Equivalent‐circuit modeling of a piece of neuronal cable by means of the cable equation

7/28/2019 Equivalentcircuit modeling of a piece of neuronal cable by means of the cable equation

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From: An anthology of developments in clinical engineering and bioimpedance:Festschrift for Sverre Grimnes, edited by . Martinsen and . Jensen,

Unipub forlag, Oslo, Norway, 2009

Neurophysics:whatthetelegrapher'sequationhastaughtusaboutthebrainKlasH.PettersenandGauteT.Einevoll

DeptartmentofMathematicalSciencesandTechnology,NorwegianUniversityofLifeSciences,1432s

[email protected],[email protected]

1 IntroductionNeurons, the shrubbery cells responsible for our mental capabilities,are utterly complexandnonlinear in their signal processing. Both their morphologies and their behavior are highlyentangled; each neuron typically receives signals from between 1000 and 10000 neuronsimpingingonitsdendrites,theinputbranchesoftheneuron.Theseinputsignalsareprocessedinthemaincellbody,thesoma,oftheneuroninsuchawaythattheneuroneitherstayssilentorfireanactionpotential.Anactionpotential isanabruptchange intheneuron'smembranepotential,i.e.,thedifferenceinpotentialbetweentheinsideandoutsideofthecellmembrane,lastingafewmilliseconds.Wheninitiatedinthesoma,theactionpotentialwillpropagatedown

theneuron's

axon,

the

neuron's

output

channel,

and

convey

information

through

synapses

to

otherneurons.Foraschematicoverviewof thebasicconstituentofaneuron,seeFig.1.Thegeneration of action potentials is a 'binary' allornothing process: either a single actionpotential with a standardized shape is produced and propagated down the axon, or nothinghappensatall.

Figure 1: Schematicillustrationofaneuron(nervecell)anditssynapticconnections.


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Oneof themostprizedachievements in theoreticalbiology is theestablishmentover the lasthundredyearsorsoofamathematical theory for thesignalprocessing in individualneurons.

Themost

spectacular

event

is

maybe

the

Nobel

prize

winning

work

of

Alan

Hodgkin

and

Andrew

Huxley intheearly1950swheretheydescribedthepropagationofactionpotentialsalongthesquid giant axon by a modified electrical circuit where the charge carriers are sodium,potassium, calcium, chloride and other ions flowing through and along the neuronal cellmembrane [14]. This mathematical formulation could not only account for the results fromtailored experiments used to construct the model and fit the model parameters; from theirmodeltheycouldalsopredicttheshapeandvelocityoftheactionpotentialwhilemovingdowntheaxon.Theycalculatedthepropagationvelocityoftheactionpotentialintheirexperimentalsystemtobe18.8meterspersecondwhichwasroughly10%offtheexperimentalvalueof21.2meters per second [3]. Such quantitatively accurate model predictions are rare in theoretical

biology.Duetoitsstunningsuccessindescribingactionpotentials,theHodgkinHuxleyapproach

waslatergeneralizedtoincludemodelingofthesignalprocessingpropertiesofentireneurons,socalledcompartmentalmodeling[57],andalsomodelingofelectricallyexcitablecells intheheart [8]. With the advent of compartmental modeling of neurons, computationalneuroscientistsnowhavearelativelyfirmstartingpointformathematicalexplorationsofneuralactivity.Thusneuroscience ispresentlyamong thebiologicalsubdisciplineswhere theuseofmathematicaltechniquesismostestablishedandrecognized.

AtthecoreofHodgkinHuxleytheoryandcompartmentalmodelingofneurons liesthesocalledcableequationdescribinghow themembranepotentialdynamicallyspreadsalongadendriticbranchoranaxon.Thisequationhasalongandhonorablehistorywhichcanbetracedbacktothe'telegrapher'sequation'exploredbythe(later)LordKelvinasearlyas1855.

In this chapter we will briefly outline the origin of recordings of biological electricalactivity and, in particular, the origin of the cable equation as used in computationalneuroscience.The roleof thecableequation indetermining thesignalprocessing inneurons,i.e., how input signals are converted into trains of action potentials, has received lots ofattention [26].Herewe will instead focus on recent work fromourgrouponhow the cableequationdeterminestheextracellularpotentialsrecordedaroundneurons[9,10].

