialofK+ionsVr≈−75mV.Thepassivecurrentthroughpermanently
openchannelsiscalledleakcurrent.
Incontrast,electricsignals,calledactionpotentialsorspikes,aregeneratedbyvoltage-
gatedNa+andK+channelsthatopenandcloseactivelywithacharacteristictiming.
ThrougharapidinfluxofNa+ions,themembranevoltageisabletoincreaseupto30mV
foraboutamillisecond.ThenadelayedoutflowofK+ionsleadstoarepolarizationofthe

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ductiontroIn1.

Figure1.6:Temporaldevelopmentofanactionpotential(spike).Theplotshowsthe
memexternalbraneinput,potenthetialVneuron(t)re(top)mainsininrespitsonseequilibriutoanmstateappliedwherecurrenthet(bmemottom).branepotenWithouttial
ThecorrespNa+ondsandtoK+therevrestingersalppotenotentialtialsVr=of−EN70a=mV.50Na+mVandandEactivKe=K+−90channmVelsarearecldepicted.osed.
Foraweakinputcurrentthemembranepotentialonlyslightlydepolarizesandreturnstoits
restingpotential.Incontrast,assoonasexternalinputincreasesthemembranepotential
abtheovcelleaandcertaindepolarizthreshold,esthemNa+emcbranehannelspotenoptial.en.AAsalargenumconsequence,beroftheNa+memionssbranetrepamsotenintialto
quicklyincreasesuptoaround30mV,anactionpotentialisgenerated.Atthesametime
moreandmoreactiveK+channelsopensothattheinfluxofK+ionsstartstodecrease
thethereforememfirstbranereppotenolarized,tialtowthenardevtheenhrevypersalerppolarized,otentiali.e.,EK.decreaseThedmembelowbranethepotenmemtialbraneis
pbutotenmoretial.Durexcitatoryingthecurrenphasetofishnypeerpededinolarizationcomparisonanothertotheactionprestingotentialstate.canThisbeperiogenerateddis
thereforedenotedasrelativerefractoryperiod.Theperiodduringtheactionpotential
fromwhere[97no,p.spike40].generationispossibleiscalledabsoluterefractoryperiod.Figuremodified

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ductiontroIn1.

Figure1.7:Synapsesconnectpre-andpost-synapticneuronsandallowactionpotentials
fromthepre-synapticneurontoinfluencethemembranepotentialofthepost-synaptic
neuronwithasynapse-specificefficiency.

membranepotential.Afterarefractoryperiod,themembranepotentialreachesitsoriginal
restingpotentialandthenextactionpotentialcanbegenerated;fordetailsseeFig.1.6.
Inthenextstep,anactionpotentialgeneratedinonespecificneuronistransmittedto
theconnectedneurons.Sinceneuronsareindividualcells,specializedstructures,called
synapsesfromGreek“syn-”togetherand“haptein”toclasp,areneededtopassinformation
fromapre-synaptictoapost-synapticcell;cf.Fig.1.7.Themajorityofthesynapsesare
chemicalsynapsesthatfunctionthroughareleaseofneurotransmitterfromthepre-synaptic
cellintothesynapticcleft,a20nmthingapbetweenpre-andpost-synapticcell.These
neurotransmitterscanthendiffusethroughthecleftandreachneurotransmitterreceptors
ofthepost-synapticcell.Eachsynapseischaracterizedbythesynapticstrength,i.e.,the
efficiencyoftransmissionofapre-synapticstimulustothepost-synapticcell.Depending
ontheireffectonthemembranepotential,i.e.,depolarizing(exciting)orhyperpolarizing
(deactivating)acell,synapsesarecalledexcitatoryorinhibitory.Synapticstrengths
indicatethedegreetowhichneuronsareconnected.
Toquantitativelydescribeneuronalbehaviorinresponsetoincoming(synaptic)currents,
neuronsaredescribedwithmathematicalmodels.Thesemodelsarealwayscompromises
betweendetailedandgeneralizeddescription.Togainintuitiveunderstandingofneuronal
behavior,themodelusedshouldincludeallprocessesneededtoexplaintheobservedeffect
andshouldleaveoutallothers.TheHodgkinandHuxleymodel[89]isanexampleofa
detailedneuronmodelthatincludesthedynamicbehaviorofvoltage-dependedNa+and
K+channels.Thismodel,however,ischaracterizedbyalargenumberofvariablesthatare
notabletobefullydeterminedexperimentally.Furthersimplificationoftheneuronmodel
sothatitonlyreproducesbasicphenomenologicalandelectricpropertiesofaneuronis
thereforereasonable.Asaconsequence,mathematicallytractableneuronmodelssuchas
theleakyintegrate-and-fireandthePoissonmodelhavebeendeveloped.

8

ductiontroIn1.

Figure1.8:Equivalentcircuitforaleakyintegrate-and-fireneuron.Behavioroftheneuron
isdescribedbythemembranepotentialV(t),thatis,thepotentialbetweenintra-and
extra-cellularspace.Themembraneofaneuroncanbecharacterizedascapacitorwith
leak,i.e.,acapacitorCminparallelwitharesistanceRmandtherestingpotentialVr.
ThemembranepotentialismodifiedbythecurrentIinj(t),i.e.,synapticcurrentsorions
injectedartificiallythroughelectrodes.

neurontegrate-and-fireinLeaky1.3.2

Theleakyintegrate-and-fireneuronmodelreducestheneurontoapointneuronwhose
behaviorisdescribedbythemembranepotentialV(t).Inresponsetoincomingcurrents,
theactionmempotenbranetialp(spikotene)tialisvariesgenerated,untilitandreacthehesmemabranecertainpotenthresholdtialisVresetabotovethewhichrestingan
potential.Sincethemodelfocusesonthesubthresholdmembraneproperties,itexcludes
themechanismsresponsibleforgeneratingtheactionpotentialsitself,i.e.,thevoltage-
dependentsodiumandpotassiumchannels;fordetailsseereview[16].Thespikeeventis
itsthereforefiringthonlyreshold.registeredTheasexactaformal,formofthediscretespikeveendoteswhennotethentermemthebranemodel.potentialreaches
remarkedSubthresholdthatthememmembranebranepofropaertiesneuronhavecanlongbecbhareenacterizedrecognized.asIncapacitor1907,withLapicqueleak,
i.e.,acapacitorCminparallelwitharesistanceRm;see[135]forLapicque’soriginaland
[the13,14mem]forbranethe[Vr−translatedV(t)]/Randm,thecommenmemtedbranemanpuscript.otentialcanBesidesbethemodifiedOhmicbycurrentexternallyover
injectedcurrentsI(t),i.e.,currentsinjectedthroughsynapsesorintra-cellularelectrodes.
Kirchhoff’snodalinjrulefortheequivalentcircuitofaleakyintegrate-and-fireneuron(cf.
ofFig.the1.8no)de.requiresApplicationthatallofcurrenthetslawflowingyieldsinthetoanodifferendearetialequalequationtothecurrentsflowingout

CmV˙(t)=[Vr−V(t)]/Rm+Iinj(t).

(1.1)

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ductiontroIn1.

Foratime-dependentinputcurrentIinjthecorrespondingmembranepotentialisgivenby
t−t01ts−t0
V(t)=Vr+exp−CmRmCmt0dsexpCmRmIinj(s)(1.2)
whichassumesthattheneuronhasfiredandthemembranepotentialwasresettoits
restingpotentialVrattimet0.ToprovethattheaboveV(t)fulfillsEq.(1.1),theinitial
conditionfort=t0isfirstverifiedasV(t0)=Vr.Second,thetimederivativeofV(t)can
ascalculatedeb1t−t01ts−t0
V˙(t)=−CmRmexp−CmRmCmt0dsexpCmRmIinj(s)
t−t01t−t0
+exp−CmRmCmexpCmRmIinj(t)
11=−CmRm[V(t)−Vr]+CmIinj(t).(1.3)
MultipliedwithCm,theresultingequationcorrespondsexactlytothedifferentialequa-
tion(1.1).ToillustrateEq.(1.2),inthecaseofaconstantcurrent,thesolutionreduces
toV(t)=Vr+RmIinj1−exp−Ct−Rt0.(1.4)
mmWithroundtime,brackthetsetememndsbraneto1.pIfotenthistialvalueapproaclieshesbVelorw+RthemIinjasthresholdtheV,expressionthewneuronithindothees
notspikeatall,otherwiseitspikesregularlywithaconstantinterspikeintervalTgiven
bytheequationV(t=T)=V.Takentogether,theabovecalculationsillustratethat
adescriptionofneuronalbehaviorthroughleakyintegrate-and-fireneuronsispossible
butlengthy.Forcalculationsincludingalargenumberofneuronsinparticular,itis
advantageoustorepresentfiringofeachneuronstochastically.

neuronoissonP1.3.3ThePoissonneuronreducesthefiringofaneurontoastochasticprocess,inotherwords,
tohomoangeneinhomoousPgeneoissonousproPcess,oissonthisproratecesswithfunctionaisctime-deponstanetnovdenertratetime).Anfunctionν(tinhomogeneous)(fora
Poissonprocessfulfillsthreeaspects:
•disjointintervalsareindependent,
•theprobabilityofgettingasingleeventattimetinaninterval[t,t+Δt)isgivenby
ν(t)Δt,
•theprobabilitypmultofgettingmultipleeventsfollowso(Δt),thatis,
pmult/Δt→0forΔt→0.(1.5)

10

ductiontroIn1.

FordetailsofPoissonprocessesandPoissonneuronssee[110]andAppendixBof[216].
TheadvantageofPoissonneuronsliesinacompactmathematicaldescriptionallowing
exactsolutions.Forexample,theprobabilitythataneuronfiresktimesinaninterval
[t0,t)canbecalculatedexplicitly.Tothisend,theintervalissplitupintoNdisjoint
intervalsoflengthΔt=(t−t0)/N.First,theprobabilityp0thataneuronremainssilent
overtheentireinterval[t0,t)iscalculatedas
1−Np0=[1−ν(tl)Δt](1.6)
=0lwheretldenotest0+lΔt.TermsofhigherorderinΔtareneglected.Totransformthis
probabilityintoamoreconvenientformapplicationoftheequalityx=exp[ln(x)]leadsto
N−1
p0=expln[1−ν(tl)Δt].(1.7)
=0lForxsmallthelogarithmln(1+x)=∞(−1)n+1xn/ncanbeapproximatedasln(1+x)≈x.
=1nForasufficientlysmallΔt,thisapproximationappliessothattheprobabilityp0transforms
toN−1t
p0≈exp−ν(tl)Δt≈exp−dtν(t).(1.8)
=0lt0Theprobabilityp0ofaneuronremainingsilentconsequentlydecreasesexponentially
withincreasinglengthoftheconsideredintervalandincreasingfiringrate.Basedonthis
probabilityp0,theprobabilityofgettingkspikesintheinterval[t0;t)isnowcalculated.
ForonespecificrealizationofkspikesinNin∗terv∗als,thetimestl∗,l∈N,0≤l<∗k
denotethekchosensucceedingspikeintervalstl<tl+1;cf.Fig1.9.Theprobabilitypkof
thisspecificrealizationcanbecalculatedastheproductoftheprobabilitiesofspikesin
intervalstl∗andthenon-firingprobabilityintheinterspikeintervals
1−kpk∗=[ν(tl∗)Δt]∙
=0lfiringprobabilityattl∗
t0∗k−2tl∗+1t(1.9)
exp−dtν(t)exp−dtν(t)exp−dtν(t);
t0l=0tl∗+Δttk∗−1+Δt
non-firingprobabilityininterspikeintervalsaccordingtoEq.(1.8)
seeFig.1.9foranillustration.AsΔt→0theaboveequationsimplifiesto
k−1t
pk∗=[ν(tl∗)Δt]exp−dtν(t).(1.10)
=0lt0

(1.10)

11

ductiontroIn1.

Figuredisjoint1.9:intervalsDistributionoflengthofkΔ=t=3(tspik−tes)in/N.theTheintervplotal[tsho0;tws).aThespinecifictervalisrealizationsplitupofinkto=N3
spikesinthechronologicallyorderedin0tervals{t0∗,t1∗,t2∗}.

Theequationgivestheprobabilityofgettingkspikesatthespecificchronologicallyordered
spiketimest∗<t∗<...<t∗.Sincespikesareindistinguishableanotherarrangement,
i.e.,permutation,0of1thespikektimes−1correspondstothesameeventwithidenticalprobability.
Inanextstep,thekspiketimesarerandomlychosenfromtheNpossibleintervals.
Forreasonsofmathematicalconveniencearandomarrangementofspiketimesisallowed
thatisaccountedforbydevisionthroughalladditionalpermutationsk!ofthesamek
spikes.AltogetherthegeneralprobabilityforkspikesinNintervalscorrespondsto
tpk=exp−dtν(t)∙
t0non-firingprobability
1N−1N−1N−1
k!ν(tn1)Δtν(tn2)Δt...ν(tnk)Δt.
n1=0n2=0nk=0
n2=n1nk=n1,...nk−1
firingprobabilitiesofkspikesindifferentunorderedintervals
(1.11)AgainasΔt→0,theprobability(1.11)canbewritten
ttk
pk=exp−dtν(t)k1!dtν(t).(1.12)
tt00ityHence,togetkfundamenspikestalstwithinochaasticgivenpropinertiesterval.yieldSimilaracompactcalculationsdescriptioninvofolvingthePfiringoissonprneuronsobabil-
cansinceleadneuronstoanareintuitivpartseoflargerunderstandingarraysofofneurons,complicatede.g.,ofneuronalneuproronalcesses.maps.Thisisessential

12

ductiontroIn1.

1.4Neuronalrepresentationofsoundstimuli
mapsNeuronal1.4.1Torepresentsensorystimuliwithinthebrain,neuronsdonotfunctionautonomously
butareoftengroupedinwell-orderedarrayscalledneuronalmaps[124,217].Theidea
thatamap-likearchitectureunderliescertainaspectsofsensoryprocessingwassuggested
in1879byHelmholtz[86]whoremarked“DassdurchdasEntlangf¨uhrendestastenden
FingersandenObjectendieReihenfolgekennengelerntwird,indersichihreEindr¨ucke
darbieten,dassdieseReihenfolgesichalsunabh¨angigdavonerweist,obmanmitdiesem
oderjenemFingertastet,dasssiefernernichteineeinl¨aufigbestimmteReiheist,deren
Elementenmanimmerwiedervor-oderr¨uckw¨artsinderselbenOrdnungdurchlaufen
muss,umvoneinemzumanderenzukommen,alsokeinelinienf¨ormigeReihe,sondern
einfl¨achenhaftesNebeneinander.”Thatis,Helmholtzalreadyrecognizedatopographic
neuronalrepresentationofatwo-dimensionalsurface.Aneuronalmapisconstructed
fromanarrayofneuronsinwhichneighboringneuronsrespondtosimilarsensorystimuli
[23,43,91,114,116,119,151,202,207].Intheauditorysystem,mapsrepresentfrequency,
interauraltimedifference,interauralamplitudedifference,andevenamplitudemodulation
[62,147,167,172,194,209].
Withingivenlimitations,e.g.,noiseandnumberofavailableneurons,neuronalmaps
reconstructasensorystimulusaspreciselyaspossible,i.e.,optimally;seeChap.5fora
generalmathematicalframeworktocalculateoptimalconnectivity.Theneurophysiological
conditionsforsuchaprecisetopographicarrangementareanorderedarrangementofthe
connectingaxonsandafine-tunedsynapticpattern.Accordingly,mapcalibrationconsists
oftwosteps,apre-wiringthroughgrowingaxonsandafine-tuningthroughactivity-based
synapses.theoflearning

mapaPre-wiring1.4.2Forthepre-wiringofmaps,moleculargradientsandaxon-axoninteractionplayakey
rolebrain.[140,Concen213].trationWhenthegradienbtsrainofdevspecificelops,axonsmoleculesgrosucwhandasephrinsconnectvactariasousguidancepartsofcuesthe
ingraduallyattractingalongorrepanarraellingyaxofons.neurons,Whengrothewingmoleculesaxonsandarrangetheiralongrectheeptorsaxisareofthedistributedarray
andInsoformaddition,acoarsegrowingpre-wiringaxonsoftendatomap.fillupavailablespacesmoothlybecausetheyare
competingforoneormorelimitingmoleculesinthetarget.Asaconsequence,axonsthat
growfromatopographicallyarrangedstructure,e.g.,theretinaorthecochleainteract
witheachother.Theaxonsthereforemaintaintheirstructuralarrangementandprojectit
area.targetthetoThroughoutthisthesis,themechanismsofmoleculargradientsandaxon-axoninteraction
areBeyreondferrethis,dtothewimplicitlyorkconcenasnectratesessaryontotheolsforsecondachievingstageofamappre-wiringformation,ofthesethatnsis,oryonmaps.the
fine-tuningofmapsthroughactivity-basedlearning.

13

ductiontroIn1.

Figure1.10:Left:AneuronreceivesinputfromNneurons.TheinputspiketrainsSiin
with1≤i≤NaretransmittedthroughsynapseswithsynapticstrengthJitothesoma.
Inreactiontothecombinedinputs,theneuronproducesanoutputspiketrainSout.Right:
Theincomingandoutgoingspikesmodifytheefficiencyofasynapsebyvalueswinandwout.
AccordingtothetimingbetweenincomingspiketiandoutgoingspikeTj,thelearning
windowW(s)withs=ti−Tjadditionallyincreasesordecreasesthesynapticstrength.
Asynapsethattransmitsaninputspikeslightlybeforeanoutputspikeislikelytobe
importantforthegenerationoftheoutputspikeandisenhanced.Theoppositeisthecase
forasynapsethattransmitsaninputspikeslightlyafteranoutputspike.Experimentally,
thislearningwindowhasbeenmeasuredinvariousforms,e.g.,inrathippocampus[240].

mapaFine-tuning1.4.3Theprecisebehaviorofaneuronalmapisdeterminedbyafine-tunedconnectivity,i.e.,
synaptic,pattern.Anadulthumanbraincontainsfrom1014to5∙1014synapses.Each
synapsehasaspecificstrengththatcanbemodifiedduetothefiringactivityoftheneuron.
Theresultingamountofinformationisgiganticandcannotbeencodedgeneticallybut
hastobelearnedduringthedevelopmentofananimalorhuman.Asaconsequence,
fine-tuningofamapisbasedonexperience,i.e.,learning,thatmodifiesthesynaptic
patternofamap.TheCanadianpsychologistDonaldOldingHebb(1904-1985)wasthe
firsttounderstandhowtheactivityofneuronscontributestothelearningprocess.Inhis
book“Theorganizationofbehavior”[79]Hebbwrote:
•“Whenonecellrepeatedlyassistsinfiringanother,theaxonofthefirstcelldevelops
synapticknobs(orenlargesthemiftheyalreadyexist)incontactwiththesomaofthe
63)(p.cell.”second•“Thegeneralideaisanoldone,thatanytwocellsorsystemsofcellsthatarerepeatedly
activeatthesametimewilltendtobecome’associated’,sothatactivityinonefacilitates
activityintheother.”(p.70)
Dependingonthesynchronybetweenfiringofpre-andpost-synapticneurons,asynapse
canundergolong-termpotentiation(LTP),along-lastingenhancementofthesynapse,or
theopposite,thatis,long-termdepression(LTD).Mathematically,bothprocessescan
bedescribedintermsofspike-timing-dependentplasticity(STDP)[6,7,36,66,109,149,
199,216,240].Figure1.10inillustratestheunderlyingmechanism.Hereaneuronreceives
inputs,i.e.,spiketrainsSfromNinputneuronsviasynapseswithstrengthsJiwhere
i∈N,1≤i≤N.Moidificationofsynapticstrengthsinresponsetoactivityoftheinput

14

ductiontroIn1.

neuronsisthreefold.First,eachincomingspikeattimetichangesthecorresponding
synapsebyanamountwin.Second,eachoutputspikeattimeTichangesallsynapsesby
theamountwout.Third,theprecisetimingbetweeninputandoutputspikechangesthe
correspondingsynapseviathelearningwindowW(s=ti−Ti);seeFig.1.10,right,middle
row.Forthelastmechanism,synapticstrengthsincreaseifapre-synapticfiringprecedesa
spike,anddecreaseotherwise.Assumingthelearningprocesstobemuchslowerthanthe
neuronaldynamics,synapticchangescanbedescribedbythedifferentialequations

∞dJi(t)=winνiin(t)+woutνout(t)+dsW(s)Ci(s,t)(1.13)
dt−∞withνiin=<Siin>andνout=<Sout>themeanfiringratesofinputneuronsandneuron
andconsidered,Ci(s,t)=<Siin(t+s)Sout(t)>(1.14)
thecorrelationtermbetweenthei-thinputspiketrainattimet+sandtheoutputspike
trainattimet.Here<f(t)>denotestheensembleaverageofanarbitraryfunctionf(t)
andf(t):=T−1tt+Tdtf(t)itstimeaverageinanintervalT.
Forthepurposeoffine-tuning,STDPisappliedtothesynapticpatternofamapthat
getsadditionalinputfromanalreadycalibratedteachermap.Theensuingsupervised
learningcansuccessfullyexplainmapformation[52,56].Inthefirststageofbrain
development,however,allavailablemapsshowabadresolution[223,225].Onemap
neverthelesshastofunctionastheteachermaptocalibratetheothersensorymaps.Given
thatthemultimodalmapascombinationofallunimodalinformationfromtheavailable
sensorysystemsisthemostreliableandprecisemap,Chap.6suggestsmapformation
basedontheintrinsicmultimodalteacher.Asaconsequence,mapformationcorresponds
toamutualimprovementwithinallsensorymaps.

15

Heartheotherside.(Audipartemalteram.)
SaintAugustine(354-430AD)

2.ModelingInternallyCoupledEars:
delmoICEThe

Lizards,frogs,alligators,andmanybirdspossessaspecializedhearingmechanism:inter-
nallycoupledears(ICE)wherethetympanicmembranesconnectthroughalargemouth
cavitenhancesysothethatpthehasevibrationsdifferencesofandthetcreatesympanicmamplitudeembranesdifferencesinfluenceineacthehtother.ympanicThismemcouplibraneng
ofinvibrations.ternallyBothcoupledcuesears,showthestrongICEmodeldirectionalit,thaty.consistsThiscofhaptertwopresenparts.tsOnathegeneralonemohand,del
athree-dimensionalmodelofinternallycoupledearswithasimplifiedcylindricalmouth
cahand,vitynallowsumericalforsimcalculationulationsofofathecompeigenfunletectvibrationionsinprofileexemplaryofthe,memrealisticallybranes.Onthereconstructedother
themouthcacylindricalvitiesmouthfurthercavitestimateytheadditionallyeffectsproofvidesthethecomplexopportunitgeometryy.toTheincorpmooratedelwiththe
effectoftheasymmetricallyattachedextracolumellaofthemiddleear,whichleadsto
themodelactivcanationexplainofhighermeasuremenmembranetstakenvibrationfrommothetdes.Inympaniccorpmemoratingbranethisofaeffect,livingthelizard,ICE
forexample,datademonstratinganasymmetricalspatialpatternofmembranevibration.
Astheanalyticalcalculationsshow,theinternallycoupledearsincreasethedirectional
inrespternalonse,timeappearingdifferencesinlarge(iTDdirecs).tionalinternalamplitudedifferences(iADs)andinlarge

ductiontroIn2.1Incontrasttomammals,wherethetympanicmembranesareindependentofeachother,
lizardspossessinternallycoupledearswherethetympanicmembranesconnectthroughlarge
Eustachiantubes,asillustratedbyFig.2.1.Thus,asignalarrivingatoneofthetympanic
membranescanpropagatethroughtheinternallyinterconnectedcavitiesandinfluence

17

2.ModelingInternallyCoupledEars:TheICEmodel

thevibrationoftheother.Consequently,onetympanicmembraneshowsadirectional
responsetoincomingsoundsignals,aso-calledpressure-gradientreceivercharacteristic
asfirstdescribedbyAutrum[4]andlaterbyMichelsen[150]forlocusts.Similarinternal
couplingispresentinmanybiologicalsystemssuchasfrogs[e.g.,[29,49,103]],birds[e.g.,
[88]quailsand[33]barnowlsbelow3kHz],andlizards[inparticular,[27,28]].Reviews
byCarretal.[19,20]covertheevolutionaryaspectsofthiscoupling.

Figure2.1:Left:PictureofaTokaygecko’shead(snouttotheleft).Alightsourceonthe
othersideoftheheadilluminatesthetympanicmembranefromtheback.Thecartilaginous
elementattachedtothemembraneispartofthemiddleear,calledextracolumella.Right:
Castimprintofthemouthcavityofagecko.Upperjawofthelizardwiththemouth
pointingtothetopofthepicture.Thefiguresillustratethecouplingofthetympanic
membranesthroughlargeinternalcavitiesgivingrisetointernallycoupledears(ICE).
PhotographscourtesyofJakobChristensen-Dalsgaard.

ForICE,phasesandamplitudesofthetympanicmembranevibrationsvarywiththe
directionofthesoundsource,andtheydosoinaratherpronouncedway.Foranimals
ofsmallheadsizeamplitudedifferencesbetweentympanicvibrationsexclusivelyarise
fromICE.Thisisbecausethesmallheadresultsinnegligiblesounddiffractioneffects
suchthatsignalsfromthesamesoundsourcearriveatthetwoearsatslightlydifferent
timesbutwithsimilarintensities.Incontrasttosystemshavingindependentears,the
couplingofthetympanicmembranescreatesdirectionalcuesthroughtheamplitudesof
thetympanicmembranevibrations.ICEthereforetranslatetheexteriorphaseshifted
signalsintotympanicmembranevibrationsthatvarybothphaseandamplitudewith
thesubtractiondirectionofofthethesoundlogarithmicsource.vibrationAssuggestedamplitudesbyfromJørgensenthet[wo103t],aympanicpossiblememneuronalbranes,
theinternalamplitudedifference(iAD),couldfurthersharpendirectionality.
Belowisathree-dimensionalmodelwithacylindricalmouthcavityandadjoined
membranes,developedtoelucidatetheprincipalmechanismofinternallycoupledears.

18

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.2:ConstructionandfunctioningofthemiddleearasreproducedfromManley[143].
Left:laginousSchematicelementbviewetwofeentheAtandCympanicisattacmemhedbranetothewithttheympanfolloumwingandmiddlecalledear.Thextracolumellaecarti-.
Thevibrationscolumellaofthethentympanicdirectlymemconnectsbranetoviatheitscofochlea.otplateTBoptoleft:theovalPhotographwindowofandthettransmitsympanic
membranewiththeattachedextracolumelladuetoJakobChristensen-Dalsgaard.Right
cantop:onlForyhatchfrequenciesaroundbelothew4p−oin5tC.kHzTh[143us,]thethedisplacemenextracolumellatofbettheweenAextracolumellaandCissantiffdandthe
ICEattacmoheddeltincympanicorporatesmemthisbraneisrestrictionlinearasinradialadditionalbdirectionoundaryoftheconditioncircularforttheympanmemum.braneThe
reprodisplacemenducesttheovmoervtheemenlinetofofthethetipoftheextracolumellaextracolumellaattachmenAt;incfan.Fig.amplified2.5.Themanner.footplateRighBt
bottom:Forfrequenciesabove4−5kHz(notconsidered∗inthefollowing)theextracolumella
ThestartsfotootplatebendBandisnoalongeflectionroactivccuratedsatbythethepoinmotvCemenintofthethetipofthecolumella-extracolumellaextracolumellalinA.k.
Asaconsequence,theappearanceofflectionlimitsthehearingrangeoftheanimal.

ThisfirstpartoftheICEmodelismathematicallytractableandcontainsthemost
importantaspectofICE,viz.,thecouplingofthemembranesthroughtheinternalmouth
cavity.Thevibrationsofthemembranesarecalculatedasafunctionoftheforcedifference
bairetwineenthetheinexternalternalmouthprescasurevity.fromInthethecasoundvitysourceitself,aandwavtheepressureequationfromwiththetempmoorallyving
varyingboundaryconditionsattheeardrumsdescribestheairmovement.Consequently,
thethethreeanalytically-dimensionalsolubleICEstructuremoofdelisICE.theFmosturthermore,fundamenonetal,canphadaptysicallythiscorrectgeneralmomodeldelforto
anyspecificanimal,inthiscasethelizard,andincorporateadditionalfeatures.
onlyOneonenotablmiddleespearbecificationone,theinctheolumelICEla,mothatdelforcontactslizardstheistheeardrummiddleviaear.aLizardscartilaginoushave
element,theextracolumella.Asaconsequence,thetympanicmembranecannotvibrate
freelybutisloadedbytheaccessorystructureofthemiddleear;seeFig.2.2.The
extracolumellamovesasacompletelystiffbar.Flectioncanonlyoccurinthecolumella-

19

2.ModelingInternallyCoupledEars:TheICEmodel

extracolumellaattachmentfromfrequenciesaboveapproximately4−5kHz[143]andis
notconsideredinthefollowing.Thestiffnessoftheextracolumellaforcesthemembrane
intoalineardisplacementoverthelineoftheattachment.Thesolutionthereforebecomes
alinearcombinationofeigenmodesforthefree,unloadedmembrane.Asaconsequence,
highermodesemergeinadditiontothefundamentalmode.Thesemodescauseacomplex,
asymmetricspatio-temporalpatternoverthemembraneasmeasuredbyManley[144].
Giventhattheextracolumellaisattachedtotheeardrum,theICEmodelrepresentsthe
vibrationofthemembrane,eithertotherightortotheleft,asalinearcombinationofthe
externalpressureexcitationofthetwotympanicmembranes.Thecoefficientsare,inthis
case,functionsreflectingtheinternalresponseofthesystemtoipsilateralandcontralateral
localstimulation.ForHemidactylusfrenatusandTokaygeckoamplitudesofthemembrane
vibrations,iADs,andiTDsarecalculatedandcomparedtoexperimentaldata.
Aconsistent,realisticevaluationofICEfurtherrequiresconsiderationofsoundtrans-
missionvariationduetothemouthcavityshape.TheICEmodelthereforefurthercontains
anumericaleigenfrequencyanalysisoftherealisticmouthcavities.Three-dimensional
scansofacastimprintfromthemouthcavityofHemidactylusfrenatusandasetofscans
fromslicesthroughtheheadofTokaygecko(cf.Fig.2.1,right)allowareconstructionof
exemplaryrealisticmouthcavities.Numericalsimulationprogramscanloadtheresulting
meshesandcalculatetheeigenfunctionsandthecorrespondingeigenfrequencies.
Inpreviouswork,lizardearmodelsconsistedofathreeimpedanceelectricalcircuit[27,
28],basedonFletcher[50,p.164].Intheimpedancemodel,abstractimpedancesdetached
fromthegeometricpropertiesofthesystemrepresentboththetympanicmembranesand
themouthcavity.Thoughelectricalcircuitmodelscanexplainthegeneralpatternand
magnitudeofdirectionality,theycanbeseenasatrulycrudeapproximationtothereal
acousticsoftheear,forexample,bytreatingthecavityvolumeasasingleimpedance.
Neglectinginterestingphenomena,suchelectricalcircuitmodelsreduceacomplicated
systemtoamodelwithasmallnumberoflumpedparameters.Forexample,theattached
extracolumellaofthemiddleearonlyinfluencesthevalueofthemembraneimpedance;an
averagevalueforthewholemembranereplacesthetwo-dimensionalvibrationprofileof
themembrane;thecomplexformofthemouthcavityisrepresentedbyitsvolume.
Incontrast,theICEmodel,asitcontainsageometricrepresentationofinternally
coupledears,offersthepossibilityofanalyzingthemembranevibrationinspatialdetail.
Thisiscrucial,forexample,tounderstandingtheeffectoftheattachedextracolumellaof
themiddleearonmembranevibration;cf.theresultsbelow.Inaddition,theinfluence
oftherealisticmouthcavitycanbeestimatedbycalculatingitslowesteigenfrequency.
Furthermore,an“acoustical”modelbasedonelectricalcircuitsisvalidonlywhenthe
dimensionsofthecircuitelementsaresmallascomparedtothewavelength[159].Given
sucharestrictedpointofview,amoregeneralmodelwillrepresentabroaderrangeof
systemsfarbetterandwillalsoapplytolargeranimalshavingICE.

20

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.3:Geometricrepresentationofinternallycoupledearsasusedforthefirstpartof
theICEmodel.Theinternalcavityisrepresentedbyacylinderusingparameters(x,r,φ)
withanatomicallengthLdataandtheradiuscirclesasoegmenthatt−0β≤≤xφ≤≤Lβandrepresen0≤rts≤thea.conWheretactβissurfacebestimatedetweenfromthe
membraneandtheextracolumella.Arrowsmarktheorientationsofthecoordinatesystems
forthedisplacementsu0anduLofthemovingmembranesandforthevelocityvxofthe
movingairinthemouthcavity.

2.2DerivationoftheICEmodelwithcylindricalmouthcav-
yit

Theequationsderivedinthefollowingsectiondescribethesteadystatesolutionforthe
tympanicmembranes’vibrations,neglectingonsetphenomena.Thederivationstep-by-step
considersthedifferentelementsparticipatingintheprocess.ThecylindricalICEmodel
consistsofelementaryacousticelements:membranesandanairfilledtubeasdescribedin
acoustictextbooks,suchasRschevkin[187,p.107],Temkin[211,p.247],andFletcher[50,
p.73].AllusedfunctionsandparametersofthemodelarecollectedinTable2.1.
AsshowninFig.2.3,themodel’sgeometryconsistsofacylinderrepresentinga
simplifiedmouthcavity,usingcylindricalcoordinates(x,r,φ)astheparameters.The
lengthofthecylinderisdenotedbyLanditsradiusbya.Thetwocircularsurfacesat
x=0andx=Lrepresentthetympanicmembranes.Thesegment−β≤φ≤βofthe
membranemodelstheextracolumellaasacontactsurfacebetweenthemembraneandthe
attachedextracolumella.
Inthefirststepofsoundprocessing,theexternalsoundpressureexcitesthetympanic
membranes.Consequently,thefollowingSec.2.2.1givesaformulationforthesound
pressurearrivingatthetympanicmembranesfromagivensoundsource.Afterarrival,
thesoundprocessingconsistsoftwosteps,viz.,outsideactivationofthemembranesand
excitationoftheairintheinternalmouthcavity.Section2.2.2describesthefluctuating
airintheinternalcavityusingawaveequation.
Whileequationsdescribingaloadedforce-drivenmembranearepresentedinSec.2.2.3here
thederivedequationsaccountforageneralforce-drivenmembranewithoutspecifyingthe
eigenmodes.Thelatterarethencalculatedinasecondstep,thoughnotably,theattached
extracolumellarequiresadditionalmodificationsviaboundaryconditions.
Finally,Sec.2.2.3showsthatadaptationoftheboundaryconditionsleadstocouplingofthe

21

2.ModelingInternallyCoupledEars:TheICEmodel

Table2.1:FunctionsandparametersusedintheICEmodelwithcylindricalmouthcavity.

θHorizontalangleofthesourcewithrespecttotheheadmidline
ωAngularexcitationfrequencyoftheincomingsoundwave
pex(θ,ω;t)Incomingsoundwave
|pex|Amplitudeoftheincomingsoundwave
Ψ(r,φ;t)Drivingforceofthemembrane
u(r,φ;t)Displacementfunctionoverthemembrane,subscriptsdenote
theipsilateral(x=0)andcontralateral(x=L)membrane
cMPropagationvelocityonthemembrane
fmn(φ,r)Generaleigenmodeofthemembranedisplacement
Jm(kmnr)Besselfunctionofdegreemwithzeron
−β≤φ≤βContactsurfacebetweentympanicmembraneandextracolumella
gl(φ,r)Eigenmodeoftheloadedmembranedisplacement
χ[(Cmn)(m,n)]Integratedquadraticerrorbetweenalineardisplacement
andthefunction(m,n)Cmnfmn(r,β)
(r,φ,x)Cylindercoordinates
p(x,r,φ;t)Pressurefunctioninthemouthcavity
vx(x,r,φ;t)Velocityfunctioninthemouthcavity
kWavenumberk=ω/cwiththesoundvelocityc=343m/s
Gipsi(r,φ)Ipsilateralfilter
Gcontra(r,φ)Contralateralfilter

twodomains.Takentogether,themembranevibrationresultsfromalinearcombination
oftheexternalsoundinputswithanipsi-andacontralateralfilter,definedastheresponse
ofaneardrumtocontralateralandipsilaterallocalstimulation(soonlyonesoundinput).

inputsoundExternal2.2.1Letpexbeaharmonicexternalpressurewavewithangularfrequencyωandamplitude
|pex|.Thesourceislocatedatahorizontalangleθmeasuredfromthecentralaxisofthe
head.Forsmallheadsizethesoundsignalsarrivingatthetwotympanionlydifferin
phaseduetovaryingarrivaltimeskL/2[sin(θ)+θ]withwavenumberk=ω/candcas
soundvelocityinair;seeFletcher[50,p.154]fordetails.Assumingaflatheadform,the
effectofsoundtravelingaroundtheheadvanishesandthephasedifferenceforalizard
reducestokLsin(θ);cf.Fig.2.4.Christensen-Dalsgaard[27,28]experimentallyverified
thatdifferencesinamplitudeandphaseduetodiffractionaroundtheheadandbodyof

22

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.4:Illustrationofinterauraltimedifference(ITD)asitarisesfromasoundsourceat
angleθwithinterauraldistancebetweenthetympaniLandvelocityofsoundc.Depending
ontheangleofthesoundsourceθthedistancebetweensoundsourceandcontralateral
earislongerthanthedistancebetweensoundsourceandipsilateralear.Thesoundsignal
thereforereachesthecontralateralearlaterthantheipsilateralear.Thedifferencebetween
thearrivaltimesatthetympaniisdenotedbyITD=L/csin(θ).

thelizardarenegligibleforthefrequenciesusedhere.Thepressurefunctionsatthetwo
tympanicmembranes(x=0andx=L)arethereforegivenby
pex0(θ,ω;t)=|pex|eikLsin(θ)/2eiωt,
(2.1)pexL(θ,ω;t)=|pex|e−ikLsin(θ)/2eiωt
wherethesubscriptsdenotethecorrespondingmembrane.

