Intrinsic dendritic filtering gives low-pass power spectra of local field potentials

J Comput Neurosci (2010) 29:423–444DOI 10.1007/s10827-010-0245-4

Intrinsic dendritic filtering gives low-pass power spectraof local field potentials

Henrik Lindén · Klas H. Pettersen · Gaute T. Einevoll

Received: 31 August 2009 / Revised: 30 March 2010 / Accepted: 6 May 2010 / Published online: 26 May 2010© Springer Science+Business Media, LLC 2010

Abstract The local field potential (LFP) is among themost important experimental measures when probingneural population activity, but a proper understandingof the link between the underlying neural activity andthe LFP signal is still missing. Here we investigate thislink by mathematical modeling of contributions to theLFP from a single layer-5 pyramidal neuron and asingle layer-4 stellate neuron receiving synaptic input.An intrinsic dendritic low-pass filtering effect of theLFP signal, previously demonstrated for extracellularsignatures of action potentials, is seen to strongly affectthe LFP power spectra, even for frequencies as low as10 Hz for the example pyramidal neuron. Further, theLFP signal is found to depend sensitively on both therecording position and the position of the synaptic in-put: the LFP power spectra recorded close to the activesynapse are typically found to be less low-pass filteredthan spectra recorded further away. Some recordingpositions display striking band-pass characteristics of

Action Editor: Abraham Zvi Snyder

H. Lindén · K. H. Pettersen · G. T. EinevollDepartment of Mathematical Sciences and Technology,Norwegian University of Life Sciences,PO Box 5003, 1432, Ås, Norway

H. Lindéne-mail: [email protected]

K. H. Pettersene-mail: [email protected]

G. T. Einevoll (B)Center for Integrative Genetics,Norwegian University of Life Sciences,Ås, Norwaye-mail: [email protected]

the LFP. The frequency dependence of the propertiesof the current dipole moment set up by the synaptic in-put current is found to qualitatively account for severalsalient features of the observed LFP. Two approximateschemes for calculating the LFP, the dipole approxima-tion and the two-monopole approximation, are testedand found to be potentially useful for translating resultsfrom large-scale neural network models into predic-tions for results from electroencephalographic (EEG)or electrocorticographic (ECoG) recordings.

Keywords Local field potential · Single neuron ·Forward modeling · Frequency dependence · EEG

1 Introduction

Extracellular recordings have been, and still are, amongthe most used methods for probing neural activity.This popularity mainly stems from the spike-countingabilities of sharp electrodes: when placed sufficientlyclose to a particular neuronal soma, such electrodeswill measure a sequence of standardized extracellularpotential signatures, each signalling the presence of anaction potential in that particular neuron. Informationabout spiking is commonly extracted from the highfrequency band (� 500 Hz) of the recorded extracel-lular potentials. The interpretation of the local f ieldpotential (LFP), i.e., the low-frequency part (� 500 Hz)of extracellular potentials, is generally not so straight-forward. The LFP appears to be dominated by dendriticprocessing of synaptic inputs, not firing of action poten-tials (Nunez and Srinavasan 2006; Einevoll et al. 2007;Pettersen et al. 2008), and the LFP measured at any

424 J Comput Neurosci (2010) 29:423–444

point will have sizable contributions from neuronslocated many hundred micrometers away (Kreimanet al. 2006; Liu and Newsome 2006; Berens et al. 2008;Lindén et al. 2008, 2009a; Katzner et al. 2009; Xinget al. 2009).

The advent of new silicon-based multicontact electri-cal probes in various geometrical arrangements, such as‘multi-shank’ (Buzsáki 2004) or ‘needlepad’ (Normannet al. 1999), offers new exciting opportunities for mas-sively parallel recordings of LFP. Thus LFP certainlyhas the potential of becoming one of the most im-portant experimental measures when probing neuralpopulation activity (Mitzdorf 1985; Arieli 1992; Di et al.1990; Einevoll et al. 2007; Kreiman et al. 2006; Nauhauset al. 2009). This will require, however, a substantialimprovement in our understanding of the link betweenthe underlying activity in neurons and the recordedLFP signal. The present model study aims to add to thisunderstanding by investigating the contribution to theLFP signal from individual neurons receiving synapticstimulation. Due to the linearity of electromagnetism, arecorded LFP signal will be built up by a linear sum ofsuch contributions from individual neurons located inthe vicinity of the electrode contact. The insight gainedby studying the LFP signals generated by individualneurons will thus be of great help when embarking onthe larger project of linking measured LFPs to activityin populations of neurons or comprehensive neuralnetworks (Lindén et al. 2009a, b).

In the present study we particularly address thequestion on the origin of observed frequency spectrain LFP and EEG (electroencephalography) recordings(Pritchard 1992; Freeman et al. 2003; Bedard et al.2006b; Buzsaki 2006; Bedard and Destexhe 2009;Milstein et al. 2009; Miller et al. 2009). In Pettersen andEinevoll (2008) we described an unavoidable low-passfrequency-filtering effect of the extracellular action-potential signature due to the electrical cable propertiesof the neuronal dendrites. In the present paper we findfor our example layer-5 pyramidal neuron that the sameintrinsic dendritic filtering effect also strongly affectsfrequencies down to about 10 Hz, i.e., well into the typ-ical LFP and EEG frequency bands. Moreover, we findthe detailed frequency-filtering effects to vary stronglywith recording position: for apical synaptic stimulationthe low-pass filtering effects are most prominent forrecording positions near the soma, and vice versa. Wealso consider a spatially more compact layer-4 stellateneuron. The same low-pass filtering effect is observed,although with a higher cut-off frequency than for thespatially more extended layer-5 pyramidal neuron.

The use of dipole and other multipole momentsin the modeling of bioelectric signals has a long his-

tory (Plonsey 1969; Plonsey and Barr 2007). The cur-rent dipole concept has been particularly importantin the interpretation of EEG signals, but there onehas typically considered ‘mesoscopic’ current dipolesrepresenting the collective effect from large numberof activated neurons (Nunez and Srinavasan 2006).In a recent study, however, we found the current di-pole momentfrom a single neuron to be a very use-ful concept for gaining thorough understanding of thecharacteristics of extracellular signatures of action po-tentials (Pettersen and Einevoll 2008). Likewise, wehere find the concept to be very useful for obtainingbetter understanding of the results from our numeri-cally comprehensive calculations of LFPs generated bysynaptic activation of individual neurons with complexdendritic morphologies. Further, used in combinationwith the standard far-field dipole approximation fromelectrostatics (Jackson 1998; Plonsey and Barr 2007),this quantity is even found to provide quantitativelyaccurate predictions of the LFP a millimeter or moreaway from the neuron. This dipole approximation, aswell as a two-monopole approximation also exploredhere (Freeman 1980), may even find its use in ambitiouslarge-scale neural network modeling schemes aspir-ing to predict results from EEG recordings or ECoGrecordings, i.e., recordings done at the cortical surface.

Preliminary results from this project were presentedearlier in poster format (Lindén et al. 2008).

2 Methods

2.1 Forward modeling of extracellular potentials

Extracellular potentials are generated by transmem-brane currents, and in the presently used volumeconductor theory the system is envisioned as a 3-dimensional smooth extracellular continuum with thetransmembrane currents represented as volume currentsources (Nunez and Srinavasan 2006). In volume con-ductor theory the fundamental formula for the contri-bution to the extracellular potential φ(r, t) from theactivity in an N-compartment neuron model is givenby (Nicholson and Freeman 1975; Holt and Koch 1999;Pettersen and Einevoll 2008; Pettersen et al. 2008)

φ(r, t) = 1

4πσ

N∑

n=1

In(t)|r − rn| . (1)

Here In(t) denotes the transmembrane current in com-partment n positioned at rn, and σ is the extracellularconductivity. This formula relies on a set of assump-tions and approximations: The first is the use of the

J Comput Neurosci (2010) 29:423–444 425

quasistatic approximation of Maxwell’s equations. Thisamounts to neglecting the terms with time derivativesof the electric field E and magnetic field B from theoriginal Maxwell’s equations so that the electromag-netic field effectively decouples into separate ‘quasi-static’ electric and magnetic fields (Hämäläinen et al.1993). Then the electric field E in the extracellularmedium is related to the extracellular potential φ viaE = −∇φ. For frequencies inherent in neural activity,i.e., less than a few thousand hertz, the quasistaticapproximation seems to be well fulfilled (Hämäläinenet al. 1993).

Further, the formula assumes an infinite volume con-ductor where the electrical properties of the extracellu-lar medium are assumed to be (1) linear and frequencyindependent (Logothetis et al. 2007), i.e., j = σE wherej is the current density, (2) ohmic, i.e., no imaginarypart of σ (Nunez and Srinavasan 2006; Logothetis et al.2007), (3) position-independent, i.e., σ is the same ev-erywhere, and (4) isotropic, i.e., same σ in all directions(Logothetis et al. 2007). For a more comprehensivediscussion of these assumptions regarding the extracel-lular medium see Pettersen et al. (2010).

In a numerical scheme based on compartmentalneuron models, the formula in Eq. (1) can be useddirectly with the position rn set to correspond to acharacteristic ‘mean’ position of compartment n, e.g.,the center of a spherical soma compartment or themid-point of a cylindrical dendritic compartment. Thisscheme corresponds to the so called point-source ap-proximation since all transmembrane currents leavingor entering the extracellular medium from a particu-lar compartment are assumed to go through a singlepoint (Holt and Koch 1999; Pettersen and Einevoll2008). Another scheme, the line-source approximation,assumes the transmembrane currents from each cylin-drical compartment to be evenly distributed along aline corresponding to the cylinder axis (Holt and Koch1999; Pettersen and Einevoll 2008; Pettersen et al.2008). A line-source formula, analogous to the point-source formula in Eq. (1), can be found in Eq. (2) ofPettersen and Einevoll (2008) . All forward-modelingcalculations presented in this paper use the line-sourceapproximation with the extracellular conductivity set toσ = 0.3 S/m (Hämäläinen et al. 1993).

2.2 Compartmental neuron modeling

All simulations were carried out using the NEURONsimulation environment (Carnevale and Hines 2006)which was controlled via a Python interface (Hineset al. 2009). We used two different reconstructed cellmorphologies, a layer-5 pyramidal cell and a layer-

4 stellate cell from cat visual cortex (Mainen andSejnowski 1996), both downloaded from ModelDB athttp://senselab.med.yale.edu/. To assure sufficient nu-merical precision the length of each compartment of themodel neurons was chosen to be maximum one tenthof the electrotonic length at 100 Hz. This gave a totalof 1,072 compartments for the layer-5 cell and 343 com-partments for the layer-4 cell for our default choice ofpassive membrane parameters (see below). Simulationsin NEURON were performed at a time resolution of0.0625 ms. The default passive membrane parametersof the cell models were: specific membrane resistanceRm=30 k� cm2, specific axial resistance Ra=150 � cm,and specific membrane capacitance Cm=1.0 μF/cm2

(Mainen and Sejnowski 1996). In the testing of the de-pendence of LFP power spectra on model parameterswe also considered specific membrane resistances Rm

and specific axial resistances Ra corresponding to 1/10and 10 times the default values.