2 HistoryofelectricalrecordingsinbiologyFor

several

centuries

it

has

been

known

that

mechanisms

within

the

body

both

react

to

and

create electricity. Already in 1786 the italian Luigi Galvani began investigating the action ofelectricityuponthemusclesoffrogs[11].Thiswasthestartoftheresearchonwhathecalledanimalelectricity,butittookacenturybeforeAugustusWallerinLondonwasabletorecordthefirsthumanelectrocardiogram (ECG) in1887 [12].Onereasonwhy this tooksucha long timewas the lack of measuring devices with the desired sensitivity to measure the weak surfaceelectricityofthebody,inthiscaseabovetheheart.Waller'sECGexperimentwasmadepossibleby a breakthrough in techniques for measuring electrical potentials: around 1873 GabrielLippmanninParishadinventedthemercurycapillaryelectrometerwithasufficientsensitivity.Thecapillaryelectrometerhadaratherlongadjustmenttimewhichresultedinapoortemporal


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resolution,but in1901thedutchWillemEintoven inventedthestringgalvanometer[13].Thisdidnotonlyhavetherequiredsensitivity,italsohadanexcellenttemporalresolution.WithhisnewmeasuringdeviceEinthovenwasabletomeasureanddescribethehumanECGindetailforwhich

he

received

aNobel

prize

in

medicine

in

1924.

As early as 1875 the englishman Richard Caton measured stimulusevoked electricpotentialsatbrainsurfaces.Heusedapredecessorofthestringgalvanometer,adeviceknownasamirrorgalvanometer,andputelectrodesdirectlyontothesurfacesofthebrainsofrabbits.Heobservedthattherecordedpotentialsvariedwhentheretinawasstimulatedwithdifferentlight intensities [14].However,electrical recordings from thebrain firstbecamepopularaftertheGermanHansBerger in1929publishedhisfindingsonmeasuredelectricfieldsoriginatingfrom the human brain, recorded through the intact skull. Berger named his recordings'electroenkephalogram',whichtodayareknownaselectroencephalograms(EEG).

3 OriginofcableequationThe firstelectricalbrain recordingsoccurredata timewhen itwas stilldebatedwhether theneuronswerephysicallyconnectedinajointmeshworkoriftheywereseparatecomputationalentities.The latterview,called theneurondoctrine,wasproven tobe right.Actually,FridtjofNansen, theNorwegianexplorer, scientistanddiplomat,wasoneof the pioneersarguing forthisdoctrine.Nansenstartedhisworkinneurosciencein1882,andin1887thisresultedinthefirst Norwegian doctoral thesis in neurobiology titled 'The structure and combination of thehistological elements of the central nervous system'. Today the neuron doctrine is firmlyestablished, and the neuron is generally accepted to be the basic computational unit in the

brain. Theoriginofthecableequation,thecoreingredientofcompartmentalneuronmodels,isevenolder. Intheearly1850sthequestionofa transatlantic telegraph linewasraised,andthe question appealed so much to the physicist William Thomson, later Lord Kelvin, that hestarteddevelopingamathematical theory forsignaldecay inunderwater telegraphcables. InDecember1856when theAtlanticTelegraphCompanywasformed,Thomsonwas in factalsoonitsboardofdirectors.

Thompson'smathematicalmodelforthesignalconductionthroughcableswasbasedonFourier's equations for heat conduction in a wire. This resulted in the socalled telegraph ortelegrapher's equation describing the variation of voltage V along an electrical cable asfunctionoftimeandposition,

.)(=2

2

2

2

RGVt

VCRLG

t

VLC

z

V+

++

(1)

HeretheresistanceR andtheinductanceL representseriesimpedancealongthecable,whilethe capacitance C and the leakage conductance G form the shunt admittance across thecable.

Theinductivetermsreflectsocallededdycurrents.Suchcurrentsaretypicallylargestinthick, highly conductive cables, especially for high frequencies. Since neuronal cables have arelatively low inner (axial) conductivity and are very thin (certainly compared to the firsttransatlanticcable!),theinductivetermscansafelybeneglectedforthetypicalfrequencies


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Figure 2: Illustrationofequivalentcircuitmodelingofapieceofneuronalcablebymeansofthecableequation in Eq. (2). For figure clarity a discretized version is illustrated, and the cable equation isobtainedwhenthedistancebetweenneighboringcircuitelements 1n and n approacheszero[24].