2.2.2Internalcavity
TheCommonnextsectionacousticspmoecifiesdelsthe(cf.equationsacousticthattextbodescriboks,esucthehasmovinRscgairhevkinin[the187,inp.ternal107]cavitandy.
Temkin[211,p.247])assumethattheairismovingduetopressurep(x,r,φ;t)described
bythewaveequationincylindricalcoordinates(cf.Fig.2.3),
1∂2p(x,r,φ;t)=1∂r∂p(x,r,φ;t)+1∂2p(x,r,φ;t)+∂2p(x,r,φ;t)(2.2)
c2∂t2r∂r∂rr2∂φ2∂x2
withthereisthenosoundsoundveloinputcitycthroughasthethepropagationmouth.Thatvelois,citythe.Themouthaboisveclosed,equationwhichimplisietsypicalthat
forawaitinganimal.Adaptationoftheboundaryconditions(below)furtherincorporates
thevibrationofthetwotympanicmembranes.
TosolveEq.(2.2),aseparationansatz
p(x,r,φ;t)=f(x)g(r)h(φ)eiωt(2.3)

(2.3)

23

toleads

2.ModelingInternallyCoupledEars:TheICEmodel

k2f(x)g(r)h(φ)+f(x)h(φ)∂2g(2r)+1∂g(r)
∂rr∂r(2.4)
22+f(x)g(r)12∂h(2φ)+g(r)h(φ)∂f(2x)=0
x∂φ∂rwithk:=ω/c.Separatingtheparametersyieldsthesetofequations
d2f(2x)+kx2f(x)=0(2.5)
dxd2h(φ)+q2h(φ)=0(2.6)
2dφ22∂g(2r)+1∂g(r)+(k2−kx2)−q2g(r)=0(2.7)
∂rr∂r=:kq2sr
withseparationconstantsqandkx.ThelastequationisknownasBessel’sdifferential
equation[35,p.313]thatdefinestheBesselfunctionsJq(kqsr)ofthefirstkind,oforderq,
andwithsomeconstantkqs.Thesolutionsarethengivenby
f(x)=e±ikxx,h(φ)=e±iqφ,andg(r)=Jq(kqsr).(2.8)
Inotherwords,thepressurepisgivenbyalinearcombinationofwavespropagatingalong
thex-axiswithwavenumberkx=ω/cmultipliedbyfunctionsofrandφ.Furthermore,
thelinearconservationofimpulseforsmalldisplacements
v∂−p=ρ∂t(2.9)
allowsforthedirectcalculationoftheparticlevelocityinthexdirection,
vx(x,r,φ;t)=−kxAq+seiqφ+Aq−se−iqφeikxx
∞∞
ρωq=0s=0
+kρωxBq+seiqφ+Bq−se−iqφe−ikxxJq(kqsr)eiωt.(2.10)
Thecoefficientskqs,Aq+s,Aq−s,Bq+s,andBq−smustbeadjustedinsuchawaythatthe
vibrationprofilesoftheinternalairandthetwomembranesatx=0andx=Lareequal.
Asthemouthcavityisaclosedvolume,thereisnomasstransport.Thevelocityis
consequentlypurelyoscillatoryandmathematicallypossibleadditiveconstants[cf.Eq.(2.9)]
aresettozero.Thecouplingoftheinternalcavitywiththemembranesnecessitatesan
analysisofthevibratingmembranesbeforeadjustingthecoefficients.

24

2.ModelingInternallyCoupledEars:TheICEmodel

2.2.3Vibrationofthemembrane
Theaimofthefollowinganalysisis,givenanexternalsoundpressurepex,tofindfunctions
u0(r,φ;t)anduL(r,φ;t)forthedisplacementofthetwotympanicmembranes.The
temporalderivativesu˙0/L(r,φ;t)thenyieldthevibrationvelocity;cf.Fig.2.3.Thedriving
forceΨ(r,φ;t)isthedifferencebetweentheexternalpressureandtheforceexertedby
thefluctuatingairintheinternalcavity.Themembranesarefurtherloadedbythe
extracolumella,whichisanadditionalconditionincalculatingthemembranes’vibrational
eigenmodes.Asdiscussedpreviously,thederivationfirstconcentratesontheforce-driven
membranebeforecalculationoftheeigenmodesinasecondstep.

Force-drivenmembraneLetu(r,φ;t)bethedisplacementofoneofthemembranes
inx-direction.ForanappliedforceΨ(r,φ;t)thefunctionu(r,φ;t)hastofulfillthe
two-dimensionaldampedwaveequationinpolarcoordinates[50,p.78],
−∂2u(r,2φ;t)−2α∂u(r,φ;t)+c2M∂2u(r,2φ;t)
∂t∂t∂r
+1∂u(r,φ;t)+1∂2u(r,φ;t)=1Ψ(r,φ;t)(2.11)
r∂rr2∂φ2ρmd
withαbeingthedampingcoefficientofthemembrane,cMthewavepropagationvelocity
onthemembrane,ρmthemembranedensity,andditsthickness.
Herethemembraneisfixedattheboundaryr=a,whichmeansu(a,φ;t)=0.The
firstAgainasteptoseparationsolve(2.11ansatz)isleadsthetocalculationequationsofofththeeform(homogeneous2.6)and(2.7solution)sothat(thattheis,sΨolu=tion0).
ascalculates∞∞u(r,φ;t)=fmn(r,φ)eiωmnt(2.12)
=1n=0mwithωmn=kmncMtheresonancefrequencyoftheeigenmodes
fmn(r,φ)=M+mneimφ+M−mne−imφJm(kmnr),m,n∈N,(2.13)
whereJmdenotesaBesselfunctionofordermandkmnguaranteesthatJm(kmna)=0is+the
n-thzeroofJ.Inaddition,theboundaryconditionslaterdeterminetheparametersM
andM−mn.Themeigenmodesfmn(r,φ)areorthogonalwhenintegratedoverthemembranes,mn
Sfmn(r,φ)fij(r,φ)dS=δmiδnj(2.14)
withSdSbeingtheintegraloverthesurfaceofthemembrane.
ThenextstepconsistsofthecalculationoftheinhomogeneoussolutiontoEq.(2.11).
isInthegeneral,(withtherespforcectetoactingφandonr)theconstanmemtbranesexternalconsistspressureoftwoforceparts.pexOnfromonethehandacousticthere

25

2.ModelingInternallyCoupledEars:TheICEmodel

excitement,andontheotherhandtheforcefromtheoscillatoryairintheinternalcavity
thatdependsonangularandradialcoordinatesφandr.Whereastheexternalpressureis
givenbyEq.(2.1),theinternalforceresultsfromthevibrationvelocityvx(r,φ;t)shownin
Eq.(2.10).Butfirst,theboundaryconditionshelptosimplifythefollowingcalculations.
Ingeneral,theboundaryconditionsrequirethatthevelocitiesinthecylinderandofthe
vibratingmembraneatx=0andx=Lareequal[211,p.131],thatis,theairinthe
cavitydirectlyatthemembranemoveswiththesamevelocityasthemembraneitself
u˙0(r,φ;t)=−vx(0,r,φ;t),(2.15)
u˙L(r,φ;t)=vx(L,r,φ;t).
Theminussignintheupperequationensuresthatthedirectionsdefinedforthedisplacement
functionsofthemembranesu0anduLareinlinewiththecoordinatesystemofthecylinder;
cf.Fig.2.3.Thefunctionsu˙0/Landvxarelinearcombinationsoforthogonalfunctions
fromthesameset.Furthermore,onlytheexternalforcefromthesoundsourcecanactivate
movementsoftheinternalcavity.FromEq.(2.15),thepressurefunctionpintheinternal
cavitycanonlycontainthevibrationmodesdeterminedbythemembranes.Thus,the
pressurephastobealinearcombinationofthemodesofthemembranesleadingto
p(x,r,φ;t)=Amneikxx+Bmne−ikxxfmn(r,φ)eiωt(2.16)
mnwherethefmnaretheeigenmodesofthemembranesasdefinedinEq.(2.13)andωisthe
angularfrequencyoftheexternalsoundsource.TheabovefunctionstillsatisfiesEq.(2.2).
Theappliedforceisthengivenby
Ψ(r,φ;t)=pex0−p(0,r,φ;t)forx=0,(2.17)
pexL−p(L,r,φ;t)forx=L.
ThisforcecanbeinsertedintoEq.(2.11)leadingforx=0to
−∂2u(r,φ;t)−2α∂u(r,φ;t)+c2M∂2u(r,φ;t)+1∂u(r,φ;t)+1∂2u(r,φ;t)
1∂t2∂t∂r2r∂rr2∂φ2(2.18)
=ρdpex0−(Amn+Bmn)fmn(r,φ)eiωt.
mmnForasolutionoftheform
∞u0(r,φ;t)=Cmnfmn(r,φ)eiωt(2.19)
mnwithcoefficientsCmntodetermine,Eq.(2.18)thentransformsto
Cmnω2−2iαω−c2Mk2mn−m2+m2fmn(r,φ)eiωt
∞22
mnrr(2.20)
=ρ1dpex0−(Amn+Bmn)fmn(r,φ)eiωt.
mmn

26

(2.19)

(2.20)

2.ModelingInternallyCoupledEars:TheICEmodel

Multiplyingtheequationbyfmnandintegratingoverthesurfaceyieldsthesolution
u0(r,φ;t)=1Ω−mn1−pex0fmn(φ,r)dS+(Amn+Bmn)fmn(φ,r)eiωt
iωSmn(2.21)tiondefinithewithΩ−1:=1iω(2.22)
mnρmd(ω2mn−ω2)+2iωα
whereωmn=cMkmn.Ananaloguederivationforx=Lleadstothesolution
1uL(r,φ;t)=
Ω−mn1−pexLfmn(φ,r)dS+AmneikxL+Bmne−ikxLfmn(φ,r)eiωt.
iωSmn(2.23)InthenextsectiontheboundaryconditionsdeterminetheremainingcoefficientsAmn
.Bandmn

arycAdaptingonditionsthevrequireelocitythattothethevelobcitiesoundaryintheconditionscylinderandofAccordingthetovibrating(2.15),memthebbraneound-at
x=0andx=Lbeequal.Equation(2.15)with(2.21)or(2.23)ontheleftsideand(2.9)
and(2.16)ontherightsideleadsfororthogonalfunctionsfmnandx=0to
−pex0fmn(φ,r)dS+Amn+Bmn=Ωmn(Amn−Bmn)(2.24)
ρcSwithkx=ω/candfortheotherside,x=L,to
−pexLfmn(φ,r)dS+AmneikxL+Bmne−ikxL=−ΩρcmnAmneikxL−Bmne−ikxL.
S(2.25)ThetwoequationsallowforacalculationofthecoefficientsAmnandBmnandaconsequent
representationofthesolutionsintheform
u˙0(r,φ;t)=Gipsi(r,φ)pex0+Gcontra(r,φ)pexL,
u˙L(r,φ;t)=Gipsi(r,φ)pexL+Gcontra(r,φ)pex0,(2.26)
filteripsilateralthewithGipsi(r,φ)=(2.27)
−ρcicot(kxL)+Ωmn
−Sfmn(φ,r)dS[Ωmn−ρcicot(kxL)]2+ρ2c2sin−2(kxL)fmn(φ,r)
mnfiltertralateralcontheandGcontra(r,φ)=(2.28)
−ρci[sin(kxL)]−1
mnSfmn(φ,r)dS[Ωmn−ρcicot(kxL)]2+ρ2c2sin−2(kxL)fmn(φ,r).

27

2.ModelingInternallyCoupledEars:TheICEmodel

LoadedmembraneInafinalstep,themodelhastoaccountforthefactthatthe
tympanicmembranescannotvibratefreelybutareloadedbytheextracolumella;cf.Fig.2.2.
ThemodelassumestheadditionalloadtobeattachedtoasegmentofthecircleSCwhere
−β≤φ≤β;cf.Fig.2.3.Thatis,theeigenmodeshavetofulfillthehomogeneouspart
ofEq.(2.11)withadditionalboundaryconditionsforφ=−βandφ=β.Theform
oftheeigenmodesfmnthereforeremainsasdefinedin(2.13)butisusingthereduced
membraneareaS\SC.Astheattachedloadisrigidintheφ-direction,thevelocityand
thedisplacementatacertainradiusarethesameforφ=βandφ=−βsimplifyingthe
eigenmodesin(2.13)to
fmn(r,φ)→Cmncos(mφ)Jm(kmnr),(2.29)
withm,n∈NandsomecoefficientsCmn.ThechoiceofpossibleexcitableBesselfunctions
reducesbecausetheextracolumella,thecartilaginouselementofthemiddleearattached
tothemembrane,isstiffintherdirection;cf.Fig.2.2,topright.Thismeansthatthe
displacementamplitudeforφ=±βhastobelinearwithrespecttorleadingto
Cmncos(mβ)Jm(kmnr)=a−r.(2.30)
am,nThatis,thesolutionfortheloadedmembranebecomesalinearcombinationofeigenmodes
oftheunloadedmembrane.Figure2.5illustrateshowasumofeigenmodesoftheunloaded
membranecanensurealineardisplacementofthemembraneoverthelineofattachment
withtheextracolumella.Atthisstepinthederivation,normalizationlimitsthemaximal
amplitude(atr=0)ofthelineardisplacementto1.However,theappliedexternalforce
determinesthemaximalamplitudefurtherbelow.
Tosolveanequationlike(2.30)foracoefficientCij,usuallyonemultipliestheequationby
thefunctionfijandintegratesoverthemembrane.Thecoefficientscouldbecalculated
easilygiventhefmnareorthogonal;cf.Eq.(2.14).Unfortunately,thegivenrelation
(2.30)istrueonlyfortheanglesφ=±β,whichmeansthatthefunctionsfmnarealready
evaluatedatthisangleandintegrationoverthesurfaceSisimpossible.Asaconsequence,
thecoefficientsCmnshouldminimizetheerrorfunction
2rχ[(Cmn)(m,n)]:=Cmncos(±mβ)Jm(kmnr)−a−rdr(2.31)
a0m,nthatmeasuresthedegreeofsimilaritybetweentheconstructedtestfunctionandthelinear
function.MinimizingtheerrorwithrespecttothecoefficientsCmnleadsto∂χ/∂Cij=0,
where(i,j)isonespecificpairfromthepossiblecombinations(m,n).Interchangingthe
orderofderivationandintegration,theminimizationconditionleadstothelinearsystem
equationsofrCmncos(mβ)cos(iβ)Jm(kmnr)Ji(kijr)dr
0m,n(2.32)r1=acos(iβ)Ji(kijr)(a−r)dr
0

28

2.ModelingInternallyCoupledEars:TheICEmodel

moFiguredesof2.5:theunloadedDisplacemenmemtofabraneloaded(top);memcf.Eq.brane(2.33(b).ottom)Theasfigurelinearillustratescombinationcompofositioneigen-of
thesurfaceeigenmobetwdeeeng0memfromTbraneableand2.2.Theextracolumella;omittedscf.egmentcircle(bluesegmencurvtes)−β≤represenφ≤tsβthineFig.contact2.3.
Anunloadedfreemembraneusuallyshowsasymmetricaldisplacementcorrespondingto
thefundamentalmode(topleft).Thedisplacementofthefundamentalmodeisnonlinear
inhowtheevrer,adialforcesdirection.thememBelobranew4in−to5akHzlineartheattacdisplacemenhedstifftovertheextracolumellalineofofthetheattacmiddlehmenear,t;
cf.Fig.2.2,topright.Themodelincorporatesthisrestrictionasadditionalboundary
memconditionbraneatbtheecomeslineaoflinearthecomattacbhmeninationt.ofAsaeigenmodesconsequence,forthethefree,solutionunloadedforthememloadedbrane.
Sincehighermodesdonothavetobesymmetrical(see,e.g.,toprighteigenmode)the
thememlinearbranedisplacemendisplacementtovcanershothewlineaofcomplex,attachmentasymmetricwiththespatio-tempextracolumella.oralpatterndespite

29

2.ModelingInternallyCoupledEars:TheICEmodel

forall(i,j)thatoccurinEq.(2.29).Thissystemisinfinite,asisthenumberofeigenmodes,
requiringrestrictiontoacertainnumberoffunctionsthatsufficetoapproximatelinear
displacement.ThelimitedsetofeigenmodesthataretakenintoaccountisdenotedbyZ.
Asanexample,thecalculationsbelowusethesetZ={(m,n)|0≤m≤5and0≤
n≤5}ofselectedeigenfunctions.Highermodesresultinastronglyfluctuatingsurface
thathasnotbeenobservedinexperimentssofar.

Table2.2:Theloadedmembrane:twopossibleeigenmodesg0andg1oftheloadedmembrane
withtheirrelativeerrorχwhencombiningthreeeigenmodesofthefreemembranetofulfill
theconditionoftheattachedextracolumellawithanerrorχ≤0.05%.Thethreefunctions
couldbechosenoutofapoolof25eigenmodesgivenbythesetZ.Thefunctionsare
orthogonalizedbyaGram-Schmidtmethod.Theeigenmodeg0anditscomponentsare
.2.5Fig.invisualized

ngn

[%]χ

0425.6J0(k01r)−51.3J0(k02r)0.0008
−183.7cos(φ)J1(k11r)
16.8J0(k01r)+369.4J0(k02r)0.0275
−160.0cos(φ)J1(k11r)+768.3cos(2φ)J2(k21r)

Furthermore,onlyacertainnumberofeigenfunctionsfromthissetZarenecessary
tobuildalinearfunctionwithminimalerror.Likelythecombinationoftwoorthree
eigenfunctionsissufficient,thusinafirststepdifferentnumbersofcombinedfunctionsfmn
with(m,n)∈Zaretested.Theresultsshowthatthreeeigenmodesfromthepossibleset
Zalreadygeneratealinearprofilewithanerrorbelow0.05%.Thefollowingcalculations
thereforecenteraroundcombinationsofthreeeigenmodeswithindicesZlwithl∈Nfrom
theabovesetZ.Solvingtheresultinglinearequationsystem(2.32)forthecoefficients
Cmngivestheeigenmodes
gl(r,φ)=alCmncos(mφ)Jm(kmnr)(2.33)
(m,n)∈Zl
withascalingfactoral.Allchoseneigenmodesglmustyieldanerrorχbelow0.05%.
EachofthesefunctionsisacombinationofthreeBesselfunctions.Twoeigenmodesglcan
containthesamefunctionfmn(onlyacompleteagreementofallthreemodesisexcluded).
Thustheneweigenmodesglarenotnecessarilyorthogonal.Onlysubsequentapplicationof
anappropriatemethodsuchasGram-Schmidt(describedintextbooks,suchasCohen[32,
p.82]orReedandSimon[179,p.46])leadstoorthogonaleigenmodes.Takentogether,
thismethodproducesasetoforthogonaleigenmodesthatfulfillthemembraneequation

30

2.ModelingInternallyCoupledEars:TheICEmodel

andtheadditionalboundaryconditionsinducedbytheattachedextracolumella.
ForloadedthememabovebranesetandZ,Ttheablerelativ2.2eshoerrorwsχthe≤t0.wo05%.possibleTheeigenmoorthogonaldeg0andeigenmoitsdescompglonenfortsa
arevisualizedinFig.2.5.Ingeneral,theeigenmodescorrespondtothefundamental
Theeigenmoresonancedef01withfrequenciesmoofdificationstheeigenmofromthedesglasadditionalneededforhigherEq.mo(des2.21)f02are,fgiv11,enanbdyfthe21.
resonancefrequenciesofthecontributingmodesfmndependingontheirexcitedfractions
inthelinearcombination.

2.3Numericalsimulationoftheeigenfunctions
vitiescamouthrealisticofAsthecomplexshapeofarealisticmouthcavitypreventsananalyticaldescription,the
ICEmodelcontainsanumericaleigenfrequencyanalysistoestimatetheeigenfrequenciesof
theinternalvolume.Theeigenfrequencyanalysisisnowperformedintwosteps.Thefirst
consistsoftheconstructionofsimplegeometriesandthecalculationoftheireigenfrequencies
withthesimulationprogramCOMSOL(seehttp://www.femlab.de/).Indoingso,the
influenceofanarrowingorwideningofageometrycanbeestimated;seeFig.2.13for
results.

Figure2.6:Hemidactylusfrenatus(toprow)andTokaygecko(bottomrow)andcasts
oftheirmouthcavitiesproducedbyJakobChristensen-Dalsgaard.Asindicatedonthe
right-handside,theinterauraldistanceis10mmforHemidactylusfrenatusand22mmfor
Tokaygecko.Inadditiontodifferentinterauraldistancesthecavitiesvaryinshapeand
volumesothattheycoveralargerangeofpossiblerealisticmouthcavities.

frenatusInaandsecondTokaystep,geckothe(cf.Fig.eigenfrequencies2.6)areofcalculatedrealisnticumericallymouth.caWithvitiesfordifferentinHemidactylusteraural

31

2.ModelingInternallyCoupledEars:TheICEmodel

distancesof10mmforHemidactylusfrenatusand22mmforTokaygeckotheanalysisof
thetwolizardscoversalargerangeofpossiblerealisticmouthcavities.Theprerequisitefor
anumericaleigenfrequencyanalysisisaclosedthree-dimensionalmeshofthecorresponding
geometry.ForHemidactylusfrenatusthree-dimensionalscans[154]ofacastimprintofthe
mouthcavityallowforareconstructionoftherealisticmouthcavity.Figure2.7illustrates
thethreemajorstepstoconstructthespatialmeshfromthescannedcastofthemouth
cavity.Thefirstone(left)isalignmentoftheindependentlyscannedprofilesfromthe
differentsidesofthegeometry.Inasecondstep(middle),thealignedprofilesarecombined
inaclosedgeometrythatpermits,inathirdandfinalstep(right),thecreationofa
three-dimensionalmeshofthemouthcavityinterior.

Figure2.7:Constructionofathree-dimensionalmeshfromthescannedcastofthemouth
cavityofHemidactylusfrenatus.Left:Alignmentofthescannedprofilesofthegeometry
fromdifferentsides.Middle:Constructionofaclosedgeometry.Right:Constructionof
theBLENDERthreeweredimensionalused.Inmesh.anextForthestep,theeditingofthethree-dimensionalmeshthemeshprogramsoftheICEMrealisticCFDmouandth
cavitycanbeimportedintothenumericalsimulationprogramCOMSOLforeigenfrequency
.2.14Fig.cf.analysis;

ForTokaygecko,spatialreconstructionsofthemouthcavityhavebeenreceivedfrom
asCatherineautomaticallyCarrandBrucereconstructedYoung.fromaFiguresetof2.8scanned(left)shosliceswsofthetheoriginallizard’sreceivhead.edThegeomemesh,try
however,containsholesanddisconnectedorundefinedelementswhichpreventsadirect
importintonumericalsimulationprograms.Inseveralpost-processingstepsthegeometry
thereforehastobeclosed,simplifiedandsmoothened(Fig.2.8,middle)tofinallygenerate
aFclosedortheeditingthree-dimensionalofbothmmesheshesofththeegeometryprograms(Fig.TGRID2.8,andright).ICEMCFDfromANSYS
(seehttp://www.ansys.com/products/)andBLENDER(seehttp://www.blender.org/)
wereused.ForbothHemidactylusfrenatusandTokaygecko,theclosedmeshesofthe
mouthcavitiesallowforaneigenfrequencyanalysisinthenumericalsimulationprogram
COMSOL.IncorporationofthetympanitosetupacompletenumericalICEmodelcould
beisolatedrealizedmouthinacanextvities,futuhorewevstep.er,allowNumericalforananalysiscalculationsoftheofthemouthcaviteigenfrequenciesy’seffectforonththee
behaviorofthewholesystemofinternallycoupledears.

32

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.8:ConstructionofarealisticmouthcavityofTokaygecko.Left:Originalmeshas
automaticallygeneratedfromscannedslicesofthemouthcavityreceivedfromCatherine
CarrundefinedandBruceelemenYtsoung.severalSincepost-prothemescessinghatthisstepsarestageconnecessarytains.holesMiddle:andCloseddisconnectedgeometryor
oftherealisticmouthcavityasobtainedfromtheoriginalgeometryusingwrappingand
thesmoothingrealisticmouthalgorithmscavityfromcanbeTGRIDimpandortedintoBLENDER.thenumericalRight:simTheulationobtainedprogramclosedmeshCOMSOLof
foraneigenfrequencyanalysis;cf.Fig.2.19.

2.4Evaluationandresults

RatingthesignificanceoftheICEmodelnecessitatesacomparisonofitspredictions
withmeasurements.Tothisend,theICEmodelisevaluatedusingparametersfrom
usetheHemidactylusICEmofrdelenatuwithsa(gekkonid)cylindrical[28]asmouthcasummarizedvity.inTable2.3.Allcalculatedresults
First,ThelaserevaluationdopplerofthevibrometrymodelformeasuremenHemidactylustsofearfrdrumenatusprovibrationsceeds[28in]areseveralcomparedsteps.
withthecalculateddatabasedontheICEmodel.Theanalysiscoversthevibration
amplitudeamplitudeofthedifferencesmembrane,(iAD),theandgaintheofintheternalintimterauraledifferencestransmission(iTD).pathInwaay,thesecondinterstep,nal
therealisticeigenfunctionmouthcasvitareynofumericallyHemidactylusfrcalculatedenatusfor.Fsimpleurthermore,artificialtheinfluencegeometriesofandtheforlowtheest
eigenfrequencyoniTDsandiADsisestimated.
ToillustratethattheICEmodelisauniversalmodelfordescribinginternallycoupled
earsdescribandedabnotoveonlyareapsperformedecializedasmowelldelforforTokaygeHemidactyluscko;cf.frSec.enatus2.4.3.theessenWhereastialHemidactyluscalculations
frenatuswithaninterauraldistanceof10mmisarathersmalllizard,Tokaygeckoisthe
secondthereforelargestunderlinesGeckospthatecieths.eAnICEmoadditionaldelcandescribconfirmationeinofternalltheyICEcoupledmodelearsforTforokayanimalsgecko

33

2.ModelingInternallyCoupledEars:TheICEmodel

Table2.3:GeometryparametersforHemidactylusfrenatus(gekkonid),obtainedfrom
].28[Christensen-Dalsgaard

L=10mmLengthofthecylinder(interauraldistance)
a=1.2mmRadiusofthetympanicmembrane
α=1000Hz/(2∙1.32)Dampingcoefficientofthemembrane
ρm=3.2mg/mmDensityofthemembrane
d=10µmThicknessofthemembrane
c=343m/sVelocityofsound

withlargelyvaryinginterauraldistanceandmouthcavities.
Theaimofthethirdstepistoprovethatthehighermodesincludedinthecylindrical
ICEmodelareresponsibleforthecomplexpatternofthevibrationamplitudeoverthe
membraneasmeasuredbyManley[144]forTokaygecko.Therefore,calculationofa
membraneprofileofthevibrationamplitudeusingthecorrespondingparameterset2.4
allowsforacomparisonwiththeexperimentaldata[144].

2.4.1Directionalityofthemembranevibrationpattern
BymeansoftheICEmodelwithacylindricalmouthcavity,itispossibletoestimatethe
dependenceofthemembranevibrationamplitudeuponthespecificfrequencyanddirection
ofasoundsource.Tothisend,itisnecessarytoevaluatethevibrationamplitudesata
specifiedpointonthemembrane,whichisusuallythetipoftheextracolumella.IntheICE
model(cf.Fig.2.3),thespecificpointisatthemiddleofthecircularmembrane,where
0.=rTheleftsideofFig.2.9showsthelogarithmicvibrationamplitudes,dependentonthe
soundsourcedirectionandfrequencyasgivenbytheICEmodelforHemidactylusfrenatus.
Therefore,onecandirectlycomparethecalculated(Fig.2.9,left)andexperimental
(Fig.2.9,right)data.Theexperimentaldata[28]consistofmeasuredeardrumvibrations
atthetipoftheextracolumellaofHemidactylusfrenatususinglaservibrometry.The
plotshowsvibrationamplitudeasafunctionoffrequency(y-axis)anddirection(x-axis).
Foreverymeasuredfrequency,themeansquareerrorbetweenmeasuredandcalculated
dataaveragedoverthetestedangulardirectionsquantifiesthegoodnessofthefit.The
resultingdeviationasillustratedinFig.2.10liesaround1.3dBre1mm/(sPa).However,
forfrequenciesabove5kHztheerrorfunctionsystematicallyincreases.Inthisfrequency
region,theexperimentaldatashowasystematicdeclinewithinthedirectionalresponse
pattern.Incomparison,thecalculateddatapredictamuchstrongerresponse.Apossible
explanationforthisoverestimationcouldbetheoccurrenceofalimitingprocessatthis
frequency,suchasbendingoftheextracolumella(cf.Fig.2.2,rightbottom)oraneffect
oftheeigenfrequenciesattherealisticmouthcavityofHemidactylusfrenatus.TheICE
modeldoesnotconsiderabendingoftheextracolumella,theinclusionofthelattercould
thereforebeafuturetestapplicationforthemodel.Numericalcalculationsbelowfurther
detailtheeigenfrequenciesoftherealisticmouthcavity.

34

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.9:Calculated(left)andexperimental(right)amplitudeoftympanicvibrationfor
Hemidactylusfrenatus(gekkonid)indBre1mm/(sPa),i.e.,thevibrationamplitudewith
regardtoareferencevibrationvelocityof1mm/sat1Pa.Theamplitudesdependon
thesoundsourcedirection(x-axiswithdirectionsvaryingfrom−180◦to180◦;negative
directionsarecontralateral,0isfrontal,andpositivedirectionsareipsilateral)andfrequency
(y-axis).ThecalculatedresultsarebasedontheICEmodelwithacylindricalmouthcavity.
MeasuredeardrumvibrationamplitudesforHemidactylusfrenatusarefromChristensen-
Dalsgaard[28].ThevibrationamplitudepatternascalculatedbytheICEmodelnicely
reproducesexperimentaldata.Themodel,however,overestimatesthevibrationamplitudes
above6kHzandunderestimatestheresponseofthesystemforlowfrequencies.A
quantitativecomparisonispresentedinFig.2.10.

Ingeneral,forbothexperimentalandmodeldata,atfrequenciesbelow1kHzthe
eardrumshowsanidenticalvibrationamplitudeforallanglesθ.Localizationofasound
sourcebasedonthevibrationamplitudeisthereforenotpossibleforthesefrequencies.In
contrast,above1kHztheamplitudevarieswithangleθ.Consequently,forthesefrequencies
thelocationofthesoundsourcecouldbeextractedfromtheamplitudevariations.Itis
remarkablethat,becauseofthespecialconstructionofICE,thesedirectionalhintsare
alreadypresentatthelevelofthetympanicmembranes.
Next,onecananalyzetheadvantagesofthetransmissionthroughtheinternalcavity
inmoredetailbyevaluatingthetransmissiongain.Transmissiongainisdefinedasthe
responseratioofeardrumvibrationwithcontralateralandipsilaterallocalstimulation
(onlyonesoundinput).Forlocalstimulation,oneoftheinputsiszero.Figure2.11shows

35

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.10:Meansquareerrorbetweenmeasuredandcalculatedamplitudesoftympanic
vibrationsforHemidactylusfrenatusaveragedoverthetestedangulardirectionsforevery
measuredfrequency.Theerrorisaround1.3dBre1mm/(sPa);onlyforfrequenciesabove
5kHztheerrorfunctionsystematicallyincreasesindicatingapossibleattenuationofthe
biologicaldatathroughlowereigenfrequenciesoftherealisticmouthcavityincomparison
totheusedcylindricalmouthcavity.

thecalculateddataaslightlinesandtheexperimentalresultsasblacklines.Thecomplex
gainNotably,functionexperimenconsiststalofdatatwofromparts,amplitudeChristensen-Dalsgaard(Fig.2.11[,28]left)presenandtphaseeardrum(Fig.2.11vibrations,righoft).
conHemidactylustralaterallofrcalenatusstimusingulation.aloThecalsoundqualitativesourceformanofdthosecomparedcalculatedtherespgainonsetofunctionsipsi-ovander
frequencyissimilarfordatafromtheICEmodelandexperiments.Thecalculateddata
furthershowasmootherbehaviorandlessextrememinimaandmaxima.Thisanalytically
solublesignificanparttlyofthesmallerICEthanmodel,thehowrealisticever,one,includesthusaaddcylindricalitionalmomouthdificationcavityiswithpaossiblevolumefor
thedirectionalresponsethatresultsfromtheeigenfrequenciesoftherealisticallyshaped
.yvitcamouthForHemidactylusfrenatusthecomparisonofthedatafromtheICEmodelwiththe
ofapcylindricalossiblemouthneuronalcavityproandcessingthebeasxpederimenontthealindataternal[27]amplitudeconcludesdiffewithrencetheev(iAD);aluationsee
Fig.2.12,top.Thelattermeasuresthevibrationamplitudedifference(indB)betweenthe
inputfromtheipsi-andcontralateralearbysubtractingthefreefieldeardrumvibration
is(indB)assumedofthetotworeflecttheeardrums.neuronalInfact,processingsubtractionbofecause,dBvwithinaluesacorrespneuron’sondstodynamicdivisionrange,but

36

2.ModelingInternallyCoupledEars:TheICEmodel

Figureradians)2.11:definedTastheransmissionresponsegainratioofamplitudeeardrum(leftvibrationsideinwithdB)conandtralateralphaseand(rightipsilateralsidein
localtransmissionstimulationgain(soasonlycalculatedonesoubyndithenput)ICEformodelHemidactwithayluscylindricalfrenatus(gekkmouthonid)ca.vityTheis
depictedaslightline,themeasuredtransmissiongain[28]asblackline.Calculatedand
expagreemeneriment.tallyOnlyformeasuredfrequenciestransmissionabove4gainskHzisshowtheretheagapsamebetweentendenciestheandcalculatedareinaandgoothed
gain.measured

thespikerateisalinearfunctionofthesoundlevelindB.TheiADisthereforedefinedas
|u˙0|
iAD:=20log|˙uL|.(2.34)
TheprocessingfunctioniADcorrespondstotheIVADfunctionofJørgensenetal.[103]
andismeasuredindB.AscomparedtoexperimentaldatafromChristensen-Dalsgaard[28],
thecalculatedfunction(positivevaluesforanglesexceedingzero,negativeforthosebelow
zero)showsthesamebehavior.Inbothcasestheresponseofthesystemisdirectional.
InadditiontoiADs,thevibrationsofthetympanicandifferintimeaswellleadingto
internaltimedifferences(iTDs)definedas
iTD:=argu˙0/ω.(2.35)
u˙LCalculatediTDsareshowninFig.2.12,bottomleft.Forsoundsourcedirections±π/2
andresultingfromtheinternallycoupledears,themaximaliTDsarearound±86µsfor
frequenciesbelow2.5kHz.ThesedifferencesillustratetheenhancingeffectofICE,asa
systemwithindependenttympanicmembranesandheadsizeofHemidactylusfrenatus
would,incomparison,producemaximalinterauraltimedifferences(ITDs)of±30µs.The
enhancementfactorofmaximaliTDsforinternallycoupledearsincomparisontomaximal
ITDsforindependentearsindependenceuponsoundfrequencyisillustratedinFig.2.12,
bottomright.Forfrequenciesbelow3kHztheinternalcouplingresultsiniTDsthat
areafactor3higherthanITDsforindependentears.LocalizationduetoiTDscould
thereforebepossibleatthesignallevelevenforanimalswithasmallinterauraldistance.