In one application the synaptic input current Is(t)was modeled as an α-function, that is,

Is(t) = I0tτs

e1−t/τs θ(t) , (2)

where θ(t) is the Heaviside unit step function. Note thatan excitatory synaptic input corresponds to a negativevalue of the current amplitude I0. Simulations withthis stimulus was run for 100 ms prior to onset of theα-current to avoid any possible inaccuracies from up-start effects.

The input currents used for calculating LFP powerspectra were created as a sum of sinusoidal currents,

Is(t) = I0

1000∑

f=1

sin(2π f t + γ f ) (3)

where γ f represents a random phase for each frequencycontribution. This current contains all frequencies be-tween 1 Hz and 1,000 Hz in steps of 1 Hz, has a flatfrequency spectrum, and is for the purpose of this studyequivalent to a ‘white-noise’ input current. The simula-tions were run for 1,200 ms. The calculated LFP signalsfrom the last 1,000 ms were used in the evaluation ofthe power spectra which were obtained by squaringthe Fourier amplitudes of the calculated LFP for eachfrequency.

Since the cable equation describing the presentpassive neuron is linear, the assumption of currentsynapses and passive neuronal membrane model makesthe overall model linear. This means that all calcu-lated extracellular and intracellular potentials will beproportional to the current amplitude I0. Further, inthe case of a sinusoidal input current, all extracellular

http://senselab.med.yale.edu/


and intracellular potentials will also be sinusoidallyoscillating with the same frequency, although in generalwith a different phase. The linearity of the model makesit easier to analyze. However, we expect that most ofthe observed qualitative features would remain if we,e.g., instead had considered excitation by conductance-based synapses.

2.3 Current dipole moments

Current dipole moments in the x-, y-, and z-directionswere calculated as follows:

px(t) =N∑

n=1

xn In(t) , py(t) =N∑

n=1

yn In(t) ,

pz(t) =N∑

n=1

zn In(t) . (4)

where xn, yn, and zn are the x, y and z positions ofcompartment n. When the synaptic input current is si-nusoidal, i.e., Is(t) = I0 sin(2π f t), these dipole momentswill due to the linearity of the system be of the form

px(t) = p0x( f ) sin(2π f t + γ f x) ,

py(t) = p0y( f ) sin(2π f t + γ fy) ,

pz(t) = p0z( f ) sin(2π f t + γ f z) . (5)

In the special case when the phases are identical, i.e.,γ f x = γ fy = γ f z ≡ γ f , the system simplifies to a sinu-soidally oscillating dipole along a f ixed axis,

pfa(t) = p0( f ) sin(2π f t + γ f ) (6)

where p0( f ) is a frequency-dependent vector given by

p0( f ) ≡ p0x( f )ex + p0y( f )ey + p0z( f )ez . (7)

(Note that the system also reduces to a fixed-axisoscillating dipole when one of γ f x, γ fy, or γ f z is offby a factor π compared to the two others; then thefactor π can be eliminated by changing the sign of the‘amplitude’ in front of the particular sinus function.)

In the application of the dipole approximation forLFP generation in Sections 3.5 and 3.6 we assume thephases of the three cartesian dipole-moment compo-nents to be the same (modulo a factor π) so that a lineardipole is obtained. In order to specify the direction ofthis linear dipole, one also has to decide the directionsof the three components relative to each other. Herewe set these directions, for all frequencies considered,to correspond to the current-dipole component direc-tions found in the low frequency limit (1 Hz). Withsuch slowly oscillating input currents the phase shiftsbetween the dipole components are negligible, and the

assignment of directions follows directly from evalu-ating the sign of, say, the x and y components of thecurrent dipole at the first peak for the z component.

3 Results

3.1 Example local field potentials

In Fig. 1 we show examples of patterns of extracellularpotential traces around reconstructed neurons. Theseextracellular potentials are generated by single synapticinputs for various synaptic positions and neuronal den-dritic morphologies. The synaptic input currents are inall examples chosen as α-functions (cf. Eq. (2)) with atime constant of 2 ms and a peak amplitude of 1 nA.In the following we will denote such synaptically gen-erated extracellular potentials as local f ield potentials(LFPs).

In Fig. 1(a) we show results for a reconstructed layer-5 pyramidal neuron, taken from Mainen and Sejnowski(1996), receiving a single excitatory synaptic input atan apical branch. The form of the injected synapticcurrent is illustrated by the inset panel. An importantfeature which is immediately apparent is that the shapeand amplitude of the LFPs depend strongly on the re-cording position. Near the apical synaptic input theLFP signature is generally negative, reflecting that theexcitatory current synapse providing a current sinkdominates the sum in the forward-model formula, cf.Eq. (1). At positions close to the soma the LFP is alwayspositive, reflecting that return currents in the soma areadominate the sum.

This characteristic dipolar structure is further illus-trated by the contour plots shown in the figures asgrey solid and dashed lines. These logarithmic contourplots show how the maximum value for LFP signalamplitude, i.e., maximum positive or negative deviationof LFP from baseline following the synaptic input,varies with position. For spatial positions above and tothe left of the synaptic input in Fig. 1(a), this largestdeviation corresponds to a negative LFP peak, and thedashed contour lines in this region thus correspondsto ‘equi-LFP’ lines with negative numerical values ofthe LFP. For spatial positions around the soma, posi-tive LFP peaks dominate, and the solid contour linescorrespondingly represent ‘equi-LFP’ lines with posi-tive values of the LFP. Regardless of the sign of themaximum LFP amplitude, we generally observe a rapiddecay of the LFP signal amplitude with distance: whenmoving away from the neuron, each new contour linerepresents a reduction of the LFP amplitude by a factortwo compared to the previous contour line.


0.2 mm

50 ms

0.1 µV

0.1 µV

0.1 µV

(b)

(d)(c)

0.1 µV

(a)1 nA

x

z

Fig. 1 Examples of calculated local field potentials (LFPs) fol-lowing an excitatory synaptic input into purely passive neu-ron models. The synapse is current-based and modeled as anα-function (Eq. (2)) with an amplitude of 1 nA, i.e., I0 = −1 nA,and a time constant τs = 2 ms; see panel inset in (a) for il-lustration of synaptic current. Neuron models are passive withthe following specific electric parameters: membrane resistivityRm = 30 k� cm2, axial resistivity Ri = 150 � cm, membranecapacitance Cm = 1 μF/cm2. Extracellular potentials (thick solidlines) at selected spatial positions (marked with dots on tracestarting points) are shown in 50 ms windows. Grey contour linesillustrate maximal extracellular potential (LFP) amplitudes, i.e.,maximum positive or negative deviation of LFP from baselinefollowing the synaptic input. Contour plots are logarithmic and

based on log2, i.e., the LFP amplitude decays by a factor 2 be-tween each contour line. Solid contour lines are ‘equi-LFP’ linescorresponding to positive values for the LFP amplitude, dashedcontour lines to negative values of the LFP amplitude. (a) Resultsfor reconstructed L5 pyramidal neuron (solid branch structure)from Mainen and Sejnowski (1996) with single excitatory synapse(solid dot) on apical branch. The neuron has been rotated so thatboth the synapse and soma are in the 2D plane of the plot. (b)Corresponding results as in (a) for a single excitatory synapsein soma. (c) Results for reconstructed L4 stellate neuron fromMainen and Sejnowski (1996) with single excitatory synapse ondistal branch. (d) Corresponding results for a single excitatorysynapse in soma

An excitatory synaptic input onto the basal dendritesof the pyramidal neuron will typically give the oppositepattern: negative LFPs around the basal synaptic inputand positive LFPs around the apical dendrites (resultsnot shown). However, due to the asymmetric dendriticbranching with a dominant apical dendrite, even an ex-citatory input onto the soma gives a prominent dipolar

spatial pattern for the extracellular potentials. This isdemonstrated in Fig. 1(b). Here we show results frominjecting the same current used in Fig. 1(a) into thesoma instead of into the apical synapse, and a spatialLFP pattern similar to what might be expected for anexcitatory basal input is indeed observed. In Fig. 1(b),however, we see that the amplitudes of the LFP signal


are generally much smaller than for the situation withapical excitation depicted in Fig. 1(a). A qualitativeexplanation is that presumably more of the injectedsynaptic current returns to the extracellular mediumcloser to the injection point in the case with somaticinput, than for input onto the apical dendrite. There willthus be more cancelation of the contributions from thevarious terms in the sum in Eq. (1).

A close inspection of the equipotential contour linesin Fig. 1(b) reveals a deviation from the simple dipo-lar pattern. In a region inside the basal dendritic treejust below the soma, the LFP signals with the largestmagnitudes are seen to be positive, not negative asexpected from the dipolar structure. In this region theearly positive peak prominently seen in the potentialtrace to the right of and below the soma, has a largermagnitude than the peak of the following negative LFPsignal. If we instead had plotted equipotential contourlines for the dominant, i.e., late, LFP peak, a cleandipolar pattern would have been seen.

The layer-5 pyramidal neuron in Fig. 1(a) and (b) hasa characteristic ‘open-field’ dendritic structure wherethe synaptic input currents and the bulk of the re-turn currents may be substantially separated in space(Johnston and Wu 1995). This will imply a sizablecurrent dipole and, in turn, a sizable contribution to theexperimentally recorded LFPs. For example, in panel(a) the distance between the position of the apicalsynaptic input and the soma, where a large part of thereturn current exits, is about a millimeter.

In Fig. 1(c) and (d) we show corresponding resultsfor a layer-4 stellate neuron where the dendritic struc-ture has a spherically symmetric appearance, implyinga more ‘closed-field’ structure (Lorente de Nó 1947;Rall 1962; Johnston and Wu 1995). As shown in panel(c), a distal synaptic input onto a dendritic branch willnevertheless produce a dipolar LFP pattern around thestellate neuron where the dipole axis will be orientedalong a line between the synaptic input and the soma.Only when the set of distal inputs are distributed sym-metrically around the soma can significant cancelationeffects on the resulting LFP be expected. However, thecancelation can never be expected to be complete dueto unavoidable asymmetry in the dendritic structure.The dendritic asymmetry as seen from the soma for thepresent stellate neuron is illustrated in Fig. 1(d) wherethe LFP pattern following a synaptic input into thesoma is shown. Also for this situation a dipolar patternof LFPs is observed, albeit generally with reduced am-plitudes compared to the distal-excitation situation in(c). A comprehensive investigation of the dependenceof sizes and shapes of the LFP patterns on the neuronalmorphologies and synaptic input positions is beyond

the scope of the present paper, but the results in Fig. 1caution us that one must be careful before a priorineglecting contributions to experimentally recordedLFP from populations of neurons with stellate dendriticstructures.