Theneuron

depicted

on

the

right

is

an

anatomically

reconstructed

pyramidal

neuron

from

cat

visual

cortex[15].

inherentinneuronalactivity(


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handsideofthecableequationinEq.(2).Atthetimethephysicalsubstrateofthesecurrentswasunknown,andphenomenologicalmodelsextractedfromexperimentswereused.Todayitisknownthatthesespecializedionchannelscorrespondtovariousmembranespanningproteins

whose

structure

in

turn

is

encoded

in

the

DNA.

So

from

amathematical

point

of

view

onemightsaythatevolutionhasfiddledaroundwiththerighthandsideofthecableequationformillionsofyearstoprovideuswiththebasicelementofthinking.

WilfridRall,amongthefirstpropercomputationalneuroscientists[16],wasapioneerintheapplicationofthecableequationtounderstandthesignalprocessingpropertiesofneurons,inparticularhowthedendrites integratesynaptic inputs.Hewas trainedasaphysicistduringthe second world war and worked on the Manhattan project. After the war he moved toUniversityofChicagotoattendabiophysicsprogramorganizedbyKennethS.Cole,knownforthe Cole impedance and ColeCole permittivity equations [17], and others, and Rall soonbecamea leading figure in themathematicalneurosciencecommunity. Inapaper in1959he

describesthe

historical

development

of

the

cable

equation

[18]:

The mathematical treatment of axonal electrotonus [alteration in excitability

andconductivityofanerveormuscleduringthepassageofanelectriccurrent

through it] began in the 1870s with the work of Hermann (1872,1879)

supportedbyWeber's(1873)mathematicalanalysisoftheexternalfield inthe

surroundingvolumeconductor.Hermannrecognizedthemathematicalanalogy

of this problem with the analog in heat conduction, but the analogy with

Kelvin's (1855) treatment of the submarine telegraph cable in the 1850s was

first recognized by Hoorweg in 1898. This cable analogy was developed

independentlybyCremer(1899,1909)andbyHermann(1905)earlyinthe20th

century and has been widely used since that time. These mathematical

analogies are important because of the extensive literature devoted to both

generalmathematicalmethodsandspecialsolutionsapplicabletoproblemsof

this kind (Carslaw andJaeger, 1939). Importantpapers on the steadystate

distributionsofaxonalelectrotonusarethoseofRushton(1927,1934)andCole

and Hodgkin (1939)published in the 1920s and 1930s. The two most useful

mathematicalpresentationsofaxonalelectrotonus (includingconsiderationof

transients)are thoseprovidedbyHodgkinandRushton (1946)andDavisand

LorentedeNo(1947)inthe1940s.

In presentday compartmental modeling the neuronal cables are divided into compartmentswhereeachcompartmentessentiallyismodeledbyadiscretizedversionofthecableequation

withvarious

transmembrane

currents

accounting

for

the

action

of

the

various

ion

channels

[4

7], see Fig. 2. Mathematically the neuron is expressed as a system of coupled differentialequations,and freesimulation toolssuchasNEURON [19]andGenesis [20]havebeen tailormadetosolvetheseequationsefficiently.

4 ModelingofextracellularsignaturesofactionpotentialsMostofwhatweknowaboutthefunctioningofneuronsandneuralnetworkshascomefromelectrophysiological recordings, i.e., recordings of electrical potentials in the brain usingelectrodes.Inintracellularrecordingsanextremelythinelectrodeispokedthroughtheneuronal


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membranesothatthemembranepotential, i.e.,thevariableV inthecableequation,canbemeasured directly. Intracellular recordings are technically challenging, in particular in livinganimals, and the workhorse of brain electrophysiology in vivo has been extracellularrecordings.

In

such

recordings

the

potential

in

the

extracellular

medium

is

typically

measured

relative to a distant reference electrode. Extracellular potentials are much smaller thanintracellularpotentials;whileatypicalmembranepotentialis6080millivolts,theextracellularpotential is typically much less than a millivolt. The extracellular potentials stem from aweighted sumoverall transmembranecurrents in thevicinityof theelectrode tipandare ingeneralmuchhardertointerpretthanintracellularlyrecordedmembranepotentials.