37

38

2.ModelingInternallyCoupledEars:TheICEmodel

Top:FigureInter2.12:nalOvamplitudeerviewofpossdifferencesiblelo(iADs)calizationindBcues[cf.forEq.(Hemidact2.34)]asylusafrenfunctionatusof(gekkdirectiononid).
(x-axis,negativedirectionscontralateral,0frontal,andpositivedirectionsipsilateral)and
frequency(y-axis)resultingfromtheICEmodelwiththecylindricalmouthcavity(left)
andfromexperiments[28](right).TheobtainediADssystematicallyvarywithdirectionof
soundsimilarsourceprofile.forThefrequenciescylindricalaboICEve1model,kHz.howevCalculateder,ovanderestimatesexperimeniADstalforiADshigh[28]frequenciesshowa
andunderestimatesiADsforlowfrequencies.Bottomleft:Internaltimedifference(iTDs)
iniTDsµsv[cf.aryEq.with(2.35)direction]asaoffunctionsoundofsourcefdirectionorlow(x-axis)frequencies.andBfrequencyottom(yright:-axis).EnhanceCalculatedment
factoranimalsofhavingmaximaliniTDsternallyresultingcoupledfromearstheinICEcomparisonmodeltowiththemaximalcylindITDrical=kLmouthsin(cθa=vityπ/for2)
ofanimalshavingindependentearsindependenceupondifferentsoundfrequencies.For
frequenciesbelow3kHztheinternalcouplingleadstoiTDsthatareafactor3higherthan
loITDscalizationforcuesanimalsinwithcomplemenindeptaryendentfrequencyears.TakrangesenhintogethertingatiADsseparatedandiTDspathwacouldysfordeliviTDer
andiADprocessing;seeSec.3.2fordetails.

2.ModelingInternallyCoupledEars:TheICEmodel

Asthevectorstrength(seeFig.3.3fordetails)decreasessubstantiallyforfrequencies
above1kHz[148],thisfrequencyseemstobeanaturallimitforevaluationoftheiTD
cues.

topFigureandb2.13:ottomCalculateddiameterfirstof0.5eigenmocm.Indesofcomparisonsimplifiedtothegeometriescylindricalwithgeometrylength2cm(middle)and
thesurfacestwoofotherequalgeometriespressurearefromeitherthenarronegativweerminim(left)umorb(blacroaderk)to(righthet).positivTheeplotsmaximshoumw
The(white).obtainedThelowabsoluteestvalueeigenfrequenciesoftheareextrema4.0iskHzequal(left),and8.6indicatedkHzb(middle),elowtheand13.1geometries.kHz
the(righlot)wasestindicated.eigenfrequencyAnarroofwingtheofthegeometry.geometrythereforedecreases,awideningincreases

2.4.2Eigenmodesofarealisticmouthcavity
ThedirectionaldifferencespredictedbythecylindricalICEmodelforHemidactylusfrenatus
above5kHzcontradictexperimentaldata(seeFig.2.10)andmayresultfromthestrongly
simplifiedcylindricalmodelofthemouthcavity.Eigenfrequenciesoftherealisticmouth
cavitycouldmodifytheresponseofthesystem.Totestthishypothesis,twoquestions
havetobeaddressed.First,whataretheeigenfrequenciesforcomplexgeometries,in
influenceparticular,offorthelothewestrealisticeigenfrequencymouthcaonvitytheofresponseHemidactylusofthefrsystem,enatus.thatSecond,is,onwhatamplitudesisthe
oftympanicvibration,iTDs,andiADs.
theToeffectaddressofathenarrofirstwingorquestion,wideningnumofericalthesimgeometryulationsofonitssimplifiedlowestgeometrieseigenfrequencyinv.estigateThe
analyzedgeometriesaremodificationsofacylinderoflength2cmandradius0.25cm.
Modificationofthiscylinderresultsingeometriesthateitherwidenornarrowinthe

39

40

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.14:Calculatedeigenfunctionsoftherealisticmouthcavity(snouttobottom)
forHemidactylusfrenatus(gekkonid)withthecorrespondingeigenfrequenciesandtheir
maximalamplitudes;seeFig.2.7(right)fortheusedthree-dimensionalmeshoftherealistic
mouthcavity.Theestimatedpositionsofthetympanicmembranesaremarkedbyblue
circles.Theplotsshowsurfacesofequalpressurefromthenegativeminimum(black)to
thepositivemaximum(white).Theabsolutevalueoftheextremaisequalandgiveninthe
tableforeacheigenfrequency.Showninthebottom-rightcorneraretheeigenfrequenciesof
thecylinderthatcorrespondstothespacebetweenthetwoeardrumsusedforcalculations
inthecylindricalICEmodel;cf.Fig.2.3.Theeigenfrequenciesofthecylinderliehigh
abovethetreatedregionandthereforehavenoeffectonthesystem’sresponse.Incontrast,
theeigenmodesoftherealisticmouthcavityhaveeigenfrequenciesstartingat5.1kHzand
mayinfluencethedirectionalresponseofthetympanicmembranes.

2.ModelingInternallyCoupledEars:TheICEmodel

middle;cf.Fig.2.13.Calculationoftheeigenfrequenciesrevealsthatanarrowingofthe
geometrydecreases,awideningincreasesthelowesteigenfrequencyofthegeometry.
Inthenextstep,numericalsimulationsbasedontheconstructedthree-dimensional
meshoftherealisticmouthcavityofHemidactylusfrenatus(seeFig.2.7,right)allowfor
thecalculationandanalysisofthecorrespondingeigenfrequencies.Figure2.14contrasts
thecalculatedeigenfunctionsoftherealisticmouthcavitywiththeeigenfunctionsof
thecorrespondingcylinderusedfortheICEmodel.Indetail,theeigenfunctionofthe
realisticmouthcavitywiththelowesteigenfrequency,5.1kHz,showsahorizontalpattern.
Intermediateeigenfunctionsoccurataround12.2kHzand13.0kHzwiththehighest
pressureamplitudes,andweakerverticaleigenfunctionsarearound14.1kHz.Incontrast,
thelowesteigenfrequencyforthecorrespondingcylinderusedincalculatingtheICE
modelliesat17.2kHz,therefore,outsidetheregionofinterest.Furthermore,strong
intermediateeigenmodesaremissing.Tosummarize,theeigenfrequenciesoftherealistic
mouthcavityarelowerthanthoseofthecorrespondingcylinder.Therefore,theycan
modifythedirectionalresponse,calculatedusingthecylindricalICEmodel,inparticular,
inthefrequencyregionabove5kHz.
Inthefollowing,thesecondquestionisaddressed,thatis,howthelowesteigenfrequency
influencestheresponseofthesystem,inparticular,amplitudesofthetympanicvibration,
iADs,andiTDs.Hereare-calculationoftympanicvibrationswithparametersasgivenin
Table2.3andcorrespondinginputITDsbutwithvaryinglengthoftheinternalcylinder
providesthepossibilitytoestimatetheinfluenceofdifferentlowesteigenfrequencieson
thedirectionalresponseofanICEsystem.ThecylindervariesbetweenL=1cm,2cm,
and3cmcorrespondingtolowesteigenfrequenciesof17.2kHz,8.6kHz,and5.7kHz.
Figure2.15summarizestheresultingamplitudesofthetympanicvibrationforthethree
differentinterauraldistancesindependenceuponsounddirectionandfrequency.With
decreasinglowesteigenfrequencyoftheusedcylinderthefrequenciesofmaximalamplitudes
ofthetympanicvibrationareshiftedtowardslowerfrequencies.Asaconsequence,vibration
amplitudesforlowerfrequenciesaround1kHzincrease.Whensoundfrequencycorresponds
tothelowesteigenfrequencyoftheinternalcavitytheinternalpressurereachesalocal
maximum.AccordingtoEq.(2.17)theforceactingonthetympaniisthereforedominated
bythepressureinthemouthcavitythatisindependentofthedirectionofthesoundsignal.
Theexternalinputshardlyinfluencethetympanicvibrationssothattympanicvibrations
forthelowesteigenfrequencyshowalmostidenticalamplitudesforalldirectionsofsound
source.AsillustratedinFig.2.16,iADsandiTDsareattenuatedwithdecreasinglowest
eigenfrequencyofthemouthcavity.Whenthelowesteigenfrequencydecreasesfrom
17.2kHzto5.7kHz,themaximaliTDdecreasesatthesametimefrom86µstoaround
50µs,themaximaliADfrom15dBto8dB.Whensoundfrequencycorrespondstothe
lowesteigenfrequency,tympanicvibrationsdonotvaryanymorewithdirectionofthe
soundsourcesothatiADsandiTDsvanish.LocalizationbasedoniTDsoriADstherefore
becomesimpossiblewhenreachingthelowesteigenfrequency.Inparticular,arapidand
completedecayofiADshasbeenmeasuredexperimentallyforHemidactylusfrenatusat
5kHz[28]whichfitsnicelytothecalculatedlowesteigenfrequencyofthemouthcavity;
cf.Fig.2.14.Asystematiccalculationoftheeigenfrequenciesofseveraldifferentmouth
cavitiestogetherwiththecorrespondingexperimentaldataforiADscouldverifywhether

41

2.ModelingInternallyCoupledEars:TheICEmodel

Figuredirection2.15:ofsoundCalculatedsource(x-axis)amplitudesandofthefrequencytympanic(y-axis)forvibrationvaryingindloepwestendenceupeigenfrequencyonthe
ofcorrespthemouthondingcainputvity.FITDsortheonlytheparameterslengthofoftheHemidactinternalyluscylindfrenatuservasariesgivandeninTconsequenable2.3tlyandso
doestheeigenfrequencyofthemouthcavityvarybetween17.2kHz(leftcolumn),8.6kHz
(middle),and5.7kHz(rightcolumn).Withdecreasinglowesteigenfrequency(stepping
toshiftedthetorighwt)ardsthelowfrerequencfrequenciesiesof.theVibratimaximalonamplitudesamplitudesforloofwethertfrequenciesympanicaroundvibration1kHzare
increase.Whensoundfrequencycorrespondstothelowesteigenfrequencyoftheinternal
caactingvityontheinthetternalympanipressureisthereforereachesalodominatedcalbmaximytheum.AcpressurecordiinngthetoEq.mouth(2.17cavit)ythethatforceis
identicalforeverydirectionofthesoundsignal.Theexternalinputshardlyinfluencethe
tympanicvibrations.Forthelowesteigenfrequencytympanicvibrationsthereforeshow
almosthearingbidenecomesticalimpamplitudesossibleforforallthelowdirectionsest,orofwhicsoundhever,source.Ineigenfrequencyother.words,directional

thelowesteigenfrequencylimitssoundprocessinginthedescribedway.ForTokaygecko
thisanalysisisperformedbelow.
Inconclusion,forHemidactylusfrenatusthesystematicincreasewithintheestimation
(cf.errorFig.betw2.10ee)ncouldcalculatedwellandresultexpfromerimenamissingtallyfirstmeasuredameigenfrequencyplitudesofaroundt5ympanickHz.vibrationsWithout
suchaneigenfrequency,theICEmodelwiththecylindricalmouthcavitysystematically
overestimatesiADsinthecorrespondingfrequencyregion.

2.4.3GeneralizationoftheICEmodel
ThepreviouscalculationsareperformedforHemidactylusfrenatusthatis,withaninteraural
distanceof10mm,oneofthesmallestlizards.ToillustratethattheICEmodelcangenerally
describetympanicvibrationsofanimalshavinginternallycoupledears,inparticular,all

42

2.ModelingInternallyCoupledEars:TheICEmodel

inFiguredep2.16:endenceupCalculatedonthediADsirection(top)ofs[cf.oundEq.(source2.34)](x-andaxis)iTDsand(middle)frequency[cf.E(y-axis)q.(2.35an)d]
enhancementfactorofmaximaliTDforvaryinglowesteigenfrequencyofthemouthcav-
ity;foradescriptionofthesetupseecaptionofFig.2.15.Thebottomrowshowsthe
calenhancemenmouthcatvityfactorforofanimalsmaximalhavingiTDsinresultingternallyfromcoupledtheearsICEinmodelcomparisonwiththetomaximalcylindri-
ITDsound=kLfrequenciessin(θ=πand/2)lowofestanimalshaeigenfrequencyvingindepoftheendentmouthearscainvity.depWithendenceupdecreasingonlodifferenwestt
theloweigenfrequencyesteigenfre(steppinquengcy.toFtherequenciesright)ofiADsandmaximaliTDsiADsareareattenfurtheruatedforshiftedtofrequencieswardsbloelwower
decreasesfrequencies.forThedecreasingenhancemenlowesttfactoreigenfrequencyof,maximale.g.,foriTDsainnarrowcomparisonmouthcatovity;cfmaximal.Fig.ITDs2.13.
Whensoundfrequencycorrespondstothelowesteigenfrequencytheamplitudesofthe
tympanicvibrationsarealmostidenticalforalldirectionsofsoundsource;cf.Fig.2.15.
Asacompletelyconsequence,forthedloiffwesterencesbeteigenfrequencyweent.Loympaniccalizationvibrations,basedoni.e.,iTDsiADsorandiADsiTDsvthereforeanish
becomesimpossiblewhenreachingthelowest,orwhichever,eigenfrequency.

43

2.ModelingInternallyCoupledEars:TheICEmodel

kindsoflizards,themodelisevaluatedaswellforTokaygecko,thesecondlargestGecko
species.AnagreementofexperimentalandanalyticalresultsforTokaygeckothenshows
thattheICEmodelcoversalargerangeofanimalswithdifferentinterauraldistances
havingICE.Chapter4furthershowsthattheICEmodelmakesreasonablepredictions
forthebestfrequency,i.e.,thefrequencywiththelowesthearingthreshold,fordifferent
speciesofanimalshavingICE.Takenthesetwoargumentstogether,theICEmodelcan
beseenasuniversalmodelfordescribinginternallycoupledears.
Table2.4:GeometryparametersforTokaygecko.Lengthofthecylinder,i.e.,interaural
distance,istakenfrom[27],theotherparametersarelinearlyscaledfromtheparameters
ofHemidactylusfrenatus;cf.Table2.3.Inparticular,theinterauraldistanceL=10mm
ofHemidactylusfrenatusisusedforthescalingbelow.

L=22mmLengthofthecylinder(interauraldistance)
a=3.5mm∙(22mm/10mm)Radiusofthetympanicmembrane
α=1000Hz/(2∙1.2)∙(10mm/22mm)Dampingcoefficientofthemembrane
ρm=3.2mg/mm3Densityofthemembrane
d=10µm∙(22mm/10mm)Thicknessofthemembrane

Thefollowingcomparisonofcalculatedandexperimentaldataanalyzesvibration
amplitudesofthetympanicmembranes,iTDs,iADs,andtheeigenfrequenciesforthe
realisticmouthcavityofTokaygecko.Allcalculationsareperformedwithparameters
asgiveninTable2.4.Thelengthofthecylinder,i.e.,theinterauraldistance,istaken
from[27],theotherparametersarescaledfromtheparametersofHemidactylusfrenatus
undertheassumptionthatthegeometryscaleslinearlywithsize.Indetail,theradiusa
andthethicknessdofthemembraneincreaselinearly,whereastheeigenfrequenciesofthe
tympanidecreaselinearly,withsize;cf.[50,p.74]fordetails.
Figure2.17contrastscalculatedandexperimentallymeasuredamplitudesoftympanic
vibrationforTokaygeckoindependenceupondifferentdirectionsandfrequenciesof
sound.Bothprofilesshowsimilarvibrationamplitudepatternswithidenticalmaximum
of10dBre1mm/s/Pa.TheICEmodelwiththecylindricalmouthcavity,however,
overestimatesthedirectionalresponseofthesystemabove2.5kHzandunderestimatesthe
directionalresponseforlowfrequencies.Alowesteigenfrequencyoftherealisticmouth
cavityatround3kHzcouldprovokesuchamodification;cf.Fig.2.15.Atthesametime,
suchalowesteigenfrequencywouldmodifyaswellinternalamplitudedifferences(iADs)
andinternaltimedifferences(iTDs)thatarecalculatedinthefollowing.
Figure2.18summarizesiADsandiTDsforTokaygeckoindependenceupondifferent
directionsandfrequenciesofsound.Forfrequenciesabove1kHzincomingITDsare
translatedintodirectionaliADsupto15dB.AsalreadyobservedforHemidactylusfrenatus,
theobtainediADssystematicallyvarywithdirectionofsoundsourceandreproduce
experimentaliADs[27].Forfrequenciesabove2.5to3kHz,however,experimentally
measurediADsvanishmorequicklythanthecalculatediADshintingatamodification
duetoalowesteigenfrequencyoftherealisticmouthcavityaround3kHz;cf.Fig.2.16.

44

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.17:Calculated(left)andexperimental(right)amplitudeoftympanicvibration
forTokaygeckoindBre1mm/s/P◦a,dep◦endentonthesoundsourcedirection(x-axis
withdirectionsvaryingfrom−180to180;negativedirectionsarecontralateral,0is
frontal,andpositivedirectionsareipsilateral)andfrequency(y-axis).Thecalculated
resultsarebasedontheICEmodelwithacylindricalmouthcavityandparametersasgiven
inTable2.4.MeasuredeardrumvibrationamplitudesforTokaygeckoarefromChristensen-
Dalsgaard[27].TheICEmodelreproducesthemeasuredvibrationamplitudepatternbut
overestimatesthedirectionalresponseofthesystemabove3kHz.Inaddition,calculated
amplitudesoftympanicvibrationarelowerforfrequenciesbelow1kHzthanexperimentally
measuredvibrationamplitudes.AccordingtoFig.2.15alowesteigenfrequencyofthe
realisticmouthcavityaround3kHzcouldprovoketheobservedmodifiedresponseofthe
measuredvibrationamplitudes.Anumericalcalculationofthelowesteigenfrequencyof
themouthcavityisthereforenecessary.

45

46

2.ModelingInternallyCoupledEars:TheICEmodel

Figuredifferences2.18:Ov(iADs)erviewindBof[cf.possibleEq.(lo2.34)]calizationasacuesfunctionforofTokaydirectiongeck(ox.T-axis,op:Innegativternaledirectionsamplitude
confromthetralateral,ICE0mofrondeltal,withandthepositivcylindricaledirectionsmouthcavitipsilateral)y(left)andandfromfrequencyexp(yerimen-axis)ts[as27](rresultinight).g
TheobtainediADssystematicallyvarywithdirectionofsoundsourceforfrequenciesabove
1inµskHz[cf.andEq.repro(2.35duce)]asexpaerimenfunctiontalofiADs[direction27].(xBottom-axis)left:andInterfrequencynaltime(y-axis).differenceCalculated(iTDs)
iTDsfactorvofarywithmaximaldirectioniTDsofresultingsoundfromsourcetheforICElowmodelfrequencies.withtheBcylindottomricalright:mouthEnhcaancevitymenfort
ofanimalsanimalshahavingvinginternallyindependencoupledtearsearsinindepencomparisondenceuptoondifferenmaximaltITDsound=kLsinfrequencies.(θ=πF/or2)
of3frequencieshigherbethanlow1.those5kHzfortheanimalsinternalwithcouplingindependenprotduceears.timeThedifferencesadditionalthatareenhancemenafactort
couldmeansofguaraniTDsteeforthatanfrequenciesanimalbelowithwa1smallkHz.TheadakensizetogethercouldstilliADsloandcalizeiTDssoundcouldsignalsdelivbery
localizationcuesincomplementaryfrequencyranges.

2.ModelingInternallyCoupledEars:TheICEmodel

ForlowfrequenciescalculatediTDsareupto±225µsforthelargeTokaygeckoandvary
withdirectionofsoundsource.Forfrequenciesbelow1kHztheinternalcouplingenhances
iTDsincomparisontoITDsbyafactor3;cf.enhancementfactorinFig.2.18,bottom
right.SimilartothedirectionallocalizationcuesforHemidactylusfrenatus(cf.Fig.2.18)
iADsandiTDscouldtogetherdeliverlocalizationcuesincomplementaryfrequencyranges.

Inafinalstep,theeigenfrequenciesoftherealisticmouthcavityofTokaygecko
(cf.Fig.2.6,bottom)arecalculatedtoaccountforamodificationoftympanicvibrations
duetothelowesteigenfrequencyoftherealisticmouthcavity(Figs.2.15and2.16).
Duetothecomplexshapeofthecavitytheeigenfrequenciescanonlybecalculated
numerically.Fortheeigenfrequencyanalysisthereconstructedthree-dimensionalmeshof
therealisticmouthcavity(Fig.2.8,right)isloadedintothesimulationprogramCOMSOL.
Figure2.19illustratestheobtainedeigenmodesandthecorrespondingeigenfrequencies.
AsalreadyobservedforHemidactylusfrenatus(cf.Fig.2.14)theeigenfunctionofthe
realisticmouthcavitywiththelowesteigenfrequency,here3.2kHz,showsahorizontal
pattern.Eigenfunctionsofhighereigenfrequencies5.9kHz,6.0kHz,and7.4kHzwiththe
highestpressureamplitudesshowintermediatetoverticalpatterns.Incomparison,the
lowesteigenfrequencyforthecorrespondingcylinderusedincalculatingtheICEmodel
withthecylindricalmouthcavityliesat7.8kHz.Thelowesteigenfrequencyoftherealistic
mouthcavitycouldthereforemodifythedirectionalresponseascalculatedusingthe
cylindricalICEmodel,inparticular,inthefrequencyregionaround3kHzandbelow
1kHz;cf.Figs.2.15and2.16.
TakenresultsfromHemidactylusfrenatusandTokaygeckotogether,theICEmodel
canexplainvibrationpatternsasobservedinexperiments.Inthisway,theICEmodel
presentsitselfasuniversalmodeltogenerallydescribetympanicvibrationsofanimals
havinginternallycoupledears.

47

48

2.ModelingInternallyCoupledEars:TheICEmodel

Figure2.19:CalculatedeigenfunctionsoftherealisticmouthcavityforTokaygeckowith
thecorrespondingeigenfrequenciesandtheirmaximalamplitudes;seeFig.2.8(right)forthe
usedthree-dimensionalmeshoftherealisticmouthcavity.Theestimatedpositionsofthe
tympanicmembranesaremarkedbybluecircles.Theplotsshowsurfacesofequalpressure
fromthenegativeminimum(black)tothepositivemaximum(white).Theabsolutevalueof
theextremaisequalandgiveninthetableforeacheigenfrequency.Theeigenmodesofthe
realisticmouthcavityhaveeigenfrequenciesstartingat3.2kHz.SimilarlytoHemidactylus
frenatus,therealisticeigenfrequenciesliebelowthelowesteigenfrequencyofthecylinder
usedbytheICEmodel.AccordingtoFigs.2.15and2.16,thelowesteigenfrequencyofthe
realisticmouthcavitycanmodifythedirectionalresponseofthesystemaround3.2kHz.

2.ModelingInternallyCoupledEars:TheICEmodel

2.4.4Spatialvibrationpatternofthemembrane
Thefinalstepoftheevaluationconsistsoftheanalysisofthespatialpatterninthe
membranewithrespecttotheexcitingfrequency.Thetympanicvibrationpatternwas
measuredexperimentallybyManley[144]foraTokaygeckowiththestrongestvibration
responsearound1kHz.Manleydeterminedthevibrationamplitudeforeightlocations
onthemembraneandmeasuredacomplexpattern(seeFig.2.20,right)thatdoesnot
8correspkHz,theondtopatternashosymmetricalwstwomaxima,eigenmodehinwithtingonthatlyonehighermaximmodesum.areBetlikwelyeeninv4olvkHzed.and

Figure2.20:Calculated(left)andexperimental(right)vibrationamplitudepatternoverthe
membrane,dependentonsoundfrequencyfrom0.5to10kHzforTokaygecko.Experimental
datahavebeentakenfromManley[144].ThecalculateddataresultfromtheICEmodel
withacylindricalmouthcavity.Inbothcases,asimilarcomplexvibrationalpatternvaries
withfrequency.TheinclusionofhighereigenmodesintheICEmodelresultsincomplex
amplitudepatternsoverthemembranethatareingoodagreementwithexperimentaldata.

IntheICEmodelwithacylindricalmouthcavity,highermodesarenecessarytosatisfy
theboundaryconditionsofthemembraneresultingfromtheattachedextracolumellaof
themiddleear.Figure2.20(left)illustratesthevibrationamplitudepatternoverthe
Tmemable2.4brane,forTokaycalculatedgeckousing.TheEq.(omitted2.26)forsegmendifferentretpresfreenqutsenctheiesattacwithhedtheparametersextracolumella.in
Thecomparisonofthecalculateddatawiththeexperimentaldataisqualitative,asthe
parametersusedareonlyestimatedvalues.Theaimofthecalculationistoshowthat

49

2.ModelingInternallyCoupledEars:TheICEmodel

reproductionofacomplexvibrationalpatternoverthemembraneispossible.Asmeasured
toexptheerimeninclusiontally(coff.Fig.higher2.20mo,desright)in,tothetheICEasymmetricalmodelthepatternspatialvariesvibrationwithfrequpatternency.ofDuethe
tympanumhasacomplexshapethatclearlydiffersfromasymmetricalpattern.Depending
oronthewithtwofrequencyneighbtheoringcalculatedmaxima.profileThatis,cantheeithermosdelhowresaultspatterncorrespwithondonlyqualitativoneelymaximwithum
experimentalresultssuchthatthevibrationpatternisnolongerrotationsymmetricand
tconympanictainsmorememthbraneanonecouldmaximinfluenceum.itsConsequenvibrationtly,thepattern.appearanceofhighermodesforthe

Conclusion2.5

cf.FigureFig.2.122.21:andOv[221erview].ofThisthetfigurewoproassumescessingapathsmallwainysteraasuralresultindistancegfromsothethatICEamplitudemodel;
differencesbetweenthesoundwavesatthetwoearscanbeneglected.Incominginteraural
timemouthcavitdifferencesyand,(ITDs)hence,aregivproingcriseessedtobinytheternallyvibratingcoupledmemearsbrane(ICE).sForcoupledlowthroughfrequenciesthe
ITDsaretranslatedintointernaltimedifferences(iTDs),forhighfrequenciesintointernal
(iADs).differencesamplitude

Insummary,thischapterhaspresentedageneralmodelofinternallycoupledears,
thevibrationICEmopatterndel,thatoftwoconsistsloadedofamembranescylindrical,coupledanalyticallybytheinsolubleternalcamovitdelytoandancalculateumericalthe
eigenfrequencyanalysisoftherealisticmouthcavities.Toincludetheeffectoftheattached

50

2.ModelingInternallyCoupledEars:TheICEmodel

extracolumellaofthemiddleear,asfoundinlizards,highermodesofthemembrane
vibrationarepartofthesolution.Thecalculatedresultstogetherwiththenumerical
calculationsagreewellwithexperimentaldata.Evaluatingmembranevibrationatthetip
ofcanthelargelyextracolumellareproduceforthedifferendirectionaltanglesrespofonsestheofsoundICEsourcesystemsandasfoundfrequencies,intheHemidactylusmodel
frFenatusorandHemidactylusTokaygefrckoenatus.theICEsystemcreatesiTDsupto±86µsforlowfrequencies
whichreflectsanenhancementoftheincomingITDsduetotheinternalcouplingbya
factor3.IncomingITDsaretranslatedintodirectionaliADsupto20dBforhighfrequencies.
ForcalculatedthelargeriADsTareokayupgetocko15dBforcalculatedhighiTDsfrequencies.areupTtoak±en225µtogether,sforloiADswandfrequencies,iTDscouldand
deliversummarizeslocalizationthecuesfrequency-depinencomplemendenttaryresponsefrequencyofanICErangessystemforbtoothincominganimals.ITDs.Figure2.21
Tofurtheranalyzetheeffectoftheinternalmouthcavity,three-dimensionalmeshesof
thefromrealisticscanneddmouthata.caThevitiesofeigenfrequenciesHemidactylusforfrtheenatusinteandrnalTcaokayvitiesgeckoarearethenreconstrucalculatedcted
ntheumericallylargerT.FokayorgeckoHemidactylusat3kHz;frseeenatusFig.the2.19.eigenmoInbdesothstartcasesat5thekHzlowest(seeFig.eigenfrequency2.14),for
maywellprovokeamodifiedresponseoftherealisticICEsystemincomparisontothe
ICEmodelbasedonasimplifiedcylindricalmouthcavity.Inconcreteterms,thelowest
eigenfrequencycouldlimittherangeofdirectionalhearinginsuchawaythatiADsand
TiTDsokayvgeanishckoexpforerimensoundtallysignalsmeasuredaroundiADsthis[fr27e,qu28e]ncy.confirmForthishypHemidactylusothesisfbryvenatusanishingand
aroundthecalculatedlowesteigenfrequencyofthecorrespondingrealisticmouthcavity.A
systematiccalculationofeigenfrequenciesforrealisticmouthcavitiesincombinationwith
measurediADscouldfinallyconfirmthehypothesisandwouldbeaninterestingtopicfor
.studyfurtherTheresultsofSec.2.4.1couldbeobtainedbymeansofasimplemodelincludingthree
impexplainsedancesmuchformorethetofwothememexpbraneserimentalanddatathecaandvitycan[27b,e28ge].nerHoalizewevder,fortheallnewkindsICEofspmoecies.del
Anew,interestingfeatureofthemodelistheactivationofhighermodes.Comparedtothe
fundamentaleigenmodethatisrotationallysymmetricwithonemaximumatthemiddle
ofthemaxima.circularIncludingmemhigherbrane,mohigherdesinthevibrationsolutionmodesthereforeexhibitbreaksathpatternewithrotationalanumbsymmetryerof
andreproducesthecomplexvibrationpatternsobservedexperimentally.

51

3.

theTheonemostthatexcitingheraldsphrnewasetodischearoveries,inissciencnote,
’Eureka!’(Ifoundit!)but’That’sfunny...’
IsaacAsimov(1920-1992)

NeuronalprocessingofiTDsand
iADs

ductiontroIn3.1

TheneuronalprocessingensuingfromiTDsandiADsisexemplarilyanalyzedbymeansof
datafromHemidactylusfrenatus.ForthelargerTokaygeckomaximaliADsareinthe
samerangeandmaximaliTDsareevenlargerthanthoseofHemidactylusfrenatus.As
aconsequence,theneuronalprocessinghastocopewiththesmallerspeciesasthemore
difficultcase.ForHemidactylusfrenatus,analyticalresultsoftheICEmodelbasedona
simplifiedcylindricalmouthcavity(cf.Fig.2.12)andexperimentaldata[27,28]showthat
thetympanioflizardsrespondtosoundsignalsinamannerdependentonthehorizontal
directionofthesoundsource.Moreprecisely,internaltimedifferences(iTDs)andinternal
amplitudedifferences(iADs),astheyappearatthelevelofthetympanicmembranes,vary
systematicallywiththeangleofthesoundsource.Inthenextstepofsoundprocessing,
theiTDandiADsoundlocalizationcueshavetobeextractedfromthevibrationsofthe
ympani.tVibrationsofthetympanicmembranesarefirsttransmittedbythemiddleear(through
thecolumella)tothecochleaandtotheembeddedbasilarmembrane;cf.Fig.1.3.Dueto
thesystematicallyvaryingstiffness,thebasilarmembraneistonotopicallyorganizedin
suchawaythateachofitshaircellsismaximallyexcitedbystimulationatonespecific
characteristicfrequency(CF).Haircellsalongasmallregionofthebasilarmembrane
inturnenervatelocalauditorynervefibers.Asaconsequence,thefrequency-specific
tonotopicrepresentationisalsopreservedatthelevelofneuronalprocessing.Accordingly,
thesimulationsbelowhavebeenperformedforonespecificfrequency.
Themechanicaltransductionprocessfromtympanitothehaircellsofthebasilar

53

3.NeuronalprocessingofiTDsandiADs

toFigurethefirst3.1:nucleiDifferenthtenervneuronalefibproersjectionsencountefromrinlow-theandhauditoryigh-frequencybrainstem.auditoryFigurenervmoefibdifieders
fromSzpir[208]forAlligatorlizard.Left:Low-frequencyfibersprojecttothelateral
nNMM).ucleusRighangularist:Incon(NAL)trast,andthehigh-frequencylateralandfibersmedialpronjectucleusalmostmagnoexclusivcellulariselytothe(NMLmedialand
nnervucleusefibersangularissuggestan(NAM).indepTheendentdifferenprotprocessingjectionoflow-pathandwaysoflohigh-frequencyw-andsignals.high-frequency

membranetotheauditorynerveissimplifiedinsuchawaythattheresponseoftheauditory
nerveisproportionaltothecorrespondingtympanicvibration.Thissimplifiedmodelmight
notbethecaseintherealworld,theassumption,however,ensuresthattheiTDandiAD
propertiesareconserved.Thatis,ifmechanicalprocessingofthemembranevibration
reducesamplitudeorshiftsthephaseofthesignal,themodificationsapplyatbothsides
inthesamemanner.Modificationsmadealongthepathcanceleachotherout,andiADs
andiTDsremainunchangedbythespecifictransductionfromtympanicvibrationtothe
responseoftheauditorynerve.Low-frequencyauditorynervesarethereforeassumedto
reproducethetemporalprofileofthemembranevibrations.Spikeratesofhigh-frequency
auditorynervesareassumedtobelinearfunctionsofthetympanicvibrationlevelindB.
Inthenextstepsofneuronalprocessing,lowandhighfrequencies,i.e.,iTDsandiADs,
areassumedtobetreatedseparately.

3.2SeparatedpathwaysforiTDandiADprocessing

ResultsoftheanalyticalcalculationsforHemidactylusfrenatus(gekkonid)asillustratedin
Figs.2.21and2.12showfundamentaldifferencesbetweeniADsandiTDsforfrequencies
belowandabove1kHz.Forthelowfrequencyrange,thereareiTDsavailablerangingup
to86µs(Fig.2.12,bottomleft),whereasintensitydifferencesarenegligible.Incontrast,
forhighfrequencies,Fig.2.12(top)illustratesthatthereareiADsupto20dBbetweenthe
vibrationsofthetwotympanicmembranes.Consequently,separatedprocessingpathways

54

3.NeuronalprocessingofiTDsandiADs

foriTDsinthelowfrequencyandforiADsinthehighfrequencyregionsareprobable.For
mammals,sucha“duplextheoryofsoundlocalization”hasbeensuggestedmorethan100
yearsagobyThompson[212]andRayleigh[178].Theystatedthatphaseandamplitude
cuesareusedforsoundlocalizationincomplementaryfrequencyranges.
Forlizards,thehypothesisoftwoseparatedprocessingpathwaysforiTDsandiADsis
supportedbyanumberofexperimentalstudies.Manley[148],forinstance,showedthat
auditorynervefiberscannotreliablyrepresent,i.e.,theydonotphase-lockto,periodic
signalsabove1kHz;seeFig.3.3andnextsectionforadetailedexplanation.Thus,neuronal
processingofiTDsbecomesimpossible.
Furthermore,thebasilarmembraneoflizards,whichfollowsthetympanicmembrane
inthemechanicalsoundprocessingchain,differsfromthemammalianbasilarmembrane.
Itisdividedintotwosegments:theapicalsegmentandthebasalsegment[127,145].
Whereashaircellsontheapicalsegmentreactbesttolowfrequencies,haircellsonthe
basalsegmentarebesttunedtofrequenciesabove1kHz.Thisseparationoflowandhigh
frequenciesisconservedwithintheneuronalprojectionsfromthehaircellstodifferent
brainareas;seeFig.3.1and[208].Basedonthesedata,lowandhighfrequencies,i.e.,
iTDsandiADs,arelikelyprocessedindependentlyinthebrain.
Incontrasttotheabovemeasurements,thesubsequentneuronalprocessingpathwaysof
iTDsandiADs,i.e.,theinvolvedbrainareasandtheirconnections,arenotexperimentally
determinedforlizards.Therefore,theparallelexistenceoftwowidespreadprocessing
strategieshasbeenanalyzedsuggestingaJeffress-likemodelforlow-frequencysignalsand
excitatory-inhibitory(EI)processingforhigh-frequencysignals.

iTDsofcessingPro3.3ThemostpopularandstrikinglysimplemodelforneuronallocalizationthroughiTDsis
theJeffressmodel[99]assuggestedbyLloydJeffressin1948;see,e.g.,[73,104]forfurther
information.Figure3.2illustratestheJeffress-likemodelthatisequivalenttotheJeffress
model(see[57])andusedforiTDprocessinginlizards.TheJeffress-likemodelconsistsof
thefollowingthreeelements.
•Phase-lockedandphase-shiftedinputsfromthetwosides
Vibrationsofthetympaniaretranslatedintophase-lockedneuronalspikepatternsof
theauditorynerve.Thatmeansthatspikesalwaysoccurataspecificphaseofthe
incomingwave,e.g.,atthemaximum;seeFig.3.3(left)forillustration.Thistranslation
ofthesoundwavesintospikesisnotprecisebutshowsacertainvariationarounda
perfectsynchronization.Forthefirsttime,GoldbergandBrown[69]havequantified
synchronizationbycomputingacyclehistogramofspikesplottedrelativetotheirphase
withinthecycle.Thelengthoftheaveragevector,calledvectorstrength,corresponds
tothedegreeofsynchronization.Asaconvexcombination,thevectorstrengthliesin
theinterval[0,1].Avectorstrengthof1representsperfectsynchronizationsinceall
spikeshavetopointatexactlythesamedirectioninthecyclehistogram.Foravector
strengthof0allspikesarerandomlydistributedwithinthecyclehistogram,there
isnosynchronizationatall.Figure3.3(right)showsvectorstrengthsforlow-and
high-frequencyfibersofthelizardTiliquarugosaindependenceuponsoundfrequency.