Close inspection of Fig. 1(a) reveals another qualita-tive feature: the LFP signals close to the synaptic inputspositions have ‘narrower’ temporal traces compared tosignals further away. In other words, the LFPs recordedclose to the synaptic inputs have more power at highertemporal frequencies than LFPs recorded further away.This position-dependent temporal filtering of the LFPis also revealed by the shift in the times of the peaksof negative LFPs compared to the peak times for thepositive LFPs in Fig. 1(a): we find, for example, that thepeak of the purely negative LFP trace depicted immedi-ately above the synaptic current-injection point, occurs5 ms before the maximum of the purely positive LFPtrace depicted immediately below the soma. This effectmay at least partially underlie observed asynchronouspositive and negative peaks of LFPs found in depth-resolved intracortical recordings in humans; in Fig. 2 inGodey et al. (2001) the negative LFP peak at around45 ms in electrode H’1 is, for example, seen to occurabout 5 ms prior to the positive LFP peak seen inelectrode H’3.

This intrinsic dendritic filtering effect is a genericand unavoidable feature of spatially extended neuronalstructures (Pettersen and Einevoll 2008), but the de-tailed characteristics of it will be determined by den-dritic morphology, electrical cable properties of thedendrites, positions of the current-injecting synapses, aswell as the recording position.

3.2 LFP power spectra depend on recording positions

In Fig. 2 we show calculated LFP power spectra, i.e.,the square of Fourier amplitudes, for extracellular po-tentials in the frequency range between 1 and 1,000 Hz.Note the double-logarithmic axes. The same passivelayer-5 pyramidal neuron as in the upper panels ofFig. 1 is considered, and we show LFP power spectrafound from white-noise current injection (Eq. (3)) intothe position of the synapse on the apical branch inFig. 1(a). For easy comparison we have also drawn linescorresponding to 1/ f and 1/ f 2 power laws of the LFPin an auxiliary panel.

An immediate observation is that the shape of thespectra is very different at the different recording po-sitions and consequently that no general power-lawbehavior of the LFP spectra is found. The highest fre-quencies are to a large extent retained at positions closeto the synaptic input, where the frequency at which


0.2 mm

(a) (b)

1 Hz100 Hzx

z

1 10 100

0.10.01

0.0010.0001

1/f

1/f 2

frequency [Hz]

pow

er

1

(c)

norm

aliz

ed p

ower

norm

aliz

ed p

ower

membrane potential

membrane currents

Fig. 2 (a) Calculated power spectra of local field potentials(LFPs) following injection of white-noise current into a purelypassive layer-5 pyramidal neuron (grey solid branch structure)from Mainen and Sejnowski (1996). The current injection pointon the apical branch is marked with a circle. Same passiveparameters as in Fig. 1. Doubly logarithmic power spectra in thefrequency range 1–1,000 Hz of extracellular potentials (thick solidlines) are shown at selected spatial positions (marked by f illed ar-rows). The power spectra are normalized to the maximum valuein the range 1–1,000 Hz. See separate legend box in the upperright part of panel (a) for description of the logarithmic frequency

and power axes. Logarithmic contour plot with grey solid linesillustrate the decay of the 1-Hz power, i.e., square of Fourieramplitude, of the extracellular potential with distance from neu-ron. The power decays by a factor 4 between each contour line.Grey dashed contour lines correspondingly illustrate the decayof the 100-Hz power. (b) Normalized doubly-logarithmic powerspectra of transmembrane currents at three different positions(synaptic current-injection point, circle; intermediate position,triangle; soma, square) marked with corresponding symbols in(a). (c) Same as (b), but for membrane potential

the power is reduced to one hundredth, is seen to besubstantially above 100 Hz. In fact, the position imme-diately below the synaptic input shows a very modestlow-pass filtering effect, and the power at 1,000 Hz isstill as large as one tenth of the 1 Hz amplitude. Thelow-pass filtering effect is seen to be much stronger atthe recording position closest to the soma where the‘one-hundredth cut-off frequency’ is seen to be slightlyless than 100 Hz. At some positions outside the mainstalk of the apical dendrite one even observes a strikingband-pass effect: the largest LFP power is found forfrequencies between 10 and 100 Hz.

The above observations can be qualitatively un-derstood by inspection of the corresponding powerspectra of the membrane currents in Fig. 2(b). Themembrane current at the synaptic injection point hasby construction a flat (white-noise) spectrum (markedwith a circle), and LFPs recorded in the vicinity ofthis current-injection point will thus retain substantialcontributions from high frequencies. Due to the passiveelectrical properties of the dendritic cable structure, thesomatic membrane current will be low-pass filtered asseen in the power spectrum for the soma (marked witha square). LFPs recorded in the vicinity of the soma


will be dominated by this soma membrane current andwill thus exhibit similar power spectra. This is indeedconfirmed by visual comparison with the power spectraof the LFPs recorded closest to the soma which showsessentially the same power spectrum.

The membrane current at the intermediate posi-tion on the dominant apical dendrite (marked with atriangle) has a smaller low-pass filtering effect com-pared to the soma membrane current, and the low-passfiltering effect of the LFPs recorded in the vicinity ofthis point is correspondingly smaller as well. Here asomewhat poorer agreement between the LFP spec-trum and the corresponding membrane-current spec-trum is observed. This reflects that the membrane area,and thus membrane current, around this intermediateposition is small compared to both the basal and apicaldendritic regions. Thus the recorded LFP signal willget significant contributions from the apical and basalparts of the dendritic tree, and the LFP power spectrumwill be determined by a mixture of membrane-currentcontributions.

For comparison we also show in Fig. 2(c) the powerspectra for the membrane potential at the same threepositions on the pyramidal neuron. A much strongerlow-pass filtering effect is observed for the membranepotentials compared to the membrane currents. Sincethe LFPs are determined by a weighted sum overmembrane currents and not membrane potentials, themembrane potential is expected to be a poor predictorof the LFP. This is indeed confirmed by comparison ofthe power spectra of the membrane potential and theLFPs.

3.3 LFP power spectra depend on synaptic position,neuronal morphology and neuron parameters

In Fig. 3(a) we show the corresponding LFP powerspectra for the situation where the synapse is at thesoma of the layer-5 neuron instead of at an apicalbranch, cf. Fig. 1(b). Here the situation is reversedcompared to Fig. 2(a): a strong low-pass filtering effectis now seen at the recording positions outside the apicaldendrite, while the high frequencies are to a large ex-tent retained at the recording position immediately out-side the soma. This confirms the rule that LFP powerspectra will exhibit less low-pass filtering close to thesynaptic input position than further away (in analogywith the observation in Pettersen and Einevoll (2008)that the extracellular spike signature will become morelow-pass filtered when moving away from soma).

In Fig. 3(b) we show LFP power spectra for the layer-4 stellate neuron receiving synaptic input at a distalpart of a dendrite, in analogy to Fig. 1(c). Also here we

see prominent low-pass filtering effects, although witha higher cut-off frequency and less variation of powerspectra with recording positions compared to the layer-5 neuron. This illustrates that the dendritic morphologyof the neuron strongly affects their contribution tothe LFP signal, and in particular that the length ofthe dendrites is an important parameter determiningthe frequency dependence of the intrinsic dendriticfiltering (Pettersen and Einevoll 2008).

While Figs. 2 and 3 together illustrate the strongdependence of the intrinsic dendritic filtering effect ofthe LFP on recording position, synaptic input positionand dendritic morphology, Fig. 4 illustrates how thisfiltering effect is affected by the passive electrical pa-rameters in the neuron model. To highlight the qualita-tive effects we look at the extreme and likely unrealis-tic situations where the specific membrane resistanceRm and specific axial resistance Ra are increased orreduced by an order of magnitude compared to thedefault parameter values.

In Fig. 4(a) and (b) we show how the power spectrashown in Fig. 2(a) for the layer-5 neuron with api-cal synaptic input are changed when Rm is reduced(Fig. 4(a)) or increased (Fig. 4(b)) by a factor ten,respectively, compared to the default value Rm=30k� cm2 used in Fig. 2(a). In Fig. 4(a) where the mem-brane resistance Rm is reduced by a factor ten, wesee that the low-pass cut-off frequencies are shifted tohigher values compared to the results for the defaultparameters in Fig. 2(a). This can be qualitatively under-stood since the reduction of the membrane resistanceimplies a corresponding reduction of the membranetime constant τm = RmCm by a factor ten. This inturns implies that the characteristic frequencies of thesystem, such as low-pass cut-off frequencies, will beincreased. We also observe that the characteristic band-pass spectra for positions outside the apical dendriticstalk in Fig. 2(a) is absent in Fig. 4(a). This illustratesthat this resonance-like phenomenon depends on a de-tailed interplay between the neuronal morphology andits electrical cable properties. The inverse relationshipbetween the membrane resistance Rm, and thus mem-brane time constant τm, and the characteristic frequen-cies of the system is further illustrated by observationsof the depicted spatial pattern of equipotential contourlines in Fig. 4(a). For this reduced membrane-resistancecase the 100-Hz contour line is much more spatiallyextended and thus similar to the 1-Hz contour lines,than for the default situation in Fig. 2(a).

The opposite effect is observed in Fig. 4(b) where thespecific membrane resistance is increased by a factorten, and the typical cut-off frequencies as expected areshifted towards smaller values compared to the default


(a) (b)

x

z0.2 mm 1 Hz

100 Hz

1 10 100

0.10.01

0.0010.0001

1/f

1/f 2

frequency [Hz]

pow

er

1

Fig. 3 Calculated power spectra of local field potentials (LFPs)following injection of white-noise current into (a) the soma ofthe layer-5 pyramidal neuron considered in Fig. 2 and (b) and a

dendritic synapse of the layer-4 stellate cell (cf. Fig. 1(c)). Thecurrent injection points are marked with circles. See caption ofFig. 2 for detailed explanation of the plots

situation. But the change compared to the results inFig. 2(a) based on the default value of Rm, is lesscompared to decreasing the value of Rm by a factor often, cf. Fig. 4(a). This illustrates the fact that a changeof the membrane resistance not only changes the mem-brane time constant, but also the frequency-dependentlength constant of the current dipole (cf. Eqs. 10–11 inPettersen and Einevoll 2008). The overall change of theLFP spectra will thus depend on the detailed interplaybetween these variables and morphological quantitiesof the neuron and thus not follow a simple scalingrule suggested by considering changes in the membranetime constant alone.