However, if the electrode is placed very close to the soma of a neuron firing actionpotentials, the recorded extracellular potential will largely be dominated by the strong andcharacteristicsomacurrentsaffiliatedwiththeactionpotentials.Eachactionpotentialwillthenberecognizedbyacharacteristicspikyvoltagetraceintherecordedextracellularpotential.The

countingof

these

extracellular

spikes

can

be

used

to

record

the

train

of

action

potentials

from

thisneuron. Ingeneral, however, spikes frommanyactive neuronsmaybepicked up by theelectrode,andseveral issuesarisewhensuch recordingsare interpreted,e.g.,which typesofcellsaremost likelytobeseen intherecordings,whichcellparametersare important forthespike amplitude and shape, and which parameters are important for the decay of the spikeamplitude with increasing distance from the neuron? It turns out that the cable equation isessentialforunderstandinghowintracellularactionpotentialsare'translated'intoextracellularspikes,andinthissectionwewilloutlineresultsfromapreviousstudybyuswherethisquestionwasinvestigatedindetail[9].

4.1Forward

modeling

scheme

Neuronalactivitycanbecomputedanalytically from thecableequationonly for the simplestneuron models. For more complicated neuron models, compartmental simulation tools likeNEURON[19]orGenesis[20]mustbeusedtocalculatethetransmembranecurrentsactingassources for the extracellular potential. With all transmembrane currents and their spatialpositions known, the extracellular potential at any point in the brain can in principle becomputedusingMaxwell'sequations.However,thispresupposesthattheelectricalpropertiesofthesurroundingmediumareknown.Mathematically,thiscanbedonebynumericallysolvinga variant of Poisson's equation [21] using finiteelement methods (FEM). Here, however, amathematicallyandconceptuallysimplerforwardmodelingschemewillbeused[9,10,22].

Inour

compartmental

modeling

scheme

aneuron

is

divided

into

N

compartments,

and

thetransmembranecurrentfromeachcompartmentisdenoted )(tIn .Onecanthenderivethe

followingformulafortheextracellularpotential ),( tr duetoactivityinthisparticularneuron

[21,22],

,||

)(

4

1=),(

1= n

nN

n

tIt

rrr

(5)

where is theextracellularconductivity,andcompartment n ispositionedat nr . Inderiving

thisformula,thefollowingassumptionsandapproximationsareused:


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1. QuasistaticapproximationofMaxwell'sequations.ThisamountstoneglectingthetermswithtimederivativesoftheelectricfieldEandmagneticfieldBfromtheoriginalMaxwell'sequationssothattheelectromagneticfieldeffectivelydecouplesinto

separate

'quasistatic'

electric

and

magnetic

fields

[23].

Then

the

electric

field

E

intheextracellularmediumisrelatedtotheextracellularpotential viaE=.

Forfrequenciesinherentinneuralactivity,i.e.,lessthanafewthousandhertz,thequasistaticapproximationseemstobewellfulfilled(seeargumentonp.426of[23]).

2. Extracellularmediumisassumedtobe linear,i.e.,j=E,wherej isthecurrentdensity, ohmic,i.e.,noimaginarypartof[24,25],positionindependent,i.e.,isthesameeverywhere[25],and isotropic,i.e.,sameinalldirections[25].

Foramorecomprehensivediscussionoftheseassumptionsregardingtheextracellularmedium,

andalso

ways

of

generalizing

Eq.

(5)

when

the

assumptions

do

not

apply,

see

Ref.

[26].

4.2 EffectofdendriticfilteringonextracellularpotentialIn Fig. 3A we show a typical shape of an intracellular action potential calculated by thesimulation tool NEURON using a model pyramidal neuron constructed and made publiclyavailable by Mainen and Sejnowski [15]. This model has several types of active ion channelsspread across the neuronal membrane. (For simulation details see Ref. [10].) The membranevoltage tracehasacharacteristicshapewitha fastdepolarizingphase (fromabout 55mV toalmost20mVinafractionofamillisecond),followedbyanalmostequallyfastrepolarization,

andthen

alonger

hyperpolarizing

phase

(membrane

potential

more

negative

than

the

resting

potential).Thecorrespondingextracellularspikepatternsatdifferentspatialpositionsareshownin

Fig.3B.These extracellularpotentials are found fromevaluatinga sum of the type inEq. (5)where )(tIn corresponds to the transmembranecurrents found foreachcompartment in the

NEURONsimulation2.Severalfeaturesarenotable: Theextracellularspikehasamuchloweramplitudethantheintracellularaction

potential.Evenclosetothesomatheamplitudeislessthanafewtensofmicrovolts,morethanafactorthousandsmallerthantheintracellularamplitude.

Notonlythesize,butalsotheshapeoftheextracellularpotentialvarysignificantly

with

position.