55

56

3.NeuronalprocessingofiTDsandiADs

Figure3.2:Visualizationofsoundlocalizationbymeansofinterauraltimedifferences
throughaJeffress-likemodelillustratedfortwodifferentpositionsofasoundsource;
figuremodifiedfrom[126].Inputsignalsfromtheipsilateraleararriveatallmapneurons
(coincidencedetectors)simultaneouslywhereasinputsignalsfromthecontralateralear
arriveateachmapneuronswithacertaindelaythatsystematicallyvariesoverthearrayof
mapneurons.Left:Soundsignalsfromasourcedirectlyinfrontoftheheadarriveatthe
twotympaniatthesametime.Theneuronthatfiresthemostistheonewithdelaylines
thatshowanequaldelayfortheipsi-andcontralateralside.Right:Soundwavesfroma
soundsourceatthecontralateralsideoftheheadarriveearlieratthecontralateralthanat
theipsilateraltympanum.Theneuronthatfiresthemostistheonewithdelaylinesthat
compensateforthistimeshift,thatis,thedelayoftheipsilateralaxonislongerthanthe
delayofthecontralateralaxon.

Foraspecificfrequencyωsmallerthan1kHz,thefiringratefipsioftheipsilateral
auditorynervehastoreflectbothtiminginformationofthetympanumandcoding
precisionoftheauditorynervefiber.AccordingtoFriedeletal.[57]thefiringrate
canthereforebedescribedbyasumofGaussianprofileswithmaximaatωt=2πnfor
n∈N0leadingto
2
fipsi(t)=Aexpmin(|ωt−2πn|,n∈N0)2σ2(3.1)
∈[0,π]
withstandarddeviationσandAasmaximalamplitude.Thevalueofσcanbe
determinedfrommeasuredvectorstrengths(Fig.3.3B)astheabsolutevalueofthe
ratiobetweenfirstandzero-thFouriercoefficient;see[57]fordetails.Forσ→0,the
firingrateconvergestoasumofdeltafunctionswithfrequencyω.Smallvaluesofσ
thereforecorrespondtoaverypreciseneuronalrepresentationoftheincomingwave,
andlargevaluesresultinablurredrepresentation.
Giventhattheinputsarephase-lockedtheyalsohavetobephase-shifted.Thisis
thecasesincethefiringratefcontraofthecontralateralauditorynerveisdelayedby
theinternaltimedifference(iTD)withrespecttotheipsilateralfiringratefipsi.The

3.NeuronalprocessingofiTDsandiADs

Figure3.3:(A)Illustrationofphase-locking.Theneuronalfiringpatternofaphase-locking
neuron(top)reproducestheformoftheincomingsoundwave(bottom).Thatis,aneuron
alwaysfiresatonespecificphaseofthesoundwave(e.g.,atitsmaxima).Figuretakenfrom
Grothe[73].(B)Thequalityofphase-lockingismeasuredbythevectorstrength.Avector
strengthof1correspondstoperfectsynchronizationbetweenincomingwaveandneuronal
spikepattern;avectorstrengthof0denotescompletelyuncorrelatedsignals.Theplot
showsaveragedvectorstrengthsofauditory-nervefiberswithlowandhighcharacteristic
frequencies(CFs)forthelizardTiliquarugosa[148].Whereaslow-frequencyfibersshowa
vectorstrengthnear1forfrequenciesbelow1kHz,high-frequencynervesarenottuned
tophase-locktoincomingsignalswithahighreliability.Theanimalcanthereforeexploit
iTDsforsoundlocalizationonlybelowabout1kHz.Figuretakenfrom[148].

iTDsresultfromtheITDsoftheincomingsoundsignalandtheirprocessingbythe
internallycoupledearsanddependontheangleθofthesoundsource.Thefiring
ratefcontraofthecontralateralauditorynerveisthereforegivenbyEq.(3.1)witha
timearrivingdifiedmo2
fcontra(t)=Aexpmin(|ωt+ωiTD−2πn|,n∈Z)2σ2.(3.2)
∈[0,π]
Takentogether,firingprofilesfromipsi-andcontralateralauditorynervesprovide
phase-lockedandphase-shiftedinputs.
•Anarrayofcoincidencedetectionmapneurons
Inthenextstep,thephase-lockedandphase-shiftedsignalsfromtheipsi-andcontralat-
eralauditorypathwaysarriveatanarrayofcoincidencedetectionneuronsreferredtoas
mapneurons.Eachneuronfiresmaximallywhenspikesfromthetwosidestemporally
coincide,i.e.,arrivesimultaneously.
•Systematicallyvaryingdelaylines
Themapneuronsareconnectedtoleftandrightinputneuronsbyaxonsthattransmit
inputsignalswithacertaindelay.ForaJeffress-likemodelthisdelayisidenticalfor
allaxonsfromtheipsilateralsidewhereasthedelaysofaxonsfromthecontralateral
sidevarysystematicallyoverthearrayofmapneurons.Thatis,aspecificspikefrom
thecontralateralsidearrivesateachmapneuronataslightlydifferenttimecompared

57

3.NeuronalprocessingofiTDsandiADs

totheneighboringmapneurons.Thecoincidencedetectionmapneuronsthenreact
mosttosoundinputwiththespecificiTDthatcompensatesexactlyforthedifference
betweenthedelaytimesoftheaxonsfromthetwosides.GiventhatiTDscorrespond
todifferentanglesofthesoundsource,thearrayofiTD-detectingneuronscorresponds
toaspatialmap.Figure3.2illustratestheJeffress-likemodelfortwoexamplesof
soundsourcesatdifferenthorizontaldirections.Asoundsourcedirectlyinfrontofthe
listenermaximallyexcitestheneuroninthemiddleofthearray,thatis,theneuron
connectedbyaxonswithanidenticaldelaytotheipsi-andcontralateralside.In
contrast,asoundsourceattherightsideofthelistenermaximallyexcitesaneuronat
theleftedgeofthearray.

ThefollowingnumericalsimulationsoftheJeffress-likemodelrefrainfromspecifying
theexactbrainareasandauditorypathways(possibleprojectionsthroughintermediate
brainareas).Instead,themodelincludesrightandleftinputneuronsaswellasmap
neuronsinanabstractway.WithregardtoFig.3.1(left),theinputneuronsmightbe
neuronsofthenucleusmagnocellularis,themapneuronsofthesuperioroliveorthe
nucleuslaminaris.Inanycase,inputneuronsaremodeledasstochasticallyfiringPoisson
neurons(seeSec.1.3.3fordetails)withfiringratefunctionsdefinedbyEqs.(3.1)and(3.2).
Theirnumberisestimatedbyabout30,whichcorrespondstothemeasurednumberof
low-frequencyauditorynervefibers[146].
Themapneuronsaremodeledasleakyintegrate-and-fireneurons;seeSec.1.3.2and[67]
fordetails.Eachmapneuronisdirectlyconnectedthroughaxonstotheinputneurons.
ThetotalcurrentIntotarrivingatthen-thmapneuronisgivenbythesumofipsi-and
contralateralcurrents.Accordingtotheassumptionofthetwoprocessingpathwaysfor
lowfrequenciesthesecurrentsonlydifferinphase,whereasthetemporalcourseandthe
strengthofthecurrentsareidentical.Basedontheseconsiderations,theipsilateralcurrent
isdefinedtobeI(t).Thecontralateralinputthencorrespondstotheipsilateralcurrent,
exceptitisshiftedintimebytwoeffects.Ontheonehand,thevibratingtympanigenerate
iTDsthatdependontheangleofthesoundsourceθ.Ontheotherhand,theneuronal
responseisdelayedthroughthemapneuron-specificaxonaldelayΔtn.Takentogether,
thecontralateralcurrentisthegivenbyI[t+Δtn+iTD]sothatthetotalcurrentarriving
atthen-thmapneuronequals

Intot(t)=I(t)+I(t+Δtn+iTD).(3.3)

Asacoincidencedetector,then-thmapneuronfiresmaximallyforinputfromapre-
iTDferredm=iTDiTDnnthatdoesexactlynotpcomperfectlyensatescompforensateitsforneuronalthedelaneuronaly,i.e.,delaΔy.tn=−Therefore,iTDn.theAnothermore
thethatinputneuroniTDmdecreases.differsThfromusawithinneuron’sanarraspyecificofaxonalcoincidencedelay,detectionthemoreneuronsthewithfiringvratearyingof
delaotherylinesneuronsneuronsfireinwithanattenpreferreduatediTDsmannercorrespdepondingendingtoonthehowmincominguchtheiriTDfirepreferredmost,iTDand
iTD.incomingthefromdiffers

58

3.NeuronalprocessingofiTDsandiADs

(greenFigure3.4:line)forFiringapatternJeffress-likofeiTDmodelsmapwithoutneuronsand(left)withindepadditionalendenceuponGABAergicincominginhibitioniTDs
(right).Theintensityofthegraycolorindicatesthenumberofthespikesfromthemap
neuronsasdenoted.YellowdotsdenotetheestimatediTDasresultingfromarate-weighted
likmeanemoofdelthe(righfiringt)profilewithoftheparametersiTDmap;givencf.inEq.T(able3.4).3.1T,op:inSimparticular,ulationwithresultsaofamemJeffress-brane
constantτm=500µsandasynapticstrengthJ=0.015.Theestimationerrorforthefiring
profileiscalculatedtobeE=39µsoverarangeof[−86µs,86µs]fortheangles[−π/2,π/2]◦.
Thehorizontalangleofmisestimationisthenapproximately39µs/86µs∙(π/2)≈41.
(righBottom:t)asSimfoundulationinbirdsresults[58,of96,a238].Jeffress-likAdditionalemodelwithGABAergicadinhditionalibitionGABAergicleadstoainreducedhibition
membraneconstantτm=250µs[37].ThesynapticstrengthissettoJ=0.021toensure
sufficientfiringactivityofthemapneurons.WithadditionalGABAergicinhibitionthe
firingprofileoftheiTDmapsharpensandthemisestimationanglereducesto≈19◦.

Giventhefiringactivityofallmapneurons,thesystemcancalculatetheestimated
iTDasarate-weightedmeanofthemap
iTD=νiiTDiνi(3.4)
iiwhereeverymapneuron“votes”withitsfiringrateνiforitsencodediTDi.Such
anevaluationcorrespondstoanevaluationofthepopulationvector;fordetailssee,
e.g.,[65,206,218].Theadvantageofasuch“democratic”evaluationisthatinformation,
i.e.,firingrates,ofallmapneuronsisconsidered,incontrasttoadeterminationofthe
estimatediTDbythemaximumoftheiTDmap.Forbroadfiringprofileswithoutaclearly

59

3.NeuronalprocessingofiTDsandiADs

definedmaximum,onlyacalculationoftherate-weightedmeanisreasonableatall.To
quantifythequalityoftheestimatestherootmeansquareerrorEcanbecalculated.That
meansforMdifferentinputsiTDiwith1≤i≤M,andcorrespondingneuronalestimates
iTDitheerrorisdefinedas
M
E:=iTDi−iTDi2/M.(3.5)
=1iiTD.estimatedThevalueoftheerrorEthengivesthemeanmismatchbetweentheinputiTDandthe
NumericalresultsofsimulationsperformedwithparametersasgiveninTable3.1are
illustratedinFig.3.4,left.TheupperplotshowsthefiringprofilesoftheiTDmapandthe
estimatedconstruction;iTDsFig.[cf.3.4Eq.,(top3.4)]righint.depGivendenencthateuponiTDsfordifferentinputHemidactylusiTDsfrforenatusaareJeffress-likverye
smallincomparisontoneuronaltimescales,e.g.,thelengthofanexcitatorypostsynaptic
firingcurrentprofile(EPSC),ofthetheiTDmemmapbraneisveryconstant,broad.orTtheoaccounrefractorytforperiothedsloofwtdecaypicalyofneurons,thefiringthe
profile,theneuronalmaphastorepresentalargerangeofiTDsthatbyfarovershootthe
physicallyaccessiblerange.Thephenomenonhasbeenmeasured,e.g.,inalligators[24],
wheretherangeoftheneuronaliTDmapof1000µsextremelyexceedsthephysically
accessiblerange.Inthesimulationstheovershootisfixedat5,whichmeansthatmap
iTDneuronswithincovertheannaturaliTDphrangeysicallyfrom[−5accessibleiTDmax,range,5iTDi.e.,maxiTD],where=86iTDµsmaxforistheHemidactylusmaximal
maxfrenatus;cf.Fig.2.12,bottomleft.
beEThe=39µsestimationoveraerrorrangeforof[the−86firingµs,86profileµs]asforshothewnanglesin[Fig.−π/3.42,π(top/2].left)Theiscalhorizonculatetaldan-to
gleofmisestimationisthenapproximately39µs/86µs∙(π/2)≈41◦.Thishugeestimation
errorresultsfromthefactthatthefiringactivitybetweenneighboringmapneuronsisvery
similar;cf.Fig.3.4,topleft.TheGaussianprofilealmostresemblesaconstantfunction.
Therate-weightedmean,i.e.,theestimatediTD[cf.Eq.(3.4)],foraconstantfunction
left,correspyelloondswtodots)theforcentheterofcalculatedthemap.almostInaflatsimilarfiringway,profiletheesoftimtheateiTDdiTDmaps(Fig.shiftto3.4wtopard
thecenterofthemapandawayfromtheinputiTDs(Fig.3.4topleft,greenline).The
neuronalsystemthereforesystematicallyunderestimatestheinputiTDs.

ApossibilityofhowthemapfiringprofilecouldbesharpenedisanadditionalGABAer-
gicinhibitionoftheiTDmapneuronsasfoundinbirds[58,96,238].HereiTD-processing
neuronsofnucleuslaminaris(NL)receivelong-lastingdepolarizationfromneuronsofthesu-
periorolivarynucleus(SON).Sinceinhibitionseemstobepresentinneuronalprocessingof
lizardsaswell(CatherineCarrandJakobChristensen-Dalsgaard,personalcommunication)
theeffectofsuchaninhibitiononiTDprocessingisincorporatedintothepresentedmodel
andevaluatedinthefollowing.TheresultingcircuitisillustratedinFig.3.4,bottomright.
Experimentaldata[37,58,156,238]unraveledthatGABAergicinhibitioncanincreasethe
precisionoftimeinformationprocessing.Moreprecisely,GABAergicinhibitiondecreases
themembraneresistanceRmofthemapneurons,whichhastwoeffectsasneatlyvisible

60

3.NeuronalprocessingofiTDsandiADs

Figurelaminaris3.5:(NL)Tempneuronsoralincoursechicofksanwithoutexcitatory(conptrol)andost-synapticwithpotenGABAetialrgic(EPSP)inhibitionofnucleus(with
GABA)duetoFunabikietal.[58].InbirdscoincidencedetectingNLneuronsreceivea
thatlong-lastingspeedsdepuptheolarizationriseandfromdecayneuronstimesofofthesupexcitatoryeriorolivparynost-synapticucleusp(SON)oten[58tials,96of,238NL]
neurons.GABAergicinhibitionthereforesharpensthefiringprofilesoftheiTD-processing
NLneuronsandmayservetoincreasecoincidencedetection.

inmoreEq.(1.4excitatory)foracurrenleakytIininjisneededtegrate-and-firetoincreaseneurontheandmemabraneconstanptoteninputtialabocurrenvet.itsfiringFirst,
threshold.Thiseffectiscalledgaincontrol[73]andpreventsmonauralsummationathigh
soundintensities.Thus,thecoincidencedetectionneuronsremainwithinanappropriate
wrange,orkingtherange.inclusionGivenofanthatsimadditionalulatedgainneuronscontrolaredoesalreadynotaltermothedeledresultswithinofthetheirwpresenorkingted
del.moeJeffress-likThesecondeffect,however,ismorepromising.Thatis,GABAergicinhibitionreduces
themembranetimeconstantτm:=RmCmandconsequentlyspeedsuptheriseanddecay
timesfunctionoftendsexcitatoryto0pmorequost-synapticicklypwhenotenτmtials;cf.decreases.Fig.In3.5.FconcreteorEq.(terms,1.4)theDasikaexpetonenal.[tial37]
showedthatthemembraneconstantofthemaphalfenswithadditionalinhibitionsothat
mapneuronscanstillreacttosmallertimedelays.
exactForfiringthenactivitumericalyissimsmearedulationsoutbyinhibitorytheneuronslong-lastingarecnotharacterexplicitlyofthemodeledinhibition.sinceAstheira
consequence,theeffectofGABAergicinhibitionisincorporatedasreducedmembrane
constantτm=250µsofthemapneurons.TheJeffress-likemodelthereforeremainsthe
sameonlythattheparametersofthemapneuronsareadapted.
Fdeterminesorathecoincidencesizeofthedetectortimewindoneuronwthewithinwidththatoftwtheospikesexcitatoryarepreceivosedt-synapassimticpoteultaneous.ntial
Asaconsequence,theneuronfireslesswhenτmdecreases.Toneverthelessensurea
sufficientfiringofthemapneuronssynapticstrengthsareincreasedforaJeffress-like
modelwithGABAergicinhibition.AllusedparametersaresummarizedinTable3.1and,
ifdifferenFiguret3.4from(btheottomonesleft)shopreviouslywstheused,obtainedarespfirecingiallyprofilemarkofed.theiTDmapindependence
upondifferentinputiTDs,includingtheeffectofGABAergicinhibition.Theestimated
iTDs[cf.Eq.(3.4)]nicelyreproduceinputiTDs.TheestimationerrorreducestoE=18µs,
andthusthemisestimationangleis◦calculatedas18µs/86◦µs∙(π/2)≈19◦.Similar
resolutionsarefoundinbirds,suchas23forgreattits[118],27forbudgerigars[169],and

61

3.NeuronalprocessingofiTDsandiADs

arameterP

maximaliTD(Hemidactylusfrenatus)
auditoryauditorynervnerveevfiringectorratestrength
numberofinputneurons
numberofmapneurons
neuronsmapofrangeITDstrengthsynapticpost-synapticwithinhibitioncurrentwidth
membraneconstantofmapneuron
]37[inhibitionwithmapneuronrefractiontime
capacitanceneuronmapmapneuronrestingpotential
sholdethrneuronmap

alueV

iTDmaxA==86300µsHz
NVS==300.95
in100=NmapiTDmap∈[−430µs,430µs]
JJ==00..015021
τepscτ==250500µµss
mτm=250µs
τrefr=1000µs
1=CVthreshVo==01

Table3.1:ParametersusedincomputersimulationsofaJeffress-likemodel.Ifnotmarked
otherwise,allparametersaretakenfromFriedeletal.[57].Parametersthatchangeasa
consequenceofanassumedadditionalGABAergicinhibitionaremarkedandwrittenin
blue.Withdecreasingmembraneconstantthemapneuroncandistinguishmorereliable
betweennon-coincidinginputspikesandfiresless.Toensureasufficientfiringofthemap
neuronsdespiteadditionalinhibitionthesynapticstrengthJisincreased.

◦29proforvidescanariesreason[169able].TheresultspropforosediTDJeffresprocesss-likieng.modelExactwithtimeGABconstanAergictsinforhibitionHemidactylustherefore
frhowenatusever,haandvethetobeexistenceexperimenofatallylong-lvastinerified.gdepolarizationthroughGABAergicinhibition,

iADsofcessingPro3.4SimilartoneuronalprocessingofiTDsinlizards,neuronalprocessingofiADsisnot
yetexperimentallyclarified.Aneuronalconstructiontoestimateamplitudedifferences,
however,areEIneurons,thatis,neuronsthatreceiveinhibitoryinputfromtheipsilateral
sideandexcitatoryinputfromthecontralateralside.Experiments[59,69,155,173,191]
showthatEIneuronsaresensitivetoamplitudevariation.Inthefollowing,firingactivity
ofEIneuronsaresimulatedinresponsetoinputsdifferinginamplitudebetweenthe
twosidesbythetympaniciADascalculatedfromtheICEmodelbasedonasimplified
cylindricalmouthcavity;cf.Chap.2.AsillustratedbyFig.3.6themodelconsistsof
twostages.First,tympanicvibrationsexciteneuronsofthenucleusangularis(NA);cf.

62

3.NeuronalprocessingofiTDsandiADs

Figurenatural3.6:stimulusSuggestedEq.(2.26neuronal)ofthecircuitICEtomoprodelcessalloiADs.wsforFortheboththecalculationexpoferimenthetaltandympanicthe
vibrationamplitudes|u˙0|and|u˙L|indB.Thelatterareusedasinputfiringratesforthe
PoissonipsilateralneuronsNAproofthejectsipsviai-andinhibitorycontralateralsynapsnesucleus(whiteangularissphere)and(NA).theIntheconnexttralateralstep,NAthe
projectsviaexcitatorysynapses(blacksphere)toEIneuronsthataremodeledasleaky
integrate-and-fireneurons;seeSec.1.3.2fordetails.

high-frequencyprojectioninFig.3.1.Inthesecondstep,thecontralateralNAprojects
viaexcitatorysynapses,andtheipsilateralNAprojectsviainhibitorysynapsestoEI
neurons.GiventhatneuronalprojectionsfromtheNAremaintobeelucidated,theexact
locationoftheseEIneuronsandtheexactprojectingpathwayfromtheNAtotheseEI
neurons,e.g.,throughintermediateneurons,arenotspecified.Inmammals,EIneurons
aretopographicallyorganizedwithintheinferiorcolliculus(IC)[59,69,173].Inchicks,EI
neuronsarefoundwithinthedorsallaterallemniscalnucleus(LLD)[191].

3.4.1Experimentalandnaturalstimulus
Toestimatethestimulusexcitingtheauditorynerves,amplitudesofthetympanicvibrations
arecalculated.Moreconcretely,Eq.(2.26)oftheICEmodelappliesforincomingpressures
pex0andpexL.Forthelattertwodifferentstimuliarepossible.
•Experimentalstimulus
Ttheotreproympani,duceexpIAD=erimen0,talbutnotsetupsinthephase,incomingITD=0.pressureHenceis,variedpressureinintensitfunctionsybetatwtheeen
twotympanicmembranes(x=0andx=L)aregivenby
IADpex0(ω,IAD;t)=|pex|exp20exp(iωt),(3.6)
pexL(ω;t)=|pex|exp(iωt)
whereIAD=20log(|pexL/pex0|)inducestheexponentialformoftheamplitudemod-
ification.TheinputIADs,however,donotnecessarilyentertheneuronalsound
prolevelofcessingthetcircuits.ympanicInconvibrations,trast,internallyresultingincouplediADsears=firstIADs.modifythestimulusatthe

63

3.NeuronalprocessingofiTDsandiADs

ulusstimNatural•ForananimalofsmallheadsizesuchasHemidactylusfrenatus(gekkonid)shielding
effectsduetotheheadcanbeneglected.Thatis,naturalstimulionlydifferinphase
betweenthetwosides,ITD=0,whereasintensitydifferencescanbeneglected,IAD=0.
SimilarlytoEq.(2.1)incomingpressurespex0andpexLarethengivenby
pex0(ω,ITD;t)=|pex|exp[iω(t+ITD)],
pexL(ω;t)=|pex|exp(iωt).(3.7)
Thereasonwhysuchastimulusisnotusedinexperimentsisobvious,thereareno
amplitudedifferencesinthestimulation,sowhyshouldanexperimentalistwholooks
forneuronalcircuitsforamplitudeprocessingexpectareaction?Theanswerlieswithin
thepropertiesofICE.GivenexternalITDs,internallycoupledearsgenerateiADsup
to20dBbetweenthevibrationsofthetwotympaniforfrequenciesabove1kHz;cf.
Fig.2.12,top.Inotherwords,neuronalcircuitsarenotcharacterizedbytheproperties
oftheinputbutbythepropertiesofthetympani.ThusneuronscanonlybeiTD-
oriAD-sensitive.Giventhatsimultaneousmeasurementsofthetympanicvibrations
arenecessarytoestimatetheinputtotheneuronalsystem,experimentswiththe
naturalstimulusare,atafirstglance,morecomplicatedthanexperimentswiththe
experimentalstimulus.Forthelatter,theexperimentalistsdonothavetobeawareof
themodificationduetotheinternallycoupledears.NeglectingthegenerationofiADs
fromITDscouldneverthelessleadtowronginterpretationsofexperimentalresults,
e.g.,wrongclassificationofiAD-sensitiveneuronsasITD-sensitiveneurons.
ThefollowingsimulationofNAneuronsdistinguishesbetweentheexperimentalandthe
naturalstimulus.Measureddataareonlyavailableforhomologousbrainareasinchicksin
responsetoanexperimentalstimulus[191].

3.4.2Simulationofnucleusangularis(NA)
Inafirststep,simulationofNAneuronsrequiresanestimationofhowtheseneuronsreact
totympanicvibrationsofvaryingintensity.K¨opplandManleyhavemeasuredresponses
ofsee[the128]andauditoryFig.nerv3.7e.fortheAccordingexptoerimenthistalstudystim,ulusauditoryinthenervbesobtailplossessizardaTiliquadynamicrugosarange;
inexpwhicerimenhthetalfiringresults,ratesthevmoarydellinearlyconcentrateswithonsoundthepressuredynamiclevrangeelinofdB.theNAExploitingneuronsthesethat
areconsequentlysimulatedasPoissonneurons;seeSec.1.3.3.Theirinputratefunctions
inaredBassasumedtocalculatedbebpropyEq.ortional(2.26)toofthetheICEamplitudesmodel.ofthetympanicvibrations|u˙0|and|u˙L|
Simulationsofcontra-andipsilateralNAneuronsinresponsetotheexperimentaland
thefiringnaturalprofilesstimforulustheofexp1erikHzmenleadtaltostimfiringulusprofi(left)lesnicelyasreproillustratedduceinexpFig.erimen3.8.talThedataobtainedfrom
thecontralateralauditorynerveofchicksinresponsetovaryingIADs;cf.Fig3.9(topleft)
bySatoetal.[191].Giventhatinternallycoupledearshavebeenexperimentallyprovenin
moseveraldelingbirdsinput[17,and33,the88,firing183],ofinNAparticularneuronsincseemhicks[95reasonable],theforanassumptionsICEsystem.madeforthe

64

3.NeuronalprocessingofiTDsandiADs

Figurefrequency3.7:(CF)Discofharge2.1kHzratesdepofendenauditorytoninnervceominfibgerssoundofTiliquapressurerugosalevel.withWithinachartheacdynamicteristic
stimrangeulusfromstrength40-80dBindB.SPL,FigurethefiringduetoofK¨theopplauditoryandManleynerve[v128aries].almostlinearlywiththe

ThefiringactivityofNAneuronsforthenaturalstimulusofITDs∈[−28µs,28µs]
arepresentedinFig.3.8,right.HerefiringactivitiesvarymonotonicallywithinputITDs.
ThefiringactivityofipsilateralNAneuronsismaximalforpositiveITDsandminimal
around−20µswhereithardlyvaries.ThefiringactivityofcontralateralNAneuronsis
exactlytheoppositewithmaximalvaluesatnegativeITDs.

3.4.3SimulationofEIneurons
Inasecondstep,neuronsoftheipsilateralNAprojectviainhibitorysynapsesandneurons
ofthecontralateralNAprojectviaexcitatorysynapsestoEIneurons;cf.Fig.3.6.EI
neuronsaremodeledasleakyintegrate-and-fireneurons;seeSec.1.3.2.Duetothe
assumptionoftwoprocessingpathways,highfrequencyinputsonlycontainamplitude
andnottiminginformation.ThefiringratesofNAneuronsarethereforeassumedto
beconstantovertime.Equation(1.4)determinesthemembranepotentialofaleaky
integrate-and-fireneuroninthecaseofaconstantinputcurrentIinjas
tV(t)=Vr+RmIinj1−exp−CmRm(3.8)
witht0=0.TheinjectedcurrentIinjisthesumofexcitatorycurrentsfromthecontralateral
NAandinhibitorycurrentsfromtheipsilateralNA.GiventhatneuronsoftheNAfire
withaprobabilityνipsiandνcontra,theinjectedcurrentcanbeexpressedas
Iinj=Jinhνipsi+Jexcνcontra(3.9)
withJinh≤0andJexc≥0beingthestrengthsofinhibitoryandexcitatorysynaptic
connections,respectively.WithJinh=−Jexcandνipsiandνcontrabeingproportionaltothe
vibrationamplitudeofthetympani,theinjectedcurrentIinjisproportionaltotheiAD.

65

3.NeuronalprocessingofiTDsandiADs

Figure3.8:Simulatedfiringprofilesofneuronsfromtheipsilateral(blue)andcontralateral
(red)nucleusangularis(NA)(cf.Fig.3.6)inresponsetotympanicvibrationamplitudes|u˙0|
andwith|u˙L|inparametersdBasgivencalculatedinTablefrom3.2.Eq.(Left:2.26)SimoftheulationsICEpmodel.erformedSimwithulationstheareexpperimenerformedtal
stimneuronsulus,havi.e.,eveitheraryingaclearinputminIADs.imumHereorthemaximfiringumratesandofcannotipsilateralrepresenandttheconinputtralateralIADsNAin
anunambiguousway.Right:Simulationsperformedwiththenaturalstimulus,i.e.,varying
withinputtheITDs.inputTheITDsfiringandratescanofthereforeipsilateralunamandconbiguouslytralateralrepresNAentneutheronsstimvaryulus.monotonically

AsanalyzedinSec.1.3.2,theneuronwillnotspikeatallifVr+RmIinjliesbelowthe
firingthresholdV.Otherwise,thespikerateisproportionalto1/T,withTbeingthe
constantinterspikeinterval.ThelatterisgivenbyEq.(3.8)andV(t=T)=V,suchthat
TV=Vr+RmIinj1−exp−CmRm.(3.10)
GiventhattheinterspikeintervalislimitedbytherefractoryperiodTrefracastheminimum
interval,theinterspikeintervalcanbecalculatedtobe
T=−CmRmln1−RVm−IVinjrforT≥Trefrac.(3.11)
TrefracforT<Trefrac
Thisformulaissimplifiedbytheprerequisiteforafiringneuron,thatis,Vr+RmIinj>V.
Asaconsequence,thefraction(V−Vr)/(RmIinj)<1.For(V−Vr)/(RmIinj)1the
logarithmcanbeapproximatedbyln(1+x)≈xwhichleadsto
T=CmV−Vr/IinjforT≥Trefrac.(3.12)
TrefracforT<Trefrac
Altogethertheneuronshowsthreedifferentfiringbehaviors.Eitheritremainssilent
(Vr+RmIinj<V),firesconstantlyatitsmaximalrate(T<Trefrac),orshowsafiringrate
1/TthatvarieswithIinj=Jinhνipsi+Jexcνcontra.

66

3.NeuronalprocessingofiTDsandiADs

Figure3.9:Top:IADprocessinginchicks[191].Auditorynervesfirstprojecttothe
nucleusangularis(NA).ThecontralateralNAthenprojectsviaexcitatorysynapsesand
theipsilateralNAviainhibitorysynapsestothedorsallaterallemniscalnucleus(LLD).
Topleft:Measuredfiringrateinimpulsespersecond[imp/s]ofintensity-sensitiveneurons
ofthecontralateralNAwithbestfrequencyBF=800Hzinresponsetotheexperimental
stimulus,i.e.,varyingIADs.Topright:ResponsesofLLDunitswithBF=1068Hzto
binauralsoundwithvaryingIAD.Rate-intensitycurvesaremeasuredinasingleLLDunit
atfivedifferentcontralateralsoundlevelsabovethethresholdofthisunit(34dBSPL),
asindicated,whiletheipsilateralsoundlevelwasvaried.ThemeasuredresponsesofNA
andLLDneuronstotheexperimentalstimulusnicelycorrespondtocalculateddatafor
theexperimentalstimulus,asillustratedinFigs.3.8(left)and3.10(left).Bottom:IAD
processinginbats[170]:measuredfiringratesofintensity-sensitiveneuronsinresponseto
varyingIADs.WhentheinputfromtheinhibitoryeargetsmoreintensetheEIneuronstops
firingataneuron-specificIADdenotedasIADofcompleteinhibition.Withinthearray
ofEIneurons,IADsofcompleteinhibitionsystematicallyvaryoverthewholephysically
accessiblerangeofpossibleIADs.ForaspecificinputIADneuronswithIADsofcomplete
inhibitionabovetheconsideredIADfire,whereastheotherneuronsremainsilent.Thus,
theIADofcompleteinhibitionoftheneuronattheedgeofthisstepfunctioncouldencode
theestimatedIAD.SincetheIADssystematicallyvarywiththedirectionofthesound
source(cf.head-relatedtransferfunction[141])thebatcanuseIADstolocalizeasound
source.Itisthereforefairtosaythattheneuronsrepresentaneuronalmapofestimated
IADsandhenceaneuronalmapofthehorizontaldirectionofthesoundsource.

67

68

3.NeuronalprocessingofiTDsandiADs

Figure3.10:SimulatedfiringprofilesofEIneuronsreceivinginhibitoryinputsfromthe
ipsilateralsideandexcitatoryinputsfromthecontralateralside;cf.Fig.3.6.Simulations
areperformedwithparametersgiveninTable3.2.Strengthsofexcitatoryandinhibitory
synapsesvarysystematicallybetween0and1.Redindicatesmoreexcitationthaninhibition,
andbluerepresentstheopposite.Theintensiveredline,forinstance,correspondstostrong
excitatorysynapsesnear1andalmostineffectiveinhibitorysynapseswithstrengthalmost
0.Left:SimulatedfiringactivityofEIneuronsinresponsetotheexperimentalstimulus.
AnarrowmarksthefiringprofileforanEIneuronforwhichexcitatoryandinhibitory
synapsesareassumedtohavethesamestrength,i.e.,Jinh=−Jexc.Thisfiringprofilenicely
fitsexperimentaldataforEIneurons,asshowninFig.3.9,topright.Theotherprofiles,
however,eitherhaveamaximumat−15dBoraminimumat15dBindicatingthatthe
presentedEIneuronsarenotoptimallyadaptedtothestimulus.Right:Simulatedfiring
profilesofEIneuronsinresponsetothenaturalstimulus.ITDsofcompleteinhibitionvary
systematicallywithstrengthofinhibitionandcoverlargepartsofthephysicallyaccessible
range.OnlyhighITDsabove15µsdonotinducefiringactivityoftheshownEIneurons.
EIneuronsfromtheothersideofthebrain,however,respondtothecorrespondingnegative
ITDs.Firingprofilesofbothipsi-andcontralateralEIpopulationsthereforeprovidean
encodingstrategyforsoundsourcelocalization(viz.,direction)thatissimilartotheone
found,e.g.,inbats;seeFig.3.9(bottom)andParketal.[170].