The other key passive parameter is the specific axialresistance Ra, and in Fig. 4(c) and (d) we correspondinglyshow how the power spectra are altered when this para-meter is reduced (Fig. 4(c)) or increased (Fig. 4(d)) bya factor ten, respectively, compared to the default valueRa=150 � cm used in Fig. 2(a). In Fig. 4(c) where Ra isreduced by a factor ten, we see that the 100-Hz equipo-tential contour lines resemble the 1-Hz contour linesmuch more than for the results for the default value of

Ra in Fig. 2(a). This can be qualitatively understood onthe basis of the length constant of the current dipolefor an infinite cable which is proportional to 1/

√Ra

(Koch 1998; Pettersen and Einevoll 2008). A reducedaxial resistance implies a longer length constant, whichin turn implies that the neuronal dendrites becomeelectrically more compact. In analogy with the smalllow-pass filtering effect observed for frequencies below100 Hz for the short-stick stellate neuron in Fig. 3(b),we see in Fig. 4(c) that the low-pass filtering cut-offin general is shifted to higher frequencies comparedto the results for the default parameters in Fig. 2(a).This general shift of salient spectral features towardshigher frequencies is also seen in the ‘band-pass’ spec-tra outside the stalk of the apical dendrite. Here theband-pass peak is shifted upwards by almost an orderof magnitude compared to the result for the defaultvalue of Ra shown in Fig. 2(a). The opposite effect isobserved in Fig. 4(d) where the specific axial resistanceRa is increased by a factor ten: since the length constantis reduced, the neurons will be electrically less compactand the typical low-pass cut-off frequencies reduced.


(c) (d)

x

z

0.2 mm

/10Ra

(a) (b)/10Rm Rmx10

Rax10

1 Hz100 Hz

1 10 100

0.10.01

0.0010.0001

1/f

1/f 2

frequency [Hz]

pow

er

1

Fig. 4 Calculated power spectra of local field potentials (LFPs)for the same situation as in Fig. 2(a), but different passive neu-ron parameters. (a) As in Fig. 2, but with specific membraneresistance reduced by a factor ten to Rm=3 k� cm2. (b) As inFig. 2, but with specific membrane resistance increased by a factor

ten to Rm=300 k� cm2. (c) As in Fig. 2, but with specific axialresistance reduced by a factor ten to Ra=15 � cm. (d) As inFig. 2, but with specific axial resistance increased by a factor tento Ra=1,500 � cm. See caption of Fig. 2 for detailed explanationof the plots


3.4 Frequency dependence of membrane-currentdistribution

The previous subsections demonstrated the ubiquityof an intrinsic dendritic low-pass filtering effect of theLFP. They also demonstrated that the power spectra

vary strongly with recording position, and that thedetails of the spectra are determined by an interplaybetween neuronal morphology, synaptic input positionand passive cable parameters. Figure 2 also illustratedthe salient role of the transmembrane return currentsin determining the characteristics of LFPs. In Fig. 5 we

0.2 mm

membranearea density

transmembranecurrent density1 Hz 100 Hz

apic

al in

put

basa

l inp

ut

(a) (b) (c)

(d) (e) (f)

x

z

Fig. 5 Illustration of frequency dependence of distribution oftransmembrane currents following sinusoidal current injectioninto apical (upper row) or basal (lower row) synapse (thick arrowin neuron f igures). The length of the rightward oriented arrowsin the neuron figures in panels (a), (b), (d), and (e) represents theamplitude of transmembrane return currents following injectionof currents at apical (a, b) and basal (d, e) positions, see thickleftward oriented arrow. The depth distributions of the trans-membrane return currents for 1 Hz (solid), 10 Hz (dash-dotted),

and 100 Hz (dashed) are shown in panels (c) and (f) for apicaland basal excitation, respectively. Note that the lengths of thearrows representing the return currents are normalized to havethe same maximum return current in panels (a), (b), (d) and (e).In reality the amplitude of maximum transmembrane returncurrents is much larger for 100 Hz than for 1 Hz, see panels (c)and (f). The depth distribution of the total membrane area is alsoshown in panels (c) and (f)


illustrate further the core physical effect underlying thisobserved intrinsic dendritic low-pass filtering. In panel(a) the spatial distribution of transmembrane returncurrents (rightward arrows) following injection of asinusoidal input current (1 Hz) in the apical dendrite(single thick leftward-oriented arrow) is shown. Thelength of the return-current arrows illustrates the am-plitude of the corresponding sinusoidal return currents.As seen in panel (a) a substantial part of the apicallyinjected current returns through the soma and basaldendrites. The corresponding return-current patternfor a 100 Hz sinusoidal current is shown in panel (b).Here, essentially none of the injected current returnsthrough the basal region, all of it returns through theapical dendrites.

The effect is more clearly quantified by the curves inthe panel (c) in Fig. 5, showing the depth distributionof total return current along the vertical extension ofthe pyramidal neuron. For the 1-Hz situation almosthalf of the return current leaves through the basalregion where the depth distribution essentially followsthe depth distribution of the total membrane area, seepanel Fig. 5(c). For the 100-Hz case, on the other hand,almost all return current leaves through the apical den-drites. For 10 Hz an intermediate situation is observed.

In terms of generation of LFP both the 1-Hz and the100-Hz situations depicted in Fig. 5(a)–(b) correspondto ‘open-field’ situations; see, e.g., Johnston and Wu(1995): the weighted mean position of the return cur-rents is for both frequencies spatially displaced from thecurrent injection point. However, the spatial distancebetween the current injection point and the mean po-sition at which it returns to the extracellular medium,i.e., the current dipole length (Pettersen and Einevoll2008), is very different in the two cases. For 1 Hz thisdipole length is about half the distance between thecurrent injection point and the soma, while for 100 Hzit is reduced to less than one tenth of this distance. Thecurrent dipole strength, which essentially correspondsto the absolute value of the current dipole momentdescribed in Eqs. (4)–(6), is determined by the injectedcurrent multiplied by these current dipole lengths. Con-sequently, the dipole strength will be much larger forthe 1-Hz situation than for the 100-Hz situation. Atpositions some distance away from the neuron, the LFPcan be expected to be approximately proportional tothis current dipole strength (Pettersen and Einevoll2008). On these grounds the large reduction of theLFP in the 100-Hz situation compared to the 1-Hzsituation seen in the power spectra of Fig. 2, is thus notunexpected.

The above example with apical synaptic inputs ontoa large pyramidal neuron is the textbook example of an

‘open-field’ situation which may generate large extra-cellular potentials (Johnston and Wu 1995). However,as seen in the panels (d–f) in Fig. 5, also a synaptic inputonto a basal dendrite provide a current dipole. As seenin panels (d) and (f) a sizable fraction of the returncurrent crosses the membrane in the apical dendritesfor a 1 Hz sinusoidal current injected into the soma. Fora 100 Hz injection current, however, essentially all ofthe return current leaves at the basal part of the neuron,cf. panels (e) and (f) in Fig. 5.

3.5 Current multipole approximations for LFPgeneration

In Pettersen and Einevoll (2008) it was found thatthe extracellular potential signature of action potentialscould be well understood qualitatively, and to someextent also quantitatively, on the basis of frequency-dependent current dipoles accounting for the gross fea-tures of the spatial pattern of transmembrane currents.For action potential signatures the dominant frequen-cies are typically a few hundred hertz. For such highfrequencies the relevant current dipole lengths are ex-pected to be governed by the electrical cable propertiesof the dendrites protruding from the soma (Pettersenand Einevoll 2008). For the lower frequencies dom-inating the LFP, the current dipole length will to alarger extent be governed by the spatial extension ofthe entire dendritic structure. In Fig. 5, for example, wesee for the apical-excitation case that roughly half ofthe injected synaptic current returns through the basalregion, while the other half returns around the synapse.This implies that the current dipole length will be abouthalf the vertical distance between soma and synapse.

Multipole expansions, where the electrical potentialsare built up as sums over contributions from the variousmultipole terms (monopole, dipole, quadrupole, ...),have a long tradition in electrostatics (Plonsey 1969;Jackson 1998). Lately, this approach has also been usedto investigate the origin of the extracellular potentialsignatures of action potentials (Pettersen and Einevoll2008; Milstein and Koch 2008). In the present sec-tion we will pursue a similar approach and investigateto what extent simplified multipole expressions canaccount for the results based on our comprehensivemulticompartment forward-modeling scheme. In panel(a) of Fig. 6 we show the LFP-power contour plotresulting from our multicompartment modeling schemefor the situation where a 1 Hz sinusoidal current isinjected into a particular apical synapse (marked withan open dot). This comprehensive multicompartmentline-source scheme will provide the ‘gold-standard’


Fig. 6 Illustration ofvarious multipole (dipole,two-monopole)approximations for LFPmodeling. (a) Contour plotillustrating the variation ofthe 1-Hz power, i.e., square ofFourier amplitude, of theLFP with distance fromneuron following sinusoidalcurrent injection into anapical synapse (circle). Fullmulticompartmentforward-modeling scheme isused, i.e., Eq. (1). The arrowsillustrate the ‘vertical’ and‘horizontal’ directionsconsidered in detail in panels(d)–(g). (b) Same as A fordipole approximation, i.e.,Eq. (8). (c) Same as (a)for two-monopoleapproximation, i.e., Eq. (10).(d) LFP power along the‘vertical’ direction (see panel(a)) for multicompartment,dipole, and two-monopolemodeling schemes,respectively. Distance ismeasured relative to apicalsynapse (circle). Results for1 Hz and 100 Hz sinusoidalcurrent injections are shown,normalized to themulticompartment value for 1Hz. (e) Same as (d) for the‘horizontal’ direction asmeasured out from soma.Results are normalized to themulticompartment value for 1Hz. (f) Frequencydependence of the LFPpower for three particulardistances from the neuron(0.1 mm, 1 mm, 10 mm) in the‘vertical’ direction. Thespectra are normalized sothat the LFP power for themulticompartment model for1 Hz at the distance 10 μm isunity. (g) Same as (f) for the‘horizontal’ direction. Notethat also here the spectra arenormalized so that themulticompartment LFPpower for 1 Hz at the distance10 μm in the ‘vertical’direction is unity

(a)

(b)

(c)

(d)

(e)

(f)

(g)

norm

aliz

ed p

ower

norm

aliz

ed p

ower

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ed p

ower

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ower

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multicompartmentdipoletwo-monopole

µ


mmmm

mm


mmmm

mm


vertical

horizontal

x

z

0.2 mm

power spectrum horizontal

power spectrum vertical

comparison horizontal

comparison verticalmulti-compartment

dipole

two-monopole


results against which our multipole approximations willbe compared.

Due to current conservation there will be no currentmonopole contribution to the LFP, and the first non-zero contribution will in general come from currentdipoles. In the far-field limit, i.e., when the distance rto the current dipole is much larger than the currentdipole length, the dipole LFP expression is given by(Pettersen and Einevoll 2008):

φd(r, t) = 1

4πσ

|p(t)| cos θ

r2. (8)

Here p(t) is a current dipole moment which in general isgiven by

p(t) = px(t) ex + py(t) ey + pz(t) ez (9)

where px(t), py(t), and pz(t) are defined in Eq. (4).Further, θ is the polar angle relative to the dipole axisand r the radial distance to the dipole position (see, e.g.,Fig 2(a) in Pettersen and Einevoll (2008) for an illus-tration). With a sinusoidal input current it follows fromthe linearity of the model that the individual compo-nents, i.e., px(t), py(t), pz(t), also will vary sinusoidallyin time, cf. Eq. (5). In general the phases of the differentcomponents (γ f x, γ fy, γ f z) in Eq. (5) will be different,and the system will then not reduce to a dipole ori-ented along a particular fixed spatial direction. In thepresent application, however, we do the approximationof assuming identical phases, i.e., γ f x = γ fy = γ f z = γ f .We then obtain a sinusoidally oscillating dipole alonga f ixed axis with an oscillatory current dipole momentgiven by pfa(t) = p0( f ) sin(2π f t + γ ), cf. Eq. (6).