The

shape

around

the

apical

(upper)

dendrites

is

typically

inverted

comparedtoaroundthebasal(lower)dendrites. Thespikewidthincreaseswithincreasingdistancesfromthesoma.Thisis

highlightedbytheinsetsshowingmagnifiedextracellularsignatures:theextracellularspikewidth,definedasthewidthofthefirstrapidphaseat25%ofitsmaximumamplitude,isseentoincreasefrom0.625msclosetosoma(bottominset)to0.75msfurtheraway(topinset).

2Eq.(5)correspondstoapointsourceapproximationwherethetotaltransmembranecurrentfromeach

compartmentisassumedtocomeoutfromasinglepoint.IntheevaluationofFig.3Bwehaveinsteadusedthelinesourceapproximationwherethetransmembranecurrentisassumedtobeevenlyspreadalongaline,see[9].


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Figure 3: Intracellularly and extracellularly recorded action potentials. The model is a reconstructedpyramidalneurontakenfromRef.[15].Asynapticstimulisimilartowhatiscalled'synapticinputpattern1' in Ref. [10] is used. (A) Soma membrane potential during an action potential. Inset shows themembranepotentialtraceinafivemillisecondtimewindowaroundtheactionpotential.(B)Calculatedextracellular potentials based on a variant of the forwardmodeling formula in Eq. (5) (i.e., the linesourceapproximation,see[10])assuminganisotropic,homogenousandpurelyconductiveextracellularmediumwith=0.3S/m.Theextracellularpotentialsareshownforthesamefivemillisecondsasinthemembranepotentialinsetin(A).Alldistancesareinmicrometers.Notethatthepotentialsintheinsetsarenottoscale.


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Extracellular vs. intracellularpotentials. Intracellular and extracellular potentials are often confused,andmodelerssometimescomparetheirmodelpredictionsofintracellularpotentials(whichareeasiertomodel) with recorded extracellular potentials (which are easier to measure). As seen in Fig. 3 theconnection between intracellular andextracellular is not trivial, however. To illustrate this furtherweconsider the above map of the Oslo subway system. With its branchy structure of different lines('dendrites')stretchingoutfromthehubatOsloCentralStation('soma'),thesubwaysystemresemblesaneuron.Ifwepursuethisanalogy,thesubwaystations(markedwithdots)maycorrespondto'neuronalcompartments'andthenetnumberofpassengersenteringorleavingthesubwaysystemateachstationto the net 'transmembrane current' at this 'compartment'. If more passengers enter than leave thesubwaysystematapoint intime, itmeans thatthenumberofpeople in thesubwaysystem, i.e.,the'intracellularmembranepotential',increases.(Ifweintroducea'capacitivecurrent'correspondingtothe

changein

the

number

of

people

inside

each

station,

we

can

even

get

a'current

conservation

law'.)

The

intracellular soma membrane potential, crucial for predicting the generation of neuronal actionpotentials (which luckilyhasnoclearanalogy innormalsubwaytraffic),wouldthencorrespondtothenumberofpassengerswithinthesubwaystationatOsloCentralStation.Theextracellularpotentialontheotherhand wouldbemore similar towhatcould be measuredby aneccentric (atbest)observercountingpassengersflowinginandoutofafewneighboringsubwaystations(withbinocularsonthetopofalargebuildingmaybe).Whiletheanalogyisnot100%,itshouldillustratethatwhileintracellularandextracellularpotentialsarecorrelatedquantities,theyarereallytwodifferentthings.

Thespikewidth increase implies that thehigherfrequenciescontained intheactionpotentialattenuate more steeply than the lower frequenciesas a functionofdistance from the soma.

Suchaspike

width

increase

has

been

seen

experimentally,

and

one

proposed

explanation

for

the effect is that the extracellular medium acts as a lowpass filter, for example throughfrequencydependentpolarizationofcellmembranes [27,28].However,directmeasurementsoftheimpedancespectrumforcorticaltissuefortherelevantfrequencieshavegivenconflictingresults:whileGabriel andcoworkers [29] claimed to find such frequencydependent filtering,Logothetisandcolleagues[25]measurednosuchfiltering.

In Fig. 3B we see that distancedependent lowpass frequency filtering of theextracellularspikes(i.e.,changeinspikewidth)isseenalsoforourhomogenous,isotropicandpurelyconductivemediumwithnoinherentfrequencyfiltering.Inaccordancewiththiswethusproposed inRef.[9]that theneuronmorphology,combinedwith itscablepropertiesgivesan


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alternativeexplanation fortheobserved increase in lowpassfilteringwith increasingdistancefromthesoma.