3.NeuronalprocessingofiTDsandiADs

AsbeforethenumericalsimulationsofEIneuronsdistinguishbetweentheexperimental
andthenaturalstimulus.TheinputratesoftheNAneuronsareproportionaltothe
tympanicvibrationamplitudes|u˙0|and|u˙L|indB;cf.Eq.(2.26).FiringratesofEI
neuronsasaresultofsimulationswiththeexperimentalstimulusareshowninFig.3.10,
left.Hereinhibitionandexcitationsystematicallyvaryinstrengths.Anarrowmarksthe
firingprofileoftheEIneuronwhereexcitatoryandinhibitorysynapseshavethesame
strengthJinh=−Jexc.Firingprofileswithsimilarstrengthofinhibitionandexcitation
varymonotonicallywithinputIADsandshowtheabovedescribedthreedifferentfiring
behaviors.FornegativeinputIADstheexcitatoryinputdominatestheinputcurrents,
andtheEIneuronfiresatmaximalrate.Around0dBthefiringratedecreaseswithinput
IADsinalinearmanner;cf.Eq.(3.12).AssoonasinputIADsreachacertainthreshold,
calledIADofcompleteinhibition,theinhibitoryinputissostrongthatthemembrane
potentialremainsbelowitsfiringthreshold,andtheneuronstopsfiring.Firingprofiles
obtainedasaresultofthenumericalsimulationsforJinh≈−Jexchavebeenmeasuredin
thedorsallaterallemniscalnucleusofchicks;cf.Fig.3.9lemniscalnucleus,topright.
FiringprofilesofEIneuronswithdifferentstrengthsofinhibitionandexcitation
(Fig.3.10,left),however,donotmonotonicallyvarywithinputIADs.Insteadtheprofiles
eitherhaveamaximumat−15dBoraminimumat15dB.Theoccurringambiguitycould
indicatethattheusedexperimentalstimulusdoesnotcorrespondtothestimulusthemap
istunedfor.IfthishypothesisholdstruefiringprofilesofEIneuronsshouldratherbe
adaptedtothenaturalstimulus.
Figure3.10(right)showstheaccordingfiringprofilesofEIneuronsinresponsetothe
naturalstimuluswithvaryingstrengthsofinhibition.HerethefiringactivitiesofEIneurons
areperfectlyadaptedtothenaturalstimulusinsuchawaythattheymonotonicallyvary
withinputITD,i.e.,horizontaldirectionofsoundsource.Inaddition,ITDsofcomplete
inhibitionsystematicallyvarywiththestrengthofinhibitionandcoverlargepartsof
thephysicallyaccessiblerange.TheITDsofcompleteinhibitionoftheEIneuronscould
thereforeencodethehorizontaldirectionofsound.ForawholearrayofEIneuronswith
varyingstrengthofinhibitoryandexcitatorysynapsesthemapfiringprofileinresponseto
aspecificinputITDlookslikeastepfunction.NeuronswithITDsofcompleteinhibition
abovetheconsideredITDfire,whereasotherneuronsremainsilent.Thus,theITDof
completeinhibitionoftheneuronattheedgeofthisstepfunctioncouldencodethe
estimatedITDandhencethehorizontaldirectionofthesoundsource.Asimilarencoding
principlebasedonIADshasbeenmeasuredinbats[51,170];cf.Fig3.9,bottom.
Conclusion3.5BasedontheresultsoftheICEmodelwithasimplifiedcylindricalmouthcavityfor
Hemidactylusfrenatus(gekkonid)andexperimentaldata,twoseparatedpathwaysforlow
andhighfrequenciesaresuggested,aJeffress-likemodelwithoutandwithGABAergic
inhibitionforprocessingofiTDsandEIneuronsforprocessingofiADs.
ConcerningtheprocessingofiTDs,thesimulationofanexclusivelyexcitatoryJeffress-
likemodelresultsinahorizontalestimationerrorof≈41◦.AnadditionalGABAergic
inhibitioncouldimprovetheangleofmisestimationto≈19◦.Bothvaluesareingood

69

3.NeuronalprocessingofiTDsandiADs

Parameter

alueV

experimentalstimulus:maximalinputIADIADmax=25dB
naturalstimulus:maximalinputITDITDmax=28µs
excitatorysynapticstrengthJexc∈[0;1]
inhibitorysynapticstrengthJinh∈[−1;0]
post-synapticcurrentwidthτepsc=500µs
mapneurontimeconstantτrelax=500µs
mapneuronrefractiontimeτrefr=1ms
mapneuroncapacitanceC=1
mapneuronrestingpotentialVo=0
mapneuronthresholdVthresh=1

Table3.2:ParametersusedforsimulationsofEIneuronsforiADprocessing.Forthe
simulationstheexperimentalstimulus(varyinginputIADs)andthenaturalstimulus
(varyinginputITDs)areusedasdenoted.

andagreementhetexistencewithofaresolutionslong-lastingfoundindepbirds.olarizationExacttimethroughconstantsGABAergicforHemidactylusinhibition,hofrwevenatuser,
havetobeexperimentallyverified.
neuronsConcerninghavebeentheprocalculated.cessingofHereiADs,twothedifferenneuronaltstimuliactivitwyereofsptheecified.NAandThetheexpensuingerimentalEI
stimobtainedulusfivringariestheratesofinputNAIADs,andEIwhereasneuronstheinrespnaturalonsestimtotheulusvexparieserimenthetalinputstimulusITDs.areThine
perfectFiringagreemenratesoftEIwithneuronstheinmeasuredresponsefiringtotheratesofnaturalcorrespstimulusondingwithbrainvaryingareasinthestrengthschicofk.
inhibitionprovidesystematicallyvaryingITDsofcompleteinhibitionandhenceapossible
encoTakdingenstrategytogether,ofthehorizonresultstalofthesoundtwosourceprocessingdirectionpathwawithinysofaniTDsarrayandofEIiADsareneurons.reason-
ablecould,andhofitwevexper,differerimenbtaletwresultseenlizardsfromandcloselychicks,relatedneurophbirds.Givysiologicalenthatauditorymeasuremenprotscessingfrom
theandiADs.lizard’sInauditoryparticular,systemstheinarevolvedneededbraintofiareasnallyvanderifypaththewayssuggestedhavetoprobecessingclarified.ofiTDs

70

3.

Neuronal

pro

cessing

of

iTDs

and

iADs

71

4.

Thesmallerthelizard,thelargeritshopeto
becomeacrocodile.
ProverbfromAbyssinia

InAuditoryternallySensitivitCoupledyEarsand

conThevtrasttoertebratetheauditoryacousticallysystemindependenexhibitstearlargesofvariationsmammals,inmanystructureotherandvmecertebrateshanics.havIne
ears(ICE).morphologicallyHereandgeneralfunctionallyrelationsbetwcoupledeentavympanicertebrate’smemheadbranes,size,calledi.e.,ininternalteraurallycdistance,oupled
andTimeitsandbestamplfrequencyitude,viz.,differencesthebetfrequencyweenwiththethememlowbraneesthearingvibrationsdthreshold,eterminareeexplored.hearing
abilitiesofanimalswithinternally−1coupledears.Thecorrespondingequationsthendeliver
anwhicinhviserseconfirmedrelationbfbyestexp∝Lerimenbtaletweendata.inExpterauralerimentaldistancebestLandfrequenciesbestoffrequencyanimalsfbwithest,
beindepexplaiendennedtbearsyproanalysisvideofantheinversecorrespdepondingendencewithacoustics.anexpApartonentfromarounddifferen-0.5tdepthatendencescannot
uponinterauraldistance,calculationsandexperimentalmeasurementsshowthatanimals
inwithterauralinternallydistancecoupledwithearsindephaveendenintgeneralears.loInwerternallybestcoupledfrequenciesearsthanthusanimalspresentofthethemselvsamees
asadistinctauditorymechanismthatassuresananimal’scapabilitiesofhearinglow
frequenciesdespiteasmallheadsize.

ductiontroIn4.1Internallycoupledearsareprobablytheleaststudiedamongthevarioushearingsystems,
despitecouplingbofeingtheleftcommonandinrighbtothmiddleinvearertebratecavities,andvtheertebratevibrationoftaxa.eacByhtmeansympanicofthememdirectbrane
showsadirectionalresponsetoincomingsoundsignals,theso-calledpressure-gradient
asreceivwellerascamplitudeharacteristic,ofasafirstsingletdescribympanicedbymemAutrumbrane[4]forvibrationlocusts.variesThiswithmeansrespectthattophasethat

73

4.AuditorySensitivityandInternallyCoupledEars

Figure4.1:Illustrationofanatomicalvariationswithintheauditorysystem.Theplots
showcross-sectionsthroughtheheadattheleveloftheearofanimalswithICE(Aand
B)andwithindependentears(CandD).Black(tympanicmembrane),white(middleear
cavity),yellow(Eustachiantube),gray(pharynx),red(directlinkage).ADirectlylinked
middleearcavitieswithEustachiantubes,e.g.,lizards,frogs,alligators,andbirds.For
lizardsallcavitiesaremergedtoonehugemouthcavitysothatthemiddleearcavitiesare
directlylinkedthroughEustachiantubesandpharynx;cf.Fig.1.2,right.BDirectlylinked
middleearcavitieswithoutEustachiantubes,e.g.,chameleons.CIndirectconnection
ofthemiddleearcavitiesthroughEustachiantubesandpharynx,e.g.,mammals.DNo
connectionbetweenthemiddleearcavities,e.g.,anumberoflizardsandsnakes.

oftheothermembranedependingonthedirectionofthesoundsource.Theexpression
“pressure-gradientreceiver”referstothefactthatsuchasystemisnotsensitivetothe
pressureamplitudeofasoundwaveatonespecificpointbutrathertothedirection
ofthesoundwavealreadyatthestageofthetympani.Moreprecisely,Autrum[4]
notedthatsuchareceiverreactstothepressuredifferencebetweenthetympani.To
avoidmisunderstandingsconcerningtechnicalterms,throughoutthethesiscavity-coupled
tympanicmembranesarereferredtoasinternallycoupledears(ICE).Descriptionsand
measurementsofICEareavailablefromanumberofdifferentanimalsincludingcrickets
[150],frogs[29,49,103],barnowlsandotherbirds[17,33,88],andlizards[27,28].
Evolutionaryaspectshavebeencoveredinseveralreviews[19,20].
Simplisticallytheearsofthetetrapodal(ornon-piscine)vertebratescanbeclassified
intooneoftheconditions(Fig.4.1),withavarietyofanatomicalvariationineachcondition.

74

4.AuditorySensitivityandInternallyCoupledEars

righThetfirstandleftauditorymiddleconditionearcavities,(Fig.4.1aswA)ellinaswhicanhindithererectisaconnectiondirectviaconnectionEustacbethianweentubthees
linkingthemiddleearcavitiestothepharynx,occursinsomefrogs,lizards,alligators,
anddirectbirds,andandindirecthasfrequenconnectiontlybareeenideninticalterpretedsinceasallancavitiesauditoryarespmergedecialization.toonehFugeormouthlizards
caandvity.pharynx;Thecf.middleFig.ear1.2ca,righvitiest.areThethereforerarestdirectlyarrangemenlinkted(Fig.through4.1B)isEustacforthehianmidtubdlees
earrepresencavteditiesbytoabesingledirectlyopeninglinkbed,etwbuteenforthethepharynxEustacandhianthetubesconnectiontobebeteitherweentheabsentrighort
andleftmiddleearcavity.Inthiscondition,bestrepresentedbytheChameleons[231],
thepharynxdoesnotplayanyroleinestablishingapressuredifference.Amorecommon
configuration(Fig.4.1C)occurswhenthemiddleearcavitiesareonlyindirectlylinkedby
earpairedbutEuisstacalsohianfrequentubestlyopeniencounngintoteredtheinpharreptiles.ynx.InThistheconditionfourthtyconfigurpifiestheation(Fig.mammalian4.1
maD)ybtheecongreatlytralateralreducedmiddleorevenearcalost.vitiesThisareconfigurationanatomicallyisisolatedencountered,fromforoneanothinstance,er,inanda
numberoflizardsandsnakes.
twotUsingympaniexpmosterimentalanimalsgroupmeasuremenashtsaofvingeidirectionaltherindepandendendeptendenearstorviICE.brationsInofgeneral,the
conditionsAandBfromFig.4.1arefoundinanimalswithICE,whileconditionsCand
Darefoundinanimalswithindependentears.Thequestionofhowdirectthecoupling
betunsolvweened.theThefollomiddlewingearcaanalysisvitieshasexplorestobtheeinorderrelationshipforbtheetweenearstoheadformsize,ICEmoreremainsexactly
intheterauralmorphometricsdistance,ofandthepeakcoupling(s)auditorythatsensitivitestablishysincetheheICE.adsizehasadirectimpacton
withinThespinveciesersewasrelationrecenbettlyweenreviewboeddy[size71].andPreviousfrequencystudiesofpeakhaveusedsensitivitayvbarietothyofamongmetrandics
foranimalsizeincludingbodymass,headwidth,andfunctionalheadwidth(interaural
distancedividedbythespeedofsoundinthesurroundingmedium).Nevertheless,the
relationshipbetweenpeaksensitivityandsizeinanimals(excludingsubterraneanspecies)
withindependentears[80,81,82,83]isclearlydifferentfromtherelationshipreportedin
purpanimalsoseofwiththeICEpresasentcdescribhaptered,istoe.g.,bexploreyWtheerner[bioph229,ysical230]andbasisofGleicthihs[68].differenceTheand,primaryby
doingso,exploreaspectsofthefunctionalperformanceofICE.

4.2Theoreticaldescriptionofmembranevibrations
Ingeneral,soundprocessingisbasedondifferencesbetweenthevibrationsofthetympanic
memmembranebranes.Invibrationsthebfolloothwingforindepequationendensfortandtimeinandternallyamplitudecoupledearsdifferencesaredberetivwed.eenThethe
bestfrequenciescorrespondtothefrequencieswherethesedifferencesbecomemaximal.
Thisisreasonablesincethebestfrequencyisdeterminedbythefrequencywiththelowest
hearingthreshold;cf.Fig.4.2.
Membranevibrationsareresponsestosoundwavesofacertainfrequencyωfroma

75

4.AuditorySensitivityandInternallyCoupledEars

Figure4.2:Visualizationofbestfrequencyattheexampleofhumanhearing[81].The
Thebaudiogramestshofrequencywsishearingdefinedasthresholdsfrequencyindepwithendentheceloupwonestthehearingfrequencythreshold.oftheInsoundthepresensignal.t
examplethebestfrequencycorrespondsto4kHz.

specificsoundsourceatdistanceDandangleθ.Becauseofthedifferentpositionsofthe
twotympaniwithrespecttothesoundsourcethesoundpressurewavespex0/Larriveat
differenttimesandwithdifferentamplitudesatthetwotympani.Thepressureatapoint
withangleθonasphericalheadcanbecalculated[see,e.g.,Kuhn[133]]tobe
pex0/L(θ,ω;t)=iρc/(4πL2)e−iωtPn[sin(±θ)]hncωD/hncωL(4.1)
nwhereListheinterauraldistance,cthevelocityofsound,andρtheairdensity.Furthermore,
PnistheLegendrepolynomialofordernandhnisthesphericalHankelfunctionofordern.
Thussoundarrivesatthecontralateraltympanumlaterandmorealleviatedthanatthe
ipsilateraltympanum.Forω/cLlessthanone,thedifferencebetweenthetwoarriving
pressurefunctionsreducestoaphaseshift.ExperimentaldatafromvertebrateswithICE
(seeTable4.1)fulfillthisassumption.Accordingly,thesimplificationgenerallyappliesfor
internallycoupledearsandforindependentearsinthelow-frequencyrange.Inthiscase
thedifferencesbetweentheinputpressuresaresufficientlydescribedbytheinterauraltime
differenceITD=(L/c)sin(θ);(4.2)
fordetailsseeSec.2.2.1.Indoingsotheeffectofsoundtravelingaroundtheheadis
neglected.Forindependentearsandhigherfrequencies,Eq.(4.1)cannotbesimplified.Interaural
amplitudedifferences(IADs)arecalculatedbythefullsolutiontobe

IAD:=20log[|pex0(θ,ω;t)|/|pexL(θ,ω;t)|].(4.3)
Forasoundsourcedirectlyinfrontofthehead,theangleθbecomeszero,andconsequently
sodothecorrespondingITDandIADvalues.

76

4.AuditorySensitivityandInternallyCoupledEars

4.2.1Membranevibrationdifferencesofindependentears
Forindependentearsthetwomembranesprocessincomingsoundindependently.The
membranevibrationsu˙0/Larethereforesolutionsofaforcedmembraneequation[50,221].
is,Thatu˙0/L(r;t)=Ω−1pex0/Lf0(r)dSf0(r)eiωt(4.4)
Swiththedefinition−11iω
Ω:=ρmdω02−ω2+2iωα.(4.5)
Theparametersarethefollowing
r,tsolutionparameters:radialpositiononthemembraneandtime,
Ssurfaceofthemembrane,
aradiusofthemembrane,
dthicknessofthemembrane,
ρmdensityofthemembrane,
ω0firsteigenfrequencyofthemembrane,and
αdampingcoefficientofthemembrane.
Thefundamentalvibrationmodeofthemembrane,anormalizedBesselfunction[35,p.
313]offirstkindoforder0,isdenotedbyf0(r):=[J1(k1a)]2J0(k1r)withthek1defined
byJ0(k1a)=0.Todeterminethespecificparametersfordifferentsystems,valuesforthe
geckoHemidactylusfrenatus(gekkonid)[221]areusedandlinearlyscaledwiththesystem
asawhole.Radiusaandthicknessdofthemembranethereforeincreaselinearly.In
contrast,theeigenfrequenciesofthetympaniωmndecreaselinearlywithheadsize,i.e.,
interauraldistanceandradius.Theinverserelationbetweenωmnandradiusisillustrated
byEq.(2.7)thatisvalidfortheradialfunctiong(r)ofinternalcavityandmembrane.
WhenthegeometryisscaledbyγtheradiusrinEq.(2.7)isreplacedbyr=γrleading
to∂2g(γr)1∂g(γr)2q2
γ2∂r2+γrγ∂r+kqs−γ2r2g(γr)=0.(4.6)
Multiplyingtheequationbyγ2leadsto
22∂g(γ2r)+1∂g(γr)+γ2kq2s−q2g(γr)=0.(4.7)
sq∂rr∂r:=(k)2r
TheresultingscaledsolutionsareagainBesselfunctions[35,p.313]butwithmodified
kqs.Sincetheeigenfrequenciesoftheoriginalsystemaregivenbyωmn=kqscMwith
cMthepropagationvelocityonthemembrane[50,p.74]theeigenfrequenciesofthe
scaledmembranearegivenbyωmn=kqscM=kqs/γcM=ωmn/γ.Inparticular,the
eigenfrequencyofthefundamentalmodescalesasω0=ω0/γforr=γr.

77

4.AuditorySensitivityandInternallyCoupledEars

ThetimeshiftbetweenthemembranevibrationsforindependentearsITDindcan
thenbecalculatedasadifferencebetweenthearguments,denotedasarg,ofthecomplex
vibrationsofthetympanileadingto
ITDind={arg[u˙0(r;t)]−arg[u˙L(r;t)]}/ω
=arg[u˙0(r;t)/u˙L(r;t)]/ω
=arg[pex0/pexL]/ω.(4.8)
TheITDindofindependentearsconsequentlyreproducesthetimeshiftoftheexciting
externalpressurewaves;cf.Eq.(4.2).Asaconsequenceofneuronalphase-locking,which
onlyoccursifthefrequencyislowenough,ITDsareonlyusedinthelow-frequencyrange
sothatapplicationofthesimplification(4.2)leadsto
ITDind=L/csin(θ).(4.9)
Thedifferencebetweenthevibrationamplitudesofthemembranes,i.e.,theIADind,can
ascalculatedebIADind=20log[|˙u0(r;t)/u˙L(r;t)|]
=20log(|pex0/pexL)|).(4.10)
AgainthevibrationamplitudesofthemembranesofindependentearsreflecttheIADsof
theincomingsoundwave;cf.Eq.(4.3).

4.2.2Membranevibrationdifferencesofinternallycoupledears
Tympaniofinternallycoupledearsdonotvibrateindependentlybutinfluenceeachother.
Thatis,theexternalstimuliandtheinternalcouplingdeterminethemembraneresponse.
Differencesbetweentimeandamplitudeofmembranevibrations(iTDandiAD)canbe
describedbytheICEmodelpresentedinChap.2and[221].Forreasonsofsimplicitythe
problemisassumedtoberotationallysymmetrical,i.e.,apossibleasymmetrywithinthe
attachmentofthemiddleearisneglected.ByareducedICEmodel[seeEqs.(2.26)-(2.28)]
membranevibrationscanbecalculatedtobe
u˙0(r;t)=Gipsi(r;t)pex0+Gcontra(r;t)pexL,
u˙L(r;t)=Gipsi(r;t)pexL+Gcontra(r;t)pex0,(4.11)
filteripsilateralthewithGipsi(r;t)
−ρcicot(ω/cL)+Ω
=−Sf0(r)dS[Ω−ρcicot(ω/cL)]2+ρ2c2sin−2(ω/cL)f0(r)(4.12)
filtertralateralcontheand−ρci[sin(ω/cL)]−1
Gcontra(r,ω;t)
=Sf0(r)dS[Ω−ρcicot(ω/cL)]2+ρ2c2sin−2(ω/cL)f0(r)(4.13)

78

4.AuditorySensitivityandInternallyCoupledEars

whereρdenotesdensityofairintheinternalcavity.Theotherparametershavebeen
introducedintheprevioussection.
Theinternaltimedifference(iTD)forinternallycoupledearsisthengivenby
iTD[ρcicot(ω/cL)−Ω]pex0−ρci[sin(ω/cL)]−1pexL
:=arg[u˙0(r;t)/u˙L(r;t)]/ω(4.14)
=arg[ρcicot(ω/cL)−Ω]pexL−ρci[sin(ω/cL)]−1pex0/ω
wherethefirstequalityisadefinitionandthesecondfollowsfrom(4.11)to(4.13).Fur-
thermore,theinternalamplitudedifference(iAD)forinternallycoupledearsisinasimilar
waygivenby

iAD:=20log[|u˙0(r;t)/u˙L(r;t)|](4.15)
=20log[ρcicot(ω/cL)−Ω]pex0−ρci[sin(ω/cL)]−−11pexL.
[ρcicot(ω/cL)−Ω]pexL−ρci[sin(ω/cL)]pex0
ForinternallycoupledearsiTDsandiADsareresponsesofthewholeacousticalsystem
andnotsimplyadirectreflectionofthevibrationsoftheincomingITDsandIADs.

analysisEmpirical4.2.3Inthenextstep,foreveryinterauraldistanceLbestfrequenciesbothforindependentand
internallycoupledearsareestimatedbymaximizingamplitudedifferencesbetweenthe
vibrations.ympanictSincebestfrequenciesforanimalswithindependentearsareabovetheregionwhere
ITDsareused,Eq.(4.10)determinesthefrequencywherecalculatedIADsbecomemaximal
foreveryinterauraldistanceL;seeFig.4.3,top.Thisfigurealsopresentsliteraturedata
forinterauraldistanceandbestfrequencyforanumberofspecieswithindependentears;
seeTable4.1.Linearregressionanalysisoftheliteraturedatarevealsaslopeofaround
−0.5withR2valueof0.4;seeTable4.2.Thetheoreticalcurvecannotreproducethe
experimentallyfoundregressionfunction,ratheritconsistentlyunderestimatesthebest
frequency.Sincethepresentedmodelonlyincludestheacousticalsystemanditsprocessing
throughthetympani,experimentaldataforanimalswithindependentearscouldreflect
substantialmiddleearandneuronalprocessing,whichcouldalsoexplainthegreaterlevel
ofvariation(lowerR2value)inthisdataset.
DespitevanishingIADswithintheinputfunction,internallycoupledearscreateiADs
forhigherfrequenciesasillustratedinFig.2.12.TheanimalcanthereforeuseiTDsand
iADsdependingonthefrequencyoftheincomingsound;cf.Fig.2.21.Similartoanimals
withindependentears,foreveryinterauraldistanceLthefrequencywhereiADs(4.15)
becomemaximalprovidesanestimationofthebestfrequency,viz.,thefrequencywith
thelowesthearingthreshold(cf.Fig.4.2),ofthesystem.Figure4.3(bottom)showsthe
resultingtheoreticalbestfrequenciestogetherwithliteraturedataforinterauraldistance

79

80

4.

Auditory

ySensitivit

and

ternallyIn

Coupled

arsE

Figure4.3:Bestfrequencies[kHz](verticalaxis)asafunctionoftheinterauraldistance[cm]
(horizontalaxis)foranimalswithindependentears(top)andinternallycoupledears
(bottom).Theregressionfunctionsascalculatedfromexperimentallymeasuredbest
frequenciesareshowninblack;cf.Table4.2.Thegraycurvesandareasshowpossible
statisticalvariationsofthecurveresultingfromdifferentslopes;cf.Table4.2.Top:For
independentearsthebluecurveshowsbestfrequenciesasresultingfrommaximizing
IADs(4.10).Themismatchbetweenpredictedandexperimentaldataprobablyresults
frommiddleearandneuronalprocessingthatarenotincludedinthepresentedpurely
acousticalmodel.Bottom:Forinternallycoupledearstheredoverlayingcurvesshowbest
frequenciesasresultingfrommaximizingiADs(4.15).Predictedbestfrequenciesnicelyfit
totheexperimentallyobtainedrangeofbestfrequencies.

4.AuditorySensitivityandInternallyCoupledEars

Figuredistance4.4:[cm]Maximal(horizontemptaloralaxis)fordifferencesanimals[µs]with(verticalindepaxisenden)astaearsfunction(dashedofredtheinline)terauraland
inindepternallyendentcoupledearsandearsEq.(solid(4.14)blueforline).iTDsVofaluesinternalarelycalcucoupledlatedbyears.Eq.(Dep4.9)endingforITonDstheof
interauraldistancethecouplingofthetympanienhancestimedifferencesstartingfrom
aheadfactorsizeofof5aroundcm.3.5Theforinanternalinterauralcouplingofdistancethetof0.ympani5cmtoathereforefactorofeffectivaroundely2causesforaa
distance.terauralinofmagnification

andanalysisbestofthefrequencyliteratureforadatanumrevberealsofaspslopecieseofwitharoundICE;−1seewithTableR2v4.1alue.ofLinear0.8.Figureregression4.3
(bottom)illustratesthegoodfitbetweenthetheoreticaldataandthisexperimentaldata
set.Comparingdataofinternallycoupledandindependentears,thecalculatedregressions
(cf.forTtheabletwo4.2),literaturereflectingdatathesets,andunderlyingforthetwdifferenottheoreticalbiophysicscurvofes,thesareetwosignificanhearingtlydiffesystems.rent
IndifferenFig.t4.3vtheerticalhorizonaxestalmayaxesobscurearetheonesameof,thekwhereaseythedifferencesverticalbetwaxeseenarethesedifftwerenot.auditTheseory
withsystems;indepanimalsendentwithears,inoftenternallybyafactorcoupledofears2−ha4.veThelowersystembestofinfrequenciesternallythancoupledanimalsears
mayensureananimal’scapabilitiesofhearinglowfrequenciesdespiteasmallheadsize.
Thepresentedtheoreticalmodelofthetympanicvibrationspredictsanenhancementof
totheindeptimeendendifferencestears.betwFigureeenthe4.4memcomparesbranevibmaximalrationsITDsforaforsystemindepwithendenICEtinearsandcomparisoniTDs
forliesbinetwternallyeen2couandpled3.5.ears.TheinDepternalendingoncouplingtheofintheterauraltympanidistance,thereforetheeffectivenhancemenelytcausesfactora

81

4.AuditorySensitivityandInternallyCoupledEars

magnificationofinterauraldistance,sothatbestfrequenciesofsystemswithinternally
coupledearsareconsistently,andsignificantly,belowthoseofsimilar-sizedorganisms
withindependentears;seeTable4.1.Calford[17]hasdocumentedthesetimedifferences
experimentallyandreportedanenhancementofthesamemagnitudepredictedbythe
presentedbiophysicalanalysis.

Conclusion4.3Inprinciple,internallycoupledears(ICE)translateanexternalsignalarrivingwith
interauraltimedifferences(ITDs)andinterauralamplitudedifferences(IADs)atbothears
intocoupledvibrationsofthetwotympanicmembraneswithinternaltimedifferences(iTDs)
andsignalinandternaltheinamplitudeternalcoupling.differences(iADs),Becauseofthethelattertwcouplingobeingofththeetresultympanicofbothmemthebranesexternalthe
toiTDstheare,incomingaccordingITDstobyaanalyticalfactordepcalculationsendingonandinexpterauraleriments[distance,17],oftenhancedwotointhreecompariforsloonw
Forfrequencies;animalsseeFig.with4.4ICE.thereisbothatheoreticallyandempiricallysignificantinverse
relationbetweenfunctionalheadsizeandbestfrequency,reflectingthelimitationsresulting
fromthewavelengthofthesignal.Theslopeofthisrelationisnearlytwiceassteepin
animalswithinternallycoupledears(−1)ascomparedtoanimalswithindependentears
(−sound0.5),proacessingdifferenceforthatanimalsissignificanwitht.indepTheendentdifferenandtinregresternallysioncoupledfunctionsears.suggestAnimalsadifferenwitht
internallycoupledearshaveenhancedcapabilitiestohearlowfrequencies.

analysisEmpiricalds:Metho4.AFortheempiricalanalysisinterauraldistanceLandbestfrequencyfbestforanumber
ofdifferentanimalsarecollected.Table4.1summarizesexperimentaldata[3,12,17,
27,44,48,87,162,192].Theanimalspresentedhaveeitherindependentorinternally
coupledears,asindicated.Thesoundpropagationvelocityusedforthecalculationofthe
functionalheadsizeL/cisc=343m/swhentransmissionisthroughtheairat20◦C
andc=1483m/swhenthetransmissionisthroughwaterat20◦C.
Tocopefortheexpectedinverserelationexp(a)Lm=fbestbetweeninterauraldistanceL
andbestfrequencyfbesttheequationislinearizedintheform
logfbest=mlogL+a(4.16)
withslopemandtheinterceptparametera.Sincethemeasureddatapointsareblurred
bynoisetheyfulfillagivenrelationonlywithinacertainerrorrange.Tofindopti-
malestimatesfortheslopeandtheinterceptaminimizationoftheexpectationvalue
log(fbest)−mlog(l)+awithrespecttomandaisnecessary.Equivalently,minimization
ofthesumofsquaredresidualsiN=1[log(fbesti)−mlogLi−a]2determinesmanda.This
methodiscalledaleast-squarelinearregressionanalysis[235].Ameasureforthequality
ofthefitistheR2oftheregression.Letfbestibetheestimatedvalueoffbestiascomputed

82

4.AuditorySensitivityandInternallyCoupledEars

throughafitbymeansof(4.16)andthesampleaveragebyfbest=N−1iN=1
R2oftheregression,sometimescalledcoefficientofdetermination,isdefined


fbasesti.TheNf−f2
R2:=iN=1bestibest2∈[0;1].(4.17)
i=1fbesti−fbest
TheR2measurestheratiooftheexplainedvariationcomparedtothetotalvariation,i.e.,
thefractionofthesamplevariationinfbestthatisexplainedbyL.TheR2equals1fora
aphyerfectpandothetical0forarelationpoorbfit.etwAneentwoadditionalvariables,t-testthatontheis,estwhetherimationthereresultsgressionfurthercoevefficienaluatests
arelikelytobezero.Thet-testthereforeanalyzeswhetherthereisastatisticallysignificant
relationbetweentheexplainingandtheexplainedvariable.Theresultofthet-test,called
p-value,givesthelikelihoodthatthecoefficientsmandaareequaltozero.Itiscommonto
definearelationtobestatistically“significant”forp-valuesof0.05or0.01,corresponding
toa5%or1%chanceofanoutcomeliketheobservedonewithoutacorrelationbetween
thevariables.Fordetailsonregressionanalysissee,e.g.,Wooldridge[235].

83

84

4.AuditorySensitivityandInternallyCoupledEars

Table4.1:OverviewofdifferentanimalswiththeirinterauraldistanceLin[m],their
functionalheadsizeL/cin[µs]withthespeedofsoundcinthesurroundingmedium,and
theirbestfrequenciesin[kHz]takenfrompapersasindicatedinthe4thcolumn.The
animalshaveindependentorinternallycoupledears,asindicatedinthe5thcolumn.Forthe
porpoiseandthealligatorinwaterthesoundpropagationvelocityistakentobe1483m/s.
Otherwisethesoundpropagationvelocityinairis343m/s.

animalindistanceteraural[m]headfunctionalsize[µs]frequencybest[kHz]incouplingternal

budgerigarswl)(ohfinczebraskinkagamidpigeonfroggrasskestrelchick(owl)
onidgekk)rate(walligatorfrogtreefrogleopardiguanidbullfrogratcottonmousegerbilratrabbitporpchincoise(whillaater)
catdogsheepumanhwcohorsetelephan

0.010.0110.0130.0130.0150190.0.020.0210.0220.0250.030.030.0320.0750.0150.0180.030.050.0950.10.120.150.200.20.210.2950.31.3

43.732.137.937.958.355.458.361.264.1167.987.587.593.3218.742.352.587.5145.8277.0291.580.9437.3568.5583.1612.2860.1874.63790.1

4.02.86[[4812]]
]27[3.23.192.0[[1727]]
]48[1.11.51.5[[17192]]
]27[1.82]87[0.81.10.9[[16248]]
]27[1.950.68.0[[4848]]
]48[15.0]48[4.08.010.0[[4848]]
]48[2.08.08.0[[348]]
]48[3.010.01.5[[4848]]
]48[8.02.1.00[[4848]]

esyesyesyesyesyesyesyesyesyesyesyesyesyesynononononononononononononono

4.AuditorySensitivityandInternallyCoupledEars

Table4.2:Regressionanalysisforthebestfrequencyindependenceuponthefunctional
headsize(i.e.,interauraldistanceLdividedbypropagationvelocityofsoundc)for
animalswithindependentandinternallycoupledearsasgiveninTable4.1.Slopemand
theinterceptparameterawithagivenconfidentialrateandstandarderrorshavebeen
calculatedbyminimizingtheexpectationvaluelog(fbest)−mlog(l)+a.Thecalculations
wereperformedonlyforinternallycoupledears(leftcolumn),foranimalswithindependent
ears(columninthemiddle),andforthetwogroupstogether(rightcolumn).Thedummy
variableCequals0forindependentearsand1forinternallycoupledears.Itindicates
towhichdegreethetwopopulationsdiffer.ThevalueR2isthefractionofthesample
variationinfbestthatisexplainedbythefunctionalheadsizeL/c;cf.Eq.(4.17).Thatis
R2=1wouldbeaperfect,R2=0apoorfit.Thep-valuefurthergivesthelikelihoodthat
thecoefficientsmandaareequaltozero,i.e.,whetherthereisastatisticallysignificant
relationbetweentheexplainingandtheexplainedvariable.P-valuesof0.05or0.01indicate
statistically“significance”,correspondingtoa5%or1%chanceofanoutcomelikethe
observedonewithoutacorrelationbetweenthevariables.Theresultsshowastatistically
significantinverserelationbetweenfunctionalheadsizeandbestfrequencyforanimals
withinternallycoupledears.Incontrastbestfrequenciesofanimalswithindependentear
showaninversedependenceuponthefunctionalheadsizewithanexponentaround-0.5.
ThehighvaluesofR2andthelowp-valuesunderlinethegoodnessandreliabilityofthefit.
Thedifferencebetweenthetwogroupsisstatisticallysignificant.

VariablesInternallyIndependentboth
EarsEarsCoupled

mCa

2R

(0.16)-0.97***(0.16)-0.46**(0.11)-0.52***
-1.8***(0.3)11.5***(0.7)(0.9)11.0***(0.7)11.4***

0.70.40.8

Standarderrorsinparentheses
***p<0.01,**p<0.05,*p<0.1

85

5.

Allourknowledgehasitsoriginsinourper-
eptions.cLeonardodaVinci(1452-1519)

mOptimalitultisensoryyinmapmono-andformation

aInmtheultitudestruggleofforsophisticatedsurvivalinasensorycomplexsystems.andTodynamicexploitenthevironmeninformationt,natureprohasvideddevbyeloptheseed
sensorysystems,highervertebratesreconstructthespatio-temporalenvironmentfrom
eachofthesensorysystemstheyhaveattheirdisposal.Thatis,foreachmodalitythe
animalcomputesaneuronalrepresentationoftheoutsideworld,amonosensoryneuronal
map.Theherepresenteduniversalframeworkallowsforthecalculationofthespecific
laviz.,youtstoofchastictheinvolvoptimalityed.Aneuronalstep-bnetwy-steporkbytutorialmeansofillustratesageneralhowtoapplymathematicthealtheoreprinciple,tical
frameworktoconcretesituations.Thatis,givenaknownphysicalsignaltransmissionand
rudimentalknowledgeofthedetectionprocess,theapproachallowstopredictneuronal
propertiesofbiologicalsystems.Finally,informationfromdifferentsensorymodalitieshas
tobeintegratedsoastogainaunifiedperceptionofrealityforfurtherprocessing,e.g.,
forthecreationofdistinctmotorcommands.Conceptsofmultimodalinteractionandthe
evolvementofamultimodalspacebyalignmentofmonosensorymapsarebrieflydiscussed.

ductiontroIn5.1

Amousehearsarustlinginthegrass,seessomeleavesmovingand–escapesfromthe
predator.Thus,theperceptionoftheoutsideworldbysensorysystemsandtheconsequent
translationoftheirresponseintoareliableneuronalrepresentationthat,forinstance,
allowsfordirectionalmotorcommandsisanessentialconceptforsurviving.Aneuronal
representationoftheexternalworldisdenotedasneuronalmap.Dependingonthemap
processinginformationfromoneormanysensorysystems,themapiscalleduni-or

87

5.Optimalityinmono-andmultisensorymapformation

Figure5.1:Thethreestepsofsensoryprocessingleadingtotheformationofaunified
multisensorymap.Anobjectintheoutsideworldgeneratesphysicalinputsignals,which
canbedetectedbydifferentsensorysystems.Toformamap,thephysicalmappingmust
be“inverted”insomesuitableway.Aftermonosensorymapformation,thedistinctmaps
arecombinedintoaunifiedmultisensorymap.

multimodal.Theadvantageousconceptofaneuronalmapisdiscussedindetailinthenext
section.Theprocessingofsensoryinformation,fromitsgenerationtomultimodalmapformation,
canbesubdividedintothe“goldenthree”ofsensoryprocessing:physicalmapping,optimal
mapformation,andmultimodalintegration;seeFig.5.1.