In panel (b) of Fig. 6 we show the LFP-power con-tour plots assuming this dipole approximation. This plotis found by applying pfa from Eq. (6) in the dipoleexpression in Eq. (8) to get the LFP signal. For this plotthe current dipole has been placed in the ‘middle’ of theneuron, i.e., straight above the soma at a height corre-sponding to halfway between the lowest and uppermostpoints of the dendritic structure. A visual comparison ofthe dipole-approximation results with the correspond-ing multicompartment results in panel (a) reveals a verygood agreement at distances far away from the neuron.Close to the neuron the agreement is as expected poor.

An obvious question to ask is how far away fromthe neuron one has to be in order for the dipole-approximation to work. As illustrated by panels (d) and(e) in Fig. 6, this depends on direction. Here we focuson two directions: (1) vertically up from the positionof the apical input current (labeled ‘vertical’, cf. panel(a)), and (2) horizontally out from the soma (labeled‘horizontal’, cf. panel (a)). The resulting LFP powerfor the ‘vertical’ direction is shown in panel (d). For

the 1-Hz situation depicted in panels (a) and (b) wesee that the dipole approximation (dotted line) givespredictions in excellent agreement with the multicom-partment model (solid line) for distances larger thanabout 1 mm. In panel (d) we also show the correspond-ing comparison when the input current is oscillating at100 Hz. Also here we observe agreement at distanceslarger than about a millimeter or two. The same com-parison for the ‘horizontal’ direction is shown in panel(e), and the same excellent agreement is observed in thefar-field limit. However, the transition to the far-fieldregime appears to occur for slightly larger distancesthan in the ‘vertical’ direction.

The results in panels (d) and (e) of Fig. 6 clearlydemonstrate the expected inadequacy of the far-fielddipole approximation (Eq. 8) for positions close to theneuron. The contour plot for the multicompartmentmodel in panel (a) shows that the 1-Hz LFP powerpeaks both around the synaptic-input position and thesoma, in accordance with the observation of the plots ofthe transmembrane currents in Fig. 5. This suggests analternative two-monopole approximation φtm(r, t) forthe LFP,

φtm(r, t) = Itm(t)4πσ

(1

|r − rsyn| − 1

|r − rsoma|)

, (10)

where the current is assumed to enter or exit the neu-ron only at two positions: the synapse and the soma.From current conservation it follows that the currentItm(t) entering the neuron at the apical synapse (rsyn)must leave at the soma (rsoma). In this two-monopoleapproximation the current Itm(t) does not correspondto the true synaptic input current. Rather, Itm(t) is setto give the correct magnitude of the current dipolemoment, i.e.,

Itm(t) ≡ |pfa(t)|/λtm . (11)

Here λtm ≡ |rsyn − rsoma|, i.e., the distance between thesynapse and soma. Since the direction of the currentdipole moment pfa(t) in general will not exactly co-incide with the direction from synapse to soma, thetwo-monopole approximation will not converge to thedipole approximation in the far-field limit. However,with this choice of Itm(t) one may still hope to obtainreasonably correct predictions for the LFP also forlarge distances.

In panel (c) of Fig. 6 we show the LFP-power contourplot for the 1-Hz situation assuming this two-monopoleapproximation, i.e., Eq. (10) with (the sinusoidally vary-ing) Itm(t) given by Eq. (11). A visual comparison ofthe contour plots for the two-monopole approxima-tion with the corresponding multicompartment plots in


panel (a) reveals qualitatively similar spatial patterns.And indeed, in panel (d) the two-monopole approxi-mation is seen to follow the multicompartment resultsclosely when one moves along the ‘vertical’ directiondepicted in panel (a). The agreement is particularlygood for current modulations of 1 Hz, but also for100 Hz the two-monopole approximation is seen togive reasonable results for distances larger than a fewhundred micrometers.

As seen in panel (e) of Fig. 6 the performance ofthe two-monopole model in the ‘horizontal’ direction ismuch poorer. For one, this model significantly deviatesfrom the multicompartment results also for the largestdistances. This error stems from the deviation betweenthe direction of the true current dipole moment and theline between the apical synapse and the soma. This de-viation can also be seen directly by close inspection ofthe contour plots in panels (a–c): the lines (in these 2Dplots) corresponding to zero LFP power have slightlydifferent directions for the dipole and two-monopoleapproximations. This ‘direction error’ of course alsoaffects the LFP predictions in the ‘vertical’ direction,but much less so.

Secondly, the two-monopole approximation predictsmuch too large LFP power close to the soma, i.e., fordistances smaller than about 100 micrometers in the‘horizontal’ direction. Such short distances correspondsto positions well inside the bush of basal dendrites, andas seen in panels (a) and (b) in Fig. 5, the transmem-brane currents in the basal region appear to be spreadout all over the basal dendrites, rather than focusedin the soma. The point-like monopolar expression inEq. (10) representing the contribution from the trans-membrane currents in the soma region thus clearlyprovides a poor approximation, as demonstrated quan-titatively by the curves in Fig. 6(e).

Further, while the soma region provides the dom-inant contribution to the return current for the 1-Hzsituation as illustrated, e.g., by the LFP power contourplots Fig. 2(a) and by the plot of the transmembranecurrents in Fig. 5(a), this is not generally the case forhigher frequencies. The corresponding results for the100-Hz situation in Figs. 2(a) and 5(a) reveal that thereturn currents are strongly shifted in the direction ofthe apical synapse so that the ‘center-of-gravity’ of thereturn current is far above the soma. This explainsthe even larger overestimation of the LFP power bythe two-monopole models seen in Fig. 6(e) for 100 Hzcompared to 1 Hz, for horizontal distances from thesoma less than a few hundred micrometers.

Finally, in panels (f) and (g) of Fig. 6 we investigatethe power spectra due to the intrinsic low-pass filteringeffect at three different distances (0.1 mm, 1 mm,

10 mm) along both ‘vertical’ and ‘horizontal’ directions.At all positions we observe sizable low-pass filteringeffects for frequencies above about 10 Hz. The exactfunctional form of the dampening of high frequenciesvary, but for the largest distances (1 mm, 10 mm) adecay of the LFP power approximately following 1/ f 2

relationships are commonly seen for frequencies above50 Hz. For the largest distance (10 mm) the dipoleapproximation is seen to predict the filtering character-istics excellently. For the two-monopole approximationthe success is more mixed: it works well in the ‘vertical’direction, but poorly in the ‘horizontal’ direction. Forthe intermediate distance (1 mm) the success of bothapproximations is more limited, and for the shortestdistance considered (0.1 mm) they essentially fail. Forexample, in the ‘horizontal’ direction a sharp dip in theamplitude spectrum is observed for frequencies around200 Hz in the full multicompartment model. This effectis likely due to details of the basal dendritic structureproviding subtle cancelation effects at these frequen-cies. Such a phenomenon can clearly not be accountedfor by our two approximate models where the detaileddendritic structure has been neglected. Further, in the‘vertical’ direction the full multicompartment modelpredicts a power-law behaviour close to 1/ f over a widefrequency range for this shortest distance (0.1 mm).This contrasts the predicted power laws for the two ap-proximate schemes which have much steeper frequencydecays.

3.6 Current dipole approximation for LFP

In the previous section we focused on the special casewhere the synaptic current was provided by a singleapical synapse. For this special case we found thatthe dipole approximation in Eq. (8), combined withthe approximate current dipole moment pfa(t) givenby Eq. (6), predicted the LFP power accurately atsufficiently large distances from the neuron, cf. Fig. 6.With the proper incorporation of frequency-dependentcurrent dipole moments, the dipole approximation alsoappeared to explain main features of the observed high-frequency dampening in this far-field limit.

In the present section we investigate to what extentthis LFP dipole approximation also applies for synapsespositioned on other parts of the dendritic tree. In panel(a) of Fig. 7 we first illustrate the magnitude and direc-tion of the current dipole moments for synapses at a fewselected places on the dendritic tree. Sinusoidal inputcurrent at three different frequencies, 1 Hz, 10 Hz,and 100 Hz, are considered. The figures illustrate sev-eral qualitative points: (1) the current dipole momentsare typically roughly directed along a straight line


(a) (c)

radial distance [µm]

0.2 mm

x

z

100 Hz

10 Hz

1 Hz

2 3

2 3

2 3

log-log

log-log

log-log

.

.

.

(b)

1/f

Fig. 7 Current dipole moments and the LFP dipole approxima-tion for various synaptic positions. (a) Illustration of directionand amplitudes of current dipole moments p0 (cf. Eq. (7)) fora set of synaptic positions (black dots) superimposed on thedendritic structure. The figure shows the projection of thesedipole moments into the 2D plane of the figure for the 1-Hz,10-Hz, and 100-Hz situations. Lengths of line segments are pro-portional to current dipole-moment amplitudes. Dot plots on theright illustrates the magnitude of the vertical component p0z,cf. Eq. (5), versus the vertical position of the synapse for thefull number of synapses (1,072; one for each compartment inthe multicompartment model). Dot plots above correspondinglyillustrate the magnitude of the horizontal component p0x for thesame group of synapses (for the 1-Hz and 100-Hz situations,only). Color code: 1 Hz sinusoidal synaptic current: light grey,10 Hz: darker grey, 100 Hz: black. (b) Illustration of frequencydependence of power of current dipole moments, i.e., square ofFourier amplitudes p0( f ) defined by Eq. (6), for four differentsynaptic positions marked in (a) distal apical dendrite (circle),main apical dendrite (triangle), soma (square), basal dendrite(star). The same synaptic current is injected for all four situations,and the results have been normalized to the power of the currentdipole moment for the distal apical-dendrite case for 1 Hz. (c)Illustration of accuracy of LFP dipole approximation for fullgroup of 1,072 synaptic input positions considered in (a). Meanvalues of logarithms of LFP values from full multicompartment

model φ (Eq. (1)) are illustrated by solid lines. Surrounding greybands above and below these lines correspond to the LFP valueswithin the standard deviation (of the logarithm of LFP) aroundthe mean. Results are shown as a function of radial distancefrom the ‘mid-point’ of the neuron (i.e., straight above the somaat a height corresponding to halfway between the lowest anduppermost points of the dendritic structure). For each radialdistance r the LFP is evaluated separately for the 1,072 synapticpositions considered. Further, for each of these synaptic positionsthe LFP is calculated at a randomly selected spatial point withthe constraint that the vertical position, i.e., z-value, does notgo above or below the vertical extension of the neuron. Themean value at each radial distance corresponds to the mean ofthe logarithm of these 1,072 calculated LFP values. Likewise,the standard deviation at each radial distance corresponds tothe standard deviation of the logarithm of these 1,072 calculatedLFP values. The dashed lines, with the surrounding grey bands,correspondingly illustrate the mean and standard deviation ofthe dipole-approximation values φd (Eq. (8)) found by usingthe same averaging procedure. The current dipole is placedin the ‘mid-point’ of the neuron as it is described above. Theinset figures show the same results with double-logarithmic axes(log10). All results are normalized to the value of the LFP powerfound at the smallest distance considered (50 μm) for the 1-Hzsituation in the full multicompartmental scheme


connecting the synaptic input position and the soma,and (2) the magnitude of the current dipole momentsis reduced from 1 Hz to 10 Hz, and even more so from10 Hz and 100 Hz. Thus also the current dipole momentexhibits a prominent intrinsic low-pass filtering effect.