4.3 PhysicaloriginofdendriticfilteringNumerical exploration of a variety of neuron models in Ref. [9] showed that the distancedependent lowpass filtering effect for the extracellular potential is a generic property ofneurons.Thephysicalorigin lies inthecableequation itself,andtheneuronal lengthconstantbrieflyintroducedinEqs.(34)turnsouttobeakeyconcept.

Let us first consider the simple infinite ballandstick neuron model consisting of asphericalsomaconnectedtoaninfinitelylongdendriticstickofconstantdiameter d describedbythecableequation[2,9],seeFig.4.Thisballandstickneuronisfurtherassumedtohaveaninward steadystate (DC) transmembrane current in the soma. Since the transmembrane

currentsat

all

times

have

to

sum

to

zero,

the

same

amount

of

current

has

to

leave

through

the

dendriticstick.Fromthesolutionofthecableequationitfollowsthatthedensityfunctionofthedendritic return current decays exponentially with distance from the soma with the length

constant RG1/= [2,9].Itiscustomarytodescribe inspecificparameters,i.e.,parameters

that only depend on the physical properties of the membrane and the intracellular medium.

Thenwehave im/4= RdR where mR isthemembraneresistivity[2cm ],and iR istheaxial

resistivity [ cm ] [3, 9]. In the DC situation the length constant also corresponds to thedendriticpositionwherethesteadystatereturncurrenthasdecreasedto e1/ ofitsvalueatthesoma,oralternatively,thepositionwherethedendriticreturncurrenthasitscenterofgravity.The centerofgravity is then defined as the mean of the normalized transmembrane currentdensity

weighted

by

dendritic

position

[9].

The length constant is not only an important measure when describing the neuron'sintrinsic qualities (for example electrotonic compactness, i.e., how much the membranepotentialvariesacrosstheneurons)[2,3].Itisalsoveryusefulforunderstandingtheneuron'sextracellularpotential.Forexample,whencomputingtheextracellularpotentialfarawayfromanactiveneuronfiringanactionpotential,theballandstickneuronmaybeapproximatedbyanevensimplermodel,thedipolemodel[9].Thenallthereturncurrentisassumedtocrossthedendriticmembrane throughasinglepointadistance above thesoma,so that thesystemeffectivelyisdescribedasa(transmembrane)currentdipoleoflength,cf.Fig.4.

TraditionallythelengthconstantisonlydefinedfortheDCsituation,andonlyforinfinite

cables.

We

here

define

a

more

general

alternatingcurrent

(AC)

length

constant

)(AC

applicablealsofordendriticsticksoffinitelength.TheDClengthconstantcanbeconsideredtobetheweightedmeanpositionoftheDCreturncurrent,and inanalogytothiswedefinethegeneralized,frequencydependentlengthconstanttobe

,d|)(|

d|)(|=)(

m0

m0AC

zzi

zziz

l

l

(6)

where f2= is the angular frequency, and |)(| m zi is the amplitude of the sinusoidally

oscillatingtransmembranecurrentatapositionz whenasinusoidalcurrentisinjectedinthe


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Figure 4: Illustrationofballandstickneuronand itstransmembranereturncurrentdensity followingcurrentinjectioninthesoma(lowerarrows).Thedipolesize(distancebetweenupperandlowerarrows)isillustratedbothforahigh(hf)andalow (lf) frequency(B).Thedashedcirclesillustratethedistanceatwhichthetransitiontothefarfieldlimitoccurs.

soma [9].Thedendritic stick isassumed tobeorientedalong thepositivezaxis from 0=z (somaposition)to lz= .ForaninfinitestickEq.(6)reducesto[2,9]

,])(12/[1=)( 2AC

++ (7)

with denoting the membrane time constant, mm=/= CRGC where mC is the specific

membranecapacitance.The main feature of the functional dependence of )(AC is that it decreases with

increasingfrequency,cf.Fig.4B inRef.[9].The intracellularactionpotentialwaveform,cf.Fig.3A,consistsofacombinationoffrequencycomponents,andeachcomponentcanbeviewedasa somatic voltage source forcing sinusoidally varying currents into the dendritic stick. Thedecreaseof )(AC withfrequencyimpliesthatthereturncurrentsonaveragewillbelocated

closer to the soma for the highest frequency components than for the lowest frequencycomponents,andthusmakesmallercurrentdipolelengths.