PhysicalMapping.Anobjectintheoutsideworldrevealsitspresencebygenerating
differentsignalsthataretransmittedalongdistinctphysicalpathways.Arunninganimal
may,forinstance,generatesoundandachangingvisualimageasitmoves,aswellas
vibrationsandaninfraredprofile.Inconcreteterms,givenanysignalthatvariesasa
functionofspatialpositionandtime,itispossibletocalculatethetime-dependentresponse
ofthereceptorneurons.Thatis,thephysicalmappingofthesignalontotheneuronal
detectorresponsecanbedescribedbyasetoftransferfunctions,indicatedbytheupper
arrowsinFig.5.1.Theresponsesofthesensorysystemsthenrepresentparticularphysical
quantitiessuchassound,lightintensity,volatilemolecules,orheatoriginatingfromthe
ct.ejob

Optimalmapformation.Fromthesensoryresponsesanobserverneedstoreconstruct
amapthatrepresentsthespatio-temporalstimulus.Forsomesensorysystemsapre-stage
mapalreadyexistsinherently,e.g.,forthevisualsystemontheretina.Forothersystems,
sucandhaastheneuronalauditorymapmsystemustb[e9,21,22constructed,111,136more],spatialexplicitly.Ininformationeitheriscase,notthereadilymapavhasailableto
representtheenvironmentasaccuratelyaspossible,thatis,optimally.Thetaskofthe
brainistoobtainanoptimalreconstructionofthesignal;cf.Fig.5.1,middlearrows.The
kandeytothesuccesscorrespistheondingchoicemaps.oftheThatrightis,theneuronalsynapticconnectionsconnectionsbetweenhavethetobesensoryadjustedsystemsin

88

5.Optimalityinmono-andmultisensorymapformation

suchawaythatthenetwork“inverts”thephysicalmappingofthesignaltothesensory
response[168,210,241].Thefiringactivityofthemapneuronsthenaccuratelyrepresents
signal.oralspatio-tempthe

Multimodalintegration.Inafinalstepofsensoryprocessingthemonosensorymaps
mergeintoasingleunambiguousmultisensorymap.Heretwodifficultiesarise.First,the
successfulfusionofunimodalmapsrequirespropermapalignmentasobserved,forexample,
1inshouldthebsupeeriorcomcbinedolliculusoptimally(SC)[[7418,,158112],to204];maximizeseeSec.the5.2.qualitSecond,yofthetheintegratedmonosensorymapmapsin
comparisontothatofthecontributingmaps.Besidesthis“integrationofinformation”the
multimodalmaptogetherwiththealignedunimodalmapsallowsforanewconceptin
mandcultimodalharacterizeproancessing,obtjecthebyitsso-calledsensory“poolinpropg”oferties.information,anefficientwaytoidentify

Thefirstprocessingstep,physicalmapping,isapurelyphysicaldescriptionofthe
signalgenerationanddetectionprocess.Forinternallycoupledears,theICEmodel
describestheprocessofphysicalmappingofauditorystimuli;cf.Chap.2and[221].This
chapterthereforefocusesonoptimalmapformationandtouchesmultimodalintegration
superficially;adetailedmodelofmultimodalmapalignmentispresentedinChap.6.
Section5.2providesareviewoftheconceptofaneuronalmap.Section5.3discussesa
generalframeworkthatdescribeshowaneuronalmapcanemergefromagivensensory
inputinastochasticallyoptimalway[160].AfterintroducinginSec.5.4a“recipe”howthe
presentedframeworkcanbeappliedtorealisticsituations,Sec.5.5treatstheintegrationof
monosensoryintomultisensorymapsandreviewsthecurrentliteraturefromtheperspective
ofmaps.Basicconceptssuchas“integration”and“pooling”ofinformationarepresented.
Thefinalsectionaddressesthequestionofhowacommonsensoryspacecandevelopatall.

5.2Fundamentalconceptofneuronalmaps
Amajorroleinsensoryprocessingisreservedformaps[124,217].Aneuronalmapisa
ofneuronalneurons.represenNeighbtatoriionngofmaptheneuronsexternalwresporldondtorealizedsimilarbyatopsensorystimographicallyuli.arrangedarray
corticalAsanlayersexample,thatarevisualorganizedinputintheaccordingmammaliantothetopbrainographisproyofcessedtheretinalthroughinputmulcellstiple
from(“retinotopicneighboringporganization”)ointsin[107space.,227].SuchHerespatialneighbmapsoringhaveneuronsbeendrespiscoondvtoeredinvisualvariousinput
sensorysystemsinmanygroupsofvertebrates[23,43,91,114,116,119,151,202,207].
Onemightarguethatneuronalmapsexistsimplybecausetheirneuronalarchitecture
onlyinstance,followsthevithesusallaensoryyersaresurfaceofretinotopicallytheirinputmoorganizeddalityb.ecFauseromtheythispreceivointeoftheirview,inpufort
fromtheretina.Similarly,afrequencymapjustreflectsthetonotopicorganizationofthe
hlea.cco1theThesuSCpiseriorcalledcollicuopticlus,tedepctumenindingonthenon-mammals.context.Here“SC”simplyreferstoeithertheoptictectumor

89

5.Optimalityinmono-andmultisensorymapformation

certainFigurep5.2:ositionTheinfiringsensoryprofilespace.ofaFomapcusingencoondesthethefiringlikelihorateoofdofonlyafindingsingleanobneuronjectatsucha
pasneuronerceptionxicanandonlyignoringbeacitshievneedighifbtheorsprevactivitenytsofathefaithfulwholeperceptionneuronalofmaprealitisy.takAenfaithfulinto
account,thatis,justcomparethe‘neighborhood’ofxiwiththerest.Consequently,even
thoughthenominalvalueofthefiringrateofneuronxiisidenticalincaseAandB,the
representedphysicalrealitydifferssignificantlyinbothcases.

Thisargument,however,doesnotholdforeverysensorymap,inparticular,notfor
auditorymapsofinterauraltimeandamplitudedifferences[24,147,167,209].Inthiscase
itiscertainlynotstraightforwardtobuildamap;cf.Chap.3.
Akeyquestion[217]thereforeis:Whatisthefunctionofaneuronalmap?That
is,whychooseamapstructureforneuronalprocessing?Oneargumentisthat,in
contrasttoarbitrarypopulationcoding,neuronalmapsensureatopographicneuronal
organization.Thisorganizationthenunderliestheneuronalprocessingandprovidesan
efficientrepresentationofacontinuouslyvaryinginputsignal.Forinstance,itallowsfor
theinterpretationofafiringpatternonaspatialmapasthelikelihoodtofindasensory
objectatacertainposition[41,98,174,195,217].AsillustratedbyFig.5.2,inthemap
perspectiveoneneedstoconsidertheactivityofthecompletemapinordertoretrieve
meaningfulinformationfromthefiringrateofasingleneuron.
NelsonandBower[163]suggestedajustificationofmapsbasedontheirfunctionby
comparingcomputationalprincipalsofthebrainandparallelcomputers.Accordingly,they
distinguishedthreetypesofmaps.
•Continuousmapsaretopographicallyarrangedneuronalarraysthatrepresentacontin-
uousparameter.Interactionwithinthemapmainlytakesplacebetweenneighboring
neurons.Anexemplarycontinuousmapisfoundinthesomatosensorycortexof
rats[228].Hereneuronsarelocallyinterconnectedthroughshortrangeaxonalarboriza-
tionofstellatecells.Continuousmapsareusedforspatialfilteringandlocalfeature
extraction.Thelocalconnectivityensuresabalancingoftheload.
•Scatteredordiscretemapsarecharacterizedbyalackofsystematicstructure,i.e.,

90

5.Optimalityinmono-andmultisensorymapformation

non-topographicarrangementoftheneurons.Theuseddefinitionofamaptherefore
theexcludesolfactoryscatteredinputsdmaps.onotAproneuronalvideatopexampleographicistheolforder.actoryConsequencortexoftlyratsan[76]extensivsincee
networkoffibersinterconnectsallcorticalregions.

•Patchymapsareintermediatebetweencontinuousandscatteredmaps.Theinteraction
withinthemaphasbothalocalandaglobalcomponent.Thesomatosensorymap
withinthecerebellarcortexofrats[196]constitutesaneuronalexampleofapatchy
map.Theneuronalstructureincludeslong-distanceparallelfiberconnectionsaswell
asstronglocalinfluencethroughshortrangeaxons.Thetaskofthisbrainareaisthe
analysisoflocalsensoryinformationwithinamoreglobalsensorycontext.

Insummary,structureandfunctionofamaparehighlycorrelated.Foreachcomputational
unittheload-balanceshouldbeoptimizedwithrespecttothegiventask.Withinaneuronal
contextthelimitedcapacityofthesupplynetworkrequiressuchaload-balancingsince
oxygenandglucosearetransportedtothecellsviasupplynetworksofsmallcapillaries.A
high,localincreaseintissuemetabolicactivityleadstoanundersupplyoftheneurons.
Brainareasthatperform,e.g.,localfeatureextractionarethereforeorganizedincontinuous
mapswithstronginter-neighborconnectionstodistributecomputationalloadequally.
Therealcomputationalpowerofneuronalmaps,however,canonlybeappreciatedwhen
theinterplayofseveralmapsisconsidered.HeretheSCisabeautiful,well-studiedexample
ofacollectionofdifferentmapsofsensorysystemsthatprovidespatialinformationina
map-likeform[18,204].TheSCcontainsmultisensory,predominantlymonosensory,aswell
asmotormaps,i.e.,motorneuronsorganizedinamap-likestructure.Allneuronalmapsare
mutuallyalignedtogainaunifiedmultisensoryrepresentationofsensoryspace[18,112,204].
Thecombinedsensoryinformationcanthen,forinstance,activatemotormapsandgenerate
directionalmotorresponses[138,203,219].Directevidenceforthishypothesishasrecently
beenfoundineyetrackingexperiments[77].
Moreover,externalobjectscanbeidentifiedbytheirpositionencodedthroughthe
firingpatterninaneuronalmap.Moreconcretely,thepositionservesasappropriateand
necessaryinformationfordefiningasensoryobject.Whencombiningdifferentsensory
systems,thespatialinformationisneededtobindinformationassociatedwiththesame
sensoryobjectintoonesinglemultimodalperceptforfurtherprocessing.

delmoMathematical5.3Themathematicalmodelforoptimalstimulusreconstructionderivedinthefollowingis
basedontheinitialdivisionofsensoryprocessingintothreemajorsteps(seeFig.5.1)but
focusesonoptimalmapformation.Thatis,stimuluscharacteristicshavetobeextracted
atthebestfromthesensoryresponseasdescribedinSec.5.1.Mathematicallytheinverse
transferfunctionisneededthatcanperformanoptimalreconstructionofaparticular
stimulusfromthesensoryresponse.Thisinversetransferfunctioncanthenbetranslated
intoaneuronalconnectivitypattern.

91

5.Optimalityinmono-andmultisensorymapformation

Thederivationbelowisbasedontworeasonablesimplifications.First,allsensory
mapsareassumedpurelymonosensorywhichreflectsthatmanyspatialmapsareclearly
dominatedbyasinglesensorymodality[226].
respTheonseofsecondthesensoryassumptionsystem.isaThatlinearmeansrelationthatbettheweendetectorthestimrespulusonsesandcthehangepropreceptoror-
tionallytothesignalstrength.Nonlinearrelationsbetweenthestimulusandthedetector
responses,e.g.,alogarithmicresponse[34,102,132,134,166],caninprinciplebetreated
withthepresentedmodelaswell(seeAppendix5.Afordetails)butareexcludedinthe
wing.follo

problemtheofDefinition5.3.1xAnTheobjectcorrespgeneratesondingasignalstimmaulusysb(e,t)forvaryinginstance,intimethetandtime-deppositionendenxtinsoundtheexternalpressurewatorld.a
particularlocationormaydenotethepresenceofedgesormovementsataparticular
positionwithinthevisualfield.
DepInendingtheonnextthestep,problemtheatsignalhandinducesasinglearespdetectoronserii(t)within1a≤seti≤ofNNcansensorybeacompletedetectors.
sensoryorgan,suchastheleftear,orapartofadetectorarraysuchasaspecificinterval
ofbestfrequenciesinthecochlea.Inprinciple,thedetectorcombinesinformationfrom
pastsignalswithinthewholesensoryspace.Theresponseisthereforedescribedby
tri(t)=dxdτsx(τ)hix(t−τ)(5.1)
−∞spaceallxwheredetection.thetrTheansfertransferfunctionfunctionhi(t)canbincorpedifferenoratesttheforpheachysicsofdetectorsignali.Transfertransmissionfunctionsand
withintheauditorysystem,forexample,incorporatethepositionofsoundsourceandear
withrespecttotheheadmidlineandthereforedifferbetweenrightandleftear.
Ingeneral,hix(t)=0forlargevaluesof|x|andt.Thispropertyreflectstheintuition
thatsinceevanenytsoccdetectorurringcanfaronlyawayreactortolongtemagopdonotoral-causal,influencei.e.,thepaststatesignals,ofaitissensor.hx(t)Moreo=0vforer,
itcon<v0.olutionThe[seeresp(onse5.3)andfunctionbox(for5.1)withdefinitionadaptedandinfurthertegrationinformation]limitsthenwithresptransformsecttototime,a
ri(t)=dx∞dτsx(τ)hix(t−τ)
−∞(5.2)=:dx(sxhix)(t).
Thequalitabyoofvetheequationdetectordescribrespesonsetheisinrespcononsetrastofanlimitedidealbyatsystem.leastInthreebiologicfactors.alsystemsthe

92

5.Optimalityinmono-andmultisensorymapformation

(5.3)(5.4)

Convolution()andautocorrelation(◦)
forarbitraryfunctionsa(t)andb(t)aredefinedby
∞(ab)(t):=dτa(t−τ)b(τ)(5.3)
−∞and∞(a◦b)(t):=dτa(t+τ)b(τ).(5.4)
−∞areertiespropUsefulab=ba,(5.5)
a(bc)=(ab)c,(5.6)
(a◦b)(t)=(b◦a)(−t).(5.7)
ForaFouriertransformationF,inparticular,itis
F(ab)=F(a)F(b),(5.8)
F(a◦b)=F(a)F(b).(5.9)
Compositionsofconvolutionandautocorrelationcanbecalculatedas
∞(abc)(t)=dτ(ab)(t−τ)c(τ)
−∞∞=dτdτa(t−τ−τ)b(τ)c(τ),(5.10)
−∞∞[(ab)◦c](t)=dτ(ab)(t+τ)c(τ)
−∞∞=dτdτa(t+τ−τ)b(τ)c(τ).(5.11)
−∞

First,informationmaygetlostduringthetransferfromanobjectoftheexternalworld
tothedetectingsensorysystem.Second,noiseinfluencesallstepsinthedetectionand
reconstructionprocess[47].Third,limitationsoftheneuronalhardware,forinstance,the
limiteddynamicrangeofreceptors,constrainpossiblesolutions;seeSec.5.3.5fordetails.
Themathematicalmodelincorporatesthesethreerestrictivefactorsasadditionalnoise
terms.Accordingly,atermdescribingbackgroundnoiseξx(t)mustbeaddedtothesignal.
Transferfunctionandsensoryresponsearehamperedbyadditionalnoisetermsηix(t)and
χi(t).Consequently(5.2)ismodifiedforabiologicalsystemsoastoread
ri(t)=dx[(sx+ξx)(hix+ηix)](t)+χi(t).(5.12)

93

5.Optimalityinmono-andmultisensorymapformation

Figure5.3:Illustrationoffunctionsandentitiesinvolvedintheprocessofoptimalstimulus
reconstruction.Physicalmapping:signalsx(t)withbackgroundnoiseξx(t)ismappedonto
anoisyreceptorresponseri(t)+χi(t)throughthenoisytransferfunctionhix(t)+ηix(t).
Optimalmapformation:theapplicationofthe(possiblynoisy)inversetransferfunction
lix(t)+λix(t)providesanestimatesˆx(t)oftheoriginalsignal.

Signalsx(t)+ξx(t)
Transferfunctionhix(t)+ηix(t)
Receptorresponseri(t)+χi(t)
Inversetransferfunctionlix(t)+λix(t)
Estimatedsignalsˆx(t)

Table5.1:Functionsanderrortermsdescribingdetectionandprocessingofsensory
information.

mationTomustreconstructbe“inanverted”estimatedinsomesignalfroappropriatmtheewdeteay.ctorTheresptime-deponsesri(tenden),ttheinabvoverseetrtransferansfor-
xoflxfunction(t)tobettheweenreceptordetectorrespionsesandattheimapleadsattoptheositionestimatexissˆx(t)calculatedoftheaslioriginal(t).signalApplicationsx(t)
iybengivsˆx(t)=[ri(lix+λix)](t)(5.13)
iwherethetermλx(t)representsthenoiseduetotheconcreterealizationofthetheoretical
inversetransferifunction.Incontrasttoelsewhere[168,175]thepresentmodelisnon-
iterativesothattheneuronalrealizationresultsinapurelyfeedforwardnetworkstructure
asdescribedinSec.5.3.5.
Figure5.3summarizesthewholemathematicalprocedureofsensoryinformation
processing,correspondingtothefirsttwostepsofFig.5.1.Alltherelevanttermsare
summarizedxinTable5.1.Thenextsectionindicateshowtocalculateinversetransfer
functionsli(t)thatenableoptimalsignalreconstruction.

reconstructionOptimal5.3.2Sensorysystemaretunedtooptimallyreconstructnotonlyonespecificsituationbut
thetypicalenvironment.Inotherwords,biologicallyrelevantsignalsbelongtoaclassof
signalsthataredenotedas“typical”.Consequentlyaspecificsensorysignalisaconcrete
realizationofaclassoftypical,biologicallyrelevantsignals.Thatis,itisastochastic

94

5.Optimalityinmono-andmultisensorymapformation

quantity.Anoptimalreconstructionthereforerequirestominimizetheexpectationvalue
ofthesquareddifferencebetweensignalandreconstruction.
Thisispossiblebecauseallquantitiesandfunctions(cf.Figs.5.1and5.3)involved
inboththeprocessofphysicalmapping(seeAppendix5.B)andtheneuronalprocessof
optimalmapformation(cf.Sec.5.3.5)areself-averaging.Themathematicaldefinitionof
self-averagingallowsforadescriptionintermsofexpectationvalues.
Theconsequentderivationoftheinversetransferfunctionslix(t)thatenableoptimalsig-
nalreconstructionforaclassoftypicalsignalsrequirestheminimizationoftheexpectation
valueofthesquarederrorbetweenestimatedandrealsignal
tE[lx(t),t]:=dtdxsx(t)−sˆx(t)2
t−T(5.14)
=tdtdxsx(t)−sˆx(t)2
T−twithlx(t)thevectoroflix(t).Herethebrackets.denotetheexpectationvaluewith
respecttothedifferenttypesofnoise.ThetypicalprocessingtimeisdenotedbyT.
Tobemathematicallyprecise,anexpectationvalueisanintegralonaprobabilityspace
withrespecttoaprobabilitymeasurep.Forarbitraryfunctionsfandg,if|f−g|2=0
thenf=gwithrespecttopor,physically,lookingattheworldthroughp’sglasses:what
pfindsimportantpopsupclearlywhereaswhatpfinds“irrelevant”hashardlyanyweight;
seevanderWaerden[222].
Minimizingthemeansquarederrormeanstominimizeboththevarianceoftheestimate
andthesystematicshiftbetweenestimateandexpectationvalueofthesignal.Forfurther
detailsseevanderWaerden[222].Inaddition,aquadraticformoftheerrortermhasbeen
proventobeareasonableandpracticalchoiceinmanyphysicaloptimizationproblems[152].
IncaseofindependentGaussianerrorterms,theformulationviaaquadraticerroris
undercertainconditionsidenticaltoresultsobtainedbymeansofmaximum-likelihood
estimates[101,108];seeSec.5.3.4.
Mathematically,theerror(5.14)isafunctionalassigningtoeverysetofinversetransfer
functionsonespecificvalue.Minimizationoffunctionalsintheaboveintegralformisa
centralandwell-studiedaspectofthecalculusofvariations[31,63,105,215].Forthe
presentsituationthefirstvariationwithrespecttoeveryinversetransferfunctionljx(t)is
tovanish.Thatis,
∂[sx(t)−sˆx(t)]2
∂ljx(t)=0foreveryj.(5.15)
SolvingEq.(5.15)requirestosubstitute(5.13)fortheestimatesˆx(t)andreplaceri(t)
byitsdescription(5.12).Expandingthesquareproducesexpectationvaluesofproducts
consistingofvaryingcombinationsofnoiseandsignalterms.Hereallnoisetermsaswell
asthesignalitselfareassumedtobestochasticallyindependentofeachothersothatthe
expectationofaproductofindependenttermsfactorizes,forinstance,
sx(t)ηix(t)=sx(t)ηix(t).


95

5.Optimalityinmono-andmultisensorymapformation
Foraproduxctconsistingofthesamekindofterm,theautocorrelationofanarbitrary
quantityf(t)isgivenby
fx(t)fx(t)=δ(x−x)δ(t−t)(µf2+σf2)(5.16)
withµfthemeanandσfthevarianceofthequantityfx(t).Thatis,thevaluesfordifferent
spatio-temporalpositionsareinafirststepcompletelyuncorrelated,akindofworst-case
scenario.Giventhatthemeansofallnoisetermsµfvanishtheautocorrelationsforthenoiseterms
toreduceξx(t)ξx(t)=δ(x−x)δ(t−t)σξ2,(5.17a)
χi(t)χj(t)=δijδ(t−t)σ2χ,(5.17b)
ηix(t)ηjx(t)=δijδ(x−x)δ(t−t)ση2
with|x|<xmaxand0<t<tmax.(5.17c)
Thefinalequationaccountsforthefactthatthenoiseηix(t)vxanishesforlargevaluesoft
and|x|inthesamewayasitdoesforthextransferfunctionhi(t).
Theautocorrelation(5.16)ofthesignals(t)itselfdependsontheproblemathand.
EitherthedetectorsofthesensorysystemmeasureabsolutesignalstrengthsµS,e.g.,
vision,ormodulationsofameanvalueofthesignal(deviationσS),e.g.,audition.Inany
case,equalonetozero.hastoIncthhoeosefollothewing,correspchoosingondingthebiologiabsolutecallysignalrelevantstrengthtermµandSofputthethesignalothersas
theappropriatequantityandthereforetakingσSzeroreducesEq.(5.16)to
sx(t)sx(t)=δ(x−x)δ(t−t)µs2.(5.18)
Whereas(5.17)incorporatesreasonableassumptionsforallnoiseterms,thecorre-
lationspatio-temp(5.18)oralforconthetinsignaluity,ise.g.,aobstrongjectshypandothesis.theircorrespSignalsareondingnsameignlyalscusuallyharacterizeddonotby
disappearfrommomenttothenext.AGaussiancorrelationterm
sx(t)sx(t)=Aexp−x−x2/(2σs2x)exp−t−t2/(2σs2t),(5.19)
forinstance,cantakeintoaccountcorrelationsbetweenneighboringpointsinspaceand
time.Hereσandσaretypicalspatialandtemporalcorrelationscales.Theapplication
ofsuchasxGaussianstcorrelation,however,doesnotgreatlyalterthefurtherderivationbut
onlysmoothensthefinalestimatedsignal;seeAppendix5.Dfordetails.Forreasonsof
clarity,relation(5.18)isusedinthefollowing.
Tosolve(5.15)theapplicationofthecorrelations(5.17)and(5.18)leadsto
ljx(t)σχ2+(µs2+σξ2)|y|<ymaxdydτση2
max0<τ<t
+(µs2+σξ2)dy(hiylix)◦hjy(−t)=µs2hjx(−t);(5.20)
i96

5.Optimalityinmono-andmultisensorymapformation

fordetailsonthecalculationsseeAppendix5.C.Theopencircle◦denotestheautocorre-
lationintegralasdefinedin(5.4).Tosimplify(5.20)twonoisemeasuresareintroduced.
Theparameterτrepresentsaninversesignal-to-noiseratioandisdefinedby
2στ2:=µ2ξ.(5.21)
sForprocessingofsensorysignalsitisreasonabletoassumeasmallvalueofτ.Thesecond
noiseparameterσdefinedby
2222σ2:=σ2χ+|y|<ymaxdydτση(µs2+σξ)(5.22)
µs0<τ<tmaxµs
describestheoverallmeasurementnoisebyrelatingdetectionandtransmissionnoise,σχ
andση,tothesignalmeanamplitudeµs.Apriori,thevalueofσcannotbeassumedto
besmallandhastobeadjustedaccordingtothesituationathand.
Tofurthersimplify(5.20)theequationistranslatedtoFourierspace,whereconvo-
lution(5.3)andautocorrelation(5.4)becomeordinarymultiplicationscombinedwith
letterscomplexandtheconjugations;complexcf.Eqs.conjugation(5.8)byandan(o5.9v).erline,DenotingEq.(5.20F)ourierwiththetransformsintrobducedycapnoiseital
measures(5.21)and(5.22)simplifiesto
Lixσ2δij+(1+τ2)dyHiyHjy=Hjx.(5.23)
icalculateEquationthe(inv5.23erse)isthetransfermainfunctionsresultofLxtheforpresenoptimaltedsignderivalation.Inreconstruction.principle,Aitallocalculationwsto
ofthesecondvariationconfirmsthattheiobtainedinversetransformationindeedminimizes
theerror;seeAppendix5.C,inparticular,Eq.(5.48),fordetails.

notationMatrix5.3.3Torewrite(5.23)inamoreconvenientnotation“matrices”HandLareintroducedby
H[ix]=HixandL[xi]=Lix.(5.24)
Thenotationillustratesthattransferfunctionsandinversetransferfunctionsarelinear
neuronaltransformationsmap)andfromviceavconersa.tinHuousandLspaceare(thethereforeoutsideonlyworld)formallyintoamatricesdiscretewithaspacespatial(the
mcoustordinateconsequenxvtlyaryingbeuinRndersto.Theodasmatrixaninmtegration.ultiplicationAinvdiscretizationolvingtheofspatialspace,coasisordinusualate
innumerics,leadstoatruematrixformulation.
in[In101,108addition,],isthecocalculatedvariancetobematrixC(R)ofthereceptorresponseRasdescribed,e.g.,
C(R):=(R−R)(R−R)T
=µs2σ2I+τ2HHT(5.25)

(5.25)

97

5.Optimalityinmono-andmultisensorymapformation

wherethesuperscriptTdenotesthematrixtransposeandItheidentitymatrix.Equa-
tion(5.23)nowsimplifiesto
MLT=Hwiththe‘modelmatrix’M:=µs−2C+HHT.(5.26)
GiventhatMisaninvertiblematrixthesolutionforLturnsouttobe
TL=M−1HT=HTµs−2C+HHT−1.(5.27)
Thisequationgivesauniquesolutionfortheoptimalreconstructionforanygivensetof
transferfunctionsandnoiseconstants(σ,τ).Theestimatedsignalcanbecalculatedfrom
themeasuredresponsevectorRusing(5.13)inmatrixformas
Sˆ=L∙R.(5.28)
5.3.4Relationtothemaximum-likelihoodapproach
Thechallengeofsignalreconstructionhasalongtradition,and,accordingly,theabove
formalismshouldincorporatemethodsthathavebeenestablishedinthisfield.The
followingdiscussion,inparticular,relatesthepresentedmodeltomethodsbasedonthe
maximum-likelihoodanalysis[101,108].Withinthemaximum-likelihoodschemeone
computesthestimulusthatisthemostlikelyonegivenasetofdetectorresponsesR.
Experimentshaveshownthatoptimalornear-optimalstimuluscombinationscanindeed
describeseveralphenomenaofsensoryprocessing[1,45,84,93,129,158].Amethodof
optimalstimuluscombinationlikethemaximum-likelihoodapproachisthereforehighly
relevanttoneuronalinformationprocessingandoughttobeincludedinthepresentmodel.
Themaximum-likelihoodapproachtriestofindthemostprobableinputsignalSgiven
thedetectorresponsesR,aknowntransferfunctionH,andnoaprioriknowledgeabout
thesignal(σs=∞).Thefollowinganalysisassumesalinearrelation
R=HS+χ(5.29)
withχrepresentingthenoise.ThenoisefollowsaGaussiandistributionwithzeromean
andthestandarddeviationσχ.Themaximum-likelihoodmethodminimizesthenoise
χ.Thatis,basedonthefundamentaldefinitionsofBayesianstatistics,itmaximizesthe
1conditionalprobabilitydensityfunction
p(R|S)∝exp−2σχ2(R−HS)T(R−HS)(5.30)
withrespecttothesignalS.Thisleadstoalinearsystemofequations
S=HTH−1HTR.(5.31)
=:LML
Usingontheothersidetheaboveassumptionsforthepresentedmodel,viz.,σs=∞,
η=0,andξ=0,Eq.(5.26)reducesto
HHTLT=H.(5.32)

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5.Optimalityinmono-andmultisensorymapformation

Figure5.4:Neuronalrealizationofoptimalstimulusreconstruction.Thesensoryresponseof
thedetectorneurons(herehaircellslabeledbyi)toanexternalsensoryobjectisprojected
ontomapneurons.Thesynapticconnectionshavetobeadjustedinsuchawaythatthe
network“inverts”thephysicalmappingofthesignaltothedetectorresponse.Thatis,the
synapticconnectionshavetorepresenttheinversetransferfunctionslix(t).Tothisend,
eachsensorconnectstoseveralmapneurons.Spatialprocessingisthengovernedbythe
topographicstructureofthenetwork,thatis,whichdetectoriisconnectedtowhichmap
neuron(encodingthelocationx).Temporalprocessingisdeterminedbythedistributionof
delayswithinthesetofconnections.Togethertheinversetransferfunctionslix(t)canbe
reliablyrepresentedintheneuronalnetwork.

Totestwhetherthetwofiltersareequal,LMLisinsertedinto(5.32).Applicationofthe
transpositionrulesshowsthatwiththeusedassumptionsitisL=LML.Underthemade
assumptionsthetwostrategiesarethereforeidentical;fordetailssee[101,108,182,190].

5.3.5Neuronalrealizationofthemodel
Thegeneralmathematicalalgorithmofoptimalstimulusreconstructionisnowtranslated
intoaconcreteneuronalcontext.Onethereforehastoverifyfirstwhethertheassumptions
madeinSec.5.3.2arefulfilledinneuronalprocessing.Thatis,neuronalquantitiesand
quanfunctionstitiesofandoptimfunctionalsmapwithinformationthephhavysicaletobemappingself-averprocessagingare.Asindeedmenself-ationedvberaging;efore,
seeAppendix5.B.Fortheneuronalsidethisassumptionisaswellfulfilledsinceonthe
onehandfiringofneuronsiscorrelatedwiththeself-averaginginput,andneuronalnoise
Thecanbedescribmathematicaledbyaframewstocorkhasticcanprocopceses,withe.g.,anayGaussiandistributionone;ofasneuronaldiscussedinnoiseaasminlongute.
xleasarnethedmeansynapticiszero.connectionsOnthebetotherweenhandthethemapsoptimalassoinciatedversewithdtransferifferenfuntcmotionsdalitilie(st)andare
hencereflectpropertiesoftheunderlyinglearningprocess.Effectivelearningisslow
becauseitneedsmanyindependentrepetitions.Accordinglytimescalesforlearningand
individualrealizationsofanexternalsignalcanbeseparated.Inotherwords,learningis
aproself-acessv[109eraging].Inprocesconclusion,swheretheonlyconditionsaveragedneededquantotitiesexploientterthebythevmathematicalerynatureframewoforkthe

99

5.Optimalityinmono-andmultisensorymapformation

(cf.Sec.5.3)arefulfilled.
Consequently,atranslationoftheinversetransferfunctionslix(t)intoneuronalhardware
ispossible.Inaneuronalarchitecture,theactualprocessingisperformedbythesynaptic
connectionsbetweendetectorsandneurons.Themathematicalframeworkalreadytakes
intoaccountthediscretecharacterofdetectorsandmapsthroughadiscretenumberof
inversetransferfunctions.Furthermore,thediscrete,“spiky”characterofresponseand
reconstructionbytheneuronalrealizationisalreadytakencareofbythenoisetermsχi
andλix.Spatialprocessingisthengovernedbythetopographicstructureofthenetwork,
thatis,whichdetectorisconnectedtowhichneuron.Temporalprocessingisdetermined
bythedistributionofdelayswithinthesetofconnections.Thatis,anumberofdendrites
withappropriatedelaysarechosentosamplethetimecourseoflix(t)inasufficientway[54].
Theresponseofthemapneuronisshowntoberobustwithrespecttothesamplingmethod
ofthetemporaldelays[137].
Figure5.4showsanexampleofthewholeneuronalsetup.Asillustratedafeedforward
networkofexcitatoryandinhibitoryconnectionscanimplementthepresentedunified
frameworkandcanformaunimodalmapfromarbitraryinput[54,137,198].Thequestion,
however,remainshowsuchaconnectivitypatternisestablishedinarealbiologicalsystem.
Franoschetal.[52]andFriedeletal.[56]haveshownthatcorrectsynapticconnectionscan
belearnedfromateachersuchasthevisualsystem.Non-visualmapscandevelopbymeans
of(supervised)STDP;fordetailsseeSec.5.5.2.Bymeansofthepresentmathematical
frameworkthelearnedconnectivitypatterncanbecomparedwiththeoptimaloneasgiven
byEqs.(5.23)and(5.27).
Ameaningfulcomparisonofthemathematicallyoptimalnetworkarchitecturewithan
actualbiologicalsetup,though,maynotbestraightforward.Inrealbiologicalsystems,
errorminimizationasinEq.(5.14)–thatis,realizingtheoptimalconnectivity–maynot
bepossiblebecauseofneuronallimitations.Themathematicalframeworkcannevertheless
copeforthelimitedneuronalaccuracybyreducingtheerroronlybelowacertainerror
threshold,whichmayevenvaryinspace.Abiologicalexampleofnon-uniformstimulus
reconstructionarevisualfieldsbeingsampledwithdifferentspatialandspectralresolu-
tions[92,200,239].Suchafocusonspecificspatio-temporaldomainscanmathematically
berealizedbyintroducingapositiveweightingfunctionintotheintegralofEq.(5.14).
Accordingly,whenreducingtheglobalerrorbelowacertainthreshold,theareaswithinthe
focusoftheweightfunctionhavetoreachahigherlevelofoptimization,i.e.,ofresolution,
rest.thethanTakentogether,theformalismofoptimalmapformationiscapabletodeliverconcrete
neuronalconnectivitypattern,justasillustratedinFig.5.4.Asaconsequence,the
mathematicalframeworkcanmakedirectforecastsofthefiringactivityofconcretemaps.