Above and to the right of the neuron shown inFig. 7(a) a large collection of magnitudes of the hor-izontal (p0x, cf. Eq. (5)) and vertical (p0z) compo-nents of the current dipole moment are depicted. Eachpoint corresponds to a synapse placed in one of the1,072 compartments representing the neuron in themulticompartment calculation scheme. The horizontaland vertical current dipole components are plotted asfunctions of the horizontal and vertical positions ofthe synapses, respectively. For the vertical component(p0z) a main salient feature is immediately apparent:for 1 Hz the magnitude of the current dipole momentis roughly proportional to the distance from the somaregion. This applies for synapses both above and belowthe soma. A closer inspection of the figure reveals thatfor the 1-Hz input the ‘center-of-mass’-position, i.e., thesynaptic position for which a current input gives nocurrent dipole moment, is more than 0.1 mm above thesoma on the thick apical dendrite. Thus also a synapticcurrent injected into the soma will give a sizable currentdipole moment, and consequently also a sizable LFP, cf.Fig. 1(b).

The lack of generation of a current dipole momentdoes in any case not mean that no LFP will be gener-ated since higher order multipole moments will not bezero. For example, injection of current at the synapticposition about 0.1 mm above the soma resulting in anegligible current dipole moment, gives a sizable cur-rent quadrupole moment. We confirmed this by directnumerical evaluations of an LFP power contour plotfor this situation which indeed revealed a ‘clover-like’spatial pattern characteristic for quadrupoles (resultsnot shown).

The rough proportionality of the vertical componentof the current dipole moment with distance from somaseen for the 1-Hz situation in Fig. 7(a), suggests that thecurrent dipole qualitatively can be pictured as a fixedamount of net current entering in the synapse region,i.e., the synaptic input current minus the return currentin the vicinity of the synapse, with the same net currentleaving from the soma region. Then the vertical compo-nent of the dipole moment will be roughly proportionalto the vertical distance between synapse and soma, incorrespondence with what is observed in Fig. 7(a).

Qualitatively similar results are observed for thecurrent dipole moment in the 10-Hz situation. The mag-nitude of the dipole moments are generally somewhatsmaller than for 1 Hz, reflecting that the net current

propagating from the synapse to the soma, thus settingup the current dipole, is reduced due to the electricalcable properties of the dendritic structure. For 100 Hz aquite different picture is seen. Here a very small verticalcomponent of the current dipole moment is seen forapical synapses, reflecting that most of the injectedcurrent return to the extracellular medium in the apicalregion, cf. Fig. 5(b). The distribution of the magnitudesof the horizontal components p0x, shown above the de-picted neuron in Fig. 7(a), is less characteristic. But alsohere the 100 Hz dipole components are generally seento be much smaller than the 1 Hz dipole components.

In Fig. 7(b) we illustrate the frequency dependenceof the dipole moments further. Here we show thepower of the dipole moments, i.e., |p0( f )|2 wherep0( f ) is defined by Eq. (6), for frequencies between1 and 1,000 Hz, generated by injecting currents atfour different synaptic positions. Prominent low-passfiltering effects are observed for all situations, but thedetailed form of the power spectra vary. The strongestlow-pass filtering is seen for inputs at the apical synapse(marked with a circle in panel (a)) and in the soma(square in panel (a)). The low-pass filtering effect isweaker and shifted to higher frequencies for inputs onthe main apical dendrite (triangle in panel (a)) and onthe basal dendrite (star in panel (a)). This further illus-trates that the low-pass filtering effect seen in powerspectra of extracellular potentials not only depends onthe position of the recording electrode, but also onthe distribution of the synaptic inputs generating thepotentials.

Calculated current dipole moments, such as the onesdepicted in Fig. 7(a), can now be used to calculatethe LFP at various spatial positions using the dipolarexpression in Eq. (8). In panel (c) we compare resultsfor the LFP power calculated by this dipolar expres-sion against the results from the full multicompartmentmodel for our three different frequencies (1 Hz, 10 Hz,100 Hz). The panels compare, as a function of radial dis-tance, the mean and the spread of the logarithms of theLFP power calculated using the dipole-approximationwith the corresponding values found from using the fullmulticompartment scheme. These statistical measuresare found by averaging over results from injecting cur-rent at the 1,072 different synaptic positions considered,see caption of Fig. 7 for details.

For the largest distances (∼1–2 mm) considered inthe plots in Fig. 7(c), one is approaching the far-fieldregime, and the dipole approximation can be expectedto be fairly precise. This is indeed observed in the plots.The averages of the observed dipolar LFP power areseen to approach the multicompartment LFP power forthe largest distances for all three frequencies consid-


ered (1 Hz, 10 Hz, 100 Hz). As illustrated by the insetfigures showing the same results in double-logarithmicplots, the multicompartment results for the LFP powerindeed decay as 1/r4, characteristic for a current dipole,for radial distances above ∼1 mm. Interestingly, the far-field limit appears to be reached faster, i.e., for smallerradial distances, in the 100-Hz situation than in the 1-Hz situation. This can be understood on the basis of theshorter effective current-dipole length for the 100-Hzsituation seen in Fig. 7(a): since the transition to the far-field regime depends on the ratio between the current-dipole length and the radial distance, a shorter currentdipole length implies an earlier transition to the the far-field regime when moving away from the neuron.

For the smallest radial distances considered(<0.5 mm), we observe as expected that the dipoleapproximation generally does a poor job of predictingthe LFP power. As seen for all three frequenciesconsidered in Fig. 7(c), the dipole approximationfollowing a 1/r4 dependence, tends to overestimate theLFP in this regime. Note that in panels (d–e) in Fig. 6it was found that the dipole approximation insteadunderestimated the true LFP for the shortest distances.The difference is simply due to the different origin anddirections of the axes considered, i.e., distance alongthe ‘vertical’ and ‘horizontal’ directions starting in theapical synapse and the soma, respectively, in panels(d–e) in Fig. 6 versus the radial distance measuredfrom the dipole position at the middle of the neuron inFig. 7(c).

4 Discussion

4.1 Intrinsic dendritic low-pass filtering effect

The origin of the observed frequency spectra of EEGand extracellularly recorded potentials in the brain(LFP) has attracted considerable interest (Pritchard1992; Beggs and Plenz 2003; Bedard et al. 2004, 2006a,b; Buzsaki 2006; Pettersen and Einevoll 2008; Bedardand Destexhe 2009; Miller et al. 2009; Milstein et al.2009). We previously showed that there is an unavoid-able low-pass frequency-filtering effect of the extra-cellular action-potential signature due to the electricalcable properties of the neuronal dendrites (Pettersenand Einevoll 2008). In action potentials the dominantfrequencies are typically a few hundred hertz, but amain finding in the present paper is that the sameintrinsic dendritic filtering effect also pertain to thelower frequencies dominating in typical LFP and EEGspectra, and is particularly prominent for large layer-5 pyramidal neurons. Our model results have demon-

strated an omnipresent low-pass filtering effect on theLFP spectra following synaptic activation. For our ex-ample pyramidal neuron we generally observed sizablefiltering effects for frequencies above ∼10 Hz. Themagnitude and form of this filtering effect will naturallydepend on, e.g., the morphology of the dendritic struc-ture and spatial distribution of activated synapses of theneuron in question. However, all neurons will to somedegree exhibit such filtering of their contributions tothe overall LFP, and any theory aspiring to explain thephysical origin of measured LFP or EEG spectra musttherefore also consider this intrinsic dendritic filteringeffect.

In an important study Bedard et al. (2006b) at-tempted to model observed LFP power spectra onthe basis of simultaneously recorded spiking activityspectra. They concluded that a factor 1/ f was ‘missing’to account for the observed LFP power spectra for fre-quencies less than ∼70 Hz, and they suggested that thismissing factor could be due to frequency attenuationin the extracellular medium itself. The present studysuggests as an alternative, or maybe supplementary,explanation that the intrinsic dendritic-filtering effectmay contribute to a missing factor 1/ f in their modelfor LFP power spectra. For the present large pyramidalneuron the LFP signal is expected to be dominatedby contributions from neurons less than about 0.5 mmaway (Lindén et al. 2009a). For the ‘vertical direction’in Fig. 6(f) we found that the LFP power spectra forthe shortest distance considered (0.1 mm) indeed arein qualitative accordance with a 1/ f decay, at leastbetween 10 Hz and 100 Hz. In the ‘horizontal’ directionin Fig. 6(g), however, a steeper decay with frequencywas observed for this short distance (0.1 mm), but thisobservation may have been confounded by an observedresonance phenomenon around 200 Hz for this particu-lar modeling example. In any case, the conclusion fromour study is not the prediction of a a particular power-law behaviour for the LFP, rather that the intrinsic den-dritic filtering effect must be included when attemptingto model such spectra. Experimental LFPs stem froma sum of numerous contributions from a populationof neurons in the vicinity of the recording electrode(Liu and Newsome 2006; Berens et al. 2008; Katzneret al. 2009; Xing et al. 2009). A more comprehensivenumerical investigation of the LFPs generated by suchpopulations of neurons is thus needed to draw firmerconclusions on this (Lindén et al. 2008, 2009a).

A striking observation in our study is that therecorded LFP signal from a single synaptic event variesdramatically with the recording position. The extra-cellular signature of a particular synaptic event willvary not only in magnitude and temporal shape, but


also in overall sign (cf. Fig. 1). Likewise, the form ofthe LFP power spectra will depend strongly on therecording position: a large variation in the typical low-pass cut-off frequencies is seen, and band-pass prop-erties are even observed at some spatial locations.The experimental LFP observed will correspond to aweighted sum over contributions from many differenttypes of neurons, located at different distances fromthe recording electrode, receiving numerous synapticinputs targeting different parts of dendritic trees. As thedominant contributions to the sum providing the LFPcannot be expected to be independent of cortical areaand depth, one cannot a priori expect common powerlaws for all observed LFP or EEG power spectra.

