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