5.4Arecipeofmakingmaps

Intheprevioussectionoptimalconnectivitypatternbetweensensorysystemandmapwas
Tcalculatedobring[Fig.tolife5.3theandEq.(mathematical5.23)]andframewrealizedorkofSec.neuronally5.3,[cf.aneasyFig.5.4step-b].y-step“recipe”
providestheoptimalconnectivityinarealisticbiologicalsetup:

100

5.Optimalityinmono-andmultisensorymapformation

x•itoFirst,athestimtranulussferpulsetfunctionhatohi(t)ccurredissptecifiedtimeunitsthatagodeterminesatpositiontherespx.onseofthedetector
•Next,theFouriertransformHixofthetransferfunctionhix(t)iscalculated.
•Vassumedaluesoftoτbeandmucσharecsmallerhosenthanina1sforuanitableyway.measurableThesignal.noise-to-signalIncontrast,ratioστneedscanbtoe
beestimatedindependenceuponthesituationathand[53,54,198].
•Theentriesofthe−1modelmatrixMijarecalculatedbyEq.(5.26).Inversionofthe
.toleadsmatrixM•TheinvertedmatrixM−1ismultipliedbythematrixHsoastofindtheinput
T.strengthsconnectionL•Finally,transpositionxandinverseFouriertransformationofLTprovidetheinverse
transferfunctionsli(t).

ydalitMultimo5.5InderivingthemathematicalframeworkinSec.5.3allsensorymapsarepurelymonosensory.
mInultimospiteofdal,thissensoryassumption,systems.Tanimalsofullandyhaccessumanstheperceivcompleteetheirinfenvirormationonmeofntallthroughmonosenseversoraly,
maps,theirinformationhastobecombined.Physiologicalandbehavioralexperiments
showthatthemonosensoryperceptionsarenotindependentbutmutuallyinteractwith
i.e.,eachmapsother[1,receiving11,46,input197].fromAcommorebinationthanoneleadssensorytothesystemformation[106,of158].multisensorymaps,
orevAennumbhighererofbrainbrainareas,areas,susucchhasasthetheAnmidbrainteriorinEctosylvianmammals,moreSulcusprecisely(AES)[25the],SCcon[204tain],
clearlydistinctmonosensoryaswellasmultisensoryneurons.Sincetheperceptualand
behaexamplevioralformroleofultimodalhigherinbrainteractionareasintheremainsfollovawing.guethewell-studiedSCisusedasan
TheSCfeaturesalayeredorganizationofspatialmapsfromallsensorysystemsthat
disposeoftopographic,map-likeinformation(suchasvisionbutnotolfaction)[204].All
thesecommonmaps,referuni-enceormsystemultisofensorysensory,aremspace.utuallyaligned[18,112,204]andthusprovidea

teractionindalMultimo5.5.1Ingeneral,therearetwocategoriesofmultimodalinteraction:integrationandpoolingof
information.

Integratedtegration.intoasingleCongruenmergedtspatialand,hence,informationmultimofromdaldifferenmap.tSuchsensoryaninsystemstegratedcanmap,beasin-
comparedtotheunimodalmaps,featuresincreasedinformationreliabilityandsaliencyand
animprovedsensitivityinbothspaceandtime[77,130,131,184].Forexample,ifvisual

101

5.Optimalityinmono-andmultisensorymapformation

andauditorysensorysystembothregisterasignal,e.g.,“brownahead”and“barking
ahead”,itisveryprobablethatthesignalcorrespondstoanactualobjectratherthanto
asensoryartifact.Atthesametime,theintegratedsignalisstrongerandallowsforfaster
reactions(e.g.,“escape!”).Insomecasesanintegratedsignalisevenoptimal[74,158].
Despiteincreasedreliabilityoftheintegratedmap,itsindividualinputstreamscannot
bedistinguishedanymore.Thatis,theinformationofwhichmonosensorymaphasdeter-
minedthepositionislost.Withintheaboveexamplethemultimodalmapmayindicate
amultimodaleventaheadbutthetriggeringmodality,forinstance,visionoraudition,
ed.unresolvremainsChapter6providesadetaileddescriptionofmultimodalintegration.Furtherneuronal
modelsbasedonstatisticalmethodshavebeenpresentedelsewhere[40,41,42].Concrete
theoreticalmodelsofmultimodalintegrationwithintheSChavebeendevelopedaswell
[2,142,171,185,186,214].

Pooling.Withinacommonreferencesystemallmonosensoryaswellasmultisensory
mapsarepresentinparallel.Sinceallmapsarealignedasimultaneousaccesstothe
diverseinformation,thus,signalcharacteristics,ispossible.Consequentlyanobjectatone
specificpositioninspace-timecanbeidentifiedandcharacterizedbythislocationinorder
toselect,e.g.,motorresponsesinacomplexenvironment.Forexample,arattlesnake
maydetectspatialcoherentactivityinitsvisualand/orinfraredmap.Onlyiftheencoded
objectisvisibleandwarmitwillbeidentifiedasalivingobject,inotherwords,apossible
prey.Ifitisvisibleandcoldthesnakewilldiscardtheinformation.Experimentalevidence
forsuchapoolingofinformationisprovidedbyneuronalANDandORprocessingsteps
forthecombinationofvisualandinfraredmap[164,165].Theseprominentexamplesof
poolingintheSCcouldenabletargetselectionandensureappropriatemotorcommands
inacomplexenvironment.

Insummary,integrationofinformationallowsforareliablespatialdeterminationof
anobject,thekeytaskofobjectformation.Poolingofinformationassuresanaccessto
thedetailsofanobjectnecessaryforobjectidentification.Switchingbetweenintegration
andpoolingcorrespondstoaswitchbetweenparallelandserialdataprocessingtobestfit
tasks.tdifferen

5.5.2Developmentofmultisensoryspace
Toenableefficientmultimodalinteractionsuchasintegrationandpooling,alignmentof
thedifferentmono-andmultisensorymapsisofcrucialimportance.Onlythencana
multimodalstimulusataspecificspatiallocationbeidentified.Analignmentofsensory
maps,however,isnotpresentatbirthandmustbelearned[120,205,217].Thefollowing
sectionandChap.6discussthequestionofhowacommonmultimodalspacecanevolveat
all.Anobviousstrategywouldbethedeterminationofonedominantmodalityasreference
forallothermodalities[120,122].Modificationsofthisreferencemapwouldthenauto-
maticallyleadtomodificationsofallothermaps.Experimentalandphysiologicalstudies
haveshownthat,inmanyanimals,destructionordisturbanceofthevisualpathwayleads

102

5.Optimalityinmono-andmultisensorymapformation

todisorganizedandabnormalsensorymapsinnon-visualmodalities.Thesefindingshave
beenobtainedinhamster[157],cat[224,226],clawedfrog[30],ferret[115],barnowl[123],
andsnakes[70].Psychophysicalexperimentswithcongenitallyblindandnormallysighted
humanshaveshownthatvisualinputearlyinlifeisnecessaryformultimodalinteraction
tooccur[90,176,180].Consequentlyvisionseemstobethedominantguidingmodality,
i.e.,the“teacher”,fornon-visualmodalities.
Aplausibleargumentsupportingthisideaistheintrinsictopographicorderofthe
retina.Itisknownthatlayersofneuronscanself-organizeintotopographicmaps,provided
thatinitiallyasmallsetofcorrectlyorganizedneuronsexists[213,232].Togetherwith
thesubsequentdevelopmentofdeeperlayersinthevisualcortex(formice,see[100])the
intrinsictopographyoftheretinacouldstep-by-stepdictatetheorganizationandalignment
ofhighervisualand,potentially,alsomultimodalmaps.
Anexamplewherethealignmenthasbeenstudiedindetail,bothexperimentallyand
theoretically,isaudio-visualintegrationwithintheSCofthebarnowl.Hereexperiments
[94,122]haveshownthattheauditorymapfollowssystematicchangeswithinthevisual
input.Thegeneralmechanismfacilitatingsuchanalignmentofmapsissupervised
spike-timing-dependentplasticity(STDP)[6,7,36,66,109,149,199,216,240].The
precisenatureoftheteachingsignalhasnotbeenclarifiedexperimentallybutselective
neuronaldisinhibition,orgating,seemstoplayakeyrole[75,233].Theoreticalstudies
haveconfirmedthatexcitatoryandinhibitoryteachinginputcanaccountforpropermap
formationandthusdevelopmentofmultimodalspace[38,56].Itis,however,onlyby
inhibitoryteachinginputthatanalreadyalignedmapcanbere-alignedlateron[56].
Insummary,theabovestudiessupporttheideaofvisionasteachermodalityto
alignothermonosensorymapsbuttherearecontradictingfindingsaswell.Ithasbeen
shownboththeoreticallyandexperimentallythat,althoughimprecise,amapofazimuthal
soundlocationcanbelearnedwithoutanyvisualinput[111,122]thoughadmittedly
onageneticallydeterminedsubstrate.Inaddition,non-visualmodalitiescaninfluence
eachotheraswell,e.g.,auditioncaninfluencehaptics[11].Moreover,somatosensory
mapsalreadysharpeninapostnatalphasewhenonlyauditorybutnovisualneuronsare
present[223,225].Behavioralandpsychophysicalstudiesshowthatvisualperceptioncan
evenbeinfluencedbyothermodalitiessuchashaptics[46]oraudition[55,197,201].More
importantly,visionitselfcanimprove,respectivelysharpenasfoundinthevisualsystem
ofyoungcats[223,225].
Takentogether,thepresentedexperimentalandtheoreticalfindingsputintoquestion
thecurrentpictureofvision-guidedmapalignment[113].WallaceandStein[225]have
pointedoutthatthedevelopmentofdifferentmodalitiesstartsinparallelandintemporal
coincidencewiththeappearanceofmultimodalintegration.Theyherebysuggestacommon
mechanismdrivingbothmapdevelopmentandmultimodalintegration.Aconceptof
integratedmultimodalteachingthatisbasedontheseconsiderationsispresentedin
.6Chap.

103

5.Optimalityinmono-andmultisensorymapformation

Discussion5.6

Insummary,sensoryprocessingisbasedonthreegeneralconcepts,denotedasthe‘golden
three’ofsensoryprocessing:physicalmapping,optimalmapformation,andmultimodal
integration;cf.Fig.5.1.Giventransferfunctionstodescribehowasignalstimulates
adetector,theformulatedmathematicalframeworkisabletoquantifyhowthedetec-
torresponseisprocessedsoastoleadtoa“reconstruction”oftheoriginalsignal.In
thecontextofneuronalinformationprocessingtheframeworkextendstheestablished
maximum-likelihoodmethodbylinkingitsparameterstoeasilyaccessibleexperimental
quantities.Themathematicalprincipleofstochasticoptimalityleadstoadiscreteand
optimalrepresentationoftheoutsideworld–amap.
Mostimportantly,themathematicalsetupcanbetranslatedintoneuronalarchitecture.
Thatis,adiscretizationinspace-timeofthemathematicalmodelallowsforaderivation
ofsynapticconnectionpatternsbetweendetectorandmapneurons.Toillustratethe
relationtorealbiologicalsettings,astep-by-steprecipeoffersthepossibilityofapplyingthe
mathematicalframeworktoconcretebiologicalsituations.Thegeneralityofthemethodof
optimalmapformationcannowbetestedtomodelandanalyzeexperimentalresults.In
particular,themeasurementofinternalconnections,forinstance,intheSC,andfiring
profilesforspecificsensorysystemswouldprovideapossibilitytoexperimentallyaccess
theinversetransferfunctionsasdefinedinSec.5.3.1.
Basedontheunderstandingofmonosensorymapformation,multimodalinteractionand
thedevelopmentofmultisensoryspacehavebeendiscussed.Heretheconceptofneuronal
mapsascomparedtosingleneuroneffectscandelivernewperspectivesonmultimodal
interaction,viz.,integrationandpoolingofinformation.Whileintegrationofinformation
allowsforareliablespatiallocalizationofanobject,poolingofinformationassuresthe
accesstothedetailsofanobject.Poolingofinformationcouldbethereasonastowhy
multiplemapsarefoundintheSCinsteadofasinglemultimodalmap.
Finally,theimportanceofpropermapalignmentformultimodalinteractionwas
emphasized.HereSTDPlearningalgorithmswithaninhibitoryteachersignalcanaccount
forbothinitialmapformationandevensubsequentre-alignmentofmaps.Further
experimentalstudiesoninhibitoryteacherinput,e.g.,withintheSC,arenevertheless
neededtoclarifythepreciseroleofinhibitioninthealignmentprocess.Inaddition,such
experimentscouldanswerthecrucialquestionofwhichsensorysystemsdeterminethe
formationofmultimodalspace.
Inotherwords,adeepunderstandingofhowmultimodalinteractionisrealizedand
establishedatananatomicallevelrequiresmoreexperimentalandtheoreticalstudies.For
example,throughwhichconcretemechanismcouldacollectionofalignedmapsallow
forthepoolingofinformation?Doessuchamechanismalsoincludefeatureselectionin
acommonsensoryspace?Howdoesmultimodalinteractionofmapscontributetothe
formationofacommonsensoryspace?Andtowhatextentdoessuchafindingcontradict
thecurrentpictureofvision-guidedmapalignment?Finally,towhatextentdoesthe
mathematicalframework,thathasbeensubstantiatedsofaronlybyfindingsintheSC,
applytootherareasofthebrain,suchastheAES,awell-definedmultisensorycortical
cats.inedobservareaSomeofthesequestionsareaddressedinthefollowingchapterwhereaconceptof

104

5.Optimalityinmono-andmultisensorymapformation

multimodalteachingispresented.

endixApp

5.ANonlinearitiesininformationprocessing
Thepresentedmodelassumesalinearrelationbetweenstimulusanddetectorresponse.For
anumberofsensorysystems,however,non-linearitiesappearduringsensoryprocessing.
First,thetransferfunctionhcanbeanon-linearfunctionh˜.Second,theneuronaldetector
responsecanbenonlinear,typicallylogarithmic[34,102,132,134,166].Incaseofa
nonlineartransferfunctionandalogarithmicresponsethedetectorresponsesr˜i(t)haveto
tberewrittenfrom(5.1)as
r˜i(t)=logdxdτsx(τ)h˜ix(t−τ).(5.33)
−∞spaceallToneverthelessapplythepresentedmodelanincorporationofanadditionalcomputational
stepcaninafirststepcancelthelogarithm.Inabiologicalsystemthiscanberealized,
e.g.,byneuronswithexponentialfiringbehavior.Assumingsuchaneuronalstepthe
detectorresponser˜i(t)remainswithanonlinearh˜ascomparedto(5.1).Alinearizationof
thenonlineartransferfunctioncanbeachievedbyaredefinitionofthesignals→s˜.That
is,appropriatecharacteristicsofthestimulusareidentifiedthatarelinearlyrelatedtor.
Forexample,insteadoflookingattheheatdistributionT(x,t)theintensitydistribution
ofthecorrespondingradiation∼T4(x,t)canbeconsideredduetotheStefan-Boltzmann
law.Together,theincorporationofanadditionalcomputationalstepandthereasonable
redefinitionofdetectorresponseandsignalallowforanoptimalstimulusreconstruction
bymeansofthepresentedlinearframework.

eragingvSelf-a5.BWhstandycanthisphvysicalaluablebutpropnoisyertyainputdetectorquantitiesreceivbeseexpinputectedsignalstobfeiswithelf-av1≤ieraging?≤NToasaunsumder-
NηgetNaide=1cenfitwherescalingηNbisehaaviorscalingasNfactor.increases.FortheMoreosakveer,officonarevstoceniencehasticηNisrandsetomtov1/ariablesNto
pwithositionsmeaninaispace,andinfinitebivologicalariance.realitFinallyy,theifthestoc1≤hastici≤Ncorrelationrepresent,betforweenpexample,ositionsdifferenthatt
arefarapartissmall.Hencefi=ai+φiwherethenoisetermsφi(cf.Fig.5.3)aretaken
asindependentrandomvariablesthatbyconstructionallhavezeromean.Forinputsof
theformNNN
fi=ai+φi=:AN+ΦN(5.34)
i=1i=1i=1
thedetailsnumseeb[er216A],NisAppendixdeterministic.A)applies,soRegardinastogΦNconcludethethat,strongaslaNw→of∞,largeitisnηumNbΦersN→(for0

105

5.Optimalityinmono-andmultisensorymapformation

isindepneededendenintofthepracticalspewcificorksincerealizationoneofnevtheer{knoφi}.wsThethelatterrealizationcircumsuntiltanciteisisalloexactlyver.whatThe
stronglawoflargenumbersguaranteesthatΦNvanishesasNbecomeslarge.
Theonly,minor,drawbackofallthisistwofold.First,inrealitytheφiarenever
perfectlyindependent.Nevertheless,aslongascorrelationsfallofffastenoughasthe
distance|i−j|becomeslarge,thestronglawoflargenumbersstillholds.Second,in
practicalworkNisandremainsfinite.Thenthecentrallimittheoremasdescribedin
AppendixAof[216]showsthatforarbitraryindependentφiprovidedthesecondmoment
φi2isfiniteandNlarge
N1√Ni=1φi(5.35)
hasaGaussiandistributionwithmeanzero.Thestandarddeviationgivesinformation
aboutthewidthoftheGaussian.Thesameholdstrueforweaklydependentφiasfound
inbiophysicalreality.

5.CRemainingderivationstepsleadingto(5.23)
Thefollowingsectionelaboratessomestepsskippedinthederivationof(5.23).Here
ideasduetothecalclulusofvariations[105]apply.Equation(5.15)isusedasstarting
conditiontominimizetheexpectationxvalueofthequadraticerrorwithrespecttothe
optimalinversetransferfunctionslj(t).Thisleadsto
∂[sx(t)−sˆx(t)]2
∂ljx(t)=0
x⇔sx(t)−sˆx(t)∂sˆx(t)=0foreveryj.(5.36)
∂lj(t)
Tosolve(5.36),theestimatesˆx(t)isexpandedusingEqs.(5.12)and(5.13)giving
sˆx=χilix+χiλix+dysyhiylix+syhiyλix+syηiylix+syηiyλix
i+ξyhiylix+ξyhiyλix+ξyηiylix+ξyηiyλix.
(5.37)Variationofsˆleadsto
∂sˆxx(t)=χj+dysyhjy+syηjy+ξyhjy+ξyηjy(0).(5.38)
∂lj(t)
Asbefore,allnoisetermsaswellastheexpectationoftheinputarestochastically
independentofeachother.Allnoisetermshavezeromean.Withtheseassumptions,the
xexpectationvaluess∂ˆs/∂landsˆ∂ˆs/∂lfrom(5.36)canbewritten
sx(t)∂sˆx(t)=dysx(t)(syhjy)(0)(5.39)
∂lj(t)

106

5.Optimalityinmono-andmultisensorymapformation

andcombining(5.37)and(5.38)

sˆx(t)x=(χilix)(t)χj(0)+dydy(syhiylix)(t)(syhjy)(0)
∂sˆx(t)
∂lj(t)i
+(syηiylix)(t)(syηjy)(0)+(ξyhiylix)(t)(ξyhjy)(0)

+(ξyηiylix)(t)(ξyηjy)(0).
(5.40)

Toillustratethecalculations,whichsimplify(5.39)and(5.40),twoisolatedtermsfrom(5.40)
areconsideredasconcreteexample.Theothertermsaretreatedinasimilarway.First
considerdydy(syhiylix)(t)(syhjy)(0)
i(5.41)
=dydydτdτdτsy(t−τ−τ)hiy(τ)lix(τ)sy(−τ)hjy(τ)
i

whereExploitingthethedefinitionautoofthecorrelationconvforolutionthe(signal5.3)(and5.18)thethiscompexpressionositionbrule(ecomes5.10)areapplied.

µs2dydydτdτdτδ(y−y)δ(t−τ−τ+τ)hiy(τ)lix(τ)hjy(τ)
i=µs2dydτdτhiy(t−τ+τ)lix(τ)hjy(τ)
i=µs2dy[(hiylix)◦hjy](t)
i

(5.42)

(5.11where).thFoelastcusingstepontheresultsthirdfromtermtheinthecomprighositiont-handrulesideofofcon(v5.40)olutionleadsandinaautosimilarcorrelationwayto

dydy(syηiylix)(t)(syηjy)(0)
i(5.43)
=dydydτdτdτsy(t−τ−τ)ηiy(τ)lix(τ)sy(−τ)ηjy(τ)
i

107

5.Optimalityinmono-andmultisensorymapformation

(5.44)

(5.45)

(5.46)

whichsimplifieswith(5.17)and(5.18)to
µs2ση2|y|<ymaxdydydτdτdτδ(y−y)δ(t−τ−τ+τ)
i0<τ<tmax
δijδ(y−y)δ(τ−τ)lix(τ)
(5.44)=µs2ση2|y|<ymaxdydτdτδ(−τ+t)ljx(τ)
0<τ<tmax
=µs2ση2|y|<ymaxdydτljx(t).
max<t<τ0Takentogether,thefinalexpressionsfortheexpectationvaluesbecome
xsx(t)∂sˆx(t)=µs2hjx(−t)(5.45)
∂lj(t)
andxsˆx(t)∂sˆx(t)=σχ2ljx(t)+ση2(µs2+σξ2)maxdydτljx(t)
∂lj(t)0|y<τ|<y<tmax(5.46)
+(µs2+σξ2)dy(hiylix)◦hjy(t).
iEquation(5.36)thereforetransformsto
ljx(t)σχ2+(µs2+σξ2)|y|<ymaxdydτση2
0<τ<tmax(5.47)
+(µs2+σξ2)dy(hiylix)◦hjy(−t)=µs2hjx(−t).
iInsertingtheparameterσandτandapplyingaFouriertransformationfinallyleadsto
).5.23(Eq.Totestwhethertheobtainedextremumisaminimum,thesecondvariationiscalculated,
readshwhic∂2[sx(t)−sˆx(t)]2∂x∂sˆx(t)x∂sˆx(t)

∂lj(t)jjj
x2=2∂lx(t)s(t)∂lx(t)−sˆ(t)∂lx(t)
=2σ2χ+ση2(µs2+σξ2)maxdydτ+(µs2+σξ2)dydτhjy(τ)2.
0|y<τ|<y<tmax
(5.48)Sincetheappearingsquaresandintegralsarepositive,soisthesecondderivativeandthus
theextremumisaminimum.

108

5.Optimalityinmono-andmultisensorymapformation

signalblurredGaussian5.DThissignal.Assubsecin(tion5.19),presenatsrealisticansignalequationwouldequivfulalenfillttosome(5.23kind)ofbutforGaussianaGaussianrelationblforurredthe
aluevectationexp22
sx(t)sx(t)=Aexp−|x−2x|exp−|t−2t|.(5.49)
2σsx2σst
Forthiscaseaderivationofanequationlike(5.23)ispossibleanalogouslytoAppendix5.C.
SinceforthesignaltheGaussiancorrelations,however,replacethedeltafunctions,e.g.,
in(5.42)and(5.44),integralsoverspaceandtimecannotbeevaluateddirectly.Instead
theycanonlyberestrictedtotheregionwheretheGaussianisnon-negligible.Denoting
thesetemporalandspatiallimitsbytand,theanalogueto(5.23)isderivedas
22
ddtAexp−2|σ|2exp−|2σt|2hjx+(t+t)
sxst
=σχ2ljx(t)+ση2σξ2|y|<ymaxdydτljx(t)+σξ2dy(hiylix)◦hjy(t)
imax<t<τ02t2x
+ση|y|<ymaxmaxdyddτAexp−2σs2tlj(t+t)
<t<τ022
+dyddtAexp−||2exp−t2(hiylix)◦hjy+(t+t).
i2σsx2σst
(5.50)Theeffectoftheadditionalspatio-temporalintegralsascomparedto(5.23)isasmoothening
ofthefinalreconstruction.Notonlyisthevalueataspecificpointinspaceandtime(y,t)
takenintoaccountbutneighboringpointsinanearlyareasurroundingitareincludedas
ell.w

109

6.

Wastrifft,diesoAgeschiehtbbildungdereineRsolcheaumverh¨alleraltnissedingsbane-
denperipherischenNervenendenimAuge
Grundade,anabdererdochtastendennurinHautbineschr¨einemanktergewissenWeise,
dadasAugenurperspectivischeFl¨achenabbil-
dungengiebt,dieHanddieobjectiveFl¨ache
K¨auforpdererobihrerfl¨m¨acheoglichstabbildet.congruentEindirectesgestaltetenBild
Reineraumgr¨nachossedreigiebtwederDimensionendasAugeausgenochdehntendie
Hand.ErstdurchdieVergleichungderBil-
derbeiderAugen,oderdurchBewegungdes
K¨orpers,beziehlichderHand,kommtdieVor-
stellungvonK¨orpernzuStande.
p.445],85[Helmholtz

MultimoCalibrationdalofmapneuronalformation:maps
tegratedinthroughMultimodalTeaching(iMT)

Forneuronalprocessingofsensoryinformationandmotorcommands,spatiallyaligned
uni-andmultimodalmapsplayakeyrole.Theprecisealignmentofmaps,however,is
notpresentatbirthbuthastobelearnedduringthedevelopmentofananimal.Here
theunifyingconceptofintegratedMultimodalTeaching(iMT)isintroducedthatisbased
onsupervisedspike-timing-dependentplasticity(sSTDP)andmultimodalintegrationto
calibrateinputsfromdifferentsensorymodalities.Analyticalcalculationsandnumerical
experimentsdemonstratethatamultimodalteachercanensureproperintrinsicmap
formationandalignment.Newinterpretationsofexistingexperimentsarepresented
andthedominanceofvisioninguidingmapalignmentisdemonstratedtobeanatural
consequenceoftheexceptionalprecisionofthevisualsystem.Finally,newexperimental
setupsthatcanshedlightonandclarifyintrinsicdevelopmentofanalignedmultisensory
suggested.arespace

ductiontroIn6.1prTheeciseabilitinytoteractionactinofathedynamicdifferentandsensorycomplexmoendalitiesvironmenthattformreliesafirstcongruenandtpforemosterceptionontheof

111

dalMultimo6.formationmap

Figure6.1:IllustrationofintegratedMultimodalTeaching(iMT)asfundamentalconcept
forunimodalmapformationandalignment.Allunimodalmodalitiesareintegratedinto
anintegratedmultimodalmap(grayarrows).Theactivityofthemultimodalmapthen
inducesaninhibitoryteachingsignal(blackarrows)thatleadstoanadaptationofthe
unimodalmapsbymeansofsupervisedspike-timing-dependentplasticity(sSTDP).The
detailedmodelsetupemployedinthischapterisexplainedinSec.6.2.2.

theexternalworld.Thispreciseinteractionofthedifferentmodalitiesisnotpresentat
birthbuthastobelearnedduringthefirstperiodofsensoryexperienceinlife[213,226].
Theprocessofsensorycalibrationiscommonlyseenasvision-guided;cf.King’s
review[113].Severalremarkableexperimentsshowthatdestructionordisturbanceofthe
Thesevisualpathfindings,wayobleadstainedtoindisorganihamsterzed[and157],catabnormal[224,sensory226],clamapswedinfrog[non-visual30],ferretmo[daliti115e],s.
barnowl[123],andsnakes[70],supporttheideaofvisionasteachingmodality.However,
vision-guidedmapcalibrationisnotasevidentasitseemsatafirstglance.First,sensory
maps,althoughimprecise,canformwithoutanyvisualinput[111,122,213].Experiments
onthevisuallyauditorydeprivandedanimalssomatosensoryandhumanssystemshavcaneshodevwnelopthatnormally;despiteasee[missing117,177visual,181]system,for
details.Second,behavioral[10,46,55,197,201]andphysiological[158]studieshave
shownthatvisionisnotstaticbutcanbeinfluencedbyothermodalities.Third,and
mostimportantly,WallaceandStein[223,225]havepointedoutthatnon-visualsensory
mapspresenint.Fthesupurthermore,eriortheycolliculushave(SC)observalreadyedthatsharpallenmoattimesdalities,whereincludingnovisualvision,neuronsdevelopare
inatemporallycoincidingway.Butifvisionactsasteachingmodality,howcanitguide
calibrationwithoutbeingpresentandhowcanitimproveitself?
currenTtogether,picturetheofabovisionveasparadothexicalonlyexppredominanerimentaltandguidingmotheoreticaldality.findingsThatis,questionwhatisthea
coherentalternativetovisionasateachingmodality?
Forthefollowing,twofindingsserveasstartingpoints.First,theformationofdifferent
modalitiesbeginsatthetimeofappearanceofmultimodalintegration[225].Second,the
motor,homolog,i.e.,theamopticultimodaltectummap(OT),inthehassupbeeneriorsuggestedcolliculusto(SC)proandvideinaitsteacnhinon-mgsignalammalianfor
auditoryspacealignment;cf.[139]andFig.6.2.
Therefore,anintegratedMultimodalTeachingconcept(iMT)(cf.Fig.6.1)issuggested
respthatectcantounieacfyhconotherastradictorywellasfindingsexplainasthetohowwell-prodiffverenentmodominancedalitiesofcanthebevisualalignedsystem.with

112

formationmapdalMultimo6.

Figure6.2:SchematicdrawingoftheSCanditsconnectivity.TheSCisalayered
arrangementofaligneduni-andmultisensorymapsthatareintrinsicallyconnected.Inputs
totheSCcomefromallsensorysystemsthatprovideinformationinamap-likestructure.
TheSCproducesoutputformotorresponsesandpartiallytoinfluencethealignment
processofassociatedsensorysystems[139].AltogethertheSCisabeautifulexampleto
studyunimodalmapformationandalignment.

Theintroductionofanintegratedmultimodalteacherrequiresacarefuldiscussionofits
intrinsiccharacteristicsandimplicationsonaconceptuallevel.Herethefocusliesonmap
formationwithintheSC/OTbymeansoflong-termpotentiation(LTP)andlong-term
depression(LTD)ofsynapsesthatcanbedescribedbysupervisedSTDP.Thequestion
ofdynamicadaptationisomittedasappearinginmanyanimalsthatcanmovetheir
sensoryorgans(eyes,ears,whiskers,etc.)independentlyoftheirbody.Themechanism
underlyingthisadaptationisnotknown,butretinotopiccoordinatesystemsseemtoplay
akeyrole[61].Attemptstomodelthisissuecanbereviewedin[60,72,188,189].Within
theSC/OT,however,multisensoryandpredominantlymonosensorylayersaremutually
alignedtogainaunifiedmultisensoryrepresentationofsensoryspace[18,112,204].The
combinedsensoryinformationcanthenbeusedtogeneratedirectionalresponsesinthe
SC/OTmotormap[130,131,138,203,219].Inotherwords,atthisstageallsensorymaps
shareacommonrepresentationsystem.Similartoothertheoreticalmodels[56,234],the
suggestediMTconceptconcentratesonthequestionofmapcalibrationappearinginthe
contextoftheverycommonsensoryspace.
BesidesthederivationofthegeneralconceptforiMT,applicationsofthemodelto
concreteexamplesarepresented.Inparticular,theexperimentsofKnudsenetal.[123]
arereanalyzedindetailandcontrastedwithfindingsofWallaceandStein[223,225]in
thelightoftheiMTconcept.Finally,theconcludingsectiongivesexperimentallytestable
predictionsofthemodel.

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formationmapdalMultimo6.

6.2Theintegratedmultimodalteacher
TheiMTconceptstructurallyconsistsofastackofunimodalmaps,eachconnectedwith
themultimodalmap;cf.Fig.6.1.Inafirststep,differentunisensorymodalitiesaremerged
intothemultimodalmap.Suchamultimodalintegrationofunimodalmapswasfound
withintheexperimentallywelldescribedSC/OT[204,225].Inthenextstep,theintegrated
multimodalmapprojectsbacktotheunimodalmaps.Thus,themultimodalmapinduces
ateachingsignalthatguidesthecalibration(formationandalignment)processofthe
differentunimodalmaps.Theteacherinputthereforedoesnotoriginatefromasingle
modality,forexample,vision,butfromacommonintegratedmultimodalmap.
Ingeneral,mapcalibrationcanbedividedintotwodifferentclasses.Thefirstdescribes
thesituationinwhichsensorymapshavetobelearnedfromscratch.Connectionsbetween
sensorysystems(inputmaps)andrepresentingmapscontainhardlyanyinformationand
arerandomlydistributedorcoarselypre-wired;seeSec.1.4.Theconnectionshavetoevolve
duringthecalibrationprocessbymeansofLTPandLTDofthesynapses(inmathematical
termsdescribedbySTDP).Thiscaseisdenotedasmapformation.
Thesecondclassrequiresacollectionofalreadypre-wiredmaps(seeSec.1.4and[213])
andcorrespondstoanalignmentofmaps,forinstance,duringgrowthorinshifting
experiments[122,123].Thestimuluspositionisdisplacedincomparisontothepositionof
theteacherinput.Bymeansofactivity-basedsynapticplasticity,thesynapticpattern
ofthemapismodifiedsoastocompensateforthemisalignmentbetweenthemapand
t.inpuherteacAcarefulanalysisoftheconsequencesofapplyinganintegratedMultimodalTeaching
(iMT)signalrequiresanunderstandingofmapformationandalignmentfromaprocedural
pointofview.Ingeneral,theiMTprocess(asshowninFig.6.3)constitutesaclosed
feedbackloopthatconsistsofthreerepeatingsteps.First,duringmultisensoryintegration,
differentsensorymodalitiesareoptimallymergedintoanintegratedmultimodalmap.
Second,duringmultimodalteaching,theintegratedmultimodalmapinducesaninhibitory
teachingsignalthatguidesthecalibrationprocess,i.e.,formationandalignment,ofthe
differentsensorymaps.Third,duringunimodalmapadaption,theadaptingandadapted
unimodalmapsagainmodifytheintegratedmultimodalteacher.
Thethreephasesareinherentlyconnectedbymeansofthreeimportantquestions:How
dounisensorymapsdetermineiMT?HowdothecharacteristicsofiMTinfluencemap
adaptation?AndhowdoesiMTcalibratedifferentunimodalmaps?Takentogether,these
threequestionsprovideastructurefortheanalysisoftheiMTconceptandguidethrough
sections.wingfollothe

6.2.1HowdounisensorymapsdetermineiMT?
Tostarttheanalysis,theessentialtermshavetobeclarified.Asensorymodalityis
referredtoasitsneuronalrepresentationonaspatialmap[15,140,213,217].Onsucha
maponecaninterpretthefiringrateasthelikelihoodtofindastimulusattheposition
pthatopulationtheconeuronsdes[41enco,98de.,In174,this195,217manner,].Reflectingsensorythemapsstatisticsrepresentofptheositionfiringestimatorsprofile,theor
meanµcorrespondstotheestimatedpositionandthedeviationσdescribestheestimator’s

114

formationmapdalMultimo6.

Figuredifferent6.3:unimoMaindalpromapsceduralshowsaspcertainectscofiMT.haracteristics.ThemultiWhatmoaredalthesemapcascomharacteristicsbinationandof
howdotheyinfluencemapcalibrationbymeansofsupervisedspike-timing-dependent
feedbacplasticitkyloop,(sSTDP)thewherequestionthehasmtobultimoedanswaleredsignalhowservesdifferenastteachunimoingdalsignal?mapsToadaptclosetothea
commonteachingsignal.Takentogether,thesethreequestionsprovideastructureforthe
analysisoftheiMTaspresentedinthischapter.

ofpaccuracyosition.Theinestimatorstegration[1,45of,74,differen84,t93,sensory129,158maps].isthereforeequivalenttoacombination
Theexactneuronalmechanismastohowdifferentmodalitiesareintegratedintoamul-
timodalmap[41,42]isstilllargelyanopenquestion.ThefamilyofBayesiancombination
schemesneverthelessmathematicallydescribestheresultofoptimalmultimodalintegration.
Forthesakeofsimplicity,thefollowinganalysisisreducedtothemaximum-likelihood
combinationoftwomodalities,namely,visionVandauditionA.Thecombinationofthree
modalitiesisdescribedinAppendix6.B.Thevisualandauditorymapsarecharacterizedby
theirmeans,i.e.,thepositionestimates,µV(vision)andµA(audition).Furthermore,the
accuracyofthepositionestimatesisgivenbyσV(vision)andσA(audition)asillustrated
.6.4Fig.ybtempTheorallymultimocoincidingdalpvisualositionandestimateauditoryµMasinputsaisgimaximvenbum-liky[101eliho,o108d]combinationfrom
22σσµM=σ2+Vσ2µA+σ2+Aσ2µV.(6.1)
VAVAAsaconsequenceofoptimalintegration,themultimodalestimatorfeaturesthesmallest
possiblestandarddeviationσMasgivenby
σM=σAσV/σ2A+σV2.(6.2)
Foradetailedderivationofthetwoaboveequations,pleaseseeAppendix6.A.
ThedistancebetweenthemultimodalpositionµMandtheunimodalpositionsµA/V
bymeansof(6.1)arecomputedto
|µM−µA/V|=σ2A/V|µA−µV|/σA2+σV2.(6.3)

115

formationmapdalMultimo6.