Our study revealed a systematic dependence of theintrinsic dendritic-filtering effect on the distance fromthe recording position to the synaptic input. In general,the high-frequency attenuation of the LFP will be lessat positions close to the active synapse where the LFPgets large contributions from the synaptic input currentitself. Farther away the LFP will get stronger contribu-tions from the transmembrane return currents which ingeneral will be low-pass filtered versions of the synapticinput current (due to the electrical cable propertiesof the dendrites). In the present paper this was mostexplicitly illustrated by the set of LFP power spectra inFigs. 2–4, but the effect is generic and also applies to,for example, extracellular action potential signatures(Pettersen and Einevoll 2008; Pettersen et al. 2010).This points to a way to infer the spatial distributionof active synapses acting on a neural population fromrecordings with linear (laminar) multielectrodes in-serted perpendicularly on the cortical layers (Einevollet al. 2007): the current source-density (CSD) profileinforms about the vertical extension of an activatedpopulation of, say, layer-5 pyramidal neurons, but maynot per se be able to distinguish the situations withinhibitory synapses acting in the basal dendritic regionfrom excitatory synapses acting in the apical region. Inboth cases one will observe a sink-source dipole withthe current sink in the apical region. However, thepresent results suggest that in the former case therewill be more high-frequency components in the CSD atvertical positions inside the basal bush than at verticalpositions at the level of the apical dendrites. In thelatter case the situation will be reversed.

4.2 Dipole and two-monopole approximationsto the LFP

The use of current dipoles in the modeling of bioelec-tric signals, and in particular EEG signals, is not new

(Plonsey 1969; Freeman 1980; Nunez and Srinavasan2006; Grech et al. 2008). In the application to EEGsignals one has typically aimed to account for the ob-served signals in terms of contributions from a set of‘mesoscopic’ current dipole sources, and the distancesbetween the current dipoles and the EEG recordingposition have typically been assumed to be so largethat the far-field dipole approximation apply (Nunezand Srinavasan 2006; Grech et al. 2008). Here we havefocused on the fundamental microscopic dipoles un-derlying these mesoscopic dipole sources, namely thecurrent dipoles set up by individual neurons (Murakamiand Okada 2006; Pettersen and Einevoll 2008).

As in the analysis of the extracellular signature ofaction potentials (Pettersen and Einevoll 2008) we findthe current dipole moment to be a crucial concept forgaining both qualitative and quantitative understand-ing of the relationship between the underlying activityin the neuron and the generated extracellular poten-tials. For the action-potential signature investigated inPettersen and Einevoll (2008) this dipole moment wastypically found to be determined by the electrical prop-erties of the dendritic branches protruding out fromthe soma. Consequently the current dipole length, ameasure of the distance between the soma and the av-erage spatial position at which the return currents passthe dendritic membrane, was found to depend mainlyon dendritic parameters such as diameter, capacitance,and axial and membrane resistances. In contrast, forthe lowest frequencies considered in the present LFPspectra (<10 Hz), we found here that the main den-dritic feature determining the current dipole length,and the current dipole moment, appeared to be thespatial distance between the active synapse and thesoma (cf. Fig. 7(a)). At such low frequencies roughestimates of the direction and relative magnitude of thecurrent dipole moments for various synaptic inputs canthus be obtained by anatomical considerations alone.However, in general the frequency-dependent currentdipole moment (cf. Fig. 7(b)) will be determined by acombination of morphological and electrical membraneproperties.

Used in combination with the well-known far-fielddipole expression in Eq. (8), the current dipole momentwas shown to predict the ‘true’ LFP (found in ourcomprehensive multicompartment scheme) accuratelyat large distances, i.e., a few millimeters away fromthe neuron. We also found that this dipole approx-imation could account for salient features of the in-trinsic low-pass filtering effect, i.e., that the low-passfrequency filtering of the LFP could be traced backto low-pass frequency filtering of the current-dipolemoment.


As expected we found this far-field approximationto break down close to the neuron, partially becauseit does not explicitly represent the two dominantcontributions from the synapse and soma regions, re-spectively. We thus also investigated a two-monopoleapproximation, i.e., a 2-compartment model where alltransmembrane current is assumed to enter the neu-ron at one point (the synapse) and leave at the other(the soma). Further, the transmembrane current wasset to give the right magnitude of the current dipolemoment. For our example in Fig. 6 we found the two-monopole approximation to give very good predictionsin the vicinity of the active synapse, much better thanthe dipole approximation. However, for our examplewith excitation of an apical synapse, we found the two-monopole approximation to predict far too big LFPs forpositions inside the bush of basal dendrites. There arelikely two main reasons for this discrepancy: (1) Forsituations where the return current indeed is focusedin the soma region (such as for low-frequency apicalsynaptic inputs onto the layer-5 example neuron), thetrue transmembrane current in the soma region is,due to the extensive basal dendritic structure, muchmore diffuse than what is implicitly assumed in thetwo-monopole model. (2) For high-frequency synapticinput the cable properties of the neuron will in generalreduce the amount of return current leaving from thesoma region, and the ‘center-of-gravity’ of the returncurrent will be positioned closer to the position of thesynaptic input, cf. Fig. 5. This will naturally violatethe assumption inherent in the two-monopole approx-imation where the return monopole is set to be atthe soma.

An alternative version of the two-monopole modelwould be to move the return-current monopole fromthe soma to the position of the center-of-gravity forthe return current. This center-of-gravity position willdepend on frequency, and such a two-monopole ap-proximation would likely yield more accurate resultsthan the present two-monopole model. However, thecalculation of the center-of-gravity position will involvenumerical solution of the full compartmental modelto obtain all transmembrane currents, making the im-plementation of the approximation almost as cumber-some as using the comprehensive line-source scheme.A main advantage of the present two-monopole modelis that the position of the two monopoles is determinedsolely by the neuronal morphology and thus easy tospecify (even though specification of the magnitudeof the monopole currents requires knowledge aboutthe dipole-moment magnitude, which in turn must becalculated based on knowledge of all transmembranecurrents).

4.3 Network modeling of LFP, ECoG and EEG

A natural ambition of neural network modeling shouldbe to predict not only spike trains, but also extracellularpotentials. One approach for achieving this would beto simulate networks of multicompartmental neuronsso that the generation of action potentials and thetransmembrane currents needed to evaluate extracel-lular potentials, could be calculated simultaneously.However, spiking neural networks are more commonlymodeled by means of 1-compartment neurons whichby design do not generate an extracellular potential:a neuron model must have at least two compartmentsto generate an extracellular potential (Pettersen et al.2010). In such cases ad hoc formulas relating synapticcurrents to LFP have been used to provide LFP predic-tions from network simulations, see, e.g., Mazzoni et al.(2008).

The forward-modeling scheme offers a method fortesting of such putative add-on formulas for predict-ing extracellular potentials. Our present work unfortu-nately does not suggest any such simplifying formulafor prediction of LFPs recorded well inside the activecortical tissue. For the present pyramidal neuron weexpect the LFP signal to get most of its contributionsfrom neurons within a radius of less than 0.5 mm(Lindén et al. 2009a), an estimated spatial range of theLFP in qualitative accordance with recent experimentalestimates (Katzner et al. 2009; Xing et al. 2009). As seenin Fig. 6 both the dipole approximation (Eq. 8) and thetwo-monopole approximation (Eq. 10) generally giveinaccurate results of the LFP for such small distances.Simplifying formulas based on these approximationsthus appears ruled out. This observation thus clearlyquestions the approach of interpreting intracorticalLFP recordings in terms of contributions from dipolarsources (Church et al. 1985; Yvert et al. 2001).

The development of simplified schemes for predict-ing contributions to extracellular potentials recorded atthe cortical surface (ECoG) is maybe more feasible.As seen in Fig. 6(d) the two-monopole approximationdoes fairly well, in particular for the 1-Hz situation, inthe vertical direction for positions more than 0.1 mmor so above the apical synapse. This suggests that thistwo-monopole approximation may be useful for pre-dicting ECoG signals. In fact a similar two-monopoleapproximation was suggested by Freeman (1980) tointerpret ECoG signals recorded by electrode arraysplaced on the brain surface. However, here we onlyexplored the two-monopole approximation for the sit-uation with an apical synapse where the LFP recordedimmediately above is dominated by a rather focal sink.Even in this situation, the agreement with the correct


multicompartmental results was not more than fair.With other synaptic inputs which generate more diffusepatterns for transmembrane currents in the apical den-dritic tree, a lesser performance of the two-monopoleapproximation must be expected. A more compre-hensive investigation also considering morphologies ofother neuron types expected to give large contributionsto the ECoG signal (e.g., supragranular pyramidal neu-rons), is needed to evaluate this approximation schemefurther.

The prospect for developing simplified schemesfor predicting contributions to extracellular potentialsrecorded at the scalp (EEG) seems much better. Herethe distances from the neural origin of the electricalsignal are sufficiently large for the dipole approxima-tion described in Eq. (8) to give accurate results. Thisformula describes the contribution to the recorded elec-trical potentials at a particular electrode contact froma particular neuron receiving synaptic input currentsthrough a particular synapse, information that in prin-ciple may be calculated and predicted in large-scaleneural network modeling (Jirsa and Haken 1997; Jirsaet al. 2002; Coombes 2005). However, the magnitudeand direction of the current dipole moment for thevarious synaptic inputs, must either be assumed or cal-culated using multicompartmental modeling, in orderfor the formula to be workable. The two-monopoleapproximation in Eq. (10) is an alternative. Here theonly spatial information needed is the positions of thesoma and the active synapse, but the magnitude of thetransmembrane current Itm must still be assumed orcalculated using multicompartmental modeling.

A caveat of the present work in the context of pre-dicting EEG signals is that the extracellular conductiv-ity σ has been assumed to be constant. This assumptionis clearly violated when the electrical contacts are po-sitioned outside the cortical tissue as in EEG record-ings, but our present approach can be generalized totake into account such inhomogeneous electrical mediausing the same techniques as in the well-establishedforward modeling of EEG signals from mesoscopiccurrent dipoles (Nunez and Srinavasan 2006).

4.4 Assumptions in forward modeling scheme

Our present LFP forward modeling scheme based onEq. (1) assumes the extracellular conductivity σ to bethe same everywhere (‘homogeneous’), the same inall directions (‘isotropic’), and also purely ohmic, i.e.,no imaginary part from capacitive effects (Nunez andSrinavasan 2006; Logothetis et al. 2007). Finally, σ isassumed to be the same for all relevant frequencies.The validity of this latter assumption is still debated:

while some studies have measured negligible frequencydependence (Nicholson and Freeman 1975; Logothetiset al. 2007), other investigations have suggested oth-erwise (Gabriel et al. 1996; Bedard et al. 2004, 2006a,b; Bedard and Destexhe 2009). However, if warranted,the present modeling formalism can be generalized toinclude also a frequency-dependent electrical conduc-tivity σ( f ). More discussion on the assumptions regard-ing σ , and also ways of generalizing Eq. (1) when theydo not apply, can be found in Pettersen et al. (2010).

Acknowledgements This work was supported by the ResearchCouncil of Norway (eScience, NOTUR). We thank one of the re-viewers for bringing relevant literature on human depth-resolvedLFP recordings to our attention.