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With the dendritic stick described by the linear cable equation, each frequencycomponent of the action potential can be considered independently. Further, if the dipolemodel isconsidered for theballandstickneuron,weexpect thateach frequencycomponent

showsa r1/ decaywithdistanceclosetothesoma,whileamuchsharper 21/r decayisexpectedin thefarfield limit[9].Thekeypointregarding lowpass filtering is that thetransition tothefarfieldlimitwilldependonthecurrentdipolelength,i.e.,theAClengthconstant )(AC ,and

thusimplicitlyonthefrequency.Thehigherfrequencycomponentswillthusreachtheirfarfieldlimits, where they are strongly attenuated, closer to the soma than the lowfrequencycomponents,seeFig.4.Theneteffectwillbea lowpassfiltering, i.e.,an increase inthespikewidthwithdistancefromsoma[9].Notethatthisreasoningappliesalsotoneuronswithmorecomplicatedgeometries,e.g.,withnumerousdendriticsticksprotrudingfromthesomas,sincethe contributions to the extracellularpotentialaddup linearly.Thisgeneric lowpass filteringeffect was in fact confirmed by direct numerical calculations for several different neuronal

morphologiesin

Ref.

[9].

4.4 Whatdeterminestheneuron'shorizonofvisibility?Ourdipoleapproximationtotheballandstickmodelcanalsogiveimportantinsightsintohowtheneuronalmorphologyandmembraneparametersaffectthesize(peaktopeakamplitude)oftheextracellularspike[9].Byconsideringeachfrequencycomponentoftheactionpotentialindividually one can derive a frequencydependent transfer function T mapping theintracellular somaticmembranepotential to theextracellularpotential.This transfer functionrevealshowtheextracellularspikeamplitudewilldependonthedendriticparameters(givena

particularintracellular

action

potential).

The

derivation

is

somewhat

involved,

see

Ref.[9],

but

thefinalexpressionsnearthesomaandinthefarfieldlimitare

,11

||,11

||2

2

far3/2

nearrR

d

rR

fCd

ii

m

~~ TT (8)

respectively3.Anotablefeatureoftheseexpressions istheabsenceofthemembraneresistance mR ;

thus, the size and shape of the extracellular spike is predicted to be independent of thisquantity.Wefurtherseethatthetransferfunction,andthusthesizeoftheextracellularspike,willdecreasewith increasingaxialresistance iR insidethedendrite.However,thedominating

intrinsic

neuronal

parameter

appears

to

be

the

dendritic

stick

diameter

d.

We

see

that

the

transferfunctionispredictedtogrowas 3/2d nearthesomaandas 2d furtheraway.Thisresultalso applies to situations where one has several dendritic sticks attached to the soma [9]. Arough ruleof thumbdeduced from theseconsiderations is thataneuron'sextracellular spikeamplitudeisapproximatelyproportionaltothesumofthedendriticcrosssectionalareasofall

dendriticbranchesconnectedtothesoma.Thus,neuronswithmany,thickdendritesconnectedto soma will produce largeamplitude spikes, and will therefore have the largest radius of

3This'farfield'expressionfortheballandstickneurontransferfunctionisnotvalidwhenmovinghorizontally

awayfromthesoma.Inthisdirectionthetransferfunctionisgivenbyafarfieldquadrupolarexpression,see.Eq.(24)inRef.[9].


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visibility.ThevalidityofthisruleofthumbwasshownbydirectnumericalsimulationsinRef.[9],alsoformorphologicallyreconstructedneuronswithcomplicateddendriticgeometries.

5 ConcludingremarksThe modestlooking cable equation now has a more than 150 year long history, but will notretiresoon.Thebirthof theequationwascertainlyspectaculardescribingsignalprocessing inthetransatlantictelegraphcable,themostchallengingandprestigioustechnologicalprojectofits time.But the future ismaybeevenbrighter.Oneof themostexcitingresearchprojectsofthis century is to figure out how we think. It is difficult to know for sure whether mankindeventuallywillbeabletosortthisout,butifwedo,thecableequationwillhavetobeatcenterstage.

Acknowledgment:We

thank

Henrik

Lindn

for

help

with

making

Figure

3.

References

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Documents

Equivalent‐circuit modeling of a piece of neuronal cable by means of the cable equation