Figure6.4:Multimodalintegrationoftwounisensorymaps.Thefiringprofilesofneuronal
ofmapstheforstimauulusdition,andarevision,thereforeandthemassumedultimotobdaliteyGaucansbsianeinfunctionsterpretedwithaspmeanositionµA/V/Mestimatorsand
likstandardelihoodmethodeviationd;σcf.A/VEqs./M.(6.1The)mand(ultimo6.2).dal(A)mapisinAccordinglytegrated,thebymmultimoeansofdalamapmaximshoum-ws
thesmalleststandarddeviationσM<σV<σAandisthusthemostpreciseofallmaps;
cf.profile)(6.2).is(loB)catedIfthemoreunisensorycloselytomapsthearemoreshpreiftedciseµVmo=µdalitA,ythe(bluemulptimrofile);odalseemap(6.3µ)M.(red

pThatositionis,ofthethemmoreultimodalpreciseandmodalitthereforey,e.g.,theteacvisionhinginsignalcomparisonislotocatedaudition,moreascloselyshowntoherethe
andinTable6.1.
Inthenextsteptheintegratedmultimodalmapprovidestheguidingsignalforunimodal
mapformationandalignment.Theequationsofoptimalcombination,however,do
notdescribehowunimodalmapschangeduringthelearningprocess.Mapformation
asanddiscussedalignmentinareSec.effec6.2.2ts.ofThissynapticsectionfoplasticitcusesyonfund(mathematicallyamentalchdescribaracteedrisbtiycsofSTDP),the
multimodalmapthatfollowdirectlyfromoptimalmultimodalintegration.Theunimodal
mapsthatformthemultimodalmaparepresumedtobevariableintime.Moreprecisely,
ontheonehand,thepositionoftheunimodalmapsµA/V(t)canandwillvaryduringthe
calibrationprocess.Ontheotherhand,themapscanimproveinprecisionsuchthatσA/V(t)
anddecreases.σM(t)[seeThesetEqs.wo(effe6.1)ctsandlead(6.2to)a],vinariableparticmular,ultimoduringdalmapthewithcalibrationtime-depproendencesstofµMthe(t)
maps.dalunimo

116

formationmapdalMultimo6.

Figure6.5illustratestwovaryingunimodalmapstogetherwiththeircombinedmulti-
modalmap;seeEqs.(6.1)to(6.3).Oneoftheunimodalmapschangesinprecision(A)or
ppreciseosition(Fig.(B)6.5whileA,theblueotherprofile)remains“attracts”constanthet.pFirsositiont,aofsensorythemmapultimothatdalbmap,ecomesi.e.,morethe
sSTDPlearninglearningsignal.proSensorycesses.mapsSecond,thataincreaseshiftinginsensoryaccuracymapget(Fmoreig.6.5pB,ositionallyblueprofile)stableinducesduring
ashiftofthemultisensorymaptowardthestationarysensorymap(yellowprofile).The
teacupshothinginsignthealconandtextwillofmapthereforealignmenhardlytisadaptthatataall.morestaticsensorymapattractsthe

Figure6.5:Characteristicsofanintegratedmultimodalmap.DuringtheiMTlearning
processthedifferentmodalitiesalignwiththeteachingsignal(µA/Vchanges)andtheir
neuronalrepresentationsbecomemoreprecise(σA/Vdecreases).Giventhatthemultimodal
mapisacombinationoftheunimodalsensorymaps[cf.Eqs.(6.1)to(6.3)],itsprecision
andpositionaredeterminedbytheunisensorymapsandconsequentlychangeaswell.Here
onemodality(blueprofile)exemplarilyvariesinprecision(A)andposition(B)whereas
theothermodality(yellowprofile)remainsunchanged.Theresultingmultimodalteacher
isdepictedasredprofile.(A)Themodality(blueprofile)thatgetsmorepreciseattracts
[seeEq.(6.3)]andsharpens[seeEq.(6.2)]themultimodalteacher(redprofile).(B)The
modality(blueprofile)thatalignswiththeteacher(redprofile)inducesthemultimodal
maptoshiftaswell.Thedistancebetweenthemultimodalteacherandthestationarymap
(yellowprofile)thereforedecreasesaswell;cf.Eq.(6.3).

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6.formationmapdalMultimo

Figure6.6:Modelarchitecture.Amultimodalobjectsuchasawaterdrophittingtheskin
producessensoryinputforaudition,somatosensorymodality,andvision.Theinputis
modeledbymeansofaGaussianfiA/S/V(y)soastostimulateneuroniofthecorresponding
inputlayer.Neuronsoftheinputlayerprojectthroughall-to-allsynapticconnections
JjA/S/V(t)toneuronjofthemaplayer.Thesesynapticconnectionsarelearnedbymeans
ofsSTDPtoaligntheoutputofthemaplayertotheinhibitoryteacherinput.

Insummary,theanalysisoftheoptimallyintegratedmultimodalmaphasunraveled
threeessentialaspects.First,themultimodalmapisalwaysmoreprecisethananyunimodal
map;cf.Eq.(6.2).Second,theoptimalmultisensoryteachingsignalislocatedmore
closelytothepositionofthemoreprecisemodality;seeEq.(6.3).Third,onthebasisof
anadaptionofthemonosensorymapsthroughsSTDPlearningrules,Eqs.(6.1)and(6.2)
describehowpositionµM(t)andprecisionσM(t)ofthemultimodalteachervaryforgiven
µA/V(t)andσA/V(t)oftheinputmaps.

6.2.2HowdoiMTcharacteristicsinfluencemapadaptation?
Intheprevioussection,anintegratedmultimodalteacherhasaspecificvarianceand
positionthatbothdependonthecharacteristicsoftheinputmaps;cf.(6.1)and(6.2).In
addition,thesecharacteristicsandconsequentlytheteachervaryduringmapcalibration.
Thissectionthereforediscusseshowteacherswithdifferentaccuraciesandteachersthat
areshiftingduringlearningaffectmapalignment.
Theresultsaspresentedinthefollowingarebasedonnumericalsimulationsofan
inhibition-mediatedSTDPmapalignmentprocess.Thebasicstructureofthemodel
consistsofthethreemodalitiesauditionA,thesomatosensorymodalityS,andvision
VthatadapttothemultimodalinhibitoryteachermodalityTviaSTDPlearningrules;
seeFig.6.6.Thefundamentalprinciplesofinhibition-mediatedSTDPmapalignment
havebeendescribedelaboratelybyFriedelandvanHemmen[56].Allsensorymapsare

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formationmapdalMultimo6.

one-dimensionalmapsofNPoissonneurons[109].Eachmodalityconsistsofaninputmap
withaGaussianfiringprofile
fiA/S/V(y):=SA/S/Vexp−(y−xi)2/(2σ2A/S/V)(6.4)
for1≤i≤N,withstimuluspositiony,thei-thmapneuron’spreferredpositionxi,
standarddeviationσA/S/V,andSA/S/Vbeingthemaximalamplitudeoftheactivity.For
twodifferentmaps,e.g.,auditionAandvisijonV,theaccuraciesdifferandσA=σV.The
inputlayerprojectsviaall-to-allsynapsesJA/S/V(t)ontoneuronjofthemaplayer.In
addition,eachneuronofthemaplayerreceivesinhibitoryinputfromonecorresponding
neuronoftheteachermodalityT.Thesynapsesconnectingteacherandmaparemodeled
connections.synapticone-to-onestaticasThesimulationsincorporateateacherthatshowstheessentialcharacteristicsderived
inSec.6.2.1.Aneuronpoftheteachermapthereforerespondstoastimulusatpositiony
informofan“inverted”Gaussianfunction
fpT(y):=ST1−exp−(y−xpT)2/(2σT2),(6.5)
withxpbeingthepreferredpositionofneuronp,STbeingthemaximalfiringactivityof
theteacherneurons,andσTbeingthewidthoftheteacherinfluence.Theprocesswherea
teachersuppressesmapactivityatlocationsdifferentfromtherealobject’spositiony,is
calledselectivedisinhibition[56,94].
FornumericalexperimentsonmapalignmentashiftΔisintroducedinthesensory
positionofthemodality,e.g.,throughprismsintheKnudsen[122,123]experimenton
thebarnowl.AsexplainedinSec.6.3.1,ashiftofvisioninducesatthesametimea
displacementofthemultimodalteacher.Consequently,ashiftofallsensorymapswith
respecttothepositionofthecombinedmultimodalteacheroccurs,thatis,againstintuition,
lensesshiftingvisionhardlydisplacevisionwithrespecttotheteacherbutdisplaceallother
sensorymapswithrespecttothemultimodalteacher.Inmathematicalterms,y+ΔA/S/V
replacesyinEq.(6.5)wheretheshiftdependsonthemodalityaccordingtoEq.(6.3).
Therefore,thebiologicalexperimentsandthenumericalexperimentsagree.
Thechangeswithinthesynapticconnectivitypatternduringthealignmentprocess
canbedescribedandexplainedbySTDP[56,109].ThemapestimationerrorEasa
least-squareerrorquantifieschangeswithinthesynapticpattern.Itmeasureshowwella
mapcanreproducespatialinformationofMsystematicallyvariedincomingsignals,
ME=xi−xiT2/M(6.6)
=1iwherexidenotesthemeanpositionofthefiringprofileofthemapandxiTrepresentsthe
theoreticalposition.ThesmallerthevaluefortheerrorE,thebetterthemapreproduces
thereal(theoretical)positionofanobject.Theparametersusedinthenumericalexperi-
mentsaregivenintheTable6.2.

Multimodalteacherwithvaryingvariance
Oneoftheessentialcharacteristicsofthemultimodalteacherisitsabilitytoimprove

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formationmapdalMultimo6.

inprecisionduringthealignmentprocessasaconsequenceofdynamicunimodalmaps;
see(6.2).Afterbirth,mostanimalsfirsthavetolearnhowtointerpretthesensory
213].informationFurthermore,theyreceivWe.allaceAllaandvailablSteine[m225aps]asobservgivenedbythatgeneticsmareultisensoryveryneuroninaccuratesin[cats140,
ydooungernotshothanwm28ultidamysodalreliablyenhancemenrespondttoorinputsdepression.frommoreThesethanobservasingleations,homowevdaliter,yarebut
mapsmeasuremenandtsestimatorsonthelevareelofelusivsinglee.Tocells.copeSubforstheequenwtorstconclusionssituation,inittheiscontextnecessaryoftoneuronalshow
thataccuratelearningthestillunimowdalorksormaps,attheleastlargerstartsthewithvaariancehighlyoftheminaccurateultimoteacdalher.teacTherhe(6.2less),
thatis,theestimatedpositionandthustheteachingsignalisspatiallylessaccurate.
Consequently,theteacheraccuracyismodeledbyanGaussiandistributedrandomshift
ΔTdifferenaddedtteactohertheshiftscorrectΔT,teactheherpunimoosition.dalmapsAsaadaptresultofmoretheslonwlyumericalandwithexpaerimenlargertswithmap
estimationerrortoalessaccurateteachersignal;cf.Fig.6.7.Withinagiventimewindow,
aninaccurateteacherthereforeslowsdownbutdoesnotpreventthemapalignmentprocess.
Inthemulticonclusion,modalthineiMTtegration.conceptdoesnotdependcruciallyontheconcreterealizationof

Figure6.7:InfluenceofteacherprecisionΔTonmapalignment.Theplotpresentsthe
mapestimationerrorE[seeEq.(6.6)]formapalignmentwithamultimodalteacherof
differentprecisions.Themapestimationerrordropsfasterandtoalowerlevelwithamore
preciseteacher(lowerΔT).Theprecisionoftheteacherthereforeinfluencesthetemporal
developmentofmapalignmentbutdoesnotpreventit.

Multimodalteacherwithdynamicallychangingposition
SimilartoFig.6.5,thepositionoftheintegratedmultimodalteachercontinuouslyshiftsin
spaceduringthealignmentprocess.Asaconsequence,theinfluenceofashiftingteacher,
moreconcretely,itsshiftingvelocity,onmapalignmenthastobeanalyzed.Figure6.8
showsthesynapticpatternofsensorymapsforthetimestept=2000s.Synapsesare
modifiedbymeansofteachingsignalsshiftingwithdifferentvelocities.Thedynamicofthe

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formationmapdalMultimo6.

teacheressentiallydeterminestowhichdegreeamapadaptstotheteacher.Theslowerthe
teacherpositionshifts,thebetteramapcanadapttotheteacher.Moreover,thecritical
velocityuptowhichmapadaptionworksproperlydependsontheprecisionofthesensory
momodalitdalityycanthatisstilladaptreflectedtoinatheteacherprecisionwithaofgivtheeninpveloutcitymap,;cf.whereasEq.a(6.4precise).Amaplesscannotprecise
followanymore;cf.Fig.6.8,changefromtoptobottom.
Inconclusion,alowteacheraccuracydoesnotpreventalignmentofdifferentmaps

Figure6.8:Influenceofdynamicallychangingteacherpositiononmapalignment.Simu-
lationsperformedwithinputstandarddeviationσ=0.03(topline),σ=0.05(bottom),
andteacherstandarddeviationσT=0.02.Fordetailsregardingtheparameters,see
Appendix6.C.Theteachersignal,thatis,theinverseGaussianprofile,isshiftedinspace.
Theredlinesindicatethemeanpositionoftheteacher.Inthenumericalexperiments
theteacherpositionstartsshiftingfromthedashedredline(Δ=0.1).Thestraightred
lineindicateshowfartheteacherhasshiftedwithinthesimulatedtimeof2000s.By
steppingtotheright,theteachervelocitydoubles.Thesynapticpatternofthesensory
mapsareinitiallylocatedaroundthediagonalandaremodifiedbymeansofsupervised
spike-timing-dependentplasticity(sSTDP).Theboxatthetopleftshowsthesynaptic
patternattime2000scausedbyaslowlyshiftingteacher.Withacertaindelay,the
synapticpatternofthemaphasadaptedtotheteachermap.Withincreasingteacher
velocity,thatis,bysteppingtotheright,themapadaptsmuchless.Forthesynaptic
patterninthetoprightboxtheteachervelocityhasreachedacriticallevelwherethemap
doesnotadaptatall.Thevelocityoftheteacherthereforeessentiallydeterminestowhich
degreeamapadaptstotheteacher.Thiseffectcanbecompensatedbydecreasingmap
precisionasrealizedfromtheboxattopmiddletotheboxatbottommiddle.Herethe
synapticpatternofthelessaccuratemapstilladaptsverywelltotheteachermap.The
criticalvelocityofnon-adaptationisthereforehigherforalow-precisionmapthanfora
map.high-precision

byaccuracymeansofofthesSTDPfinalbutstate.influencesTheshiftinthegvtempeloorcitalyofdevtheelopmenteactherand,ptoositionsomedeterminesdegree,theto

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6.formationmapdalMultimo

anwhichaccuratedegreemapambapecausecanitadaptcantoadaptthetoteacaher.fasterAlessdynamicallyaccuratechanmapisgingmoreteacher.flexiblethan

6.2.3HowdoesiMTcalibratedifferentunimodalmaps?
ThethirdandlaststepoftheiMTanalysis(seeFig.6.3)addressesthequestionofhowthe
essentialcharacteristicsofthedifferentmodalities,inparticular,thedifferentaccuraciesof
thesensorysystems,influencemapformationandalignment.Resultsofthecorresponding
simulationsareshowninFig.6.9.Heremapformationandalignmentbymeansofthe
multimodalteachermaparepossibleforinputsofdifferentaccuracies.However,the
qualityandtemporalprogressofmapformationandalignmentdependontheaccuracy
ofthesensorysystem.Withinthealignmentprocess,ahigh-precisionmapreachesa
modifiedsynapticpatternthatisagainmoreprecisethanthatofalow-precisionmap.
Moreimportantly,ahigh-precisionmapadaptsmoreslowlytotheteacherinputthana
mapwithlowprecision.Inconclusion,thehighertheprecisionofasensorymap,themore
map.thestatic

6.3ApplicationsoftheiMTconcept
Afterdetailedanalysisofmultimodalteaching,theexperimentspresentedintheintro-
ductionarereviewedinthelightoftheiMTconcept.Indoingso,theaforementioned
experimentsthatcanbeexplainedbyavision-guidedconceptofmapcalibrationandthose
thatcannotaretreatedseparately.TheiMTconceptoffersexplanationsforboth.
Inthefollowingagroupofexperimentsisanalyzedthatmotivatedtheideaofvisionas
thedominantguidingmodality:theprismglassexperimentsofKnudsenandcoworkers[94,
122].Theseexperimentsareexplainedwhiledenyingavision-guidedmapalignment.A
re-interpretationofexperimentalresultsfromSteinetal.[223,225]furtherillustrateshow
theiMTconceptcouldwork.

6.3.1Experimentspro-vision-guidedmapformation
Themostprominentexperimentssupportingvision-guidedmapformationaretheshifting
expcertainerimenangle.tsbyAfterKnaudsenlearningetal.p[94erio,d122the].Hereneuronalprismsproshiftjectionsthefromvisualthesystemauditoryofowlsmapbyarea
rearrangedtocompensateforthemisalignmentbetweenvisualandauditorymaps.The
visualmap,incontrast,remainsconstantduringthewholealignmentprocess.
CantheiMTconceptreproducethisdominanceofvisionwithinthelearningprocess?
Toanswerthisquestionthealignmentofmapswithdifferentvariancesanddistances[cf.
Eq.(6.3)]totheteachermaphasbeensimulated.Thevisualsystemusuallyhasamuch
higherspatialresolution(aboutafewarcminutes)thanallothersensorysystems,e.g.,
nothewservauditoryestosystemreconsider(aboutsynapticafewplasticitdegrees);yseedurinTgaableprism6.1sforhiftexpdetails.erimTheent.iMTconcept
•Initialstate(Fig.6.10,top)
Auditoryandvisualmapsareshiftedwithrespecttoeachotherbecauseofanartificial

122

6.formationmapdalMultimo

BothFigureplots6.9:showInfluencetheofmapvaryingestimationinputerrorprecisionEasσgivonenmapbyEq.formation(6.6)b(A)etwandeenthealignmenunimot(B).dal
mapsandtheteacher.Theinputmapsdifferinprecision(inputstandarddeviationσ)as
mapindicated.alignmeThent(B).graphsTheshobwoxedthemapregionintheestimationbottomerrorplotEisformapenlargedtoformationclearly(A)andvisualizefor
slowhicwlyhtomaptheadaptsteacherfaster.inputBoththanloplotsshow-precisionwthatmapsmapsbutwithregainahighhigherpprecisionrecisionadaptaftermorethe
calibration.map

shiftofthevisualsystemthatisintroducedbyprisms.Giventhatthemultimodal
teachermapisacombinationoftheunimodalmaps,itsactivitymeanhasshiftedas
well.Anadaptationprocessofallunimodalmapsisinduced.Againstnormalintuition,
shiftingvisiondoesnotresultinashiftbetweenvisionandthemultimodalteaching
signal;itinsteadresultsinashiftofallsensorymapscomparedtothemultimodal
teachermap.Asaconsequenceofthegreaterprecisionofthevisualsystem,the

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formationmapdalMultimo6.

AnimalσAσVµµVA−−µµAMµµVV−−µµAM
Human1◦0.02◦99.96%0.04%
Barnowl2◦◦0.3◦◦97.8%2.2%
Cat80.299.93%0.06%

Tandablecat6.1:[8,26Visual].Theandvisualauditoryloresolutionscalizationarecapabilitiesoriginallyofgivhenumanin[39cycles,153p],erbarndegree,owl[i.e.,78,ho121w],
manylinescanbedistinguishedinonedegreeofthevisualfield.Theresolutionsare
mapsassumedaretocombebinedtheintovaersemofultimothesedalvteacalues.herandBythemeansdistanceoftheofveachariancesmaptothethetwomultimounimodaldal
Themapvis[cf.ualEq.map(6.3as)]ismorecalculatedprecisemapandisnearernormalizedtobythethemultimodistancedalbetmapweenthanthethetwolessmoaccurdalities.ate
map.auditory

multimodalteachermapismainlydeterminedbythecharacteristicsofthevisualmap,
thatisσM≈σVandµM≈µV;cf.Eqs.(6.1)and(6.2).Equation(6.3)illustrates
thattheshiftbetweenvisionasthemostprecisesensorysystemandthemultimodal
teachermapissmallerthanforanyothersystem.

•InSynaptictermediateconnectionsstateto(Fig.the6.10,auditorymiddlemap)startchangingtocompensatefortheshift
betpresenweentedinauditionSecs.and6.2.2theandmultimo6.2.3dalandbteacecauseher.ofBaseditsonhighertheprecision,previousthestudiesvisualas
themapadauditoryaptsmmapuchthemoreteacsloher,wlyasathancomthebinationauditoryofthemap.twoDuemodtoalities,thenevadaptationerthelessof
henceshiftstoconwtinarduesthetovisualshifttomap.wardThetheauvisualditorymap,mapwhicadaptshstaysagainalmosttothestatic;teacseeherFig.input6.5anB.d

•Finalstate(Fig.6.10,bottom)
Thewhereastwothemapsauhaditoryvebeenmaphasrealigned.shiftedThealmostpositiontheofwholethevisualdistancemapinducedhashbyardlythechanged,prisms
atthebeginningoftheexperiment.

InofKnsummaryudsen,anthediMTothers[94concept,122can].Herereprotheducetheobservdiffederentsdominancetepsofofthetheshiftingvisualexpsystemerimenonlyts
reflectsthemoregeneralconceptofadominanceofaccuracyfollowingfromtheiMT
concept.

124

6.

dalMultimo

formationmap

Figure6.10:Numericalrealignmentexperiments.Theplotillustratestheshiftingprocess
fortheauditorymapAwithvarianceσA=0.07andthevisualmapVwithσV=0.03.
ThedistancesbetweenteacherandunimodalmappositionsarecalculatedbyEq.(6.3)for
thecorrespondingvariances.Thearrayplotsshowthesynapticconnectivitypatternfor
vision(leftcolumn)andaudition(rightcolumn).Thesynapticstrengthsherebyrange
between0and1(seecolorcodeinthebottomleftcorner).Theactivityprofilesofthe
auditorymapA,thevisualmapV,andtheteachermapareshowninthemiddlecolumn;
fordetailsoftheactivityprofilesseeSec.6.2.2.Thecolorcodeisspecifiedinthebottom
rightcorner.Theplotillustrateshowsynapticplasticitydependsontheaccuracyofa
modality.Inconcreteterms,themoreprecisevisualmapVshiftsmoreslowlyandfar
lessthanthelessaccurateauditorymapA.Insummary,theiMTconceptcantherefore
reproduceexperimentspro-vision-guidedmapformation.

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formationmapdalMultimo6.

6.3.2Experimentscontra-vision-guidedmapformation
Theintroductionhasreviewedsomeexperimentsthatquestionavision-guidedlearning
process.TheanalysisoftheiMTconcepthasshownthatvisionplaysadominantrole
becauseofitshighprecisionbutthatvisionisnotstatic.Itcanbemodifiedbyadynamically
changingmultimodalteachermapandthereforebyanyunimodalmap[46,55,158,197,
].201Furthermore,theiMTconceptshouldbeabletoreproduceformationofunimodal
mapsinvisuallydeprivedanimals.Studies[117,177,181]showthatblindhumans,cats,
andferretscanlocalizesoundaspreciselyasindividualswithnormalvision.Measurements
ofWallaceandStein[223,225]supportthecriticalviewofvision-dominatedlearningby
showingthatsensorymapsalreadysharpenatatimewhennovisualneuronsarepresent.
Learningispossiblewithoutvisionbutisdominatedbyvisionincasesinwhichitis
ailable.vaInthecontextoftheiMTconceptamultimodalteachermapcanexistwithoutavisual
mapbutithasamuchlowerprecision.Formationoftheothermapsisthereforestill
possiblewithincertainlimits;cf.Fig.6.7and[111,122,213].Furthermore,intheintrinsic
learningprocessallmapsdevelopandimproveinparallel,asreportedinexperimentssuch
as[223,225].Herethereproductionofunimodalmapformationinavisuallydeprived
animalallowsforstudyingwhethersomatosensoryandauditorymapscandevelopwithout
vision,i.e.,withaveryimpreciseteacher.

•Initialstate(Fig.6.11,top)
doesnotSomatosensoryexistatandall.Theauditorymultimomapsdalaremaponlyasacoarselycombinationpre-wired,ofandthesomatosensoryvisualmapand
auditorymapsisveryimprecise;cf.Eq.(6.2).

•Intermediatestate(Fig.6.11,middle)
Evenwithaveryimpreciseteachermaptheauditoryandsomatosensorymapsstartto
improvesothattheirvariancesdecrease.Theearliercalculations(seeFig.6.7)illustrate
thatthelearningprocesswithanimpreciseteacherstartsveryslowly,corresponding
tothetopmostcurveintheplot.Nevertheless,theunimodalmapsimproveslightly,
asdoesthemultimodalmap,thatisacombinationofthetwosensorymaps,asa
consequence.Withtheimprovingteacher(seeFig.6.7),thelearningprocessaccelerates.

•Finalstate(Fig.6.11,bottom)
Thetwomapsarealignedandhavereachedtheirbest-possibleresolutions.

Together,aninterpretationofbothexperimentspro-vision-guidedmapformationand
ciMTontraisa-vision-guidedsuccessfulconcemapptforformationcalibratispingossibledifferenwithintthunimoeiMTdalmapsconcept.thatAsisainconsequence,accordance
withknownexperimentalresultsofmapformationandalignment.

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mapdalMultimo6.formation

Figure6.11:Numericalmapformationexperimentswithanimprovingteacher.Activity
profilesoftheauditorymapA(σA=0.03),thesomatosensorymapS(σS=0.045),and
theteachermapareshowninthemiddlecolumn.Colorsoftheprofilesarespecifiedin
thebottomrightcorner.Theblurredteacherprojectsontothemapswithaprecisionthat
startsatΔT=0.1andthenreduceslinearlyintimetoΔT=0.02fort=7500s.The
blackbarsbelowthefiring-activityplots(middlecolumn)denotetheregionswithinthe
teacherisprojectingforthecorrespondingtimestep.Thearrayplotsshowthesynaptic
connectionpatternJAforaudition(leftcolumn)andJSforthesomatosensorymodality
(rightcolumn).Thesynapticstrengthsrangebetween0and1;seecolorcodeinthebottom
leftcorner.Theinitialsynapticpatternisonlycoarselypre-wired.Despitetheinitially
impreciseteacher,bothmapsdevelopproperly.TheiMTconceptcanthereforereproduce
experimentscontra-vision-guidedmapformation.

127

6.4Discussion

formationmapdalMultimo6.

Insummary,thepresentchapterhasintroducedtheconceptofintegratedMultimodal
Teaching.TheapplicationofiMTtomapformationandalignmentleadstofourfundamen-
tallynewideas.First,theintegratedteachercontainsallavailableinformationofthemaps
involved.Second,theteacherisintrinsictothesystembecauseitisgeneratedonlyfrom
informationavailablewithinthesystemofmaps.Third,asaconsequence,theteacher
itselfisdynamicwiththepotencytoshiftinpositionandtoimproveinprecisionduring
thelearningprocess.Forth,themorepreciseasensorysystemis,themoreslowlyitadapts
toteachingsignals.Thisnaturalconsequence,whichisdenotedasdominanceofaccuracy,
explainsobservationswhereadominanceofvisionwithinthealignmentprocessmaybe
wronglyinterpretedaspurelyvisuallydriven.

state-of-the-artthetoarisonCompofHerespikingsimulationsneuronsofmandultimosynapticdalteacplasticithercy,namelyharacteristics,sSTDPfor,maparecalibpresenrationted.onThetheteaclevherel
numsignalberofinfluencesexperimentheunitalmfinodaldings,mapsforbyinstance,selectiveinthedisinhibitionICC-ICX-OTasproppathosedwabyy[5an].Takincreasingento-
gether,theiMTconceptthereforeconnectstwofundamentalneurophysiologicalphenomena:
inhibition-mediatedmapcalibrationandmultimodalintegration.
MysorePrevious[64,161]theoreticalproposedstudiesmodelshaveforfocusedHebbianononemaporpartscalibrationofthebasesedasponects.theassGelfandumptionand
thatvisionservesasteachersignaltomodify,forinstance,synapsesoftheauditory
nevprojecerthelesstionpathhaswatoy.beAsalreadyquestioned.pointedSteinoutandinWtheallinacetro[223duction,,225]visionthereforeasteachersuggestedsignala
studiesconnectiononbarnbetoweewlsnfmromultimoBergandalitandyandKnudsenmap[5]formationalsoshoiwnthatcats.Thecross-molatestdalexpeffectserimenexisttalin
theICXIncludingandmatheseyexpinfluenceerimenexptalfindinerience-depgs,Wittenendentcetalal.[ibration234]ofdevconelopvedergingatheoreticalrepresenmotations.del
forvisualbimomapdalaremapadaptedalignmenduet.toInHebbiantheirmolearning.del,theBythesesynapsesofmeans,boththethemodelauditorycanrepandroducethe
thedominanceofvisionsuchthatthechannelwiththeweakerorbroaderneuronal
ofrepresenWittenettational.alwisaysexclusivexhibitselybasedmostoroneallofxcitatortheyplasticitinputsy.andHowrateevcer,othedes.Thesetheoreticalconditionsmodel
areterms,bothexpquiteerimentalimplausibledatainfromtheBergancontextandofKnmapudsen[alignmen5]tsuggestinICXthatandteacOT.hingInsignalsconcreteto
theassumptionsICXaregatedunderlyingbyinhibition.rate-basedInHebbianaddition,learningKempterareetnotal.nec[109]essarilyvdemonstratedalid,inthatparticular,the
notintheauditorysystemofthebarnowl.
ofmapIncludingalignmenthesetbyexpmeanserimenoftalselectivresults,eFriedeldisinhibitionetal.on[56the]devlevelelopofedaspikingtheoreticalneurons.moThedel
momaps.delHoshowwsever,thattheanmoinhidelbitoryonlyconteactaihernsisplasessenticittialytowithinrealigntheauditoralreadyysystem,calibratedthatunimois,thedal
visualThesystempresentisiMTstaticmoanddelprocomvidesbinesthetheguidingfundamensignaltalforaspectsauditoryofthemapmodelsalignmenfromt.Witten

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6.formationmapdalMultimo

etal.[234]andFriedeletal.[56]:spike-timingdependentlearningbasedonbothselective
informationdisinhibitionisandinfulltegratedplasticitintoyamwithinultimoalldalconmaptributingatafirstmostep.dalities,Inthatais,secondallastevp,ailablethe
mtheultimoiMTdalmodmapelcanprovidesexplainateacthehingdominansignaltforrolemofapvisioncalibrationinmaptoallmoalignmendalities.t,plasticitTherefore,yin
monon-visualdalities,moplasticitdalitiesyofwithoutnon-visualvision,moplasticitdalitiesyguidedwithinbytheothervisualmonon-visualdalitymoguideddalities,byotherand
anycombinationofthesesettings.

TGivestableenthatprethedictionsiMToftheconceptiMTcanconcreproeptducebothexperimentssupportingandquestioning
tallyvvision-dominatederifiedareneeded.learning,Tosuggestionsdistinguishregardingvision-dominatedhowtheteaciMThingconceptfromcaniMT,beexpexperimenerimetaln-
setupssituationshaveintobwhicyhpassthetheprecisidominanceonofofsensoryvision.Tmapsothisisend,equal,theyinvescantigateithereanimalsstudywithartificiala
poorlydevelopedvisualsystem,orobservemapformationwhilemultimodalmapsare
ated.deactivtallyerimenexpexpToerimenfirsttswithequalizeprismsthethatprecisiondisplaceofsensandoryblurmaps,theonevisualcould,inpufortattheinstance,sameperformtime.shiftingSecond,
studyingmapformationandalignmentforanimalswithamoreprecisesensorysystemthan
thevisionmultimocoulddalalsoinidentegrationtifytheduringteachingthemolearningdality.proAcess.thirdGivpenossibilitiMT,ywdeactivouldbeationtoofdeactivthemateul-
timoalignmendaltmap(s)shouldfailcorrespinondthisstocase.aIndeactivcontrast,ationofforatheteachingvision-dominatedmodality.teacMaphingsformationcenario(e.g.,and
vision-guidedmapalignment),nonegativeeffectshouldbeobservedonthelearningprocess.

6.AOptimalcombinationoftwomodalities
Thefiringprofileofaneuronalmapcanbeinterpretedasthelikelihoodtofindanobject
ataspecificposition.Inotherwords,neuronalmapsencodepositionestimatorsbymeans
oftheirfiringprofiles.Amodality,forinstance,auditionAorvisionV,canberepresented
profilesfiringGaussianthroughfA/V(x):=SA/Vexp−(x−µA/V)2/(2σ2A/V)(6.7)
withmappositionxandestimatedposition,i.e.,meanµA/V,standarddeviationσA/V,
andmaximalamplitudeSA/Voftheactivityprofiles.Forthesakeofsimplicity,only
here.consideredaremapsone-dimensionalTocombinetwosuchestimatorstoacommonestimateµMaweightedsumofthemean
valuesµVandµAiscalculatedas

µM=gVµV+gAµA

(6.8)

129

Multimo6.formationmapdal

withtheweightsgVandgAfulfillinggV+gA=1.ReplacinggAbygV−1thevariance
σ2Mofthecombinedestimatoristhengivenas
σ2M=gV2σV2+(1−gV)2σA2.(6.9)
InthenextsteptheweightgVhastominimizethevarianceσ2M,thatis,∂σM/∂gV=0.
SolvingthelatterequationleadstogV=1−gA=σA2/(σA2+σ2V)andthereforeto
σ2M=[σA2/(σA2+σV2)]2σV2+[σV2/(σA2+σV2)]2σA2
=σV2σA2/(σA2+σV2)(6.10)
andµM=σA2/(σA2+σV2)µV+σV2/(σA2+σV2)µA.(6.11)
Formathematicalfundamentalsofthederivationsee[222].
Inthecontextofneuronalmaps,amultiplicationoftheactivityprofilesoftwomaps
realizestheaboveoptimalcombinationschemeofthecorrespondingneuronalestimators
µA/V.Themultimodalactivityprofileisthengivenby
fM(x)=SAexp−(x−µA)2/(2σA2)SVexp−(x−µV)2/(2σV2)

=SASVexp−1/2(µA−µV)2/σV2+σA2(6.12)
factorscaling2

exp−1/2σV2+σA2/σA2σV2x−σV2µA+σA2µV/σV2+σA2.
1/σ2MµM

StandarddeviationσMandmeanµMofthemultipliedmultimodalprofileareidenticalto
theoptimalvaluesderivedabove;cf.(6.10)and(6.11).

6.BOptimalcombinationofthreemodalities
TheoptimalcombinationschemefortwomodalitiesasderivedinAppendix6.Aisnow
extendedtothreemodalities.ThemultimodalestimateµMofthreemodalities,say,
auditionA,visionV,andthesomatosensorymodalitySrepresentedbytheirmeanvalues
µA,µV,andµSandstandarddeviationσA,σV,andσSisgivenby
22µ=σVσSµ
MσA2σV2+σA2σS2+σV2σS2A
σA2σS2
µ+σA2σV2+σ2AσS2+σV2σS2V
22+σAσVµ,(6.13)
σA2σV2+σA2σS2+σV2σS2S

130

(6.13)

dalMultimo6.formationmap

(6.14)

aconvexcombinationofµA,µV,andµS.Thecombinedmaximum-likelihoodestimate
µMfeaturesthesmallestpossiblestandarddeviationσMgivenby
Mσ=σAσVσS.(6.14)
σA2σV2+σA2σS2+σV2σS2
Thedistancebetweenthemultimodalestimate(6.13)and,forinstance,theauditory
estimateAisgivenby

22|µM−µA|=σA2σS|µA−µV|+σV|µA−µS|.(6.15)
σ2AσV2+σA2σS2+σV2σS2
FordistancesbetweenthemultimodalestimateandVandSonecaninterchangeAwith
.SorV

Mo6.CarametersPdel

Table6.2summarizesthemodelparametersasusedinthenumericalexperiments;see
.6.2.2Sec.

131

mapdalMultimo6.formation

parameter

aluev

numberofmapneuronsN=100
synapseslearningminimalstrengthJmin=0.0
maximalstrengthJmax=0.25
maximalinitialstrength(J0)max=0.25
strengthofteachersynapsesJT=−1
timeonserespostsynapticptoinputspikeτI=10ms
toteacherspikeτT=25ms
inputamplitudeSA/V/S=50s−1
teacheramplitudeAT=100s−1
teacherwidthσT=0.025
shiftmaps–teacherΔ=0.2
learningtriallengthT=0.5s
simulationtimestepΔt=0.5−ms6
learningparameterη=3∙10
hangecteighwuponinputspikewin=1.5
uponoutputspikewout=−10.0

Table6.2:Modelparametersasusedinthenumericalexperiments.Shapeandparameters
ofthelearningwindowaretakenfrom[56]withdifferingoradditionalparametersas
e.voabindicated

132

7.

Frequentlyusedabbreviationsand
functions

/abbreviationfunction

ICE

ITDIADiTDiADCF

(s)STDPTPLTDL

NANMSONNLLLDneuronEISCOT

iMT

description

internallycoupledears

differencetimeterauralindifferenceeamplitudterauralininternaltimedifference(atthelevelofthetympanicvibrations)
internaltimedifference(atthelevelofthetympanicvibrations)
characteristicfrequency

(supervised)spike-timing-dependentplasticity
tiationotenplong-termiondepresslong-term

angularisucleusnnsupucleuseriorolivmagnoaryncellularisucleus
laminarisucleusndorsalneuronwithlateralexcitatorylemniscalnanducleusinhibitoryinput
supopticeriorctectumolliculus

integratedmultimodalteaching

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