References

Arieli, A. (1992). Novel strategies to unravel mechanisms ofcortical function: From macro- to micro-electrophysiologicalrecordings. In A. Aertsen, & V. Braitenberg (Eds.), Infor-mation processing in the cortex. New York: Springer.

Bedard, C., Kröger H., & Destexhe A. (2004). Modeling extracel-lular field potentials and the frequency-filtering properties ofextracellular space. Biophysical Journal, 86, 1829–1842.

Bedard, C., Kröger, H., & Destexhe, A. (2006a). Model of low-pass filtering of local field potentials. Physical Review E, 73,051911.

Bedard, C., Kröger, H., & Destexhe, A. (2006b). Does the 1/f fre-quency scaling of brain signals reflect self-organized criticalstates? Physical Review Letters, 97, 118102.

Bedard, C., & Destexhe, A. (2009) Macroscopic models of localfield potentials and the apparent 1/f noise in brain activity.Biophysical Journal, 96, 2589–2603.

Beggs, J. M., & Plenz, D. (2003). Neuronal avalanches in neocor-tical circuits. Journal of Neuroscience, 23, 11167–11177.

Berens, P., Keliris, G. A., Ecker, A. S., Logothetis, N., &Tolias, A. S. (2008). Comparing the feature selectivity of thegamma-band of the local field potential and the underlyingspiking activity in primate visual cortex. Frontiers in SystemsNeuroscience, 2(2). doi:10.3389/neuro.06/002.2008.

Buzsáki, G. (2004). Large-scale recording of neuronal ensembles.Nature Neuroscience, 7, 446–451.

Buzsáki, G. (2006). Rhythms of the brain. New York: OxfordUniversity Press.

Carnevale, N. T., & Hines, M. L. (2006). The NEURON book.Cambridge, UK: Cambridge University Press.

Church, P., Leduc, A., & Beique, R. A. (1985). Sensitivity analy-sis of depth EEG electrodes to dipolar electric sources.IEEE Transactions on Biomedical Engineering, 32, 554–560.

Coombes, S. (2005). Waves, bumps, and patterns in neural fieldtheories. Biological Cybernetics, 93, 91–108.

Di, S., Baumgartner, C., & Barth, D. S., (1990). Laminar analysisof extracellular field potentials in rat vibrissa/barrel cortex.Journal of Neurophysiology, 63, 832–840.

Einevoll, G. T., Pettersen, K. H., Devor, A., Ulbert, I.,Halgren, E., & Dale, A. M. (2007). Laminar populationanalysis: Estimating firing rates and evoked synaptic activityfrom multielectrode recordings in rat barrel cortex. Journalof Neurophysiology, 97, 2174–2190.

http://dx.doi.org/10.3389/neuro.06/002.2008


Freeman, W. J. (1980). Use of spatial deconvolution to compen-sate distortion of EEG by volume conduction. IEEE Trans-actions on Biomedical Engineering, 27, 421–429.

Freeman, W. J., Holmes, M. D., Burke, B. C., & Vanthalo, S.(2003). Spatial spectra of scalp EEG and EMB from awakehumans. Clinical Neurophysiology, 114, 1053–1068.

Gabriel, S., Lau, R. W., & Gabriel, C. (1996). The dielectric prop-erties of biological tissues: III. Parametric models for thedielectric spectrum of tissues. Physics in Medicine & Biology,41, 2271–2293.

Godey, B., Schwartz, D., de Graaf, J. B., Chauvel, P., & Liegeois-Chauvel, C. (2001). Neuromagnetic source localization ofauditory evoked fields and intracerebral evoked potentials:A comparison of data in the same patients. Clinical neuro-physiology, 112, 1850–1859.

Grech, R., Cassar, T., Muscat, J., Camilleri, K. P., Fabri, S. G.,Zervakis, M., et al. (2008). Review on solving the inverseproblem in EEG source analysis. Journal of Neuroengineer-ing and Rehabilitation 5, 25.

Hämäläinen, M., Hari, R., Ilmoniemi, R. J., Knuutila, J., &Lounasmaa, O. V. (1993). Magnetoencephalography theory,instrumentation, and applications to noninvasive studies ofthe working human brain. Reviews of Modern Physics, 65,413–449.

Hines, M. L, Davison, A. P. & Muller E. (2009). NEURON andPython. Frontiers in Neuroinformatics, 3, 1.

Holt, G. R. & Koch, C. (1999). Electrical interactions via theextracellular potential near cell bodies. Journal of Compu-tational Neuroscience, 6, 169–184.

Jackson, J. (1998). Classical Electrodynamics. NJ: Wiley,Hoboken.

Jirsa, V. K., & Haken, H. (1997). A derivation of a macroscopicfield theory of the brain from the quasi-microscopic neuraldynamics. Physica D, 99, 503–526.

Jirsa, V. K., Jantzen, K. J., Fuchs, A., & Kelso, J. A. S. (2002).Spatiotemporal forward solution of the EEG and MEG us-ing network modeling. IEEE Transactions on Medical Imag-ing, 21, 493–504.

Johnston, D., & Wu, S. M.-S. (1995) Foundations of cellular neu-rophysiology, (Chapter 14). Cambridge, MA: MIT Press.

Katzner, S., Nauhaus, I., Benucci, A., Bonin, V., Ringach, D. L.,& Carandini, M. (2009). Local origin of field potentials invisual cortex. Neuron, 61, 35–41.

Koch, C. (1998). Biophysics of computation. New York, NY:Oxford.

Kreiman, G., Hung, C. P, Kraskov, A., Quiroga, R. Q., Poggio,T., & DiCarlo, J. J. (2006). Object selectivity of local fieldpotentials and spikes in the macaque inferior temporal cor-tex. Neuron, 49, 433–445.

Lindén, H., Pettersen, K. H., & Einevoll, G. T. (2008). Frequencyscaling in local field potentials: A neuron population for-ward modelling study Frontiers in Neuroinformatics. Con-ference Abstract: Neuroinformatics, 2008. doi:10.3389/conf.neuro.11.2008.01.026.

Lindén, H., Pettersen, K. H., Tetzlaff, T., Potjans, T., Denker, M.,Diesmann, M., et al. (2009a). Estimating the spatial range oflocal field potentials in a cortical population model. BMCNeuroscience, 10(1), 224.

Lindén, H., Potjans, T. C., Einevoll, G. T., Grün, S., &Diesmann, M. (2009b). Modeling the local field potentialby a large-scale layered cortical network model. Frontiers inNeuroinformatics, Conference Abstract: 2nd INCF Congressof Neuroinformatics. doi:10.3389/conf.neuro.11.2009.08.046.

Liu, J., & Newsome, W. T. (2006). Local field potential in cor-tical area MT. Stimulus tuning and behavioral correlations.Journal of Neuroscience, 26, 7779–7790.

Logothetis, N. K., Kayser, C., & Oeltermann, A. (2007). In vivomeasurement of cortical impedance spectrum in monkeys:Implications for signal propagation. Neuron, 55, 809–823.

Lorente de Nó, R. (1947). Action potential of the motoneuronsof the hypoglossus nucleus. Journal of Cellular and Compar-ative Physiology, 29, 207–287.

Mainen, Z. F., & Sejnowski, T. J. (1996). Influence of dendriticstructure on firing pattern in model neocortical neurons.Nature, 382, 363–366.

Mazzoni, A., Panzeri, S., Logothetis, N. K., & Brunel, N. (2008).Encoding of naturalistic stimuli by local field potential spec-tra in networks of excitatory and inhibitory neurons. PLoSComputers in Biology, 4, e1000239.

Miller, K. J., Sorensen, L. B., Ojemann, J. G., & den Nijs, M.(2009). Power-law scaling in the brain surface electric poten-tial. PLoS Computers in Biology, 5, e1000609.

Milstein, J. N., & Koch, C. (2008). Dynamic moment analysisof the extracellular electric field of a biologically realisticspiking neuron. Neural Computation, 20, 2070–2084.

Milstein, J., Mormann, F., Fried, I., & Koch., C (2009). Neuronalshot noise and Brownian 1/f2 behavior in the local field po-tential. PLoS ONE, 4, e4338.

Mitzdorf, U. (1985). Current source-density method and applica-tion in cat cerebral cortex: Investigation of evoked potentialsand EEG phenomena. Physiological Reviews, 65, 37–99.

Murakami, S., & Okada, Y. (2006). Contributions of principal neo-cortical neurons to magnetoencephalography and electroen-cephalography signals. Journal of Physiology, 575, 925–936.

Nauhaus, I., Busse, L., Carandini, M., & Ringach, D. L. (2009).Stimulus contrast modulates functional connectivity in visualcortex. Nature Neuroscience, 12, 70–76.

Nicholson, C., & Freeman, J. A. (1975). Theory of current source-density analysis and determination of conductivity tensor foranuran cerebellum. Journal of Neurophysiology, 38, 356–368.

Normann, R. A., Maynard, E. M., Rousche, P. J., & Warren, D.J. (1999). A neural interface for a cortical vision prosthesis.Vision Research, 39, 2577–2587.

Nunez, P. L., & Srinavasan, R. (2006). Electric f ields of the brain:The neurophysics of EEG. Oxford: Oxford University Press.

Pettersen, K. H., & Einevoll, G. T. (2008). Amplitude variabil-ity and extracellular low-pass filtering of neuronal spikes.Biophysical Journal, 94, 784–802.

Pettersen, K. H., Hagen, E., & Einevoll, G. T. (2008). Es-timation of population firing rates and current sourcedensities from laminar electrode recordings. Journal ofComputational Neuroscience, 24, 291–313.

Pettersen, K. H., Lindén, H., Dale, A. M., & Einevoll, G. T.(2010). Extracellular spikes and current-source density. InR. Brette, & A. Destexhe (Eds.), Handbook of neural ac-tivity measurements. Cambridge, UK: Cambridge UniversityPress.

Plonsey, R. (1969). Bioelectric phenomena. New York: McGraw-Hill.Plonsey, R., & Barr, R. C. (2007). Bioelectricity: A quantitative

approach. New York: Springer.Pritchard, W. S. (1992). The brain in fractal time: 1/f-like power

spectrum scaling of the human electroencephalogram. Inter-national Journal of Neuroscience, 66, 119–129.

Rall, W. (1962). Electrophysiology of a dendritic neuron model.Biophysical Journal, 2, 145–167.

Xing, D., Yeh, C.-I., & Shapley, R. M. (2009). Spatial spread ofthe local field potential and its laminar variation in visualcortex. Journal of Neuroscience, 29, 11540–11549.

Yvert, B., Fischer, C., Bertrand, O., & Pernier, J. (2001). Local-ization of human supratemporal auditory areas from intrac-erebral auditory evoked potentials using distributed sourcemodels. NeuroImage, 28, 140–153.

http://dx.doi.org/10.3389/conf.neuro.11.2008.01.026



Intrinsic dendritic filtering gives low-pass power spectra of local field potentials

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